https://wiki.swarma.org/api.php?action=feedcontributions&feedformat=atom&user=Dorr集智百科 - 复杂系统|人工智能|复杂科学|复杂网络|自组织 - 用户贡献 [zh-cn]2026-11-04T19:14:00Z用户贡献MediaWiki 1.35.0https://wiki.swarma.org/index.php?title=%E7%83%AD%E5%8A%9B%E5%AD%A6%E7%AC%AC%E4%BA%8C%E5%AE%9A%E5%BE%8B&diff=12753热力学第二定律2020-08-16T13:25:44Z<p>Dorr：/* Perpetual motion of the second kind 第二类永动机 */</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Law of physics}}<br />
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{{Thermodynamics|laws}}<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 全文审校中，heat一般统一译为“热”，caloric 或者 caloric of heat 译为“热量”。这一点是否准确，有待各位同学和专家的批评指正<br />
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The '''second law of thermodynamics''' states that the total [[entropy]] of an [[isolated system]] can never decrease over time, and is constant if and only if all processes are reversible.<ref>http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node38.html#SECTION05224000000000000000</ref> Isolated systems spontaneously evolve towards [[thermodynamic equilibrium]], the state with maximum entropy. <br />
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The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, and is constant if and only if all processes are reversible. Isolated systems spontaneously evolve towards thermodynamic equilibrium, the state with maximum entropy. <br />
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<font color="#ff8000">'''热力学第二定律 Second Law Of Thermodynamics'''</font>指出，<font color="#ff8000">'''孤立系统Isolated System'''</font>的总熵永远不会随着时间而减少，且当且仅当所有过程都是可逆时，总熵才恒定。孤立系统自发地到达到热力学平衡状态，此时为熵最大的状态。<br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in [[thermodynamic equilibrium]], or is undergoing a (fictive) [[Reversible process (thermodynamics)|reversible process]]. In all processes that occur, including [[spontaneous process]]es,<ref>Atkins and de Paula, p.78</ref> the total entropy of the system and its surroundings increases and the process is [[Irreversible process|irreversible in the thermodynamic sense]]. The increase in entropy accounts for the irreversibility of natural processes, and the [[Arrow of time|asymmetry between future and past]].<ref>{{cite book|last=Zohuri|first=Bahman|title=Dimensional Analysis Beyond the Pi Theorem|url=https://books.google.com/books?id=pRVuDQAAQBAJ|year=2016|publisher=Springer|isbn=978-3-319-45726-0|page=[https://books.google.com.ph/books?id=pRVuDQAAQBAJ&pg=PA111&dq=%22increase+in+entropy+accounts+for+the+irreversibility+of+natural+processes+and+the+asymmetry+between+future+and+past.%22&hl=en&sa=X&ved=0ahUKEwj6spb61tbaAhUFS7wKHfftDtIQ6AEIKDAA#v=onepage&q=%22increase%20in%20entropy%20accounts%20for%20the%20irreversibility%20of%20natural%20processes%20and%20the%20asymmetry%20between%20future%20and%20past.%22 111]}}</ref><br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in thermodynamic equilibrium, or is undergoing a (fictive) reversible process. In all processes that occur, including spontaneous processes, the total entropy of the system and its surroundings increases and the process is irreversible in the thermodynamic sense. The increase in entropy accounts for the irreversibility of natural processes, and the asymmetry between future and past.<br />
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系统及其周围环境的总熵在理想情况下可以保持不变，在这种情况下，系统处于热力学平衡状态，或者正在经历一个(<font color = 'red'><s>虚拟的</s></font><font color = 'blue'>假想的</font>)可逆过程。<font color = 'red'><s>在所有发生的</s></font><font color = 'blue'>所有</font>过程中，包括<font color="#ff8000">'''自发过程 Spontaneous Processes'''</font>，系统及其周围环境的总熵增加，这一过程在热力学意义上是不可逆的。熵的增加解释了自然过程的不可逆性，以及未来和过去之间的不对称性。<br />
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Historically, the second law was an [[Empirical evidence|empirical finding]] that was accepted as an axiom of [[thermodynamics|thermodynamic theory]]. [[Statistical mechanics]], classical or [[quantum thermodynamics|quantum]], explains the microscopic origin of the law.<br />
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Historically, the second law was an empirical finding that was accepted as an axiom of thermodynamic theory. Statistical mechanics, classical or quantum, explains the microscopic origin of the law.<br />
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从历史上看，第二定律是一个经验性的发现，它被认为是热力学理论中的公理。经典统计力学、量子统计力学都可以解释这个定律的微观起源。<br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist [[Nicolas Léonard Sadi Carnot|Sadi Carnot]], who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as [[Carnot's theorem (thermodynamics)|Carnot's rule]] or limit.<ref>Jaffe, R.L., Taylor, W. (2018). ''The Physics of Energy'', Cambridge University Press, Cambridge UK, pages 150, 151, 259, 772, 743.</ref><br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist Sadi Carnot, who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as Carnot's rule or limit.<br />
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<font color = 'blue'>热力学第二定律可以使用多种方法表述。它的第一个公式归功于法国科学家<font color="#ff8000">'''卡诺Sadi Carnot'''</font>，卡诺在1824年证明了在热机中将热转化为功的效率有一个上限。第二定律的这个方面也被称为<font color="#ff8000">'''卡诺规则Carnot's Rule'''</font>或<font color="#ff8000">'''卡诺限制Carnot's Limit'''</font>。</font><br />
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'''热力学第二定律 Second Law of Thermodynamics'''是热力学的四条基本定律之一，它表述了热力学过程的不可逆性——孤立系统自发地朝着热力学平衡方向（即最大熵状态）演化，另一种表述为：第二类永动机永不可能实现。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）这里是不是少了一段原文？<br />
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==Introduction引言==<br />
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[[File:Heat flow hot to cold.png|thumb|upright|Heat flow from hot water to cold water. 热总是从热水流向冷水]]<br />
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The [[first law of thermodynamics]] provides the definition of the [[internal energy]] of a [[thermodynamic system]], and expresses the law of [[conservation of energy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 40–41.</ref><ref>Munster A. (1970), pp. 8–9, 50–51.</ref> The second law is concerned with the direction of natural processes.<ref>{{harvnb|Mandl|1988}}</ref> It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such [[Phenomenon|phenomena]] are accounted for in terms of [[entropy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 79–107.</ref><ref>Bailyn, M. (1994), Section 71, pp. 113–154.</ref> If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, {{math|''S''}}, increases.<br />
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The first law of thermodynamics provides the definition of the internal energy of a thermodynamic system, and expresses the law of conservation of energy. The second law is concerned with the direction of natural processes. It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such phenomena are accounted for in terms of entropy. If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, S, increases.<br />
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'''热力学第一定律 First Law Of Thermodynamics''' <font color = 'red'><s>对</s></font> <font color = 'blue'>定义了</font>热力学系统所涉及到的'''内能 Internal Energy'''<font color = 'red'><s>进行了定义</s></font> ，并体现了能量守恒定律。热力学第二定律与'''自然过程Natural Processes'''的方向有关。它<font color = 'red'><s>设定</s></font> <font color = 'blue'>断言</font>自然过程只在一种意义上进行，且<font color = 'red'><s>是</s></font> 不可逆<font color = 'red'><s>的</s></font> 。例如，当有了传导和辐射的（传播）路径时，热<s>量</s>总是自发地从一个较热的物体流向一个较冷的物体。这种现象可以用'''熵Entropy'''来解释。<font color = 'red'><s>若一个孤立系统最初在内部不可渗透的薄膜维持内部的热力学平衡，通过一些操作使得薄膜具有渗透性，则该系统可自发地演变，最终达到一个新的内部热力学平衡，其总熵增加。</s></font><br />
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<font color = '#32CD32'>若一个'''孤立系统Isolated System'''最初在具有隔热内壁的系统内维持热力学平衡，通过一些操作使内壁透热，则该系统可自发地演变，最终达到一个新的内部热力学平衡，且其总熵{{math|''S''}}增加。If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, S, increases.</font><br />
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--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]） more permeable不知道是否需要译为更具有 impermeable wall是不可渗透 如果要使用比较级 是否应该再有一个 permeable<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） internal partitioning impermeable walls翻译为隔热的内壁，从而permeable指的是透热的。我觉得这里的渗透性应该是说系统可以和外界进行能量/热的交换。？不知道是否正确。<br />
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In a fictive reversible process, an infinitesimal increment in the entropy ({{math|d''S''}}) of a system is defined to result from an infinitesimal transfer of heat ({{math|δ''Q''}}) to a [[closed system]] (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature ({{math|''T''}}) of the system in equilibrium and the surroundings which supply the heat:<ref>Bailyn, M. (1994), p. 120.</ref><br />
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In a fictive reversible process, an infinitesimal increment in the entropy () of a system is defined to result from an infinitesimal transfer of heat () to a closed system (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature () of the system in equilibrium and the surroundings which supply the heat:<br />
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在<font color = 'red'><s>设置的虚拟</s></font> <font color = 'blue'>假想的</font>可逆过程中，系统的熵的无穷小增量<math>dS</math>被定义为由无穷小的热<s>量</s><math>δQ</math>传递到一个封闭系统(允许能量进入或出去，但不允许物质传递<font color = 'blue'>的系统</font>)除以<font color = 'blue'>该</font>平衡系统和提供热<s>量</s>的环境的共同温度 <math>T</math>:<br />
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: <math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process封闭系统中理想状态下的可逆过程).}</math><br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 这里的公式和备注貌似有误，公式应为 <math>\mathrm dS = \frac{\delta Q}{T}</math>，备注应为“closed system, idealized fictive reversible process”封闭系统，理想的可逆过程。或者？<br />
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Different notations are used for infinitesimal amounts of heat ({{mvar|δ}}) and infinitesimal amounts of entropy ({{mvar|d}}) because entropy is a [[function of state]], while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the [[Clausius Theorem|inequality]] <ref name="MortimerBook"> Mortimer, R. G. ''Physical Chemistry''. 3rd ed., p. 120, Academic Press, 2008. </ref><ref name="FermiBook"> Fermi, E. ''Thermodynamics'', footnote on p. 48, Dover Publications,1956 (still in print). </ref> <br />
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Different notations are used for infinitesimal amounts of heat () and infinitesimal amounts of entropy () because entropy is a function of state, while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the inequality <br />
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用不同的符号''δ''和''d''表示无穷小量的热<s>量</s>和无穷小量的熵，因为熵是状态函数，而热<s>量</s>和功一样<font color = 'blue'>则</font>不是状态函数。<font color = 'red'><s>第二定律要求系统熵的增量满足不等式，</s></font>对于实际上可能存在的不与环境发生<font color = 'red'><s>质量</s></font><font color = 'blue'>物质</font>交换的无穷小过程，<font color = 'blue'>第二定律要求</font>系统熵增量满足不等式：<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 根据上下文（讲到封闭系统），此处疑“mass”译为物质更佳<br />
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: <math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process 封闭系统中，实际上可能发生的可逆过程).}</math><br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature ({{math|''T''}}) and the temperature of the surroundings ({{math|''T''}}<sub>''surr''</sub>).<br />
<ref name=":0">Adkins, C.J. (1968/1983), p. 75.</ref><ref name="Munster 45"/> <br />
Note that the equality still applies for pure heat flow,<br />
<ref name="Schmidt-Rohr 14"> Schmidt-Rohr, K. (2014). "Expansion Work without the External Pressure, and Thermodynamics in Terms of Quasistatic Irreversible Processes" ''J. Chem. Educ.'' '''91''': 402-409. https://dx.doi.org/10.1021/ed3008704 </ref><br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature () and the temperature of the surroundings (<sub>surr</sub>).<br />
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<font color = 'blue'>这是因为</font>这种情况下的一般过程可能包括周围环境对系统所做的功，<font color = 'red'><s>这是因为</s></font><font color = 'blue'>这个功</font>在系统内部会产生摩擦或粘滞效应，<font color = 'blue'>或是因为</font>此时<font color = 'red'><s>是由于</s></font><font color = 'blue'>一个</font>化学反应可能正在<font color = 'blue'>系统内部</font>进行，或<font color = 'blue'>因为</font>热传递实际上是不可逆地发生，<font color = 'red'><s>通过</s></font>系统温度<math>T</math>和周围环境温度<math>T_surr</math>之间<font color = 'red'><s>存在差异而进行驱动</s></font><font color = 'blue'>的差异驱动了热传递的发生</font>。<br />
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）该句有些不太理解<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 我觉得应该是“This is because a general process……，because a chemical reaction……，or because heat transfer……surroundings (<sub>surr</sub>).”的并列结构。<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）我的理解，第一个because是对上文的解释；后两个because是对“功在系统内部产生……效应”的解释；后两个为并列，不与第一个并列，因此“粘滞效应”后面的because前不用加“或是”<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）and the temperature of the surroundings (<sub>surr</sub>)后面缺少半句原文：“Note that the equality still applies for pure heat flow,”<br />
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<font color='blue'>注意这个等式也适用于'''纯热流Pure Heat Flow''',</font><br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change 实际上可能的,不改变成分的准静态不可逆性).}</math><br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry.<ref name="Oxtoby8th">Oxtoby, D. W; Gillis, H.P., [[Laurie Butler|Butler, L. J.]] (2015).''Principles of Modern Chemistry'', Brooks Cole. p. 617. {{ISBN|978-1305079113}}</ref> <ref name="MortimerBook"></ref> Introducing a set of internal variables <math>\xi</math> to describe the deviation of a thermodynamic system in physical equilibrium (with the required well-defined uniform pressure ''P'' and temperature ''T'')<ref name="Schmidt-Rohr 14"></ref> from the chemical equilibrium state, one can record the equality<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry.[16] [11] Introducing a set of internal variables {\displaystyle \xi }\xi to describe the deviation of a thermodynamic system in physical equilibrium (with the required well-defined uniform pressure P and temperature T)[15] from the chemical equilibrium state, one can record the equality<br />
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<font color='blue'>这种方法</font>是通过测量'''热容曲线 Heat Capacity Curves'''和相变中熵的变化，来准确测定纯物质的绝对熵的基础，比如'''量热法calorimetry'''。<br />
为了描述一个热力学系统在物理平衡状态下(要求有明确定义的等压P和等温T)偏离化学平衡状态，引入一组内部变量<math>x_i</math>,可以用该等式<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process 封闭系统，实际可能的准静态不可逆过程).}</math><br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of ''the second principle of thermodynamics'' – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<ref> Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769–781.</ref><ref>{{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}</ref><br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of the second principle of thermodynamics – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<br />
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第二项代表内部变量的功，这些内部变量<font color='red'><s>可以</s></font><font color= 'blue'>可能</font>受到外部影响的干扰，但是系统不能通过内部变量做任何正功。这种说法介绍了热力学系统在时间上<font color='red'><s>不可能逆转演化的事实</s></font>演化不可能逆转的性质<font color= 'blue'></font>，并且<font color='red'><s>可以被认为是热力学第二原理的另外一种相当于熵原理的表述。</s></font><font color= 'blue'>该公式可以被看作使用'''熵Entropy'''的热力学第二定律等价表述</font><br />
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The [[zeroth law of thermodynamics]] in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body.<ref name=dugdale>{{cite book|author=J. S. Dugdale|title=Entropy and its Physical Meaning|publisher=Taylor & Francis|year=1996|isbn=978-0-7484-0569-5|page=13|quote=This law is the basis of temperature.}}</ref> For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows{{how?|date=August 2018}} a distinguished temperature scale, which defines an absolute, [[thermodynamic temperature]], independent of the properties of any particular reference thermometric body.<ref>[[Mark Zemansky|Zemansky, M.W.]] (1968), pp. 207–209.</ref><ref>Quinn, T.J. (1983), p. 8.</ref><br />
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The zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body. For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows a distinguished temperature scale, which defines an absolute, thermodynamic temperature, independent of the properties of any particular reference thermometric body.<br />
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热力学第零定律是指如果两个热力学系统<font color='red'><s>中的每一个</s></font>都与第三个热力学系统处于热平衡(温度相同)<font color= 'blue'>状态</font>，则它们彼此也必定处于热平衡<font color= 'blue'>状态</font>。热力学第零定律在<font color='red'><s>它</s></font>这个<font color= 'blue'></font><font color='red'><s>通常的</s></font>简短叙述中让人们认识到热平衡关系中的两个物体具有相同的温度，特别是当一个被测物体与一个参考测温物体具有相同的温度时，对于<font color='red'><s>一个与另一个</s></font>两个<font color= 'blue'></font>处于热平衡<font color= 'blue'>状态</font>的物体，有无限多的'''<font color = '#ff8000'>经验温标empirical temperature scales</font>'''，这通常<font color= 'blue'>分别</font>取决于特定参考温度<font color='red'><s>标度</s></font><font color= 'blue'>体</font>的性质。热力学第二定律允许区分'''<font color = '#ff8000'>温度标度temperature scale</font>'''，它定义了一个绝对的热力学温度，与任何特定的参考温度体的性质无关。<br />
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--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）补充热力学第零定律<br />
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==Various statements of the law热力学第二定律的不同表述==<br />
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The second law of thermodynamics may be expressed in many specific ways,<ref name=MIT>{{cite web|title=Concept and Statements of the Second Law|url=http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node37.html|accessdate=2010-10-07 |publisher=web.mit.edu}}</ref> the most prominent classical statements{{sfnp|Lieb|Yngvason|1999}} being the statement by [[Rudolf Clausius]] (1854), the statement by [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] (1851), and the statement in axiomatic thermodynamics by [[Constantin Carathéodory]] (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.{{sfnp|Rao|2004|p=213}}<br />
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The second law of thermodynamics may be expressed in many specific ways, the most prominent classical statements being the statement by Rudolf Clausius (1854), the statement by Lord Kelvin (1851), and the statement in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.<br />
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热力学第二定律可以用许多特定的方式来表达，最突出的经典陈述是 '''克劳修斯 Rudolf Clausius''' (1854)<font color = 'red'><s>陈述</s></font><font color = 'blue'>表述</font>，'''开尔文 Kelvin''' (1851)<font color = 'red'><s>陈述</s></font><font color = 'blue'>表述</font>，以及康斯坦丁·卡拉西奥多里 Constantin Carathéodory(1909)在'''<font color = '#ff8000'>公理化热力学axiomatic thermodynamics</font>'''中的<font color = 'red'><s>陈述</s></font><font color = 'blue'>表述</font>。这些<font color = 'red'><s>陈述</s></font><font color = 'blue'>表述</font>用一般的物理术语来描述定律，引用某些过程的不可能性。克劳修斯和开尔文表述被证明是等价的。<br />
--[[用户:厚朴|厚朴]]（[[用户讨论:厚朴|讨论]]）statement的表述需要统一，同样的statement by Rudolf Clausius这里需要统一开尔文 克劳修斯 Kelvin Clausius<br />
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===Carnot's principle卡诺原理===<br />
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The historical origin<ref>[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986).</ref> of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a [[Carnot heat engine]], fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the [[caloric theory]] of heat was seriously considered, before the recognition of the [[first law of thermodynamics]], and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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The historical origin of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a Carnot heat engine, fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the caloric theory of heat was seriously considered, before the recognition of the first law of thermodynamics, and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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热力学第二定律的历史起源是'''卡诺原理Carnot's principle'''。<br />
它指的是'''<font color = '#ff8000'>卡诺热机Carnot heat engine</font>'''的一个循环，<font color = 'red'><s>它</s></font><font color = 'blue'>卡诺热机</font>以'''<font color = '#ff8000'>准静态quasi-static</font>'''的极限慢速运转，因此热<s>量</s>和功在子系统之间进行传递，子系统总是处于它们自己内部的热力学平衡状态。<br />
卡诺热机是研究热机效率的工程师特别感兴趣的理想装置。<br />
当卡诺发现卡诺原理时，'''<font color = '#ff8000'>热量理论caloric theory of heat</font>'''还没有得到重视，热力学第一定律还没有得到承认，熵的概念还没有数学表达。<br />
根据第一定律的解释，它在物理上等同于热力学第二定律，并沿用至今。<br />
在热力学第一定律被发现之前，卡诺最初的论点是从热量理论的观点出发的。<br />
下面是他书中的一些例子：<br />
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::...''wherever there exists a difference of temperature, motive power can be produced.''<ref>Carnot, S. (1824/1986), p. 51.</ref><br />
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...wherever there exists a difference of temperature, motive power can be produced.<br />
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……只要有温差，就能产生动力。<br />
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::The production of motive power is then due in steam engines not to an actual consumption of caloric, but ''to its transportation from a warm body to a cold body ...''<ref>Carnot, S. (1824/1986), p. 46.</ref><br />
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The production of motive power is then due in steam engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body ...<br />
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动力的产生不是由于蒸汽机实际消耗的热量，而是由于它从一个较热的物体<font color = 'red'><s>运输</s></font><font color = 'blue'>转移</font>到一个较冷的物体……。<br />
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::''The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.''<ref>Carnot, S. (1824/1986), p. 68.</ref><br />
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The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.<br />
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热<s>量</s>的原动力与实现热的媒介无关，<font color = 'red'><s>热量的数量</s></font><font color = 'blue'>热的量</font>完全取决于两个物体之间的温度，最后是热量的传递。<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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用现代术语<font color = 'red'><s>来说</s></font>，卡诺原理可能更为准确:<br />
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::The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<ref>[[Clifford Truesdell|Truesdell, C.]] (1980), Chapter 5.</ref><ref>Adkins, C.J. (1968/1983), pp. 56–58.</ref><ref>Münster, A. (1970), p. 11.</ref><ref>Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998), pp.67–75.</ref><ref>Lebon, G., Jou, D., Casas-Vázquez, J. (2008), p. 10.</ref><ref>Eu, B.C. (2002), pp. 32–35.</ref><br />
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The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<br />
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准静态卡诺循环或可逆卡诺循环的效率只取决于两种热源的温度，而且无论工作物质是什么，效率是相同的。<font color = 'red'><s>只有</s></font><font color = 'blue'>使用</font>两个热源与工作物质交换热源（一个高温热源温度<math>T_1</math>和一个低温热源温度<math>T_2</math>)的卡诺热机是最有效的热机。<br />
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'''<font color="#32CD32">使用两个热源与工作物质交换热源（一个高温热源温度<math>T_1</math>和一个低温热源温度<math>T_2</math>)的卡诺热机是最有效的热机。 A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.</font>''' <br />
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--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）最后一句进行了补充 意译<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 我没太看懂卡诺表述，所以不能确定这里的补充是否正确。可以再看一下“与工作物质交换热源”部分的表述是否准确<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）如果不确定，可以考虑将这句话标注为疑难句；或直接按原文直译<br />
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===Clausius statement克劳修斯表述===<br />
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The German scientist [[Rudolf Clausius]] laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work.{{sfnp|Clausius|1850}} His formulation of the second law, which was published in German in 1854, is known as the ''Clausius statement'':<br />
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The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work. His formulation of the second law, which was published in German in 1854, is known as the Clausius statement:<br />
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1850年，德国科学家'''鲁道夫·克劳修斯 Rudolf Clausius''' 通过研究热传递和功之间的关系，为热力学第二定律<font color = 'red'><s>实验室</s></font>奠定了基础。他在1854年用德语发表的论文中所提及的热力学第二定律定义被称为克劳修斯表述:<br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.{{sfnp|Clausius|1854|p=86}}</blockquote><br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.</blockquote><br />
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不可能把热<s>量</s>从低温物体传递到高温物体而不产生其他影响。 <br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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克劳修斯的表述使用了“'''<font color = '#ff8000'>热通道Passage Of Heat</font>'''”的概念。<font color = 'red'><s>正如通常在热力学的讨论，这意味着“能量作为热的净转移” ，而不是另一种形式上的"分摊转账"或其他。</s></font><font color = 'blue'><br />
在热力学的讨论中，通常这意味着“能量作为热的形式的净转移” ，而不是指其他方式的转移。</font><br />
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）不太理解分摊转账contributory transfers <br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 我觉得应该是说，energy as '''heat''' 而不是 as其他方式的转移<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）我认为这里不要把contributory transfer看作一个专有名词去理解，从contributory的本意“A contributory factor of a problem or accident is one of the things which caused it to exist or happen.”来理解即可，感觉嘉树理解的是对的。<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of [[refrigeration]], for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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如果不对系统<font color = 'red'><s>进行外部功</s></font><font color = 'blue'>外部做功</font>，热<font color = 'red'><s>量就</s></font>不能自发地从冷区流向热区，这一点从制冷的普通经验中也可以看出。在冰箱中，<font color = 'red'><s>热量从冷到热，但只有在外部媒介——制冷系统的强制作用下才会发生变化。</s></font><font color = 'blue'>只有在外部媒介也就是制冷系统的强制作用下热才会从冷区流到热区。</font><br />
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===Kelvin statements开尔文表述===<br />
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[[William Thomson, 1st Baron Kelvin|Lord Kelvin]] expressed the second law in several wordings.<br />
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Lord Kelvin expressed the second law in several wordings.<br />
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'''<font color = '#ff8000'>开尔文勋爵 Lord Kelvin</font>''' <font color = 'red'><s>用几个字表达</s></font><font color = 'blue'>表述</font>了热力学第二定律。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 感觉”用几个字“怪怪的，所以删掉了<br />
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::It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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<font color = 'red'><s>无法从单一热源取热使其完全转化为有用功而不对环境产生影响。</s></font><font color = 'blue'>不可能从单一热源取热使之完全转换为有用的功而不产生其他影响。</font><br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 改的内容和原文内容不太对应。这个摘自百度百科，另参见 严子浚(1994). 对热力学第二定律开尔文表述的一点讨论, 大学物理: 13(2).<br />
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::It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.{{sfnp|Thomson|1851}}<br />
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It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.<br />
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<font color = 'red'><s>通过无生命物质的作用，不可能通过将物质的任何部分冷却到低于周围物体最冷的温度来产生机械效应。</s></font><font color = 'blue'>不可能通过无生命物质的作用，将物质的任何部分冷却到低于周围物体最低的温度并产生'''<font color = '#ff8000'>机械效应 Mechanical Effect</font>'''。</font><br />
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--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）无生命物质可以意译为机器吗<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 我不太懂这个，不过感觉最好不要吧。<br />
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===Equivalence of the Clausius and the Kelvin statements克劳修斯和开尔文表述的等价性===<br />
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[[Image:Deriving Kelvin Statement from Clausius Statement.svg|thumb|Derive Kelvin Statement from Clausius Statement从克劳修斯表述推导出开尔文表述]]<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed [[Carnot engine]] as shown by the figure. The [[Heat engine#Efficiency|efficiency]] of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the [[first law of thermodynamics]], as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the figure. The efficiency of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat from <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the first law of thermodynamics, as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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假设有一个热机违反了开尔文定理: 也就是说，这个热机以循环的方式吸收热<s>量</s>并将其<font color = 'blue'>完全</font>转化为功，而<font color = 'blue'>且</font>不产生任何影响。现<font color = 'blue'>在将其</font>与反向卡诺<font color = 'red'><s>发动</s></font>机<font color = 'red'><s>相对应</s></font><font color = 'blue'>比较</font>，如图所示。<br />
普通热机的效率为η，反向热机的效率为1/η。这对<font color = 'red'><s>热机的联合作用为</s></font><font color = 'blue'>联合热机的净效应和唯一效应是</font><math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> 将热<s>量</s>从较冷热源<font color = 'blue'>转移</font>到较热热源，这违反了克劳修斯表述。（这是能量守恒定律的结果，因为系统的总能量保持不变<math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>，所以<math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math>。）因此，违反开尔文表述意味着违反克劳修斯表述，即克劳修斯表述暗示了开尔文表述。我们可以用类似的方式证明开尔文表述暗示了克劳修斯表述，因此两者是等价的。<br />
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===Planck's proposition普朗克<font color = 'red'><s>假设</s></font><font color = 'blue'>命题</font>===<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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普朗克提出了直接来自经验的如下命题。这有时被认为是他对热力学第二定律的表述，但他认为这是热力学第二定律推导的起点。<br />
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::It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 86.</ref><ref>Roberts, J.K., Miller, A.R. (1928/1960), p. 319.</ref><br />
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It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<br />
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<font color = 'red'><s>建造一台发动机是不可能的，这台发动机将在一个完整的循环中工作，</s></font><font color = 'blue'>不可能建造一台发动机，使其在一个完整的循环中工作，</font>并且除了提高重量和冷却热储以外不会产生任何效果。<br />
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===Relation between Kelvin's statement and Planck's proposition 开尔文表述与普朗克命题的关系===<br />
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It is almost customary in textbooks to speak of the "[[Kelvin-Planck statement]]" of the law, as for example in the text by [[Dirk ter Haar|ter Haar]] and [[Harald Wergeland|Wergeland]].<ref>[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966), p. 17.</ref><br />
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It is almost customary in textbooks to speak of the "Kelvin-Planck statement" of the law, as for example in the text by ter Haar and Wergeland.<br />
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教科书中几乎总是用“'''<font color = '#ff8000'>开尔文-普朗克表述 Kelvin-Planck Statement</font>'''”来称呼该定律，例如'''<font color = '#ff8000'>德克·特哈尔 Diek ter Haar</font>''' 和'''<font color = '#ff8000'>哈拉尔德·沃格兰 Harald Wergeland</font>''' 就是这样表述的。<br />
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The '''Kelvin–Planck statement''' (or the ''heat engine statement'') of the second law of thermodynamics states that <br />
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The Kelvin–Planck statement (or the heat engine statement) of the second law of thermodynamics states that <br />
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热力学第二定律的开尔文-普朗克表述（或称“'''<font color = '#ff8000'>热机表述 Heat Engine Statement</font>'''”）指出<br />
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::It is impossible to devise a [[thermodynamic cycle|cyclically]] operating device, the sole effect of which is to absorb energy in the form of heat from a single [[heat reservoir|thermal reservoir]] and to deliver an equivalent amount of [[Work (physics)|work]].<ref name="Rao">{{cite book|last=Rao|first=Y. V. C.|title=Chemical Engineering Thermodynamics|publisher=Universities Press|isbn=978-81-7371-048-3|page=158|year=1997}}</ref><br />
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It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.<br />
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设计一种唯一效果是从单一热源吸收热并提供等量的功的循环运行装置是不可能的。<br />
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===Planck's statement 普朗克表述===<br />
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Planck stated the second law as follows.<br />
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Planck stated the second law as follows.<br />
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普朗克表述第二定律如下。<br />
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::Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<ref name="Planck 100">[[Max Planck|Planck, M.]] (1897/1903), p. 100.</ref><ref name="Planck 463">[[Max Planck|Planck, M.]] (1926), p. 463, translation by Uffink, J. (2003), p. 131.</ref><ref name="Roberts & Miller 382">Roberts, J.K., Miller, A.R. (1928/1960), p. 382. This source is partly verbatim from Planck's statement, but does not cite Planck. This source calls the statement the principle of the increase of entropy.</ref><br />
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Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<br />
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自然界中发生的任一过程都是沿参与其中的所有物体的熵之和增加的方向进行的。在极限情况中，即对于可逆过程，熵的总和保持不变。<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for ''irreversible phenomena''.<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for irreversible phenomena.<br />
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与普朗克表述非常相似的是'''<font color = '#ff8000'>乌伦贝克 Uhlenbeck</font>'''和'''<font color = '#ff8000'>福特 Ford</font>'''关于不可逆现象的表述。<br />
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::... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<ref>[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963), p. 16.</ref><br />
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... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<br />
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……在从一个平衡态到另一个平衡态的不可逆或自发的变化中（例如，当两个物体 A 和 B 接触时的温度平衡过程），熵总是增加。<br />
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===Principle of Carathéodory 卡拉西奥多里原理===<br />
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<!-- [[Caratheodory's principle]] redirects here --><br />
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<!-- Caratheodory's principle redirects here --><br />
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—— caratheodory 的原理在这里重新定向——<br />
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[[Constantin Carathéodory]] formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<ref>[[Constantin Carathéodory|Carathéodory, C.]] (1909).</ref><br />
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Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<br />
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'''<font color = '#ff8000'>康斯坦丁·卡拉西奥多里 Constantin Carathéodory</font>'''在纯数学公理的基础上进行了热力学'''<font color = '#ff8000'>公理化formulated</font>'''阐明。他对第二定律的陈述被称为'''<font color = '#ff8000'>卡拉西奥多里原理 Principle of Carathéodory</font>'''，可以这样表述:<br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.<ref>Buchdahl, H.A. (1966), p. 68.</ref></blockquote><br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.</blockquote><br />
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在绝热封闭系统的任意状态 S 附近，总有从 S 出发不可达的状态。<br />
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With this formulation, he described the concept of [[adiabatic accessibility]] for the first time and provided the foundation for a new subfield of classical thermodynamics, often called [[Ruppeiner geometry|geometrical thermodynamics]]. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic [[process function]], in other words, <math>\delta Q=TdS</math>.<ref name="Sychev1991">{{cite book |last=Sychev |first=V. V. |title=The Differential Equations of Thermodynamics |year=1991 |publisher=Taylor & Francis |isbn=978-1-56032-121-7}}</ref> {{clarify|date=February 2014}}<br />
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With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, <math>\delta Q=TdS</math>. <br />
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通过这个阐明，他首次描述了'''<font color = '#ff8000'>绝热可达性 Adiabatic Accessibility</font>'''的概念，并为经典热力学的一个新的子领域，即通常所说的'''<font color = '#ff8000'>几何热力学 Geometrical Thermodynamics</font>'''<font color = 'red'><s>奠定了基础</s></font>。由卡拉西奥多里原理可以推出，作为热的能量的准静态转移是一个'''<font color="#ff8000"> 完整的过程函数holonomic process function</font>'''即<math>\delta Q=TdS</math>。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）准静态转移的热量值是一个可积过程函数 存疑<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）本句的修改供讨论。可积函数的英文应该是Integrable function。但是什么是完整的函数存疑，是连续的函数？<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）这里建议标注一下'''<font color="#ff8000"> 完整的过程函数 holonomic process function</font>'''<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）奠定了基础为何要删????????????????????????????????????????<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.<ref name="Munster 45">Münster, A. (1970), p. 45.</ref>{{sfnp|Lieb|Yngvason|1999|p=49}}<ref name="Planck 1926">[[Max Planck|Planck, M.]] (1926).</ref><ref>Buchdahl, H.A. (1966), p. 69.</ref> {{clarify|date=February 2014}}<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium. <br />
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尽管在教科书几乎惯称卡拉西奥多里原理也是第二定律的一种表述，并认为其与克劳修斯表述或开尔文-普朗克表述等价，但事实并非如此。为了得到第二定律的所有内容，需要对卡拉西奥多里原理补充普朗克表述，即等量功总是增加一个最初处于自身内部热力学平衡的封闭系统的内部能量。<br />
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===Planck's principle 普朗克原理===<br />
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In 1926, [[Max Planck]] wrote an important paper on the basics of thermodynamics.<ref name="Planck 1926"/><ref>Uffink, J. (2003), pp. 129–132.</ref> He indicated the principle<br />
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In 1926, Max Planck wrote an important paper on the basics of thermodynamics. He indicated the principle<br />
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1926年，'''<font color = '#ff8000'>马克斯·普朗克 Max Planck</font>'''写了一篇关于热力学基础的重要论文。他指出了以下原理<br />
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::The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<ref name="Munster 45"/>{{sfnp |Lieb|Yngvason|1999|p=49}}<br />
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The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<br />
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一个封闭系统的内部能量因绝热过程增加，在整个过程中，系统的体积保持不变。<br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work."<ref>[[Clifford Truesdell|Truesdell, C.]], Muncaster, R.G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 15.</ref> Planck wrote: "The production of heat by friction is irreversible."<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 81.</ref><ref>[[Max Planck|Planck, M.]] (1926), p. 457, Wikipedia editor's translation.</ref><br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work." Planck wrote: "The production of heat by friction is irreversible."<br />
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这个公式没有提到热，没有提到温度，甚至没有提到熵，也不一定隐含地依赖于这些概念，但它暗示了第二定律的内容。一个密切相关的表述为，“摩擦力从来不做正功。”普朗克写道: “摩擦生热是不可逆的。”<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above.<ref>Lieb, E.H., Yngvason, J. (2003), p. 149.</ref> It is relevant that for a system at constant volume and [[Mole (unit)|mole numbers]], the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above. It is relevant that for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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这个普朗克定理没有提到熵，是用物理术语来表述的。它与上面给出的开尔文表述密切相关。相关的是，对于恒定体积和摩尔数的系统，熵是内能的单调函数。然而，普朗克的这个原理实际上并不是普朗克对第二定律的首选表述（见前面小节），而是依赖于熵的概念。<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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Borgnakke 和 Sonntag 提出的以下表述在某种意义上是对普朗克原理的补充。他们没有将其作为第二定律的完整表述:<br />
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::... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<ref>Borgnakke, C., Sonntag., R.E. (2009), p. 304.</ref><br />
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... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<br />
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……只有一种方法可以减少（封闭）系统的熵，将热从系统中转移出去。<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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与普朗克之前提出的原理不同，这一原理明确地用熵的变化来表示。从系统中去除物质也可以减少系统的熵。<br />
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===Statement for a system that has a known expression of its internal energy as a function of its extensive state variables 一个其内能有已知表达式（其扩展状态变量的函数）的系统的表述===<br />
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The second law has been shown to be equivalent to the [[internal energy]] ''U'' being a weakly [[convex function]], when written as a function of extensive properties (mass, volume, entropy, ...).<ref>{{cite book |last1=van Gool |first1=W. |last2=Bruggink |first2=J.J.C. (Eds) |url= |title=Energy and time in the economic and physical sciences |publisher=North-Holland |year=1985 |pages=41–56 |quote= |isbn=978-0-444-87748-2}}</ref><ref>{{Cite journal | last1 = Grubbström | first1 = Robert W. | doi = 10.1016/j.apenergy.2007.01.003 | title = An Attempt to Introduce Dynamics Into Generalised Exergy Considerations| journal = Applied Energy| volume = 84| issue = 7–8 | pages = 701–718 | year = 2007}}</ref> {{clarify|date=February 2014}}<br />
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The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...). <br />
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'''<font color="#32CD32">写成广泛性质（质量，体积，熵……）的函数时，第二定律被证明等价于弱凸函数内能 U。The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...).</font>'''<br />
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==Corollaries 推论==<br />
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===Perpetual motion of the second kind 第二类永动机===<br />
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{{main article|Perpetual motion}}<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of [[first law of thermodynamics]] by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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在第二定律建立之前，许多志在发明永动机的人试图通过提取环境的巨大内能作为机器动力，以突破'''热力学第一定律 First Law of Thermodynamics''' 的限制。这种机器被称为“'''第二类永动机 Perpetual Motion Machine of the Second Kind'''”。热力学第二定律表明这种机器是不可能的。<br />
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===Carnot theorem 卡诺定理===<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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'''卡诺定理 Carnot's Theorem'''（1824）是一条限制任何可能的热机的最大效率的原理。效率完全取决于热源中热和冷之间的温差。 卡诺定理指出:<br />
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*All irreversible heat engines between two heat reservoirs are less efficient than a [[Carnot engine]] operating between the same reservoirs.<br />
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*All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs.<br />
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* 所有<font color = 'blue'>在两个热源之间工作的</font>不可逆热机的效率低于在<font color = 'red'><s>两相同</s></font><font color = 'blue'>同一个</font>热源之间工作的'''<font color = '#ff8000'>卡诺热机 Carnot Engine</font>'''。<br />
* 所有<font color = 'blue'>在两个热源之间工作的</font>可逆热机的效率与在<font color = 'red'><s>两相同</s></font><font color = 'blue'>同一个</font>热源之间工作的卡诺机相等。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）改成同一个正确吗??????????????????????????<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as [[thermodynamic reversibility]]. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the [[Carnot cycle]]'s reversibility and was condemned to be less efficient.<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the Carnot cycle's reversibility and was condemned to be less efficient.<br />
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在他的理想模型中，热转化为功的过程可以通过逆转循环的运动而恢复，这个概念后来被称为'''<font color = '#ff8000'>热力学可逆性 Thermodynamic Reversibility</font>'''。然而，卡诺进一步假定，一些热<s>量</s>损失了，并没有转化为机械功。因此，没有一个真实的热机能够实现'''<font color = '#ff8000'>卡诺循环 Carnot Cycle</font>'''的可逆性，并且被认为效率较低。<br />
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Though formulated in terms of caloric (see the obsolete [[caloric theory]]), rather than [[entropy]], this was an early insight into the second law.<br />
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Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law.<br />
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该理论尽管是用热<s>量</s>表述的（见被取代的'''<font color = '#ff8000'>热质说</font>'''），而不是熵，但是它是对第二定律的早期认识。<br />
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===Clausius inequality 克劳修斯不等式===<br />
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The [[Clausius theorem]] (1854) states that in a cyclic process<br />
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The Clausius theorem (1854) states that in a cyclic process<br />
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'''<font color = '#ff8000'>克劳修斯定理 Clausius Theorem</font>'''（1854）指出，在一个循环的过程中<br />
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: <math>\oint \frac{\delta Q}{T} \leq 0.</math><br />
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The equality holds in the reversible case<ref>[http://scienceworld.wolfram.com/physics/ClausiusTheorem.html ''Clausius theorem''] at [[Wolfram Research]]</ref> and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function [[entropy]]. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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The equality holds in the reversible case and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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等号在可逆情况下成立，严格不等号在不可逆情况下成立。可逆情况下引入状态函数熵。'''<font color="#32CD32">这是因为在循环过程中，状态功能的变化相对于状态功能为零。This is because in cyclic processes the variation of a state function is zero from state functionality.</font>'''<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）末句存疑<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 我也没看懂，按照字面意思修改了一下<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）感觉修改后的句子还是怪怪的，这里标成疑难句吧 '''<font color="#32CD32">这是因为在循环过程中，状态功能的变化相对于状态功能为零。This is because in cyclic processes the variation of a state function is zero from state functionality.</font>'''<br />
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===Thermodynamic temperature 热力学温度===<br />
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{{main article|Thermodynamic temperature}}<br />
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For an arbitrary heat engine, the efficiency is:<br />
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For an arbitrary heat engine, the efficiency is:<br />
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对于任意热机，效率为:<br />
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: <math>\eta = \frac {W_n}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H} \qquad (1)</math><br />
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where ''W''<sub>n</sub> is for the net work done per cycle. Thus the efficiency depends only on q<sub>C</sub>/q<sub>H</sub>.<br />
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其中 ''W''<sub>n</sub> 表示每个循环所做的净功。因此效率只取决于 q<sub>C</sub>/q<sub>H</sub>。<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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卡诺定理指出，所有在相同热源间运行的可逆热机的效率是相同的。<br />
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Thus, any reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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Thus, any reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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因此，任何在温度''T''<sub>1</sub>和''T''<sub>2</sub>之间运行的可逆热机必须具有相同的效率，也就是说，效率只是温度的函数:<br />
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<math>\frac{q_C}{q_H} = f(T_H,T_C)\qquad (2).</math><br />
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In addition, a reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between ''T''<sub>1</sub> and another (intermediate) temperature ''T''<sub>2</sub>, and the second between ''T''<sub>2</sub> and''T''<sub>3</sub>. This can only be the case if<br />
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In addition, a reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between T<sub>1</sub> and another (intermediate) temperature T<sub>2</sub>, and the second between T<sub>2</sub> andT<sub>3</sub>. This can only be the case if<br />
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另外，在温度 ''T''<sub>1</sub> 和 ''T''<sub>3</sub> 之间工作的可逆热机必须具有与由分别在温度''T''<sub>1</sub> 和（中间）温度 ''T''<sub>2</sub>之间、在 ''T''<sub>2</sub> 和 ''T''<sub>3</sub> 之间的两个循环组成的系统效率相同。只有下式成立才会出现这种情况:<br />
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: <math>f(T_1,T_3) = \frac{q_3}{q_1} = \frac{q_2 q_3} {q_1 q_2} = f(T_1,T_2)f(T_2,T_3).</math><br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the [[triple point]] of water. Then for any ''T''<sub>2</sub> and ''T''<sub>3</sub>,<br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the triple point of water. Then for any T<sub>2</sub> and T<sub>3</sub>,<br />
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现在考虑如下情形，<math>T_1</math> 是一个固定的参考温度: 水的<font color="#ff8000">'''<font color = '#ff8000'>三相点 Triple Point</font>'''</font>的温度。则对于任意 T<sub>2</sub> 和 T<sub>3</sub>,<br />
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: <math>f(T_2,T_3) = \frac{f(T_1,T_3)}{f(T_1,T_2)} = \frac{273.16 \cdot f(T_1,T_3)}{273.16 \cdot f(T_1,T_2)}.</math><br />
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Therefore, if thermodynamic temperature is defined by<br />
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Therefore, if thermodynamic temperature is defined by<br />
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因此，如果热力学温度被定义为<br />
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: <math>T = 273.16 \cdot f(T_1,T) \,</math><br />
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then the function ''f'', viewed as a function of thermodynamic temperature, is simply<br />
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then the function f, viewed as a function of thermodynamic temperature, is simply<br />
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那么函数 ''f'' 作为热力学温度的函数，为<br />
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: <math>f(T_2,T_3) = \frac{T_3}{T_2},</math><br />
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and the reference temperature ''T''<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be used{{snd}}the choice here corresponds to the [[Kelvin]] scale.)<br />
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and the reference temperature T<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be used the choice here corresponds to the Kelvin scale.)<br />
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参考温度 ''T''<sub>1</sub> 的值为273.16。（任何参考温度和任何正值均可用——此处的选择对应开尔文标度。）<br />
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===Entropy 熵===<br />
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{{main article|Entropy (classical thermodynamics)}}<br />
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According to the [[Clausius theorem|Clausius equality]], for a reversible process<br />
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According to the Clausius equality, for a reversible process<br />
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根据克劳修斯等式，对可逆过程有<br />
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: <math>\oint \frac{\delta Q}{T}=0</math><br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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这意味着线积分 <math>\int_L \frac{\delta Q}{T}</math> 对于可逆过程是路径无关的。<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer<ref name="Schmidt-Rohr 14" /> satisfies<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer satisfies<br />
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所以我们可以定义一个叫做熵的状态函数 S，对于可逆过程或者纯热传递满足<br />
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: <math>dS = \frac{\delta Q}{T} \!</math><br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the [[third law of thermodynamics]], which states that ''S'' = 0 at [[absolute zero]] for perfect crystals.<br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the third law of thermodynamics, which states that S = 0 at absolute zero for perfect crystals.<br />
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据此，只有对上述公式进行积分，才能得到熵的差值。为了获得绝对值，我们需要<font color="#ff8000">'''热力学第三定律 Third Law of Thermodynamics'''</font>，它指出<font color="#ff8000">'''绝对零度 Absolute Zero'''</font>下完美晶体的 ''S'' = 0。<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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对于任意不可逆过程，由于熵是一个状态函数，我们总是可以将初始状态和最终状态与一个虚拟的可逆过程联系起来，并在这条路径上积分以计算熵的差值。<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the [[Clausius inequality]] on this loop,<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the Clausius inequality on this loop,<br />
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现在把可逆过程逆过来，将其与上述不可逆过程结合。把克劳修斯不等式应用到这个循环,<br />
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: <math>-\Delta S+\int\frac{\delta Q}{T}=\oint\frac{\delta Q}{T}< 0</math><br />
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Thus,<br />
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Thus,<br />
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故,<br />
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: <math>\Delta S \ge \int \frac{\delta Q}{T} \,\!</math><br />
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where the equality holds if the transformation is reversible.<br />
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where the equality holds if the transformation is reversible.<br />
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如果变换可逆，等号成立。<br />
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Notice that if the process is an [[adiabatic process]], then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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Notice that if the process is an adiabatic process, then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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注意，若该过程是一个<font color="#ff8000">'''绝热过程 Adiabatic Process'''</font>，则<math>\delta Q=0</math>，故<math>\Delta S\ge 0</math>。<br />
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===Energy, available useful work===<br />
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===Energy, available useful work===<br />
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能量，<font color = 'red'><s>有用的工作</s></font>'''<font color = '#32CD32'>可用的有用工作available useful work</font>'''<br />
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{{See also|Exergy}}<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an ''unlimited'' heat reservoir at temperature ''T<sub>R</sub>'' and pressure ''P<sub>R</sub>'' {{snd}}so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain ''T<sub>R</sub>''; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain ''P<sub>R</sub>''.<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an unlimited heat reservoir at temperature T<sub>R</sub> and pressure P<sub>R</sub> so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain T<sub>R</sub>; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain P<sub>R</sub>.<br />
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考虑将第二定律应用于孤立系统(称为整体系统或宇宙)<font color = 'red'><s>的情形，</s></font>是一个重要<font color = 'red'><s>的、</s></font><font color = 'blue'>而</font>具有启发性的理想情形。该系统由两部分组成：<font color = 'red'>感兴趣的<s></s></font>研究关心的<font color = 'blue'></font>子系统和子系统的周围环境。因为<font color = 'red'><s>环境被想象的</s></font><font color = 'blue'>想象的环境</font>非常<font color = 'red'><s>巨</s></font>大，<font color = 'blue'>以至于</font>它们可以被视为一个温度为 T<sub>R</sub> 且压力为 P<sub>R</sub> 的无限<font color = 'red'><s>蓄热器</s></font><font color = 'blue'>热源</font>，<font color = 'blue'>因此</font>无论有多少热<font color = 'red'><s>量</s></font>被转移到（或来自）子系统，周围的温度将保持T<sub>R</sub>；无论子系统的体积膨胀（或收缩）有多大，周围环境的压力都将保持不变。<br />
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Whatever changes to ''dS'' and ''dS<sub>R</sub>'' occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy ''S<sub>tot</sub>'' of the isolated total system must not decrease:<br />
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Whatever changes to dS and dS<sub>R</sub> occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy S<sub>tot</sub> of the isolated total system must not decrease:<br />
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无论子系统和周围环境<font color = 'red'><s>的 ''dS''和''dS<sub>R</sub>'' 发生</s></font><font color = 'blue'>单独地发生什么熵的变化 ''dS''和dS<sub>R</sub></font>，根据第二定律，孤立总体系统的熵S<sub>tot</sub>不能减小。<br />
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: <math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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<math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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<math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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According to the [[first law of thermodynamics]], the change ''dU'' in the internal energy of the sub-system is the sum of the heat ''δq'' added to the sub-system, ''less'' any work ''δw'' done ''by'' the sub-system, ''plus'' any net chemical energy entering the sub-system ''d ∑μ<sub>iR</sub>N<sub>i</sub>'', so that:<br />
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According to the first law of thermodynamics, the change dU in the internal energy of the sub-system is the sum of the heat δq added to the sub-system, less any work δw done by the sub-system, plus any net chemical energy entering the sub-system d ∑μ<sub>iR</sub>N<sub>i</sub>, so that:<br />
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根据<font color = 'red'><s>能量守恒定律</s></font><font color = 'blue'>热力学第一定律</font>，子系统<font color = 'red'><s>内部能量</s></font><font color = 'blue'>内能</font>的变化 dU 是<font color = 'red'><s>子系统内部能量 q 的总和</s></font>加在子系统上的热δq的和<font color = 'blue'></font>，减去子系统所做的任何功w，再加上进入子系统的任何净化学能 d ∑μ<sub>iR</sub>N<sub>i</sub>，因此<br />
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: <math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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<math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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<math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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where μ<sub>iR</sub> are the [[chemical potential]]s of chemical species in the external surroundings.<br />
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where μ<sub>iR</sub> are the chemical potentials of chemical species in the external surroundings.<br />
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其中μ<sub>iR</sub>是外部环境中<font color = 'red'><s>化学物类</s></font>'''<font color = '#ff8000'>化学形态 chemical species</font>'''的化学势。<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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现在热<font color = 'red'><s>量</s></font>离开<font color = 'red'><s>储存器</s></font><font color = 'blue'>热源</font>进入子系统是<br />
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: <math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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<math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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数学△ qtr (- dS r) le tr dS / math<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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在这个过程中，首先使用了经典热力学中熵的定义(在统计热力学中，熵变、温度和吸收热量之间的关系'''<font color = '#32CD32'>可以将其推导出来 can be derived</font>''') ，然后从上面的公式可以推导出第二定律的不等式。<br />
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It therefore follows that any net work ''δw'' done by the sub-system must obey<br />
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It therefore follows that any net work δw done by the sub-system must obey<br />
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因此，子系统所做的任何净功δw必须服从<br />
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: <math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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<math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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数学增量 w le-dU + tds + sum mu { iR } dN i，/ math<br />
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It is useful to separate the work ''δw'' done by the subsystem into the ''useful'' work ''δw<sub>u</sub>'' that can be done ''by'' the sub-system, over and beyond the work ''p<sub>R</sub> dV'' done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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It is useful to separate the work δw done by the subsystem into the useful work δw<sub>u</sub> that can be done by the sub-system, over and beyond the work p<sub>R</sub> dV done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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将子系统所做的功δw划分为子系统可以完成的有用功δw<sub>u</sub> ，除了子系统在周围外部压力下膨胀所做的功p<sub>R</sub> dV外，<font color = 'red'><s>还可以</s></font>给出以下可用功（<font color = 'red'><s>放射本能</s></font>'''<font color = 'ff8000'>有用能 exergy</font>'''）关系式:<br />
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: <math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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<math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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<math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the ''availability'' or ''[[exergy]]'' ''E'' of the subsystem,<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the availability or exergy E of the subsystem,<br />
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为方便起见，可以把右边定义为热力学势的精确导数，称为子系统的'''<font color = 'ff8000'>可用性 Availability </font>'''或<font color = 'red'><s>放射本能E</s></font>'''<font color = 'ff8000'>有用能 exergy</font>'''<br />
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: <math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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<math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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<math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
<br />
<font color="#32CD32">因此，第二定律意味着，对于任何可以简单地被视为分为一个子系统和一个与之接触的无限温度和压力<font color = 'red'><s>储存器</s></font> '''<font color="#ff8000">热源 Reservoir</font>''' 的过程，The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,</font><br />
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: <math> dE + \delta w_u \le 0 \, </math><br />
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<math> dE + \delta w_u \le 0 \, </math><br />
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<math> dE + \delta w_u \le 0 \, </math><br />
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i.e. the change in the subsystem's exergy plus the useful work done ''by'' the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done ''on'' the system) must be less than or equal to zero.<br />
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i.e. the change in the subsystem's exergy plus the useful work done by the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done on the system) must be less than or equal to zero.<br />
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也就是。子系统<font color = 'red'><s>(火用)</s></font><font color = 'blue'>有用能</font>的变化加上子系统所做的有用功(或者，子系统<font color = 'red'><s>(火用)</s></font><font color = 'blue'>有用能</font>的变化<font color = 'red'><s>不包括任何功，除了压力贮存器所做的功之外，在系统中所做的功</s></font><font color = 'blue'>减去除了'''<font color = '#ff8000'>压力热源</font>'''外任何对系统做的功，</font>)必须小于或等于零。<br />
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In sum, if a proper ''infinite-reservoir-like'' reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in ''E'' for an irreversible process and no change for a reversible process.<br />
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In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in E for an irreversible process and no change for a reversible process.<br />
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总之，如果选择一个合适的类似于无限库的参考状态作为现实世界中的系统环境，<br />
<font color = 'red'><s>那么第二定律预测的不可逆性的 e 值会减少，而可逆过程的 e 值不会变化。</s></font><font color = 'blue'>则第二定律预测不可逆过程的''E''值减少，可逆过程的''E''值不变。</font><br />
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: <math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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<math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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<math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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This expression together with the associated reference state permits a [[design engineer]] working at the macroscopic scale (above the [[thermodynamic limit]]) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (''Also, see [[process engineer]]''). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (''See [[Exergy efficiency|second law efficiency]]''.)<br />
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This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (Also, see process engineer). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (See second law efficiency.)<br />
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这个表达式和相关的参考状态允许<font color = 'red'><s>在宏观尺度（高于热力学极限）下工作的使用第二定律</s></font><font color = 'blue'>设计工程师在宏观尺度（高于热力学极限）下使用第二定律</font>，而无需直接测量或考虑整个孤立系统中的熵变。（另见'''<font color = '#ff8000'>工艺工程师process engineer</font>'''）。<font color = 'red'><s>考虑到这些变化，假设</s></font><font color = 'blue'>这些变化已经在假设中被考虑到，该假设认为</font>所考虑的系统可以在不改变参考状态的情况下与参考状态达到平衡。将其与可逆理想<font color = 'blue'>状态</font>进行比较，还可以找到一个过程或过程集合的效率（见'''<font color = '#ff8000'>第二定律效率second law efficiency</font>'''）<br />
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This approach to the Second Law is widely utilized in [[engineering]] practice, [[environmental accounting]], [[systems ecology]], and other disciplines.<br />
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This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines.<br />
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第二定律的这种方法被广泛应用于'''<font color = '#ff8000'>工程实践engineering practice</font>'''、'''<font color = '#ff8000'>环境会计environmental accounting</font>'''、'''<font color = '#ff8000'>系统生态学systems ecology</font>'''等其他学科。<br />
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===The second law in chemical thermodynamics===<br />
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化学热力学的第二定律<br />
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For a [[spontaneous process|spontaneous chemical process]] in a closed system at constant temperature and pressure without non-''PV'' work, the Clausius inequality Δ''S > Q/T<sub>surr</sub>'' transforms into a condition for the change in [[Gibbs free energy]] <br />
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For a spontaneous chemical process in a closed system at constant temperature and pressure without non-PV work, the Clausius inequality ΔS > Q/T<sub>surr</sub> transforms into a condition for the change in Gibbs free energy <br />
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对于一个<font color = 'blue'>恒温恒压</font>封闭系统中的自发化学过程，在没有 non-PV 功的情况下，<font color = 'red'><s>在恒定温度和压力下，将</s></font>克劳修斯不等式Δ''S > Q/T<sub>surr</sub>'' <font color = 'red'><s>转化为吉布斯自由能变化的条件</s></font><br />
<font color = 'blue'>由于'''<font color = '#ff8000'>吉布斯自由能Gibbs free energy</font>'''的变化而转化为：</font><br />
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: <math>\Delta G < 0 </math><br />
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<math>\Delta G < 0 </math><br />
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<math>\Delta G < 0 </math><br />
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or d''G'' < 0. For a similar process at constant temperature and volume, the change in [[Helmholtz free energy]] must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products.<ref name="Oxtoby8th"/> <ref name="MortimerBook"></ref> The chemical equilibrium condition at constant ''T'' and ''p'' without electrical work is d''G'' = 0.<br />
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or dG < 0. For a similar process at constant temperature and volume, the change in Helmholtz free energy must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products. The chemical equilibrium condition at constant T and p without electrical work is dG = 0.<br />
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或者 dG < 0。对于一个相似的<font color = 'blue'>恒温恒压</font>过程<font color = 'red'><s>，在恒定的温度和体积下</s></font>，'''<font color = '#ff8000'>亥姆霍兹自由能 Helmholtz free energy </font>'''的变化一定是负的， <math>\Delta A < 0 </math>。因此，<font color = 'blue'>一个负的</font>自由能(G 或 A)变化<font color = 'red'><s>的负值</s></font>是过程自发的必要条件。这是热力学第二定律在化学中最有用的形式，其中自由能的变化可以通过<font color = 'red'><s>列表的生成焓</s></font>'''<font color = '#ff8000'>表列生成焓 Tabulated Enthalpies of Formation</font>'''和反应物及产物的标准摩尔熵来计算。在<font color = 'red'><s>T 和 P </s></font><font color = 'blue'>温度和压力</font>不变的情况下，化学平衡条件是 dG = 0。<br />
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-- [[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） tabulated enthalpies of formation 表列生成焓？？<br />
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==History==<br />
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历史<br />
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{{See also|History of entropy}}<br />
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[[File:Sadi Carnot.jpeg|thumb|upright|Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique]].]]<br />
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Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique.]]<br />
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尼古拉·莱昂纳尔·萨迪·卡诺穿着传统的巴黎综合理工学院制服<br />
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The first theory of the conversion of heat into mechanical work is due to [[Nicolas Léonard Sadi Carnot]] in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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The first theory of the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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<font color = 'blue'>卡诺在1824年提出热</font><font color = 'red'><s>首先提出类热</s></font>转化为机械功的<font color = 'blue'>第一个</font>理论。他<font color = 'blue'>是第一个</font>正确认识到了转换效率取决于发动机和环境之间的温差<font color = 'blue'>的人</font>。<br />
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Recognizing the significance of [[James Prescott Joule]]'s work on the conservation of energy, [[Rudolf Clausius]] was the first to formulate the second law during 1850, in this form: heat does not flow ''spontaneously'' from cold to hot bodies. While common knowledge now, this was contrary to the [[caloric theory]] of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law during 1850, in this form: heat does not flow spontaneously from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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'''<font color="#ff8000">克劳修斯Rudolf Clausius </font>'''认识到'''<font color="#ff8000">焦耳James Prescott Joule</font>'''在能量守恒方面工作的重要性后，在1850年<font color = 'red'><s>第一个提出了第二定律</s></font><font color = 'blue'>提出了第二定律的第一个公式，在这个公式中</font>: 热不会自发地从冷物体流向热物体。虽然<font color = 'red'><s>现在的常识是这样的</s></font><font color = 'blue'>现在这是常识</font>，但是这与当时流行的热理论相反，当时的热理论认为热是一种流体。从这些他推断出了'''<font color="#ff8000"> 萨迪卡诺定律the principle of Sadi Carnot</font>'''和熵的定义(1865年)。<br />
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Established during the 19th century, the [[Kelvin-Planck statement|Kelvin-Planck statement of the Second Law]] says, "It is impossible for any device that operates on a [[cyclic process|cycle]] to receive heat from a single [[heat reservoir|reservoir]] and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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Established during the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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<font color = 'red'><s>19世纪提出的开尔文-普朗克第二定律(Kelvin-Planck)表示：“任何循环运行的设备都不可能从单个蓄热体接收热<s>量</s>并产生净功。”这被证明相当于克劳修斯的陈述。</s></font><font color = 'blue'>19世纪提出的开尔文-普朗克第二陈述(Kelvin-Planck)表示：“任何循环运行的设备都不可能从单个热源接收热并产生净功。”这被证明与克劳修斯的陈述等价。</font><br />
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The [[ergodic hypothesis]] is also important for the [[Boltzmann]] approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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'''<font color="#ff8000">遍历假设ergodic hypothesis</font>'''对'''<font color="#ff8000">玻尔兹曼方法Boltzmann approach</font>'''也很重要。<font color = 'blue'>遍历假设认为</font>在很长一段时间内，在具有相同能量的微观态相空间的某些区域所花费的时间与这个区域的体积成正比，即在很长一段时间内，所有可访问的微观状态<font color = 'blue'>出现/成立</font>的可能性都是一样的。<font color = 'red'><s>同样的，</s></font><font color = 'blue'>等价于说，</font>它表明时间平均值和统计集合的平均值是相同的。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 觉得加上出现/成立好一点？<br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a [[Macroscopic bodies|macroscopic system]]. This doctrine is obsolescent.<ref>Denbigh, K.G., Denbigh, J.S. (1985). ''Entropy in Relation to Incomplete Knowledge'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-25677-1}}, pp. 43–44.</ref><ref>Grandy, W.T., Jr (2008). ''Entropy and the Time Evolution of Macroscopic Systems'', Oxford University Press, Oxford, {{ISBN|978-0-19-954617-6}}, pp. 55–58.</ref><ref name=Lambert>[http://entropysite.oxy.edu Entropy Sites — A Guide] Content selected by [[Frank L. Lambert]]</ref><br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a macroscopic system. This doctrine is obsolescent.<br />
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克劳修斯还提出了一种传统的学说，他认为熵可以被理解为宏观系统中的'''<font color = '#ff8000'>分子“无序”molecular 'disorder'</font>'''，但这种学说已经过时了。<br />
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===Account given by Clausius===<br />
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作者: 克劳修斯<br />
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[[File:Clausius-1.jpg|thumb|upright|Rudolf Clausius]]<br />
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Rudolf Clausius<br />
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鲁道夫 · 克劳修斯<br />
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In 1856, the German physicist [[Rudolf Clausius]] stated what he called the "second fundamental theorem in the [[mechanical theory of heat]]" in the following form:{{sfnp|Clausius|1867}}<br />
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In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:<br />
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1856年，德国物理学家'''<font color = '#ff8000'>鲁道夫 • 克劳修斯Rudolf Clausius </font>'''阐述了他所谓的“热力学理论中的第二个基本定理second fundamental theorem in the mechanical theory of heat” ，其形式如下:<br />
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: <math>\int \frac{\delta Q}{T} = -N</math><br />
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<math>\int \frac{\delta Q}{T} = -N</math><br />
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<math>\int \frac{\delta Q}{T} = -N</math><br />
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where ''Q'' is heat, ''T'' is temperature and ''N'' is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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其中 Q 是热，T 是温度，N 是一个循环过程中所有<font color = 'red'><s>未补偿</s></font><font color = 'blue'>非补偿</font>的相变的“等价值”。后来在1865年，克劳修斯将“等价值”定义为熵。<font color = 'red'><s>也就是在</s></font><font color = 'blue'>基于这个理论，</font>同一年，第二定律最著名的版本在4月24日苏黎世哲学学会的一次演讲中被<font color = 'red'><s>宣读</s></font><font color = 'blue'>提出</font>，在演讲的最后克劳修斯总结道:<br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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宇宙的熵趋于最大<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. [[universe]], as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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这句话是第二定律最著名的<font color = 'red'><s>措辞</s></font><font color = 'blue'>陈述</font>。由于其语言松散<font color = 'blue'>模糊</font>，如“宇宙universe”，<font color = 'red'><s>与缺乏具体的条件相同</s></font><font color = 'blue'>和缺乏具体的条件</font>，如“开放open”，“封闭closed”，或“孤立isolated”，许多人认为这一简单的陈述意味着热力学第二定律几乎适用于每一个可以想象的<font color = 'red'><s>学科</s></font><font color = 'blue'>主题</font>。这不是真的；这句话只是一个更广泛和更精确的描述的简化版本。<br />
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In terms of time variation, the mathematical statement of the second law for an [[isolated system]] undergoing an arbitrary transformation is:<br />
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In terms of time variation, the mathematical statement of the second law for an isolated system undergoing an arbitrary transformation is:<br />
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就时间变化而言，<font color = 'blue'>一个</font>经历任意变换的孤立系统<font color = 'blue'>的</font>第二定律的数学表述是:<br />
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: <math>\frac{dS}{dt} \ge 0</math><br />
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<math>\frac{dS}{dt} \ge 0</math><br />
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<math>\frac{dS}{dt} \ge 0</math><br />
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where<br />
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where<br />
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这里''S'' 是系统的熵<br />
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: ''S'' is the entropy of the system and<br />
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S is the entropy of the system and<br />
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: ''t'' is [[time]].<br />
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t is time.<br />
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T是时间<br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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!-- 可逆过程需要与周围环境保持平衡。这对于一个孤立的系统是不可能的。因此，关于可逆过程的讨论已经转移到对封闭系统的分析----<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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平衡后用等号。另一种表述孤立系统第二定律的方法是：<br />
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: <math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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with <math> \dot S_{i}</math> the sum of the rate of [[entropy production]] by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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with <math> \dot S_{i}</math> the sum of the rate of entropy production by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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用<math> \dot S_{i}</math>表示系统内所有进程<font color = 'red'><s>产生熵</s></font>'''<font color = '#ff8000'>熵产生Entropy Production</font>'''的速率之和。这个公式的优点是它显示了熵产生的<font color = 'red'><s>影响</s></font>'''<font color = 'blue'>效果</font>。熵产生率是一个非常重要的概念，因为它决定（或限制）热机的效率。乘以环境温度<math>T_{a}</math>，<font color = 'red'><s>得到</s></font>'''<font color = 'blue'>它给出</font>所谓的耗散能<math> P_{diss}=T_{a}\dot S_{i}</math>。<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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封闭系统的第二定律(允许热<s>量</s>交换和移动边界，但不允许物质交换)的表达式是:<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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Here<br />
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Here<br />
这里<br />
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: <math>\dot Q</math> is the heat flow into the system<br />
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<math>\dot Q</math> is the heat flow into the system<br />
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<math>\dot Q</math> 是进入系统的热流。<br />
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: <math>T</math> is the temperature at the point where the heat enters the system.<br />
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<math>T</math> is the temperature at the point where the heat enters the system.<br />
<math>T</math> 是热<s>量</s>进入系统时的温度。<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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<font color = 'red'><s>等式符号在只有可逆过程在系统内发生的情况下成立。</s></font><font color = 'blue'>只有在系统内发生可逆过程的情况下等号才成立</font>如果发生不可逆过程<font color = 'red'><s>(</s></font><font color = 'blue'>（</font>在实际操作系统中就是这种情况）,<font color = 'red'><s>符号保持不变</s></font><font color = 'blue'>则“＞”成立</font>。如果系统有多处供热，必须求相应项的代数和。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 因为是中文环境所以统一用中文标点<br />
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For open systems (also allowing exchange of matter):<br />
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For open systems (also allowing exchange of matter):<br />
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对于开放系统(也允许物质交换) :<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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这里<math>\dot S</math> 是进入系统的熵流，与进入系统的物质流有关。它不应该与熵的时间导数混淆。如果物质在几个地方被供给，需要取这些贡献的代数和。<br />
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==Statistical mechanics==<br />
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统计力学<br />
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[[Statistical mechanics]] gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a [[microstate (statistical mechanics)|microstate]] of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/{{radic|''N''}} where ''N'' is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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Statistical mechanics gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a microstate of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/{{radic|''N''}} where N is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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<font color = 'blue'>统计力学</font>通过假设物质是由不断运动的原子和分子组成的，<font color = 'blue'>来</font><font color = 'red'><s>统计力学</s></font>对第二定律给出了解释。系统中每个粒子的一组特定的位置和速度称为系统的微观状态，由于系统的不断运动，系统不断地改变其微观状态。统计力学假设，在平衡状态下，系统<font color = 'red'><s>可能</s></font>处于<s>的</s>每个微观状态<font color = 'red'><s>发生</s></font>的可能性是相等的<font color = 'red'><s>，当这个假设被提出时，它直接导致了第二定律必须在统计意义上成立的结论。</s></font><font color = 'blue'>这个假设的提出直接导致第二定律必须子啊统计学意义上成立</font>也就是说，第二定律<font color = 'red'><s>平均成立，其中 N 是系统中粒子数的1 / 数量级的统计变化。</s></font><font color = 'blue'>在平均意义上成立，其中统计学变异取决于数量级1/{{radic|''N''}}，其中 N 是系统中粒子数</font>。在日常(宏观)情况下，违反第二定律的概率几乎为零。然而，对于粒子数量很少的系统，热力学参数，包括熵，可能显示出与第二定律预测<font color = 'red'><s>的</s></font><font color = 'blue'>结果的</font>显著的统计偏差。经典热力学理论<font color = 'red'><s>没有</s></font><font color = 'blue'>不</font>处理这些统计变量。<br />
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（by Jie XIANG) 在通过假定一种物质由不断运动的原子和分子组成的前提下，统计力学对热力学第二定律进行了解释。系统中每个粒子其特定的位置和速度组成为系统的微观状态，由于（原子和分子的）不断运动，系统也持续改变其微观状态。统计力学假设，在系统力平衡的情况下，其微观状态其实是等可能发生的，进而能够得出结论：热力学第二定律必须在统计学意义上成立。也就是说，第二定律将保持均值不变， 其统计变化为1 /√N，N为系统中的粒子数。在日常(宏观)情况下，违反热力学第二定律的概率几乎为零，但是，对于粒子数量非常少的系统，其热力学参数（包括熵）可能会显示出与第二定律所预测的明显统计偏差。然而经典热力学理论并不处理这些统计变量。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 为什么会有两个？迷茫。<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）是不是两个译者翻译的？如果第二段翻译不用，可以括号或删除线标一下<br />
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==Derivation from statistical mechanics==<br />
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从统计力学导出<br />
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{{Further information|H-theorem}}<br />
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The first mechanical argument of the [[Kinetic theory of gases]] that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to [[James Clerk Maxwell]] in 1860;<ref>{{Cite journal | last1 = Gyenis | first1 = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G }}</ref> [[Ludwig Boltzmann]] with his [[H-theorem]] of 1872 also argued that due to collisions gases should over time tend toward the [[Maxwell–Boltzmann distribution]].<br />
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The first mechanical argument of the Kinetic theory of gases that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to James Clerk Maxwell in 1860; Ludwig Boltzmann with his H-theorem of 1872 also argued that due to collisions gases should over time tend toward the Maxwell–Boltzmann distribution.<br />
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'''<font color="#ff8000">气体动力学Kinetic theory of gases</font>'''理论的第一个力学论证由'''<font color="#ff8000">麦克斯韦James Clerk Maxwell</font>'''在1860年给出，指出分子碰撞引起温度均衡<font color = 'blue'>化</font>，因此整体趋向于'''<font color="#ff8000">平衡 Equilibrium </font>'''； '''<font color="#ff8000">玻尔兹曼Ludwig Boltzmann</font>'''在1872年提出的'''<font color="#ff8000"> H 定理H-theorem</font>'''也认为，气体由于碰撞应该随着时间的推移趋向于'''<font color="#ff8000">麦克斯韦-波兹曼分布Maxwell–Boltzmann distribution</font>'''。<br />
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Due to [[Loschmidt's paradox]], derivations of the Second Law have to make an assumption regarding the past, namely that the system is [[Correlation and dependence|uncorrelated]] at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a [[boundary condition]], and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the [[Big Bang]]), though [[Boltzmann brain|other scenarios]] have also been suggested.<ref name="Hawking AOT">{{cite journal|last=Hawking|first=SW|title=Arrow of time in cosmology|journal=Phys. Rev. D|year=1985|volume=32|issue=10|pages=2489–2495|doi=10.1103/PhysRevD.32.2489|pmid=9956019|bibcode = 1985PhRvD..32.2489H }}</ref><ref>{{cite book | last = Greene | first = Brian | authorlink = Brian Greene | title = The Fabric of the Cosmos | url = https://archive.org/details/fabricofcosmossp00gree | url-access = registration | publisher = Alfred A. Knopf | year = 2004 | page = [https://archive.org/details/fabricofcosmossp00gree/page/171 171] | isbn = 978-0-375-41288-2}}</ref><ref name=Lebowitz>{{cite journal|last=Lebowitz|first=Joel L.|title= Boltzmann's Entropy and Time's Arrow|journal=Physics Today|date=September 1993|volume=46|issue=9|pages=32–38|url=http://users.df.uba.ar/ariel/materias/FT3_2008_1C/papers_pdf/lebowitz_370.pdf|accessdate=2013-02-22|doi=10.1063/1.881363|bibcode = 1993PhT....46i..32L }}</ref><br />
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Due to Loschmidt's paradox, derivations of the Second Law have to make an assumption regarding the past, namely that the system is uncorrelated at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a boundary condition, and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the Big Bang), though other scenarios have also been suggested.<br />
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由于'''<font color="#ff8000">洛施密特悖论Loschmidt's paradox</font>'''，第二定律的导出必须对过去做出一个假设，即系统在过去的某个时刻是不相关的；这样的假设允许进行简单的概率处理。这个假设通常被认为是一个'''<font color="#ff8000">边界条件boundary condition</font>'''，因此热力学第二定律最终是过去某个地方的初始条件的结果，可能是在宇宙的开始('''<font color="#ff8000">大爆炸 the Big Bang</font>''') ，<font color = 'blue'>尽管</font>也有人提出了其他<font color = 'red'><s>假设</s></font><font color = 'blue'>场景</font>。<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the [[Statistical mechanics#Fundamental postulate|fundamental postulate]], also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy.{{citation needed|date=August 2012}} The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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<font color = 'red'><s>考虑到</s></font><font color = 'blue'>基于</font>这些假设，在统计力学中，第二定律不是一个假设，而是'''统计力学基本假设Statistical mechanics#Fundamental postulate|fundamental postulate'''的一个结果，也被称为'''<font color="#ff8000">等先验概率假设 equal prior probability postulate</font>'''。这个基本假设表明，只要一个人清楚地知道，简单的概率论证只适用于未来，而对于过去，有辅助的信息来源告诉我们，它是低熵的。<font color = 'red'><s>热力学第二定律的第一部分指出，热孤立系统的熵只能增加。</s></font>如果我们把熵的概念限制在热平衡系统中，那么热力学第二定律的第一部分<font color = 'blue'>即热孤立系统的熵只能增加，</font>是等先验概率假设的一个显然结果。处于热平衡状态的孤立系统<font color = 'red'><s>包含</s></font><font color = 'blue'>且具有</font>能量<math>E</math>的熵表示为：<br />
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: <math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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<math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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其中<math>\Omega\left(E\right)</math> 是处于 <math>E</math>和<math>E +\delta E</math这个小区间内的量子态数目。这里的 <math>\delta E</math> 是一个宏观上很小的固定能量区间。严格地说，这意味着熵取决于对<math>\delta E</math>的选择。然而在热力学极限下（例如无穷大系统的极限），狭义的熵(单位体积或单位质量的熵)不依赖于 <math>\delta E</math>。<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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假设我们有一个孤立系统，其宏观状态由许多变量<font color = 'red'><s>描述</s></font><font color = 'blue'>指定</font>。这些宏观变量可以是总体积、活塞在系统中的位置等。从而<math>\Omega</math>将取决于这些变量的值。如果某个变量不是固定的（我们不会在某个位置夹住活塞) ，那么因为在平衡状态下<font color = 'red'><s>所有可到达状态的等几率到达的，</s></font><font color = 'blue'>到达所有可到达状态的可能性是相等的，</font>平衡状态下的自由变量会使 <math>\Omega</math> 最大，因为这是平衡状态下最可能的情况。<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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如果该变量最初固定到某个值然后释放，当达到新的平衡时，变量将自我调整使得 <math>\Omega</math> 最大，这意味着熵将增加或保持不变（如果变量初始固定的值恰好是平衡值）。<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's [[H-theorem]], however, proves that the quantity {{math|''H''}} increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the quantity increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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假设我们初始位于一个平衡状态，突然移除了对一个变量的约束。我们做完这件事的时候，可达到的微观状态的数为<math>\Omega</math>，但是系统还没有达到平衡，所以系统处于某些可达到的状态的实际概率还不等于先验概率 <math>1/\Omega</math>。我们已经知道，最终的平衡状态相对于之前的平衡状态，熵会增加或者保持不变。然而，玻耳兹曼的'''<font color="#ff8000">H定理H-theorem</font>'''证明系统在不处于平衡态的期间，那个量作为时间的函数单调增加。<br />
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===Derivation of the entropy change for reversible processes===<br />
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可逆过程熵变的推导<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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热力学第二定律的第二部分指出，经历可逆过程的系统的熵变为:<br />
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: <math>dS =\frac{\delta Q}{T}</math><br />
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<math>dS =\frac{\delta Q}{T}</math><br />
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where the temperature is defined as:<br />
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where the temperature is defined as:<br />
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其中温度定义为:<br />
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:<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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See [[Microcanonical ensemble|here]] for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the [[adiabatic theorem]] of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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请参阅'''此处Microcanonical ensemble|here'''查看该定义的正当性。假设系统有一些可以改变的外部参数 x。一般来说，系统的能量本征态将依赖于 x。根据量子力学的'''<font color="#ff8000">绝热定理adiabatic theorem</font>'''，在系统哈密顿量无限缓慢变化的极限下，系统将保持在相同的'''<font color = '#ff8000'>能量本征态 energy eigenstate</font>'''，因此系统的能量会随着其所在能量本征态的能量变化而变化。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 能量本征态 不懂翻译是否正确<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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对于外部变量 x 可以定义'''<font color="#ff8000">广义力generalized force</font>''' X ，因此如果 x 增加 dx，那么<math>X dx</math> 就是系统所做的功。例如，如果 x 是体积，那么 X 就是压强。一个已知处于能量本征态<math>E_{r}</math>的系统的广义力为：<br />
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: <math>X = -\frac{dE_{r}}{dx}</math><br />
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<math>X = -\frac{dE_{r}}{dx}</math><br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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由于系统可以在<math>\delta E</math> 区间内的任意能量本征态内，我们将系统的广义力定义为上述表达式的期望值:<br />
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: <math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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<math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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为了计算平均值，我们按照能量本征态来划分<math>\Omega\left(E\right)</math>，通过计算有多少能量本征态的<math>\frac{dE_{r}}{dx}</math>值在区间在<math>Y</math> 到 <math>Y + \delta Y</math>之间。把这个数字叫做为 <math>\Omega_{Y}\left(E\right)</math>，我们有:<br />
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: <math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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<math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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The average defining the generalized force can now be written:<br />
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The average defining the generalized force can now be written:<br />
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根据平均值定义的广义力现在可以写成:<br />
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: <math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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<math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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我们可以把它和由恒定能量 E 下的 x 导出来的熵联系起来。假定我们把 x 改变至 x + dx。然后因为能量本征态依赖于 x， <math>\Omega\left(E\right)</math> 将会改变，这导致能量本征态进入或超出<math>E</math> 和<math>E+\delta E</math> 之间的范围。让我们再次关注<math>\frac{dE_{r}}{dx}</math> 处于 <math>Y</math> 和 <math>Y + \delta Y</math> 之间的能量本征态。由于这些能量本征态的能量增加了 Y dx，所有这些在 E-Y dx 到 E 之间能量本征态<font color = 'blue'>都</font>从 E 以下移动到 E 以上。 因此有<br />
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: <math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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<math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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<font color = 'red'><s>这么多</s></font><font color = 'blue'>这些</font>的能量本征态。如果<math>Y dx\leq\delta E</math>，<font color = 'blue'>则</font>所有这些能量本征态将移动到 <math>E</math> 到 <math>E+\delta E</math>的范围内，使得<math>\Omega</math>增加。从<math>E+\delta E</math>以下移动到<math>E+\delta E</math>以上的能量本征态数目为 <math>N_{Y}\left(E+\delta E\right)</math>。它们的差<br />
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: <math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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<math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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因此是 <math>\Omega</math>增长的净贡献。请注意如果 Y dx 大于 <math>\delta E</math>，那么将会有能量本征态从 E 以下移动到<math>E+\delta E</math>以上。它们在<math>N_{Y}\left(E\right)</math> 和 <math>N_{Y}\left(E+\delta E\right)</math>中都有计数，因此上述表达式在这种情况下也是有效的。<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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将上面的表达式表示为对 E 的导数，并且对 Y 求和得到表达式:<br />
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: <math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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<math>\Omega</math>的对数关于x的导数由以下方式给出:<br />
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: <math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
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第一项是集约型的，<font color = 'red'><s>例如</s></font><font color = 'blue'>即</font>它不能根据系统大小进行缩放。相反，<font color = 'red'><s>最后一项的规模与逆系统的规模一样</s></font><font color = 'blue'>最后一项跟随逆系统的大小而缩放</font>，因此将在热力学极限中消失。因此我们发现:<br />
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: <math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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<math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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Combining this with<br />
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Combining this with<br />
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将这个和下面式子结合<br />
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: <math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
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<math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
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Gives:<br />
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Gives:<br />
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给出：<br />
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: <math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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<math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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===Derivation for systems described by the canonical ensemble===<br />
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正则系综系统的推导<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the [[canonical ensemble]]:<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:<br />
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如果一个系统与某个温度为T的热浴热接触，那么在平衡状态下，关于能量本征值的概率分布值由'''<font color = '#ff8000'>正则系综canonical ensemble</font>'''给出:<br />
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: <math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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<math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the [[Partition function (statistical mechanics)|partition function]]. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. <font color = 'red'>We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. </font>It follows from the general formula for the entropy:<br />
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这里 Z 是一个使所有概率之和<font color = 'red'><s>归一化</s></font><font color = 'blue'>正态化到 1 </font>的因子，这个函数被称为'''<font color = '#ff8000'>配分函数Partition function (statistical mechanics)|partition function</font>'''。现在我们考虑对温度和能级所依赖的外部参数的无限小的可逆改变。它遵循熵的一般公式:<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 能级依赖温度吗？即“现在我们考虑……的可逆改变”当中，“on which the energy levels depend”修饰（1）in the external parameters，还是（2）in the temperature and in the external parameters<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）修饰in the external parameters<br />
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: <math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
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<math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
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that<br />
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that<br />
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以及<br />
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: <math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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<math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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把正则系综的数学形式 <math>P_{j}</math> 插入，可以得到:<br />
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: <math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
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<math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
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==Living organisms==<br />
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==Living organisms==<br />
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生命体<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as [[What is Life?|Erwin Schrödinger]], [[Léon Brillouin]]<ref>Léon Brillouin [https://books.google.com/books?hl=en&lr=&id=tPXVbiw_1P0C ''Science and Information Theory''] (Academic Press, 1962) (Dover, 2004)</ref> and [[Life and Energy|Isaac Asimov]]. It is also the topic of current{{When|date=July 2019}} research.{{Citation needed|date=July 2019}}<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as Erwin Schrödinger, Léon Brillouin and Isaac Asimov. It is also the topic of current research.<br />
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表述热力学主要有两种方式：(a)描述从一种热力学平衡状态到另一种状态的路径，(b)通过<font color = 'red'><s>使系统保持不变，同时增加周围环境的总熵</s></font><font color = 'blue'>系统保持不变但周围环境的总熵增加</font>的循环过程。这两种方法有助于理解生命过程。这个话题大大超出了本文的范围，但是已经有一些研究者考虑过这个问题，比如'''<font color = '#ff8000'>薛定谔Erwin Schrödinger</font>'''、'''<font color = '#ff8000'>布里渊Léon Brillouin</font>'''和'''<font color = '#ff8000'>阿西莫夫Isaac Asimov</font>'''。这也是当前研究的主题。<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of [[metabolism]], give out breakdown products and heat. Plants [[Photosynthesis|take in radiative energy]] from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of metabolism, give out breakdown products and heat. Plants take in radiative energy from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
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从近似的角度来看，生命体可以被认为是(b)的一个例子。近似地，一只动物的身体状态每天循环，使得它几乎没有什么变化。动物吸收食物、水和氧气，经过'''<font color = '#ff8000'>新陈代谢metabolism</font>'''，输出分解的产物和热<s>量</s>。植物吸收来自太阳的辐射能量，这可以认为是热<s>量</s>，以及二氧化碳和水，然后它们释放氧气。它们就是这样生长的。<font color = 'blue'>它们</font>最终会死亡，尸体腐烂，大部分重新变成二氧化碳和水。这可以看作是一个循环过程。总的来说，阳光来自一个高温的源——太阳，它的能量被传递到一个较低的温度源，<font color = 'red'><s>例如</s></font><font color = 'blue'>即</font>向太空辐射。这个过程使得植物周围环境的熵增加。因此从循环过程的角度来看，动物和植物服从热力学第二定律。简单的热机效率概念很难适用于这个问题，因为它们假定系统是封闭的。<br />
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From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to [[Non-equilibrium thermodynamics|unsettled current debate or research]]. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to unsettled current debate or research. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
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从(a)的热力学观点考虑，即考虑从一个平衡状态到另一个平衡状态的路径，只会产生一个粗略近似的图像，因为生命体从来不会处于热力学平衡状态。生物体经常必须被认为是开放的系统，因为它们吸收营养物质并排出废弃物。开放系统热力学目前通常从一个热力学平衡状态到另一个状态的路径的角度来考虑，或者是考虑局部热力学平衡近似下的流。生命体的问题可以通过假定一个流不变的稳态<font color = 'blue'>的近似</font>来进一步简化。对于这样的近似，<font color = 'red'><s>熵产生的一般原理在'''目前有争论和研究的不确定性Non-equilibrium thermodynamics|unsettled current debate or research'''</s></font><font color = 'blue'>目前有很多关于熵产生的一般原理的未解决的争论或研究</font>。尽管如此，从热力学第二定律的这个角度出发衍生出来的想法对生物是有启发意义的。<br />
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==Gravitational systems==<br />
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==Gravitational systems==<br />
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重力系统<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive [[heat capacity]], meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it.<ref name="Grandy 151">Grandy, W.T. (Jr) (2008), p. 151.</ref> Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive heat capacity, meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it. Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
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<font color = 'red'><s>在不需要描述相对论</s></font><font color = 'blue'>在不需要其对广义相对论的解释的</font>的系统中，物体始终具有正热容，这意味着温度随能量而升高。因此，当能量从高温物体流向低温物体时，发射源温度降低而冷源温度升高；因此，温差会随着时间的流逝而减小。对于引力很重要并且<font color = 'red'><s>基于相对论的一般</s></font><font color = 'blue'>需要广义相对论的</font>系统而言，情况并非总是如此。这样的系统可以自发地<font color = 'red'><s>转变为质量和能量的不均匀分布</s></font><font color = 'blue'>向质量和能量不均匀的分布转变</font>。这适用于大规模的宇宙，因此可能很难或不可能对它应用第二定律。除此之外，广义相对论描述的系统的热力学超出了本文的范围。<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
非平衡状态<br />
{{main article|Non-equilibrium thermodynamics}}<br />
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The theory of classical or [[thermodynamic equilibrium|equilibrium thermodynamics]] is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<ref name="Grandy 151"/><ref>Callen, H.B. (1960/1985), p. 15.</ref><br />
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The theory of classical or equilibrium thermodynamics is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<br />
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经典热力学或平衡热力学理论是理想化的。其主要假定或假设（<font color = 'red'><s>往往无明确说明</s></font><font color = 'blue'>通常甚至没有明确地陈述</font>），是基于系统处在自身内部的热力学平衡状态。通常，在自然界可能发现的，在给定时间包含物理系统的空间区域，其热力学平衡状态并不是能用最严格的术语来表示。用较宽泛的术语来说，整个宇宙中没有任何东西是或曾经是真正处于精确的热力学平衡状态中。<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of [[thermodynamic equilibrium]]. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable [[thermal fluctuations|fluctuations]]. There is an exception, the case of [[critical point (thermodynamics)|critical states]], which exhibit to the naked eye the phenomenon of [[critical opalescence]]. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of thermodynamic equilibrium. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable fluctuations. There is an exception, the case of critical states, which exhibit to the naked eye the phenomenon of critical opalescence. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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为了进行物理分析，通常<font color = 'red'><s>假设热力学平衡就足以方便可行</s></font><font color = 'blue'>做一个热力学平衡的假设就足够方便可行</font>。 这样的假设可能依赖于反复试验和试错来证明其合理性。如果假设是合理的，那么它通常会非常有价值，因为它使热力学理论成为可能。平衡假设的要素是，观察到的系统会无限期地保持不变，并且因为系统中有很多粒子，以至于完全可以忽略其微粒<font color = 'red'><s>性质</s></font><font color = 'blue'>的本质</font>。通常在这种平衡假设下，<font color = 'red'><s>没有</s></font><font color = 'blue'>不存在一个</font>宏观上可检测到的波动。临界状态是一个例外，<font color = 'red'><s>它以肉眼显示出</s></font><font color = 'blue'>肉眼可以观察到</font>临界'''<font color = '#ff8000'>乳光现象opalescence</font>'''。对于临界状态的实验室研究，需要非常长的观察时间。<br />
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In all cases, the assumption of [[thermodynamic equilibrium]], once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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In all cases, the assumption of thermodynamic equilibrium, once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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在所有情况下，一旦<font color = 'red'><s>完成</s></font><font color = 'blue'>做出</font>了热力学平衡的假设，则<font color = 'blue'>作为一个结果</font>意味着假定<font color = 'blue'>中的</font>能改变系统熵的<font color = 'red'><s>“fluctuation”（待定的波动因子）</s></font><font color = 'blue'>候选波动</font>不存在。<br />
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'''<font color = '#32CD32'>则作为一个结果意味着假定中的能改变系统熵的候选波动不存在。implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.</font>'''<br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<ref>Lieb, E.H., Yngvason, J. (2003), p. 190.</ref><br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<br />
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物理系统很容易发生内部宏观变化，其速度之快足以使熵的恒定性假设失效。或物理系统中的粒子太少，以至于微粒的本质在可观测的波动中显而易见。那么热力学平衡的假设就会被放弃。对于非平衡态，熵没有绝对的通俗定义。<br />
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There are intermediate cases, in which the assumption of local [[thermodynamic equilibrium]] is a very good approximation,<ref>Gyarmati, I. (1967/1970), pp. 4-14.</ref><ref>Glansdorff, P., Prigogine, I. (1971).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003).</ref> but strictly speaking it is still an approximation, not theoretically ideal.<br />
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There are intermediate cases, in which the assumption of local thermodynamic equilibrium is a very good approximation, but strictly speaking it is still an approximation, not theoretically ideal.<br />
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还有一些中间情况，其中局部热力学平衡的假设是一个非常好的近似<font color = 'red'><s>值</s></font>，但严格地说它仍然是一个近似值，不是理论上的理想值。<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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对于一般的非平衡状态，考虑其他量的统计力学定义可能是有用的，这些量可以方便地称为“熵”，但是它们不应该与第二定律正确定义的熵混淆<font color = 'blue'>或合并</font>。这些其他量确实属于统计力学，而不属于第二定律的主要领域——热力学。<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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宏观上可观察到的波动的物理学超出了本文的范围。<br />
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==Arrow of time==<br />
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==Arrow of time==<br />
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时间之箭<br />
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{{Further information|Entropy (arrow of time)}}Thermodynamic Asymmetry in Time<br />
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Thermodynamic Asymmetry in Time<br />
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时间中的热力学不对称性<br />
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{{See also|Arrow of time}}<br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely [[CPT symmetry]], since the second law applies statistically, it is hypothesized, on time-asymmetric ''boundary conditions''.<ref>{{cite web|first=Craig|last=Callender|url=https://plato.stanford.edu/archives/fall2011/entries/time-thermo/|title=Thermodynamic Asymmetry in Time|encyclopedia=Stanford Encyclopedia of Philosophy|date=29 July 2011}}</ref><br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with symmetries observed in the fundamental laws of physics (particularly CPT symmetry) since the second law applies statistically on time-asymmetric boundary conditions.[78] The second law has been related to the difference between moving forwards and backwards in time, or to the principle that cause precedes effect (the causal arrow of time, or causality).[79]<br />
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热力学第二定律是<font color = 'red'><s>与时间方向逆转不对称的物理定律</s></font><font color = 'blue'>在时间方向上不对称的物理定律</font>。这与在物理学基本定律中观察到的对称性概念（特别是CPT对称性）<s>并</s>没有冲突，因为第二定律在统计上适用于时间非对称边界条件。第二定律与时间向前和向后移动之间的差异有关，或者与<font color = 'red'><s>导致先于效果</s></font><font color = 'blue'>原因先于结果</font>的原理（时间的因果关系或因果关系）有关。<br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect ([[Arrow of time#The causal arrow of time|the causal arrow of time]]).<ref>{{cite book | first = J.J.| last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1|display-authors=etal}} chapter 6</ref><br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect (the causal arrow of time).<br />
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第二定律被提出来<font color = 'red'><s>部分</s></font><font color = 'blue'>侧面</font>解释时间向前和向后移动的区别，例如为什么原因先于结果(因果关系的时间箭头)。<br />
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==Irreversibility==<br />
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==Irreversibility==<br />
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不可逆性<br />
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Irreversibility in [[thermodynamic process]]es is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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Irreversibility in thermodynamic processes is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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热力学过程中的不可逆性是热力学作用不对称特性的结果，而不是物体内部任何不可逆的微观性质<font color = 'blue'>的结果</font>。热力学作用是对参与体施加的宏观外部干预，并非来自其内部属性。由于没有认识到这一点，产生了被称为“悖论”的说法。<br />
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===Loschmidt's paradox===<br />
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<font color = 'red'><s>洛斯密特悖论 </s></font><font color = 'blue'>洛施密特悖论</font><br />
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{{main article|Loschmidt's paradox}}<br />
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[[Loschmidt's paradox]], also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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'''<font color="#ff8000">洛施密特悖论 Loschmidt's paradox</font>'''，也被称为'''<font color="#ff8000">可逆性悖论reversibility paradox</font>'''，<br />
<font color = 'red'><s>其目的在于不可能从描述宏观系统微观演化的时间对称动力学中推导出不可逆过程。</s></font><font color = 'blue'><br />
它反驳说，从描述宏观系统微观演化的时间对称动力学中，不可能推导出不可逆过程。</font><br />
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In the opinion of [[Erwin Schrödinger|Schrödinger]], "It is now quite obvious in what manner you have to reformulate the law of entropy{{snd}}or for that matter, all other irreversible statements{{snd}}so that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other."<ref>[[Erwin Schrödinger|Schrödinger, E.]] (1950), p.&nbsp;192.</ref> The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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In the opinion of Schrödinger, "It is now quite obvious in what manner you have to reformulate the law of entropyor for that matter, all other irreversible statementsso that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other." The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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薛定谔的认为：“现在很明显，您必须以何种方式重新制定熵定律（亦或者所有其他不可逆陈述）以使它们能够从可逆模型中得出。你不能只谈论一个孤立的系统，而应该至少谈论两个。你可以暂时认为它们与世界其它地方是孤立的，但并不总是相互孤立的。”这两个系统被隔离内壁彼此分开，直到被热力学（由热力学定律设想出来的）作用而拆除。热力学作用是从外部施加的，不受支配系统组成部分的可逆微观动力学定律的制约。这是不可逆转的原因。本文中<font color = 'red'><s>的法律声明</s></font><font color = 'blue'>第二定律的陈述</font>符合薛定谔的的建议。 因果关系在逻辑上先于第二定律，而不是由第二定律推导而来。<br />
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===Poincaré recurrence theorem===<br />
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庞加莱始态复现定理<br />
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{{Main article|Poincaré recurrence theorem}}<br />
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The [[Poincaré recurrence theorem]] considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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The Poincaré recurrence theorem considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics.Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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庞加莱始态复现定理<font color = 'red'><s>（又叫递归定理）</s></font>考虑了孤立物理系统的理论微观描述。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）又叫递归定理吗？存疑<br />
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经过热力学作用去除隔离内壁之后，便可以认为其是热力学系统的模型。在足够长的时间后，系统将返回到非常接近初始状态的微观定义状态。庞加莱复现时间是指回归前经过的时间长度。它极其漫长，可能比宇宙的寿命还要长，并且<font color = 'red'><s>敏感地依赖于被热力学作用拆除的墙体的几何形状</s></font><font color = 'blue'>对那个被热力学作用拆除的内壁的几何形状敏感</font>。复现定理可能被认为是明显与热力学第二定律相矛盾的。但是，更显而易见的是，它只是通过移除两个系统之间的内壁而形成的隔离系统中热力学平衡的微观模型。<font color = 'red'><s>对于典型的热力学系统而言，重复时间是如此之长（比宇宙的寿命长很多倍），以至于在所有的实际目的中，人们都无法观察到这种重复。</s></font><font color = 'blue'>对所有典型的热力学系统的实际目的，人们无法观察到如此之长的复现时间很长（比宇宙的寿命长很多倍）。</font>尽管如此，还是有人会想像一个机会可以等待庞加莱复现的出现，然后重新插入被热力学作用去除的内壁。<br />
然后很明显，不可逆性的出现是由于<font color = 'red'><s>庞加莱递归的完全不可预测性</s></font><font color = 'blue'>庞加莱复现完全不可预测的特性</font>，因为仅仅给出了初始状态<font color = 'red'><s>遵守热力学平衡的条件之一</s></font><font color = 'blue'>是热力学平衡之一</font>，就像宏观热力学的情况一样。<br />
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'''<font color="#32CD32">然后很明显，不可逆性的出现是由于庞加莱复现完全不可预测的特性，因为仅仅给出了初始状态是热力学平衡之一，就像宏观热力学的情况一样。It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics.</font>''' （这句话重复上面的，是为了标注疑难句的同时保留审校痕迹。生产修改后可以只保留这句，删掉上面重复部分）<br />
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即使<font color = 'red'><s>可以等待，也没有实际可操作性</s></font><font color = 'blue'>一个人可以等那么长时间，他也没有实际操作的可能性</font>来选择合适的时间重新插入内壁。庞加莱始态复现定理为洛施密特悖论提供了一个解决方案。如果一个孤立的热力学系统能以多倍于平均庞加莱复现时间的长度下<font color = 'red'><s>进行</s></font><font color = 'blue'>被</font>监控，则该系统的热力学行为在时间反转下将变得恒定。<br />
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[[File:James-clerk-maxwell3.jpg|thumb|upright|James Clerk Maxwell]]<br />
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James Clerk Maxwell<br />
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詹姆斯·克拉克·麦克斯韦<br />
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===Maxwell's demon===<br />
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麦克斯韦<font color = 'red'><s>的恶魔</s></font><font color = 'blue'>妖</font><br />
{{unreferenced|section|date=August 2018}}<br />
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{{main article|Maxwell's demon}}<br />
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[[James Clerk Maxwell]] imagined one container divided into two parts, ''A'' and ''B''. Both parts are filled with the same [[gas]] at equal temperatures and placed next to each other, separated by a wall. Observing the [[molecule]]s on both sides, an imaginary [[demon]] guards a microscopic trapdoor in the wall. When a faster-than-average molecule from ''A'' flies towards the trapdoor, the demon opens it, and the molecule will fly from ''A'' to ''B''. The average [[speed]] of the molecules in ''B'' will have increased while in ''A'' they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in ''A'' and increases in ''B'', contrary to the second law of thermodynamics.<br />
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James Clerk Maxwell imagined one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other, separated by a wall. Observing the molecules on both sides, an imaginary demon guards a microscopic trapdoor in the wall. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics.<br />
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'''麦克斯韦 James Clerk Maxwell'''设想一个容器分为A和B两部分。两部分均在相同温度下充满相同的气体，并彼此相邻由内壁隔开。<font color = 'red'><s>观察到两侧的分子，一个虚构的恶魔在墙上守着一个微观活板门</s></font><font color = 'blue'>一个虚构的妖精看守着一个内壁上的微型陷门，并观察者内壁两边的分子</font>。当来自A的速度快于平均水平的分子飞向<font color = 'red'><s>活板</s></font>门时，<font color = 'red'><s>恶魔将其打开</s></font><font color = 'blue'>妖精将门打开</font>，该分子将从A飞向B。B中分子的平均速度将增加，而A中分子的平均速度将<font color = 'red'><s>平均</s></font>降低。由于平均分子速度与温度相对应，因此与热力学第二定律相反，温度在A中降低，在B中升高。<br />
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One response to this question was suggested in 1929 by [[Leó Szilárd]] and later by [[Léon Brillouin]]. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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One response to this question was suggested in 1929 by Leó Szilárd and later by Léon Brillouin. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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1929年，Leó Szilárd和后来的 Léon Brillouin 对这个问题提出了一个答案。Szilárd指出，现实生活中的麦克斯韦<font color = 'red'><s>恶魔</s></font><font color = 'blue'>妖</font>需要一些测量分子速度的方法，而获取信息<font color = 'blue'>的过程</font>需要消耗<font color = 'red'><s>大量</s></font>能量。<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between ''A'' and ''B''. It is therefore performing [[thermodynamic operation]]s on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between A and B. It is therefore performing thermodynamic operations on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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麦克斯韦的<font color = 'red'><s>“恶魔”不断地</s></font><font color = 'blue'>“妖”重复地</font>改变 A 和 B 之间<font color = 'red'><s>墙体的渗透性</s></font><font color = 'blue'>壁的热透性</font>。因此，它是在微观尺度上执行热力学操作，而不仅仅是观察普通的自发或自然的宏观热力学过程。<br />
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==Quotations==<br />
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==Quotations==<br />
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语录<br />
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{{quote|The law that entropy always increases holds, I think, the supreme position among the [[Laws of science|laws of Nature]]. If someone points out to you that your pet theory of the [[universe]] is in disagreement with [[Maxwell's equations]] – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.|Sir [[Arthur Stanley Eddington]], ''The Nature of the Physical World'' (1927)}}<br />
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The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.<br />
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— Sir Arthur Stanley Eddington, The Nature of the Physical World (1927)<br />
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我认为，熵总是增加的定律在自然定律中占有至高无上的地位。如果有人指出您<font color = 'red'><s>的宇宙宠物理论</s></font><font color = 'blue'>杜撰的宇宙理论</font>与麦克斯韦方程不符，那么麦克斯韦方程就遇到麻烦了。如果发现<font color = 'blue'>您的理论</font>与观察不符，那么，可能是这些实验学家搞砸了。但是，如果发现您的理论违反了热力学第二定律，那么就没有希望了。除了在深深的耻辱中崩溃，别无他法。<br />
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—亚瑟·斯坦利·爱丁顿爵士，《自然世界》（1927年）<br />
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{{quote|There have been nearly as many formulations of the second law as there have been discussions of it.|Philosopher / Physicist [[Percy Williams Bridgman|P.W. Bridgman]], (1941)}}<br />
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There have been nearly as many formulations of the second law as there have been discussions of it.<br />
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— Philosopher / Physicist P.W. Bridgman, (1941)<br />
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关于第二定律的<font color = 'red'><s>阐述</s></font><font color = 'blue'>公式</font>几乎与关于第二定律的讨论一样多。<br />
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—哲学家/物理学家 布里奇曼（1941）<br />
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{{quote|Clausius is the author of the sibyllic utterance, "The energy of the universe is constant; the entropy of the universe tends to a maximum." The objectives of continuum thermomechanics stop far short of explaining the "universe", but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size.|[[Clifford Truesdell|Truesdell, C.]], Muncaster, R. G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 17.}}<br />
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Clausius is the author of the sibyllic utterance, "The energy of the universe is constant; the entropy of the universe tends to a maximum." The objectives of continuum thermomechanics stop far short of explaining the "universe", but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size.<br />
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— Truesdell, C., Muncaster, R. G. (1980). Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics, Academic Press, New York, ISBN 0-12-701350-4, p. 17.<br />
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克劳修斯（Sir Clausius）是西伯利亚文“宇宙的能量是恒定的；宇宙的熵趋于最大。”的作者。<font color = 'red'><s>不间断热力学的</s></font><font color = 'blue'>连续热力学的目标</font>远远不能解释“宇宙”，但是在该理论内，我们可以很容易地以某种类似于克劳修斯的方式得出明确的陈述，只是它仅仅指代一个简单的物体：一个有限大小的孤立物体。<br />
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— Truesdell, C., Muncaster, R. G. (1980). Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics, Academic Press, New York, ISBN 0-12-701350-4, p. 17. 特雷斯德尔，C.，蒙卡斯特，R.G.（1980）。麦克斯韦的单原子气体动力学理论的基本原理，被作为“理性力学”的一个分支，纽约学术出版社，ISBN 0-12-701350-4，第1页. 17。<br />
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==See also==<br />
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==See also==<br />
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参见<br />
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*[[Clausius–Duhem inequality]]<br />
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*[[Fluctuation theorem]]<br />
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*[[Heat death of the universe]]<br />
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*[[History of thermodynamics]]<br />
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*[[Jarzynski equality]]<br />
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*[[Laws of thermodynamics]]<br />
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*[[Maximum entropy thermodynamics]]<br />
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*[[Quantum thermodynamics]]<br />
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*[[Reflections on the Motive Power of Fire]]<br />
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*[[Relativistic heat conduction]]<br />
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*[[Thermal diode]]<br />
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*[[Thermodynamic equilibrium]]<br />
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==References==<br />
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==References==<br />
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参考资料<br />
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===Sources===<br />
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===Sources===<br />
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资料来源<br />
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*Adkins, C.J. (1968/1983). ''Equilibrium Thermodynamics'', (1st edition 1968), third edition 1983, Cambridge University Press, Cambridge UK, {{ISBN|0-521-25445-0}}.<br />
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*[[Peter Atkins|Atkins, P.W.]], de Paula, J. (2006). ''Atkins' Physical Chemistry'', eighth edition, W.H. Freeman, New York, {{ISBN|978-0-7167-8759-4}}.<br />
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*Attard, P. (2012). ''Non-equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications'', Oxford University Press, Oxford UK, {{ISBN|978-0-19-966276-0}}.<br />
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*Baierlein, R. (1999). ''Thermal Physics'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-59082-5}}.<br />
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*Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics, New York, {{ISBN|0-88318-797-3}}.<br />
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*{{Cite book|title=Concepts in thermal physics|last=Blundell|first=Stephen J.|last2=Blundell|authorlink1=Stephen Blundell|first2=Katherine M.|authorlink2=Katherine Blundell|publisher=[[Oxford University Press]]|year=2010|isbn=9780199562107|edition=2nd|location=Oxford|pages=|oclc=607907330|doi=10.1093/acprof:oso/9780199562091.001.0001 |url=http://cds.cern.ch/record/1235139}}<br />
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*[[Ludwig Boltzmann|Boltzmann, L.]] (1896/1964). ''Lectures on Gas Theory'', translated by S.G. Brush, University of California Press, Berkeley.<br />
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*Borgnakke, C., Sonntag., R.E. (2009). ''Fundamentals of Thermodynamics'', seventh edition, Wiley, {{ISBN|978-0-470-04192-5}}.<br />
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*Buchdahl, H.A. (1966). ''The Concepts of Classical Thermodynamics'', Cambridge University Press, Cambridge UK.<br />
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*[[Percy Williams Bridgman|Bridgman, P.W.]] (1943). ''The Nature of Thermodynamics'', Harvard University Press, Cambridge MA.<br />
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*[[Herbert Callen|Callen, H.B.]] (1960/1985). ''Thermodynamics and an Introduction to Thermostatistics'', (1st edition 1960) 2nd edition 1985, Wiley, New York, {{ISBN|0-471-86256-8}}.<br />
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*{{cite journal|author=C. Carathéodory|author1-link=Constantin Carathéodory<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
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|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
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2010年3月15日 | 第355-386页 | http://gdz.sub.uni-goettingen.de/index.php?id=11&ppn=ppn235181684_0067&dmdid=dmdlog_0033&l=1<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). ''The Second Law of Thermodynamics'', Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können.(p. 363) | doi 10.1007 / bf01450409}.译文可以在这里找到[ http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf ]。同时，在 Kestin，j。(1976).热力学第二定律，Dowden，Hutchinson & Ross，Stroudsburg PA. 。<br />
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*[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986). [http://www.worldcat.org/title/reflections-on-the-motive-power-of-fire-a-critical-edition-with-the-surviving-scientific-manuscripts-translated-and-edited-by-fox-robert/oclc/812944517&referer=brief_results ''Reflections on the motive power of fire''], Manchester University Press, Manchester UK, {{ISBN|0-7190-1741-6}}. [https://archive.org/stream/reflectionsonmot00carnrich#page/n7/mode/2up Also here.]<br />
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*[[Sydney Chapman (mathematician)|Chapman, S.]], [[Thomas George Cowling|Cowling, T.G.]] (1939/1970). ''The Mathematical Theory of Non-uniform gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases'', third edition 1970, Cambridge University Press, London.<br />
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*{{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Ueber Die Bewegende Kraft Der Wärme Und Die Gesetze, Welche Sich Daraus Für Die Wärmelehre Selbst Ableiten Lassen|journal=Annalen der Physik|year=1850|volume=79|issue=4|pages=368–397, 500–524|url=http://gallica.bnf.fr/ark:/12148/bpt6k15164w/f518.image|accessdate=26 June 2012|doi=10.1002/andp.18501550403|bibcode = 1850AnP...155..500C |hdl=2027/uc1.$b242250|hdl-access=free}} Translated into English: {{cite journal|last=Clausius|first=R.|title=On the Moving Force of Heat, and the Laws regarding the Nature of Heat itself which are deducible therefrom|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1851| volume=2|series=4th|issue=VIII|pages=1–21; 102–119 |url=https://archive.org/stream/londonedinburghd02lond#page/1/mode/1up|accessdate=26 June 2012<br />
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|doi=10.1080/14786445108646819}}<br />
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|doi=10.1080/14786445108646819}}<br />
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10.1080 / 14786445108646819}<br />
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* {{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie|year=1854|journal=Annalen der Physik|volume=xciii|issue=12|pages=481–506|url=http://zfbb.thulb.uni-jena.de/servlets/MCRFileNodeServlet/jportal_derivate_00140956/18541691202_ftp.pdf|accessdate=24 March 2014|doi=10.1002/andp.18541691202<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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| bibcode 1854AnP... 169. 1856年7月 | 第2卷 | 第4页 | 第86页 | https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=242014年3月<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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热力学理论及其在蒸汽机和物体物理性质中的应用 | 1867年 | 出版商 John van Voorst | 位置: 伦敦 | https://archive.org/details/mechanicaltheor04claugoog|quote=editions:pwr_sbkwa8ic。 2012年6月19日<br />
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*Denbigh, K. (1954/1981). ''The Principles of Chemical Equilibrium. With Applications in Chemistry and Chemical Engineering'', fourth edition, Cambridge University Press, Cambridge UK, {{ISBN|0-521-23682-7}}.<br />
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*[[Josiah Willard Gibbs|Gibbs, J.W.]] (1876/1878). On the equilibrium of heterogeneous substances, ''Trans. Conn. Acad.'', '''3''': 108-248, 343-524, reprinted in ''The Collected Works of J. Willard Gibbs, Ph.D, LL. D.'', edited by W.R. Longley, R.G. Van Name, Longmans, Green & Co., New York, 1928, volume 1, pp.&nbsp;55–353.<br />
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*Glansdorff, P., Prigogine, I. (1971). ''Thermodynamic Theory of Structure, Stability, and Fluctuations'', Wiley-Interscience, London, 1971, {{ISBN|0-471-30280-5}}.<br />
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*Grandy, W.T., Jr (2008). [http://global.oup.com/academic/product/entropy-and-the-time-evolution-of-macroscopic-systems-9780199546176?cc=au&lang=en& ''Entropy and the Time Evolution of Macroscopic Systems'']. Oxford University Press. {{ISBN|978-0-19-954617-6}}.<br />
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*Greven, A., Keller, G., Warnecke (editors) (2003). ''Entropy'', Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[Edward A. Guggenheim|Guggenheim, E.A.]] (1949). 'Statistical basis of thermodynamics', ''Research'', '''2''': 450–454.<br />
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*[[Edward A. Guggenheim|Guggenheim, E.A.]] (1967). ''Thermodynamics. An Advanced Treatment for Chemists and Physicists'', fifth revised edition, North Holland, Amsterdam.<br />
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*Gyarmati, I. (1967/1970) ''Non-equilibrium Thermodynamics. Field Theory and Variational Principles'', translated by E. Gyarmati and W.F. Heinz, Springer, New York.<br />
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*[[Charles Kittel|Kittel, C.]], [[Herbert Kroemer|Kroemer, H.]] (1969/1980). ''Thermal Physics'', second edition, Freeman, San Francisco CA, {{ISBN|0-7167-1088-9}}.<br />
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*Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998). ''Modern Thermodynamics: From Heat Engines to Dissipative Structures'', John Wiley & Sons, Chichester, {{ISBN|0-471-97393-9}}.<br />
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*Lebon, G., Jou, D., Casas-Vázquez, J. (2008). ''Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers'', Springer-Verlag, Berlin, {{ISBN|978-3-540-74252-4}}.<br />
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*{{cite journal|ref=harv|title=The Physics and Mathematics of the Second Law of Thermodynamics|last1=Lieb |first1=E. H. |last2=Yngvason |first2=J.|journal=Physics Reports|volume=310|issue=1|pages=1–96 |year=1999|doi=10.1016/S0370-1573(98)00082-9|arxiv = cond-mat/9708200 |bibcode = 1999PhR...310....1L}}<br />
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*Lieb, E.H., Yngvason, J. (2003). The Entropy of Classical Thermodynamics, pp.&nbsp;147–195, Chapter 8 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*{{cite book|ref=harv|first=F.|last=Mandl|title=Statistical physics|edition=second|year=1988|publisher=[[Wiley & Sons]]|isbn=978-0-471-91533-1}}<br />
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*{{cite journal | last1 = Maxwell | first1 = J.C. | authorlink = James Clerk Maxwell | year = 1867 | title = On the dynamical theory of gases | url = | journal = Phil. Trans. R. Soc. Lond. | volume = 157 | issue = | pages = 49–88 | doi = 10.1098/rstl.1867.0004 }}<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985). ''Thermodynamics'', Pitman, London, {{ISBN|0-273-08577-8}}.<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003). Entropy in Nonequilibrium, pp.&nbsp;79–109, Chapter 5 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*Münster, A. (1970), ''Classical Thermodynamics'', translated by E.S. Halberstadt, Wiley–Interscience, London, {{ISBN|0-471-62430-6}}.<br />
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*[[Brian Pippard|Pippard, A.B.]] (1957/1966). ''Elements of Classical Thermodynamics for Advanced Students of Physics'', original publication 1957, reprint 1966, Cambridge University Press, Cambridge UK.<br />
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*[[Max Planck|Planck, M.]] (1897/1903). [https://archive.org/stream/treatiseonthermo00planrich#page/100/mode/2up ''Treatise on Thermodynamics'', translated by A. Ogg, Longmans Green, London, p. 100.]<br />
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*[[Max Planck|Planck. M.]] (1914). [https://archive.org/details/theoryofheatradi00planrich ''The Theory of Heat Radiation''], a translation by Masius, M. of the second German edition, P. Blakiston's Son & Co., Philadelphia.<br />
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*[[Max Planck|Planck, M.]] (1926). Über die Begründung des zweiten Hauptsatzes der Thermodynamik, ''Sitzungsberichte der Preussischen Akademie der Wissenschaften: Physikalisch-mathematische Klasse'': 453–463.<br />
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* Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769-781.<br />
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* {{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}<br />
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*Quinn, T.J. (1983). ''Temperature'', Academic Press, London, {{ISBN|0-12-569680-9}}.<br />
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*{{cite book|ref=harv|first=Y.V.C.|last=Rao|title=An Introduction to thermodynamics|url=https://books.google.com/books?id=iYWiCXziWsEC&pg=PA213|date=2004|publisher=Universities Press|isbn=978-81-7371-461-0|page=213}}<br />
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*Roberts, J.K., Miller, A.R. (1928/1960). ''Heat and Thermodynamics'', (first edition 1928), fifth edition, Blackie & Son Limited, Glasgow.<br />
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*[[Erwin Schrödinger|Schrödinger, E.]] (1950). Irreversibility, ''Proc. R. Ir. Acad.'', '''A53''': 189–195.<br />
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*[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966). ''Elements of Thermodynamics'', Addison-Wesley Publishing, Reading MA.<br />
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*{{cite journal|ref=harv|last=Thomson|first=W.|author-link=William Thomson, 1st Baron Kelvin|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Transactions of the Royal Society of Edinburgh|year=1851|volume=XX|issue=part II|pages=261–268; 289–298|url=https://www.biodiversitylibrary.org/item/126047#page/295/mode/1up}} Also published in {{cite journal|last=Thomson|first=W.|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Philos. Mag. |date=December 1852 |volume=IV |series=4 |issue=22 |page=13|url=https://archive.org/stream/londonedinburghp04maga#page/12/mode/2up |accessdate=25 June 2012}}<br />
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*[[William Thomson, 1st Baron Kelvin|Thomson, W.]] (1852). ''On the universal tendency in nature to the dissipation of mechanical energy'' Philosophical Magazine, Ser. 4, p.&nbsp;304.<br />
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*[[László Tisza|Tisza, L.]] (1966). ''Generalized Thermodynamics'', M.I.T Press, Cambridge MA.<br />
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*[[Clifford Truesdell|Truesdell, C.]] (1980). ''The Tragicomical History of Thermodynamics 1822–1854'', Springer, New York, {{ISBN|0-387-90403-4}}.<br />
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*Uffink, J. (2001). Bluff your way in the second law of thermodynamics, ''Stud. Hist. Phil. Mod. Phys.'', '''32'''(3): 305–394.<br />
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*Uffink, J. (2003). Irreversibility and the Second Law of Thermodynamics, Chapter 7 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003), Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963). ''Lectures in Statistical Mechanics'', American Mathematical Society, Providence RI.<br />
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*[[Mark Zemansky|Zemansky, M.W.]] (1968). ''Heat and Thermodynamics. An Intermediate Textbook'', fifth edition, McGraw-Hill Book Company, New York.<br />
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==Further reading==<br />
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==Further reading==<br />
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进一步阅读<br />
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*Goldstein, Martin, and Inge F., 1993. ''The Refrigerator and the Universe''. Harvard Univ. Press. Chpts. 4–9 contain an introduction to the Second Law, one a bit less technical than this entry. {{ISBN|978-0-674-75324-2}}<br />
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*Leff, Harvey S., and Rex, Andrew F. (eds.) 2003. ''Maxwell's Demon 2 : Entropy, classical and quantum information, computing''. Bristol UK; Philadelphia PA: [[Institute of Physics]]. {{ISBN|978-0-585-49237-7}}<br />
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*{{Cite book | first = J.J. | last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1}}(technical).<br />
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*{{cite book |title=Reflections on the Motive Power of Heat and on Machines Fitted to Develop That Power |last=Carnot |first=Sadi |editor=[[Robert Henry Thurston|Thurston, Robert Henry]] |year=1890 |publisher=J. Wiley & Sons |location=New York |isbn= |pages= }} ([https://books.google.com/books?id=tgdJAAAAIAAJ full text of 1897 ed.]) ([http://www.history.rochester.edu/steam/carnot/1943/ html])<br />
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*Stephen Jay Kline (1999). ''The Low-Down on Entropy and Interpretive Thermodynamics'', La Cañada, CA: DCW Industries. {{ISBN|1-928729-01-0}}.<br />
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*{{cite book | last1 = Kostic | first1 = M | year = 2011 | title = Revisiting The Second Law of Energy Degradation and Entropy Generation: From Sadi Carnot's Ingenious Reasoning to Holistic Generalization | journal = AIP Conf. Proc. | volume = 1411 | issue = 1| pages = 327–350 | doi = 10.1063/1.3665247 | isbn = 978-0-7354-0985-9 | bibcode = 2011AIPC.1411..327K | series = AIP Conference Proceedings | citeseerx = 10.1.1.405.1945 }} also at [https://web.archive.org/web/20130420222450/http://www.kostic.niu.edu/2ndLaw/Revisiting%20The%20Second%20Law%20of%20Energy%20Degradation%20and%20Entropy%20Generation%20-%20From%20Carnot%20to%20Holistic%20Generalization-4.pdf].<br />
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==External links==<br />
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==External links==<br />
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外部链接<br />
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*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics]" – by Lawrence Sklar.<br />
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*[http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node30.html ''Second law of thermodynamics''] in the MIT Course [http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/notes.html ''Unified Thermodynamics and Propulsion''] from Prof. Z. S. Spakovszky<br />
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*[[E.T. Jaynes]], 1988, "[http://bayes.wustl.edu/etj/articles/ccarnot.pdf The evolution of Carnot's principle]," in G. J. Erickson and C. R. Smith (eds.)''Maximum-Entropy and Bayesian Methods in Science and Engineering, Vol'' 1: p.&nbsp;267.<br />
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*[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf Caratheodory, C., "Examination of the foundations of thermodynamics," trans. by D. H. Delphenich]<br />
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* [https://www.bbc.co.uk/programmes/p004y2bm The Second Law of Thermodynamics], BBC Radio 4 discussion with John Gribbin, Peter Atkins & Monica Grady (''In Our Time'', Dec. 16, 2004)<br />
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* [http://mdpi.org/entropy/papers/e6010001.pdf [[Entropy (journal)]], 2004]<br />
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* [https://www.journals.uchicago.edu/doi/abs/10.1086/663835 The Journal of the International Society for the History of Philosophy of Science, 2012]<br />
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{{DEFAULTSORT:Second Law Of Thermodynamics}}<br />
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[[Category:Concepts in physics]]<br />
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Category:Concepts in physics<br />
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分类: 物理概念<br />
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[[Category:Laws of thermodynamics|2]]<br />
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[[Category:Non-equilibrium thermodynamics]]<br />
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Category:Non-equilibrium thermodynamics<br />
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类别: 非平衡态热力学<br />
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[[Category:Philosophy of thermal and statistical physics]]<br />
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Category:Philosophy of thermal and statistical physics<br />
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类别: 热力学和统计物理学哲学<br />
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<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Second law of thermodynamics]]. Its edit history can be viewed at [[热力学第二定律/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E7%83%AD%E5%8A%9B%E5%AD%A6%E7%AC%AC%E4%BA%8C%E5%AE%9A%E5%BE%8B&diff=12751热力学第二定律2020-08-16T13:23:56Z<p>Dorr：</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Law of physics}}<br />
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{{Thermodynamics|laws}}<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 全文审校中，heat一般统一译为“热”，caloric 或者 caloric of heat 译为“热量”。这一点是否准确，有待各位同学和专家的批评指正<br />
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The '''second law of thermodynamics''' states that the total [[entropy]] of an [[isolated system]] can never decrease over time, and is constant if and only if all processes are reversible.<ref>http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node38.html#SECTION05224000000000000000</ref> Isolated systems spontaneously evolve towards [[thermodynamic equilibrium]], the state with maximum entropy. <br />
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The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, and is constant if and only if all processes are reversible. Isolated systems spontaneously evolve towards thermodynamic equilibrium, the state with maximum entropy. <br />
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<font color="#ff8000">'''热力学第二定律 Second Law Of Thermodynamics'''</font>指出，<font color="#ff8000">'''孤立系统Isolated System'''</font>的总熵永远不会随着时间而减少，且当且仅当所有过程都是可逆时，总熵才恒定。孤立系统自发地到达到热力学平衡状态，此时为熵最大的状态。<br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in [[thermodynamic equilibrium]], or is undergoing a (fictive) [[Reversible process (thermodynamics)|reversible process]]. In all processes that occur, including [[spontaneous process]]es,<ref>Atkins and de Paula, p.78</ref> the total entropy of the system and its surroundings increases and the process is [[Irreversible process|irreversible in the thermodynamic sense]]. The increase in entropy accounts for the irreversibility of natural processes, and the [[Arrow of time|asymmetry between future and past]].<ref>{{cite book|last=Zohuri|first=Bahman|title=Dimensional Analysis Beyond the Pi Theorem|url=https://books.google.com/books?id=pRVuDQAAQBAJ|year=2016|publisher=Springer|isbn=978-3-319-45726-0|page=[https://books.google.com.ph/books?id=pRVuDQAAQBAJ&pg=PA111&dq=%22increase+in+entropy+accounts+for+the+irreversibility+of+natural+processes+and+the+asymmetry+between+future+and+past.%22&hl=en&sa=X&ved=0ahUKEwj6spb61tbaAhUFS7wKHfftDtIQ6AEIKDAA#v=onepage&q=%22increase%20in%20entropy%20accounts%20for%20the%20irreversibility%20of%20natural%20processes%20and%20the%20asymmetry%20between%20future%20and%20past.%22 111]}}</ref><br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in thermodynamic equilibrium, or is undergoing a (fictive) reversible process. In all processes that occur, including spontaneous processes, the total entropy of the system and its surroundings increases and the process is irreversible in the thermodynamic sense. The increase in entropy accounts for the irreversibility of natural processes, and the asymmetry between future and past.<br />
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系统及其周围环境的总熵在理想情况下可以保持不变，在这种情况下，系统处于热力学平衡状态，或者正在经历一个(<font color = 'red'><s>虚拟的</s></font><font color = 'blue'>假想的</font>)可逆过程。<font color = 'red'><s>在所有发生的</s></font><font color = 'blue'>所有</font>过程中，包括<font color="#ff8000">'''自发过程 Spontaneous Processes'''</font>，系统及其周围环境的总熵增加，这一过程在热力学意义上是不可逆的。熵的增加解释了自然过程的不可逆性，以及未来和过去之间的不对称性。<br />
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Historically, the second law was an [[Empirical evidence|empirical finding]] that was accepted as an axiom of [[thermodynamics|thermodynamic theory]]. [[Statistical mechanics]], classical or [[quantum thermodynamics|quantum]], explains the microscopic origin of the law.<br />
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Historically, the second law was an empirical finding that was accepted as an axiom of thermodynamic theory. Statistical mechanics, classical or quantum, explains the microscopic origin of the law.<br />
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从历史上看，第二定律是一个经验性的发现，它被认为是热力学理论中的公理。经典统计力学、量子统计力学都可以解释这个定律的微观起源。<br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist [[Nicolas Léonard Sadi Carnot|Sadi Carnot]], who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as [[Carnot's theorem (thermodynamics)|Carnot's rule]] or limit.<ref>Jaffe, R.L., Taylor, W. (2018). ''The Physics of Energy'', Cambridge University Press, Cambridge UK, pages 150, 151, 259, 772, 743.</ref><br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist Sadi Carnot, who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as Carnot's rule or limit.<br />
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<font color = 'blue'>热力学第二定律可以使用多种方法表述。它的第一个公式归功于法国科学家<font color="#ff8000">'''卡诺Sadi Carnot'''</font>，卡诺在1824年证明了在热机中将热转化为功的效率有一个上限。第二定律的这个方面也被称为<font color="#ff8000">'''卡诺规则Carnot's Rule'''</font>或<font color="#ff8000">'''卡诺限制Carnot's Limit'''</font>。</font><br />
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'''热力学第二定律 Second Law of Thermodynamics'''是热力学的四条基本定律之一，它表述了热力学过程的不可逆性——孤立系统自发地朝着热力学平衡方向（即最大熵状态）演化，另一种表述为：第二类永动机永不可能实现。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）这里是不是少了一段原文？<br />
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==Introduction引言==<br />
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[[File:Heat flow hot to cold.png|thumb|upright|Heat flow from hot water to cold water. 热总是从热水流向冷水]]<br />
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The [[first law of thermodynamics]] provides the definition of the [[internal energy]] of a [[thermodynamic system]], and expresses the law of [[conservation of energy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 40–41.</ref><ref>Munster A. (1970), pp. 8–9, 50–51.</ref> The second law is concerned with the direction of natural processes.<ref>{{harvnb|Mandl|1988}}</ref> It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such [[Phenomenon|phenomena]] are accounted for in terms of [[entropy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 79–107.</ref><ref>Bailyn, M. (1994), Section 71, pp. 113–154.</ref> If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, {{math|''S''}}, increases.<br />
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The first law of thermodynamics provides the definition of the internal energy of a thermodynamic system, and expresses the law of conservation of energy. The second law is concerned with the direction of natural processes. It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such phenomena are accounted for in terms of entropy. If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, S, increases.<br />
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'''热力学第一定律 First Law Of Thermodynamics''' <font color = 'red'><s>对</s></font> <font color = 'blue'>定义了</font>热力学系统所涉及到的'''内能 Internal Energy'''<font color = 'red'><s>进行了定义</s></font> ，并体现了能量守恒定律。热力学第二定律与'''自然过程Natural Processes'''的方向有关。它<font color = 'red'><s>设定</s></font> <font color = 'blue'>断言</font>自然过程只在一种意义上进行，且<font color = 'red'><s>是</s></font> 不可逆<font color = 'red'><s>的</s></font> 。例如，当有了传导和辐射的（传播）路径时，热<s>量</s>总是自发地从一个较热的物体流向一个较冷的物体。这种现象可以用'''熵Entropy'''来解释。<font color = 'red'><s>若一个孤立系统最初在内部不可渗透的薄膜维持内部的热力学平衡，通过一些操作使得薄膜具有渗透性，则该系统可自发地演变，最终达到一个新的内部热力学平衡，其总熵增加。</s></font><br />
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<font color = '#32CD32'>若一个'''孤立系统Isolated System'''最初在具有隔热内壁的系统内维持热力学平衡，通过一些操作使内壁透热，则该系统可自发地演变，最终达到一个新的内部热力学平衡，且其总熵{{math|''S''}}增加。If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, S, increases.</font><br />
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--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]） more permeable不知道是否需要译为更具有 impermeable wall是不可渗透 如果要使用比较级 是否应该再有一个 permeable<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） internal partitioning impermeable walls翻译为隔热的内壁，从而permeable指的是透热的。我觉得这里的渗透性应该是说系统可以和外界进行能量/热的交换。？不知道是否正确。<br />
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In a fictive reversible process, an infinitesimal increment in the entropy ({{math|d''S''}}) of a system is defined to result from an infinitesimal transfer of heat ({{math|δ''Q''}}) to a [[closed system]] (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature ({{math|''T''}}) of the system in equilibrium and the surroundings which supply the heat:<ref>Bailyn, M. (1994), p. 120.</ref><br />
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In a fictive reversible process, an infinitesimal increment in the entropy () of a system is defined to result from an infinitesimal transfer of heat () to a closed system (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature () of the system in equilibrium and the surroundings which supply the heat:<br />
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在<font color = 'red'><s>设置的虚拟</s></font> <font color = 'blue'>假想的</font>可逆过程中，系统的熵的无穷小增量<math>dS</math>被定义为由无穷小的热<s>量</s><math>δQ</math>传递到一个封闭系统(允许能量进入或出去，但不允许物质传递<font color = 'blue'>的系统</font>)除以<font color = 'blue'>该</font>平衡系统和提供热<s>量</s>的环境的共同温度 <math>T</math>:<br />
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: <math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process封闭系统中理想状态下的可逆过程).}</math><br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 这里的公式和备注貌似有误，公式应为 <math>\mathrm dS = \frac{\delta Q}{T}</math>，备注应为“closed system, idealized fictive reversible process”封闭系统，理想的可逆过程。或者？<br />
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Different notations are used for infinitesimal amounts of heat ({{mvar|δ}}) and infinitesimal amounts of entropy ({{mvar|d}}) because entropy is a [[function of state]], while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the [[Clausius Theorem|inequality]] <ref name="MortimerBook"> Mortimer, R. G. ''Physical Chemistry''. 3rd ed., p. 120, Academic Press, 2008. </ref><ref name="FermiBook"> Fermi, E. ''Thermodynamics'', footnote on p. 48, Dover Publications,1956 (still in print). </ref> <br />
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Different notations are used for infinitesimal amounts of heat () and infinitesimal amounts of entropy () because entropy is a function of state, while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the inequality <br />
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用不同的符号''δ''和''d''表示无穷小量的热<s>量</s>和无穷小量的熵，因为熵是状态函数，而热<s>量</s>和功一样<font color = 'blue'>则</font>不是状态函数。<font color = 'red'><s>第二定律要求系统熵的增量满足不等式，</s></font>对于实际上可能存在的不与环境发生<font color = 'red'><s>质量</s></font><font color = 'blue'>物质</font>交换的无穷小过程，<font color = 'blue'>第二定律要求</font>系统熵增量满足不等式：<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 根据上下文（讲到封闭系统），此处疑“mass”译为物质更佳<br />
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: <math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process 封闭系统中，实际上可能发生的可逆过程).}</math><br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature ({{math|''T''}}) and the temperature of the surroundings ({{math|''T''}}<sub>''surr''</sub>).<br />
<ref name=":0">Adkins, C.J. (1968/1983), p. 75.</ref><ref name="Munster 45"/> <br />
Note that the equality still applies for pure heat flow,<br />
<ref name="Schmidt-Rohr 14"> Schmidt-Rohr, K. (2014). "Expansion Work without the External Pressure, and Thermodynamics in Terms of Quasistatic Irreversible Processes" ''J. Chem. Educ.'' '''91''': 402-409. https://dx.doi.org/10.1021/ed3008704 </ref><br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature () and the temperature of the surroundings (<sub>surr</sub>).<br />
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<font color = 'blue'>这是因为</font>这种情况下的一般过程可能包括周围环境对系统所做的功，<font color = 'red'><s>这是因为</s></font><font color = 'blue'>这个功</font>在系统内部会产生摩擦或粘滞效应，<font color = 'blue'>或是因为</font>此时<font color = 'red'><s>是由于</s></font><font color = 'blue'>一个</font>化学反应可能正在<font color = 'blue'>系统内部</font>进行，或<font color = 'blue'>因为</font>热传递实际上是不可逆地发生，<font color = 'red'><s>通过</s></font>系统温度<math>T</math>和周围环境温度<math>T_surr</math>之间<font color = 'red'><s>存在差异而进行驱动</s></font><font color = 'blue'>的差异驱动了热传递的发生</font>。<br />
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）该句有些不太理解<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 我觉得应该是“This is because a general process……，because a chemical reaction……，or because heat transfer……surroundings (<sub>surr</sub>).”的并列结构。<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）我的理解，第一个because是对上文的解释；后两个because是对“功在系统内部产生……效应”的解释；后两个为并列，不与第一个并列，因此“粘滞效应”后面的because前不用加“或是”<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）and the temperature of the surroundings (<sub>surr</sub>)后面缺少半句原文：“Note that the equality still applies for pure heat flow,”<br />
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<font color='blue'>注意这个等式也适用于'''纯热流Pure Heat Flow''',</font><br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change 实际上可能的,不改变成分的准静态不可逆性).}</math><br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry.<ref name="Oxtoby8th">Oxtoby, D. W; Gillis, H.P., [[Laurie Butler|Butler, L. J.]] (2015).''Principles of Modern Chemistry'', Brooks Cole. p. 617. {{ISBN|978-1305079113}}</ref> <ref name="MortimerBook"></ref> Introducing a set of internal variables <math>\xi</math> to describe the deviation of a thermodynamic system in physical equilibrium (with the required well-defined uniform pressure ''P'' and temperature ''T'')<ref name="Schmidt-Rohr 14"></ref> from the chemical equilibrium state, one can record the equality<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry.[16] [11] Introducing a set of internal variables {\displaystyle \xi }\xi to describe the deviation of a thermodynamic system in physical equilibrium (with the required well-defined uniform pressure P and temperature T)[15] from the chemical equilibrium state, one can record the equality<br />
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<font color='blue'>这种方法</font>是通过测量'''热容曲线 Heat Capacity Curves'''和相变中熵的变化，来准确测定纯物质的绝对熵的基础，比如'''量热法calorimetry'''。<br />
为了描述一个热力学系统在物理平衡状态下(要求有明确定义的等压P和等温T)偏离化学平衡状态，引入一组内部变量<math>x_i</math>,可以用该等式<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process 封闭系统，实际可能的准静态不可逆过程).}</math><br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of ''the second principle of thermodynamics'' – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<ref> Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769–781.</ref><ref>{{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}</ref><br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of the second principle of thermodynamics – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<br />
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第二项代表内部变量的功，这些内部变量<font color='red'><s>可以</s></font><font color= 'blue'>可能</font>受到外部影响的干扰，但是系统不能通过内部变量做任何正功。这种说法介绍了热力学系统在时间上<font color='red'><s>不可能逆转演化的事实</s></font>演化不可能逆转的性质<font color= 'blue'></font>，并且<font color='red'><s>可以被认为是热力学第二原理的另外一种相当于熵原理的表述。</s></font><font color= 'blue'>该公式可以被看作使用'''熵Entropy'''的热力学第二定律等价表述</font><br />
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The [[zeroth law of thermodynamics]] in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body.<ref name=dugdale>{{cite book|author=J. S. Dugdale|title=Entropy and its Physical Meaning|publisher=Taylor & Francis|year=1996|isbn=978-0-7484-0569-5|page=13|quote=This law is the basis of temperature.}}</ref> For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows{{how?|date=August 2018}} a distinguished temperature scale, which defines an absolute, [[thermodynamic temperature]], independent of the properties of any particular reference thermometric body.<ref>[[Mark Zemansky|Zemansky, M.W.]] (1968), pp. 207–209.</ref><ref>Quinn, T.J. (1983), p. 8.</ref><br />
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The zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body. For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows a distinguished temperature scale, which defines an absolute, thermodynamic temperature, independent of the properties of any particular reference thermometric body.<br />
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热力学第零定律是指如果两个热力学系统<font color='red'><s>中的每一个</s></font>都与第三个热力学系统处于热平衡(温度相同)<font color= 'blue'>状态</font>，则它们彼此也必定处于热平衡<font color= 'blue'>状态</font>。热力学第零定律在<font color='red'><s>它</s></font>这个<font color= 'blue'></font><font color='red'><s>通常的</s></font>简短叙述中让人们认识到热平衡关系中的两个物体具有相同的温度，特别是当一个被测物体与一个参考测温物体具有相同的温度时，对于<font color='red'><s>一个与另一个</s></font>两个<font color= 'blue'></font>处于热平衡<font color= 'blue'>状态</font>的物体，有无限多的'''<font color = '#ff8000'>经验温标empirical temperature scales</font>'''，这通常<font color= 'blue'>分别</font>取决于特定参考温度<font color='red'><s>标度</s></font><font color= 'blue'>体</font>的性质。热力学第二定律允许区分'''<font color = '#ff8000'>温度标度temperature scale</font>'''，它定义了一个绝对的热力学温度，与任何特定的参考温度体的性质无关。<br />
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--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）补充热力学第零定律<br />
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==Various statements of the law热力学第二定律的不同表述==<br />
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The second law of thermodynamics may be expressed in many specific ways,<ref name=MIT>{{cite web|title=Concept and Statements of the Second Law|url=http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node37.html|accessdate=2010-10-07 |publisher=web.mit.edu}}</ref> the most prominent classical statements{{sfnp|Lieb|Yngvason|1999}} being the statement by [[Rudolf Clausius]] (1854), the statement by [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] (1851), and the statement in axiomatic thermodynamics by [[Constantin Carathéodory]] (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.{{sfnp|Rao|2004|p=213}}<br />
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The second law of thermodynamics may be expressed in many specific ways, the most prominent classical statements being the statement by Rudolf Clausius (1854), the statement by Lord Kelvin (1851), and the statement in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.<br />
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热力学第二定律可以用许多特定的方式来表达，最突出的经典陈述是 '''克劳修斯 Rudolf Clausius''' (1854)<font color = 'red'><s>陈述</s></font><font color = 'blue'>表述</font>，'''开尔文 Kelvin''' (1851)<font color = 'red'><s>陈述</s></font><font color = 'blue'>表述</font>，以及康斯坦丁·卡拉西奥多里 Constantin Carathéodory(1909)在'''<font color = '#ff8000'>公理化热力学axiomatic thermodynamics</font>'''中的<font color = 'red'><s>陈述</s></font><font color = 'blue'>表述</font>。这些<font color = 'red'><s>陈述</s></font><font color = 'blue'>表述</font>用一般的物理术语来描述定律，引用某些过程的不可能性。克劳修斯和开尔文表述被证明是等价的。<br />
--[[用户:厚朴|厚朴]]（[[用户讨论:厚朴|讨论]]）statement的表述需要统一，同样的statement by Rudolf Clausius这里需要统一开尔文 克劳修斯 Kelvin Clausius<br />
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===Carnot's principle卡诺原理===<br />
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The historical origin<ref>[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986).</ref> of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a [[Carnot heat engine]], fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the [[caloric theory]] of heat was seriously considered, before the recognition of the [[first law of thermodynamics]], and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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The historical origin of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a Carnot heat engine, fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the caloric theory of heat was seriously considered, before the recognition of the first law of thermodynamics, and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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热力学第二定律的历史起源是'''卡诺原理Carnot's principle'''。<br />
它指的是'''<font color = '#ff8000'>卡诺热机Carnot heat engine</font>'''的一个循环，<font color = 'red'><s>它</s></font><font color = 'blue'>卡诺热机</font>以'''<font color = '#ff8000'>准静态quasi-static</font>'''的极限慢速运转，因此热<s>量</s>和功在子系统之间进行传递，子系统总是处于它们自己内部的热力学平衡状态。<br />
卡诺热机是研究热机效率的工程师特别感兴趣的理想装置。<br />
当卡诺发现卡诺原理时，'''<font color = '#ff8000'>热量理论caloric theory of heat</font>'''还没有得到重视，热力学第一定律还没有得到承认，熵的概念还没有数学表达。<br />
根据第一定律的解释，它在物理上等同于热力学第二定律，并沿用至今。<br />
在热力学第一定律被发现之前，卡诺最初的论点是从热量理论的观点出发的。<br />
下面是他书中的一些例子：<br />
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::...''wherever there exists a difference of temperature, motive power can be produced.''<ref>Carnot, S. (1824/1986), p. 51.</ref><br />
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...wherever there exists a difference of temperature, motive power can be produced.<br />
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……只要有温差，就能产生动力。<br />
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::The production of motive power is then due in steam engines not to an actual consumption of caloric, but ''to its transportation from a warm body to a cold body ...''<ref>Carnot, S. (1824/1986), p. 46.</ref><br />
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The production of motive power is then due in steam engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body ...<br />
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动力的产生不是由于蒸汽机实际消耗的热量，而是由于它从一个较热的物体<font color = 'red'><s>运输</s></font><font color = 'blue'>转移</font>到一个较冷的物体……。<br />
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::''The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.''<ref>Carnot, S. (1824/1986), p. 68.</ref><br />
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The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.<br />
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热<s>量</s>的原动力与实现热的媒介无关，<font color = 'red'><s>热量的数量</s></font><font color = 'blue'>热的量</font>完全取决于两个物体之间的温度，最后是热量的传递。<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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用现代术语<font color = 'red'><s>来说</s></font>，卡诺原理可能更为准确:<br />
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::The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<ref>[[Clifford Truesdell|Truesdell, C.]] (1980), Chapter 5.</ref><ref>Adkins, C.J. (1968/1983), pp. 56–58.</ref><ref>Münster, A. (1970), p. 11.</ref><ref>Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998), pp.67–75.</ref><ref>Lebon, G., Jou, D., Casas-Vázquez, J. (2008), p. 10.</ref><ref>Eu, B.C. (2002), pp. 32–35.</ref><br />
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The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<br />
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准静态卡诺循环或可逆卡诺循环的效率只取决于两种热源的温度，而且无论工作物质是什么，效率是相同的。<font color = 'red'><s>只有</s></font><font color = 'blue'>使用</font>两个热源与工作物质交换热源（一个高温热源温度<math>T_1</math>和一个低温热源温度<math>T_2</math>)的卡诺热机是最有效的热机。<br />
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'''<font color="#32CD32">使用两个热源与工作物质交换热源（一个高温热源温度<math>T_1</math>和一个低温热源温度<math>T_2</math>)的卡诺热机是最有效的热机。 A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.</font>''' <br />
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--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）最后一句进行了补充 意译<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 我没太看懂卡诺表述，所以不能确定这里的补充是否正确。可以再看一下“与工作物质交换热源”部分的表述是否准确<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）如果不确定，可以考虑将这句话标注为疑难句；或直接按原文直译<br />
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===Clausius statement克劳修斯表述===<br />
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The German scientist [[Rudolf Clausius]] laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work.{{sfnp|Clausius|1850}} His formulation of the second law, which was published in German in 1854, is known as the ''Clausius statement'':<br />
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The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work. His formulation of the second law, which was published in German in 1854, is known as the Clausius statement:<br />
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1850年，德国科学家'''鲁道夫·克劳修斯 Rudolf Clausius''' 通过研究热传递和功之间的关系，为热力学第二定律<font color = 'red'><s>实验室</s></font>奠定了基础。他在1854年用德语发表的论文中所提及的热力学第二定律定义被称为克劳修斯表述:<br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.{{sfnp|Clausius|1854|p=86}}</blockquote><br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.</blockquote><br />
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不可能把热<s>量</s>从低温物体传递到高温物体而不产生其他影响。 <br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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克劳修斯的表述使用了“'''<font color = '#ff8000'>热通道Passage Of Heat</font>'''”的概念。<font color = 'red'><s>正如通常在热力学的讨论，这意味着“能量作为热的净转移” ，而不是另一种形式上的"分摊转账"或其他。</s></font><font color = 'blue'><br />
在热力学的讨论中，通常这意味着“能量作为热的形式的净转移” ，而不是指其他方式的转移。</font><br />
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）不太理解分摊转账contributory transfers <br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 我觉得应该是说，energy as '''heat''' 而不是 as其他方式的转移<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）我认为这里不要把contributory transfer看作一个专有名词去理解，从contributory的本意“A contributory factor of a problem or accident is one of the things which caused it to exist or happen.”来理解即可，感觉嘉树理解的是对的。<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of [[refrigeration]], for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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如果不对系统<font color = 'red'><s>进行外部功</s></font><font color = 'blue'>外部做功</font>，热<font color = 'red'><s>量就</s></font>不能自发地从冷区流向热区，这一点从制冷的普通经验中也可以看出。在冰箱中，<font color = 'red'><s>热量从冷到热，但只有在外部媒介——制冷系统的强制作用下才会发生变化。</s></font><font color = 'blue'>只有在外部媒介也就是制冷系统的强制作用下热才会从冷区流到热区。</font><br />
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===Kelvin statements开尔文表述===<br />
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[[William Thomson, 1st Baron Kelvin|Lord Kelvin]] expressed the second law in several wordings.<br />
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Lord Kelvin expressed the second law in several wordings.<br />
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'''<font color = '#ff8000'>开尔文勋爵 Lord Kelvin</font>''' <font color = 'red'><s>用几个字表达</s></font><font color = 'blue'>表述</font>了热力学第二定律。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 感觉”用几个字“怪怪的，所以删掉了<br />
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::It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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<font color = 'red'><s>无法从单一热源取热使其完全转化为有用功而不对环境产生影响。</s></font><font color = 'blue'>不可能从单一热源取热使之完全转换为有用的功而不产生其他影响。</font><br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 改的内容和原文内容不太对应。这个摘自百度百科，另参见 严子浚(1994). 对热力学第二定律开尔文表述的一点讨论, 大学物理: 13(2).<br />
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::It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.{{sfnp|Thomson|1851}}<br />
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It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.<br />
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<font color = 'red'><s>通过无生命物质的作用，不可能通过将物质的任何部分冷却到低于周围物体最冷的温度来产生机械效应。</s></font><font color = 'blue'>不可能通过无生命物质的作用，将物质的任何部分冷却到低于周围物体最低的温度并产生'''<font color = '#ff8000'>机械效应 Mechanical Effect</font>'''。</font><br />
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--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）无生命物质可以意译为机器吗<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 我不太懂这个，不过感觉最好不要吧。<br />
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===Equivalence of the Clausius and the Kelvin statements克劳修斯和开尔文表述的等价性===<br />
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[[Image:Deriving Kelvin Statement from Clausius Statement.svg|thumb|Derive Kelvin Statement from Clausius Statement从克劳修斯表述推导出开尔文表述]]<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed [[Carnot engine]] as shown by the figure. The [[Heat engine#Efficiency|efficiency]] of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the [[first law of thermodynamics]], as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the figure. The efficiency of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat from <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the first law of thermodynamics, as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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假设有一个热机违反了开尔文定理: 也就是说，这个热机以循环的方式吸收热<s>量</s>并将其<font color = 'blue'>完全</font>转化为功，而<font color = 'blue'>且</font>不产生任何影响。现<font color = 'blue'>在将其</font>与反向卡诺<font color = 'red'><s>发动</s></font>机<font color = 'red'><s>相对应</s></font><font color = 'blue'>比较</font>，如图所示。<br />
普通热机的效率为η，反向热机的效率为1/η。这对<font color = 'red'><s>热机的联合作用为</s></font><font color = 'blue'>联合热机的净效应和唯一效应是</font><math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> 将热<s>量</s>从较冷热源<font color = 'blue'>转移</font>到较热热源，这违反了克劳修斯表述。（这是能量守恒定律的结果，因为系统的总能量保持不变<math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>，所以<math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math>。）因此，违反开尔文表述意味着违反克劳修斯表述，即克劳修斯表述暗示了开尔文表述。我们可以用类似的方式证明开尔文表述暗示了克劳修斯表述，因此两者是等价的。<br />
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===Planck's proposition普朗克<font color = 'red'><s>假设</s></font><font color = 'blue'>命题</font>===<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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普朗克提出了直接来自经验的如下命题。这有时被认为是他对热力学第二定律的表述，但他认为这是热力学第二定律推导的起点。<br />
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::It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 86.</ref><ref>Roberts, J.K., Miller, A.R. (1928/1960), p. 319.</ref><br />
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It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<br />
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<font color = 'red'><s>建造一台发动机是不可能的，这台发动机将在一个完整的循环中工作，</s></font><font color = 'blue'>不可能建造一台发动机，使其在一个完整的循环中工作，</font>并且除了提高重量和冷却热储以外不会产生任何效果。<br />
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===Relation between Kelvin's statement and Planck's proposition 开尔文表述与普朗克命题的关系===<br />
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It is almost customary in textbooks to speak of the "[[Kelvin-Planck statement]]" of the law, as for example in the text by [[Dirk ter Haar|ter Haar]] and [[Harald Wergeland|Wergeland]].<ref>[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966), p. 17.</ref><br />
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It is almost customary in textbooks to speak of the "Kelvin-Planck statement" of the law, as for example in the text by ter Haar and Wergeland.<br />
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教科书中几乎总是用“'''<font color = '#ff8000'>开尔文-普朗克表述 Kelvin-Planck Statement</font>'''”来称呼该定律，例如'''<font color = '#ff8000'>德克·特哈尔 Diek ter Haar</font>''' 和'''<font color = '#ff8000'>哈拉尔德·沃格兰 Harald Wergeland</font>''' 就是这样表述的。<br />
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The '''Kelvin–Planck statement''' (or the ''heat engine statement'') of the second law of thermodynamics states that <br />
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The Kelvin–Planck statement (or the heat engine statement) of the second law of thermodynamics states that <br />
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热力学第二定律的开尔文-普朗克表述（或称“'''<font color = '#ff8000'>热机表述 Heat Engine Statement</font>'''”）指出<br />
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::It is impossible to devise a [[thermodynamic cycle|cyclically]] operating device, the sole effect of which is to absorb energy in the form of heat from a single [[heat reservoir|thermal reservoir]] and to deliver an equivalent amount of [[Work (physics)|work]].<ref name="Rao">{{cite book|last=Rao|first=Y. V. C.|title=Chemical Engineering Thermodynamics|publisher=Universities Press|isbn=978-81-7371-048-3|page=158|year=1997}}</ref><br />
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It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.<br />
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设计一种唯一效果是从单一热源吸收热并提供等量的功的循环运行装置是不可能的。<br />
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===Planck's statement 普朗克表述===<br />
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Planck stated the second law as follows.<br />
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Planck stated the second law as follows.<br />
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普朗克表述第二定律如下。<br />
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::Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<ref name="Planck 100">[[Max Planck|Planck, M.]] (1897/1903), p. 100.</ref><ref name="Planck 463">[[Max Planck|Planck, M.]] (1926), p. 463, translation by Uffink, J. (2003), p. 131.</ref><ref name="Roberts & Miller 382">Roberts, J.K., Miller, A.R. (1928/1960), p. 382. This source is partly verbatim from Planck's statement, but does not cite Planck. This source calls the statement the principle of the increase of entropy.</ref><br />
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Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<br />
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自然界中发生的任一过程都是沿参与其中的所有物体的熵之和增加的方向进行的。在极限情况中，即对于可逆过程，熵的总和保持不变。<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for ''irreversible phenomena''.<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for irreversible phenomena.<br />
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与普朗克表述非常相似的是'''<font color = '#ff8000'>乌伦贝克 Uhlenbeck</font>'''和'''<font color = '#ff8000'>福特 Ford</font>'''关于不可逆现象的表述。<br />
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::... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<ref>[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963), p. 16.</ref><br />
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... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<br />
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……在从一个平衡态到另一个平衡态的不可逆或自发的变化中（例如，当两个物体 A 和 B 接触时的温度平衡过程），熵总是增加。<br />
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===Principle of Carathéodory 卡拉西奥多里原理===<br />
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—— caratheodory 的原理在这里重新定向——<br />
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[[Constantin Carathéodory]] formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<ref>[[Constantin Carathéodory|Carathéodory, C.]] (1909).</ref><br />
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Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<br />
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'''<font color = '#ff8000'>康斯坦丁·卡拉西奥多里 Constantin Carathéodory</font>'''在纯数学公理的基础上进行了热力学'''<font color = '#ff8000'>公理化formulated</font>'''阐明。他对第二定律的陈述被称为'''<font color = '#ff8000'>卡拉西奥多里原理 Principle of Carathéodory</font>'''，可以这样表述:<br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.<ref>Buchdahl, H.A. (1966), p. 68.</ref></blockquote><br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.</blockquote><br />
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在绝热封闭系统的任意状态 S 附近，总有从 S 出发不可达的状态。<br />
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With this formulation, he described the concept of [[adiabatic accessibility]] for the first time and provided the foundation for a new subfield of classical thermodynamics, often called [[Ruppeiner geometry|geometrical thermodynamics]]. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic [[process function]], in other words, <math>\delta Q=TdS</math>.<ref name="Sychev1991">{{cite book |last=Sychev |first=V. V. |title=The Differential Equations of Thermodynamics |year=1991 |publisher=Taylor & Francis |isbn=978-1-56032-121-7}}</ref> {{clarify|date=February 2014}}<br />
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With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, <math>\delta Q=TdS</math>. <br />
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通过这个阐明，他首次描述了'''<font color = '#ff8000'>绝热可达性 Adiabatic Accessibility</font>'''的概念，并为经典热力学的一个新的子领域，即通常所说的'''<font color = '#ff8000'>几何热力学 Geometrical Thermodynamics</font>'''<font color = 'red'><s>奠定了基础</s></font>。由卡拉西奥多里原理可以推出，作为热的能量的准静态转移是一个'''<font color="#ff8000"> 完整的过程函数holonomic process function</font>'''即<math>\delta Q=TdS</math>。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）准静态转移的热量值是一个可积过程函数 存疑<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）本句的修改供讨论。可积函数的英文应该是Integrable function。但是什么是完整的函数存疑，是连续的函数？<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）这里建议标注一下'''<font color="#ff8000"> 完整的过程函数 holonomic process function</font>'''<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）奠定了基础为何要删????????????????????????????????????????<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.<ref name="Munster 45">Münster, A. (1970), p. 45.</ref>{{sfnp|Lieb|Yngvason|1999|p=49}}<ref name="Planck 1926">[[Max Planck|Planck, M.]] (1926).</ref><ref>Buchdahl, H.A. (1966), p. 69.</ref> {{clarify|date=February 2014}}<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium. <br />
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尽管在教科书几乎惯称卡拉西奥多里原理也是第二定律的一种表述，并认为其与克劳修斯表述或开尔文-普朗克表述等价，但事实并非如此。为了得到第二定律的所有内容，需要对卡拉西奥多里原理补充普朗克表述，即等量功总是增加一个最初处于自身内部热力学平衡的封闭系统的内部能量。<br />
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===Planck's principle 普朗克原理===<br />
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In 1926, [[Max Planck]] wrote an important paper on the basics of thermodynamics.<ref name="Planck 1926"/><ref>Uffink, J. (2003), pp. 129–132.</ref> He indicated the principle<br />
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In 1926, Max Planck wrote an important paper on the basics of thermodynamics. He indicated the principle<br />
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1926年，'''<font color = '#ff8000'>马克斯·普朗克 Max Planck</font>'''写了一篇关于热力学基础的重要论文。他指出了以下原理<br />
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::The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<ref name="Munster 45"/>{{sfnp |Lieb|Yngvason|1999|p=49}}<br />
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The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<br />
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一个封闭系统的内部能量因绝热过程增加，在整个过程中，系统的体积保持不变。<br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work."<ref>[[Clifford Truesdell|Truesdell, C.]], Muncaster, R.G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 15.</ref> Planck wrote: "The production of heat by friction is irreversible."<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 81.</ref><ref>[[Max Planck|Planck, M.]] (1926), p. 457, Wikipedia editor's translation.</ref><br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work." Planck wrote: "The production of heat by friction is irreversible."<br />
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这个公式没有提到热，没有提到温度，甚至没有提到熵，也不一定隐含地依赖于这些概念，但它暗示了第二定律的内容。一个密切相关的表述为，“摩擦力从来不做正功。”普朗克写道: “摩擦生热是不可逆的。”<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above.<ref>Lieb, E.H., Yngvason, J. (2003), p. 149.</ref> It is relevant that for a system at constant volume and [[Mole (unit)|mole numbers]], the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above. It is relevant that for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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这个普朗克定理没有提到熵，是用物理术语来表述的。它与上面给出的开尔文表述密切相关。相关的是，对于恒定体积和摩尔数的系统，熵是内能的单调函数。然而，普朗克的这个原理实际上并不是普朗克对第二定律的首选表述（见前面小节），而是依赖于熵的概念。<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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Borgnakke 和 Sonntag 提出的以下表述在某种意义上是对普朗克原理的补充。他们没有将其作为第二定律的完整表述:<br />
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::... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<ref>Borgnakke, C., Sonntag., R.E. (2009), p. 304.</ref><br />
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... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<br />
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……只有一种方法可以减少（封闭）系统的熵，将热从系统中转移出去。<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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与普朗克之前提出的原理不同，这一原理明确地用熵的变化来表示。从系统中去除物质也可以减少系统的熵。<br />
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===Statement for a system that has a known expression of its internal energy as a function of its extensive state variables 一个其内能有已知表达式（其扩展状态变量的函数）的系统的表述===<br />
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The second law has been shown to be equivalent to the [[internal energy]] ''U'' being a weakly [[convex function]], when written as a function of extensive properties (mass, volume, entropy, ...).<ref>{{cite book |last1=van Gool |first1=W. |last2=Bruggink |first2=J.J.C. (Eds) |url= |title=Energy and time in the economic and physical sciences |publisher=North-Holland |year=1985 |pages=41–56 |quote= |isbn=978-0-444-87748-2}}</ref><ref>{{Cite journal | last1 = Grubbström | first1 = Robert W. | doi = 10.1016/j.apenergy.2007.01.003 | title = An Attempt to Introduce Dynamics Into Generalised Exergy Considerations| journal = Applied Energy| volume = 84| issue = 7–8 | pages = 701–718 | year = 2007}}</ref> {{clarify|date=February 2014}}<br />
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The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...). <br />
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'''<font color="#32CD32">写成广泛性质（质量，体积，熵……）的函数时，第二定律被证明等价于弱凸函数内能 U。The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...).</font>'''<br />
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==Corollaries 推论==<br />
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===Perpetual motion of the second kind 第二类永动机===<br />
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{{main article|Perpetual motion}}<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of [[first law of thermodynamics]] by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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在第二定律建立之前，许多志在发明永动机的人试图通过提取环境的巨大内能作为机器动力，以突破'''热力学第一定律 First Law of Thermodynamics''' 的限制。这种机器被称为“'''第二类永动机 Perpetual Motion Machine of the Second Kind'''”。<font color = 'blue'>热力学</font>第二定律表明这种机器是不可能的。<br />
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===Carnot theorem 卡诺定理===<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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'''卡诺定理 Carnot's Theorem'''（1824）是一条限制任何可能的热机的最大效率的原理。效率完全取决于热源中热和冷之间的温差。 卡诺定理指出:<br />
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*All irreversible heat engines between two heat reservoirs are less efficient than a [[Carnot engine]] operating between the same reservoirs.<br />
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*All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs.<br />
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* 所有<font color = 'blue'>在两个热源之间工作的</font>不可逆热机的效率低于在<font color = 'red'><s>两相同</s></font><font color = 'blue'>同一个</font>热源之间工作的'''<font color = '#ff8000'>卡诺热机 Carnot Engine</font>'''。<br />
* 所有<font color = 'blue'>在两个热源之间工作的</font>可逆热机的效率与在<font color = 'red'><s>两相同</s></font><font color = 'blue'>同一个</font>热源之间工作的卡诺机相等。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）改成同一个正确吗??????????????????????????<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as [[thermodynamic reversibility]]. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the [[Carnot cycle]]'s reversibility and was condemned to be less efficient.<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the Carnot cycle's reversibility and was condemned to be less efficient.<br />
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在他的理想模型中，热转化为功的过程可以通过逆转循环的运动而恢复，这个概念后来被称为'''<font color = '#ff8000'>热力学可逆性 Thermodynamic Reversibility</font>'''。然而，卡诺进一步假定，一些热<s>量</s>损失了，并没有转化为机械功。因此，没有一个真实的热机能够实现'''<font color = '#ff8000'>卡诺循环 Carnot Cycle</font>'''的可逆性，并且被认为效率较低。<br />
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Though formulated in terms of caloric (see the obsolete [[caloric theory]]), rather than [[entropy]], this was an early insight into the second law.<br />
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Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law.<br />
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该理论尽管是用热<s>量</s>表述的（见被取代的'''<font color = '#ff8000'>热质说</font>'''），而不是熵，但是它是对第二定律的早期认识。<br />
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===Clausius inequality 克劳修斯不等式===<br />
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The [[Clausius theorem]] (1854) states that in a cyclic process<br />
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The Clausius theorem (1854) states that in a cyclic process<br />
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'''<font color = '#ff8000'>克劳修斯定理 Clausius Theorem</font>'''（1854）指出，在一个循环的过程中<br />
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: <math>\oint \frac{\delta Q}{T} \leq 0.</math><br />
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The equality holds in the reversible case<ref>[http://scienceworld.wolfram.com/physics/ClausiusTheorem.html ''Clausius theorem''] at [[Wolfram Research]]</ref> and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function [[entropy]]. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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The equality holds in the reversible case and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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等号在可逆情况下成立，严格不等号在不可逆情况下成立。可逆情况下引入状态函数熵。'''<font color="#32CD32">这是因为在循环过程中，状态功能的变化相对于状态功能为零。This is because in cyclic processes the variation of a state function is zero from state functionality.</font>'''<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）末句存疑<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 我也没看懂，按照字面意思修改了一下<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）感觉修改后的句子还是怪怪的，这里标成疑难句吧 '''<font color="#32CD32">这是因为在循环过程中，状态功能的变化相对于状态功能为零。This is because in cyclic processes the variation of a state function is zero from state functionality.</font>'''<br />
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===Thermodynamic temperature 热力学温度===<br />
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{{main article|Thermodynamic temperature}}<br />
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For an arbitrary heat engine, the efficiency is:<br />
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For an arbitrary heat engine, the efficiency is:<br />
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对于任意热机，效率为:<br />
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: <math>\eta = \frac {W_n}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H} \qquad (1)</math><br />
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where ''W''<sub>n</sub> is for the net work done per cycle. Thus the efficiency depends only on q<sub>C</sub>/q<sub>H</sub>.<br />
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其中 ''W''<sub>n</sub> 表示每个循环所做的净功。因此效率只取决于 q<sub>C</sub>/q<sub>H</sub>。<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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卡诺定理指出，所有在相同热源间运行的可逆热机的效率是相同的。<br />
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Thus, any reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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Thus, any reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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因此，任何在温度''T''<sub>1</sub>和''T''<sub>2</sub>之间运行的可逆热机必须具有相同的效率，也就是说，效率只是温度的函数:<br />
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<math>\frac{q_C}{q_H} = f(T_H,T_C)\qquad (2).</math><br />
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In addition, a reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between ''T''<sub>1</sub> and another (intermediate) temperature ''T''<sub>2</sub>, and the second between ''T''<sub>2</sub> and''T''<sub>3</sub>. This can only be the case if<br />
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In addition, a reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between T<sub>1</sub> and another (intermediate) temperature T<sub>2</sub>, and the second between T<sub>2</sub> andT<sub>3</sub>. This can only be the case if<br />
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另外，在温度 ''T''<sub>1</sub> 和 ''T''<sub>3</sub> 之间工作的可逆热机必须具有与由分别在温度''T''<sub>1</sub> 和（中间）温度 ''T''<sub>2</sub>之间、在 ''T''<sub>2</sub> 和 ''T''<sub>3</sub> 之间的两个循环组成的系统效率相同。只有下式成立才会出现这种情况:<br />
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: <math>f(T_1,T_3) = \frac{q_3}{q_1} = \frac{q_2 q_3} {q_1 q_2} = f(T_1,T_2)f(T_2,T_3).</math><br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the [[triple point]] of water. Then for any ''T''<sub>2</sub> and ''T''<sub>3</sub>,<br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the triple point of water. Then for any T<sub>2</sub> and T<sub>3</sub>,<br />
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现在考虑如下情形，<math>T_1</math> 是一个固定的参考温度: 水的<font color="#ff8000">'''<font color = '#ff8000'>三相点 Triple Point</font>'''</font>的温度。则对于任意 T<sub>2</sub> 和 T<sub>3</sub>,<br />
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: <math>f(T_2,T_3) = \frac{f(T_1,T_3)}{f(T_1,T_2)} = \frac{273.16 \cdot f(T_1,T_3)}{273.16 \cdot f(T_1,T_2)}.</math><br />
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Therefore, if thermodynamic temperature is defined by<br />
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Therefore, if thermodynamic temperature is defined by<br />
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因此，如果热力学温度被定义为<br />
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: <math>T = 273.16 \cdot f(T_1,T) \,</math><br />
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then the function ''f'', viewed as a function of thermodynamic temperature, is simply<br />
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then the function f, viewed as a function of thermodynamic temperature, is simply<br />
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那么函数 ''f'' 作为热力学温度的函数，为<br />
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: <math>f(T_2,T_3) = \frac{T_3}{T_2},</math><br />
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and the reference temperature ''T''<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be used{{snd}}the choice here corresponds to the [[Kelvin]] scale.)<br />
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and the reference temperature T<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be used the choice here corresponds to the Kelvin scale.)<br />
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参考温度 ''T''<sub>1</sub> 的值为273.16。（任何参考温度和任何正值均可用——此处的选择对应开尔文标度。）<br />
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===Entropy 熵===<br />
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{{main article|Entropy (classical thermodynamics)}}<br />
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According to the [[Clausius theorem|Clausius equality]], for a reversible process<br />
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According to the Clausius equality, for a reversible process<br />
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根据克劳修斯等式，对可逆过程有<br />
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: <math>\oint \frac{\delta Q}{T}=0</math><br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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这意味着线积分 <math>\int_L \frac{\delta Q}{T}</math> 对于可逆过程是路径无关的。<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer<ref name="Schmidt-Rohr 14" /> satisfies<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer satisfies<br />
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所以我们可以定义一个叫做熵的状态函数 S，对于可逆过程或者纯热传递满足<br />
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: <math>dS = \frac{\delta Q}{T} \!</math><br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the [[third law of thermodynamics]], which states that ''S'' = 0 at [[absolute zero]] for perfect crystals.<br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the third law of thermodynamics, which states that S = 0 at absolute zero for perfect crystals.<br />
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据此，只有对上述公式进行积分，才能得到熵的差值。为了获得绝对值，我们需要<font color="#ff8000">'''热力学第三定律 Third Law of Thermodynamics'''</font>，它指出<font color="#ff8000">'''绝对零度 Absolute Zero'''</font>下完美晶体的 ''S'' = 0。<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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对于任意不可逆过程，由于熵是一个状态函数，我们总是可以将初始状态和最终状态与一个虚拟的可逆过程联系起来，并在这条路径上积分以计算熵的差值。<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the [[Clausius inequality]] on this loop,<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the Clausius inequality on this loop,<br />
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现在把可逆过程逆过来，将其与上述不可逆过程结合。把克劳修斯不等式应用到这个循环,<br />
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: <math>-\Delta S+\int\frac{\delta Q}{T}=\oint\frac{\delta Q}{T}< 0</math><br />
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Thus,<br />
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Thus,<br />
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故,<br />
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: <math>\Delta S \ge \int \frac{\delta Q}{T} \,\!</math><br />
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where the equality holds if the transformation is reversible.<br />
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where the equality holds if the transformation is reversible.<br />
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如果变换可逆，等号成立。<br />
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Notice that if the process is an [[adiabatic process]], then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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Notice that if the process is an adiabatic process, then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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注意，若该过程是一个<font color="#ff8000">'''绝热过程 Adiabatic Process'''</font>，则<math>\delta Q=0</math>，故<math>\Delta S\ge 0</math>。<br />
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===Energy, available useful work===<br />
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===Energy, available useful work===<br />
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能量，<font color = 'red'><s>有用的工作</s></font>'''<font color = '#32CD32'>可用的有用工作available useful work</font>'''<br />
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{{See also|Exergy}}<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an ''unlimited'' heat reservoir at temperature ''T<sub>R</sub>'' and pressure ''P<sub>R</sub>'' {{snd}}so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain ''T<sub>R</sub>''; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain ''P<sub>R</sub>''.<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an unlimited heat reservoir at temperature T<sub>R</sub> and pressure P<sub>R</sub> so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain T<sub>R</sub>; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain P<sub>R</sub>.<br />
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考虑将第二定律应用于孤立系统(称为整体系统或宇宙)<font color = 'red'><s>的情形，</s></font>是一个重要<font color = 'red'><s>的、</s></font><font color = 'blue'>而</font>具有启发性的理想情形。该系统由两部分组成：<font color = 'red'>感兴趣的<s></s></font>研究关心的<font color = 'blue'></font>子系统和子系统的周围环境。因为<font color = 'red'><s>环境被想象的</s></font><font color = 'blue'>想象的环境</font>非常<font color = 'red'><s>巨</s></font>大，<font color = 'blue'>以至于</font>它们可以被视为一个温度为 T<sub>R</sub> 且压力为 P<sub>R</sub> 的无限<font color = 'red'><s>蓄热器</s></font><font color = 'blue'>热源</font>，<font color = 'blue'>因此</font>无论有多少热<font color = 'red'><s>量</s></font>被转移到（或来自）子系统，周围的温度将保持T<sub>R</sub>；无论子系统的体积膨胀（或收缩）有多大，周围环境的压力都将保持不变。<br />
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Whatever changes to ''dS'' and ''dS<sub>R</sub>'' occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy ''S<sub>tot</sub>'' of the isolated total system must not decrease:<br />
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Whatever changes to dS and dS<sub>R</sub> occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy S<sub>tot</sub> of the isolated total system must not decrease:<br />
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无论子系统和周围环境<font color = 'red'><s>的 ''dS''和''dS<sub>R</sub>'' 发生</s></font><font color = 'blue'>单独地发生什么熵的变化 ''dS''和dS<sub>R</sub></font>，根据第二定律，孤立总体系统的熵S<sub>tot</sub>不能减小。<br />
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: <math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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<math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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<math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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According to the [[first law of thermodynamics]], the change ''dU'' in the internal energy of the sub-system is the sum of the heat ''δq'' added to the sub-system, ''less'' any work ''δw'' done ''by'' the sub-system, ''plus'' any net chemical energy entering the sub-system ''d ∑μ<sub>iR</sub>N<sub>i</sub>'', so that:<br />
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According to the first law of thermodynamics, the change dU in the internal energy of the sub-system is the sum of the heat δq added to the sub-system, less any work δw done by the sub-system, plus any net chemical energy entering the sub-system d ∑μ<sub>iR</sub>N<sub>i</sub>, so that:<br />
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根据<font color = 'red'><s>能量守恒定律</s></font><font color = 'blue'>热力学第一定律</font>，子系统<font color = 'red'><s>内部能量</s></font><font color = 'blue'>内能</font>的变化 dU 是<font color = 'red'><s>子系统内部能量 q 的总和</s></font>加在子系统上的热δq的和<font color = 'blue'></font>，减去子系统所做的任何功w，再加上进入子系统的任何净化学能 d ∑μ<sub>iR</sub>N<sub>i</sub>，因此<br />
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: <math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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<math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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<math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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where μ<sub>iR</sub> are the [[chemical potential]]s of chemical species in the external surroundings.<br />
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where μ<sub>iR</sub> are the chemical potentials of chemical species in the external surroundings.<br />
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其中μ<sub>iR</sub>是外部环境中<font color = 'red'><s>化学物类</s></font>'''<font color = '#ff8000'>化学形态 chemical species</font>'''的化学势。<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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现在热<font color = 'red'><s>量</s></font>离开<font color = 'red'><s>储存器</s></font><font color = 'blue'>热源</font>进入子系统是<br />
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: <math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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<math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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数学△ qtr (- dS r) le tr dS / math<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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在这个过程中，首先使用了经典热力学中熵的定义(在统计热力学中，熵变、温度和吸收热量之间的关系'''<font color = '#32CD32'>可以将其推导出来 can be derived</font>''') ，然后从上面的公式可以推导出第二定律的不等式。<br />
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It therefore follows that any net work ''δw'' done by the sub-system must obey<br />
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It therefore follows that any net work δw done by the sub-system must obey<br />
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因此，子系统所做的任何净功δw必须服从<br />
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: <math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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<math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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数学增量 w le-dU + tds + sum mu { iR } dN i，/ math<br />
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It is useful to separate the work ''δw'' done by the subsystem into the ''useful'' work ''δw<sub>u</sub>'' that can be done ''by'' the sub-system, over and beyond the work ''p<sub>R</sub> dV'' done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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It is useful to separate the work δw done by the subsystem into the useful work δw<sub>u</sub> that can be done by the sub-system, over and beyond the work p<sub>R</sub> dV done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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将子系统所做的功δw划分为子系统可以完成的有用功δw<sub>u</sub> ，除了子系统在周围外部压力下膨胀所做的功p<sub>R</sub> dV外，<font color = 'red'><s>还可以</s></font>给出以下可用功（<font color = 'red'><s>放射本能</s></font>'''<font color = 'ff8000'>有用能 exergy</font>'''）关系式:<br />
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: <math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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<math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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<math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the ''availability'' or ''[[exergy]]'' ''E'' of the subsystem,<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the availability or exergy E of the subsystem,<br />
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为方便起见，可以把右边定义为热力学势的精确导数，称为子系统的'''<font color = 'ff8000'>可用性 Availability </font>'''或<font color = 'red'><s>放射本能E</s></font>'''<font color = 'ff8000'>有用能 exergy</font>'''<br />
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: <math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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<math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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<math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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<font color="#32CD32">因此，第二定律意味着，对于任何可以简单地被视为分为一个子系统和一个与之接触的无限温度和压力<font color = 'red'><s>储存器</s></font> '''<font color="#ff8000">热源 Reservoir</font>''' 的过程，The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,</font><br />
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: <math> dE + \delta w_u \le 0 \, </math><br />
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<math> dE + \delta w_u \le 0 \, </math><br />
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<math> dE + \delta w_u \le 0 \, </math><br />
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i.e. the change in the subsystem's exergy plus the useful work done ''by'' the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done ''on'' the system) must be less than or equal to zero.<br />
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i.e. the change in the subsystem's exergy plus the useful work done by the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done on the system) must be less than or equal to zero.<br />
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也就是。子系统<font color = 'red'><s>(火用)</s></font><font color = 'blue'>有用能</font>的变化加上子系统所做的有用功(或者，子系统<font color = 'red'><s>(火用)</s></font><font color = 'blue'>有用能</font>的变化<font color = 'red'><s>不包括任何功，除了压力贮存器所做的功之外，在系统中所做的功</s></font><font color = 'blue'>减去除了'''<font color = '#ff8000'>压力热源</font>'''外任何对系统做的功，</font>)必须小于或等于零。<br />
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In sum, if a proper ''infinite-reservoir-like'' reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in ''E'' for an irreversible process and no change for a reversible process.<br />
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In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in E for an irreversible process and no change for a reversible process.<br />
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总之，如果选择一个合适的类似于无限库的参考状态作为现实世界中的系统环境，<br />
<font color = 'red'><s>那么第二定律预测的不可逆性的 e 值会减少，而可逆过程的 e 值不会变化。</s></font><font color = 'blue'>则第二定律预测不可逆过程的''E''值减少，可逆过程的''E''值不变。</font><br />
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: <math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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<math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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<math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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This expression together with the associated reference state permits a [[design engineer]] working at the macroscopic scale (above the [[thermodynamic limit]]) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (''Also, see [[process engineer]]''). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (''See [[Exergy efficiency|second law efficiency]]''.)<br />
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This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (Also, see process engineer). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (See second law efficiency.)<br />
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这个表达式和相关的参考状态允许<font color = 'red'><s>在宏观尺度（高于热力学极限）下工作的使用第二定律</s></font><font color = 'blue'>设计工程师在宏观尺度（高于热力学极限）下使用第二定律</font>，而无需直接测量或考虑整个孤立系统中的熵变。（另见'''<font color = '#ff8000'>工艺工程师process engineer</font>'''）。<font color = 'red'><s>考虑到这些变化，假设</s></font><font color = 'blue'>这些变化已经在假设中被考虑到，该假设认为</font>所考虑的系统可以在不改变参考状态的情况下与参考状态达到平衡。将其与可逆理想<font color = 'blue'>状态</font>进行比较，还可以找到一个过程或过程集合的效率（见'''<font color = '#ff8000'>第二定律效率second law efficiency</font>'''）<br />
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This approach to the Second Law is widely utilized in [[engineering]] practice, [[environmental accounting]], [[systems ecology]], and other disciplines.<br />
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This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines.<br />
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第二定律的这种方法被广泛应用于'''<font color = '#ff8000'>工程实践engineering practice</font>'''、'''<font color = '#ff8000'>环境会计environmental accounting</font>'''、'''<font color = '#ff8000'>系统生态学systems ecology</font>'''等其他学科。<br />
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===The second law in chemical thermodynamics===<br />
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化学热力学的第二定律<br />
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For a [[spontaneous process|spontaneous chemical process]] in a closed system at constant temperature and pressure without non-''PV'' work, the Clausius inequality Δ''S > Q/T<sub>surr</sub>'' transforms into a condition for the change in [[Gibbs free energy]] <br />
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For a spontaneous chemical process in a closed system at constant temperature and pressure without non-PV work, the Clausius inequality ΔS > Q/T<sub>surr</sub> transforms into a condition for the change in Gibbs free energy <br />
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对于一个<font color = 'blue'>恒温恒压</font>封闭系统中的自发化学过程，在没有 non-PV 功的情况下，<font color = 'red'><s>在恒定温度和压力下，将</s></font>克劳修斯不等式Δ''S > Q/T<sub>surr</sub>'' <font color = 'red'><s>转化为吉布斯自由能变化的条件</s></font><br />
<font color = 'blue'>由于'''<font color = '#ff8000'>吉布斯自由能Gibbs free energy</font>'''的变化而转化为：</font><br />
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<math>\Delta G < 0 </math><br />
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<math>\Delta G < 0 </math><br />
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or d''G'' < 0. For a similar process at constant temperature and volume, the change in [[Helmholtz free energy]] must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products.<ref name="Oxtoby8th"/> <ref name="MortimerBook"></ref> The chemical equilibrium condition at constant ''T'' and ''p'' without electrical work is d''G'' = 0.<br />
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or dG < 0. For a similar process at constant temperature and volume, the change in Helmholtz free energy must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products. The chemical equilibrium condition at constant T and p without electrical work is dG = 0.<br />
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或者 dG < 0。对于一个相似的<font color = 'blue'>恒温恒压</font>过程<font color = 'red'><s>，在恒定的温度和体积下</s></font>，'''<font color = '#ff8000'>亥姆霍兹自由能 Helmholtz free energy </font>'''的变化一定是负的， <math>\Delta A < 0 </math>。因此，<font color = 'blue'>一个负的</font>自由能(G 或 A)变化<font color = 'red'><s>的负值</s></font>是过程自发的必要条件。这是热力学第二定律在化学中最有用的形式，其中自由能的变化可以通过<font color = 'red'><s>列表的生成焓</s></font>'''<font color = '#ff8000'>表列生成焓 Tabulated Enthalpies of Formation</font>'''和反应物及产物的标准摩尔熵来计算。在<font color = 'red'><s>T 和 P </s></font><font color = 'blue'>温度和压力</font>不变的情况下，化学平衡条件是 dG = 0。<br />
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-- [[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） tabulated enthalpies of formation 表列生成焓？？<br />
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==History==<br />
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历史<br />
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{{See also|History of entropy}}<br />
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[[File:Sadi Carnot.jpeg|thumb|upright|Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique]].]]<br />
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Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique.]]<br />
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尼古拉·莱昂纳尔·萨迪·卡诺穿着传统的巴黎综合理工学院制服<br />
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The first theory of the conversion of heat into mechanical work is due to [[Nicolas Léonard Sadi Carnot]] in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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The first theory of the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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<font color = 'blue'>卡诺在1824年提出热</font><font color = 'red'><s>首先提出类热</s></font>转化为机械功的<font color = 'blue'>第一个</font>理论。他<font color = 'blue'>是第一个</font>正确认识到了转换效率取决于发动机和环境之间的温差<font color = 'blue'>的人</font>。<br />
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Recognizing the significance of [[James Prescott Joule]]'s work on the conservation of energy, [[Rudolf Clausius]] was the first to formulate the second law during 1850, in this form: heat does not flow ''spontaneously'' from cold to hot bodies. While common knowledge now, this was contrary to the [[caloric theory]] of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law during 1850, in this form: heat does not flow spontaneously from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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'''<font color="#ff8000">克劳修斯Rudolf Clausius </font>'''认识到'''<font color="#ff8000">焦耳James Prescott Joule</font>'''在能量守恒方面工作的重要性后，在1850年<font color = 'red'><s>第一个提出了第二定律</s></font><font color = 'blue'>提出了第二定律的第一个公式，在这个公式中</font>: 热不会自发地从冷物体流向热物体。虽然<font color = 'red'><s>现在的常识是这样的</s></font><font color = 'blue'>现在这是常识</font>，但是这与当时流行的热理论相反，当时的热理论认为热是一种流体。从这些他推断出了'''<font color="#ff8000"> 萨迪卡诺定律the principle of Sadi Carnot</font>'''和熵的定义(1865年)。<br />
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Established during the 19th century, the [[Kelvin-Planck statement|Kelvin-Planck statement of the Second Law]] says, "It is impossible for any device that operates on a [[cyclic process|cycle]] to receive heat from a single [[heat reservoir|reservoir]] and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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Established during the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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<font color = 'red'><s>19世纪提出的开尔文-普朗克第二定律(Kelvin-Planck)表示：“任何循环运行的设备都不可能从单个蓄热体接收热<s>量</s>并产生净功。”这被证明相当于克劳修斯的陈述。</s></font><font color = 'blue'>19世纪提出的开尔文-普朗克第二陈述(Kelvin-Planck)表示：“任何循环运行的设备都不可能从单个热源接收热并产生净功。”这被证明与克劳修斯的陈述等价。</font><br />
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The [[ergodic hypothesis]] is also important for the [[Boltzmann]] approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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'''<font color="#ff8000">遍历假设ergodic hypothesis</font>'''对'''<font color="#ff8000">玻尔兹曼方法Boltzmann approach</font>'''也很重要。<font color = 'blue'>遍历假设认为</font>在很长一段时间内，在具有相同能量的微观态相空间的某些区域所花费的时间与这个区域的体积成正比，即在很长一段时间内，所有可访问的微观状态<font color = 'blue'>出现/成立</font>的可能性都是一样的。<font color = 'red'><s>同样的，</s></font><font color = 'blue'>等价于说，</font>它表明时间平均值和统计集合的平均值是相同的。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 觉得加上出现/成立好一点？<br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a [[Macroscopic bodies|macroscopic system]]. This doctrine is obsolescent.<ref>Denbigh, K.G., Denbigh, J.S. (1985). ''Entropy in Relation to Incomplete Knowledge'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-25677-1}}, pp. 43–44.</ref><ref>Grandy, W.T., Jr (2008). ''Entropy and the Time Evolution of Macroscopic Systems'', Oxford University Press, Oxford, {{ISBN|978-0-19-954617-6}}, pp. 55–58.</ref><ref name=Lambert>[http://entropysite.oxy.edu Entropy Sites — A Guide] Content selected by [[Frank L. Lambert]]</ref><br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a macroscopic system. This doctrine is obsolescent.<br />
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克劳修斯还提出了一种传统的学说，他认为熵可以被理解为宏观系统中的'''<font color = '#ff8000'>分子“无序”molecular 'disorder'</font>'''，但这种学说已经过时了。<br />
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===Account given by Clausius===<br />
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作者: 克劳修斯<br />
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[[File:Clausius-1.jpg|thumb|upright|Rudolf Clausius]]<br />
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Rudolf Clausius<br />
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鲁道夫 · 克劳修斯<br />
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In 1856, the German physicist [[Rudolf Clausius]] stated what he called the "second fundamental theorem in the [[mechanical theory of heat]]" in the following form:{{sfnp|Clausius|1867}}<br />
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In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:<br />
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1856年，德国物理学家'''<font color = '#ff8000'>鲁道夫 • 克劳修斯Rudolf Clausius </font>'''阐述了他所谓的“热力学理论中的第二个基本定理second fundamental theorem in the mechanical theory of heat” ，其形式如下:<br />
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: <math>\int \frac{\delta Q}{T} = -N</math><br />
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<math>\int \frac{\delta Q}{T} = -N</math><br />
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<math>\int \frac{\delta Q}{T} = -N</math><br />
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where ''Q'' is heat, ''T'' is temperature and ''N'' is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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其中 Q 是热，T 是温度，N 是一个循环过程中所有<font color = 'red'><s>未补偿</s></font><font color = 'blue'>非补偿</font>的相变的“等价值”。后来在1865年，克劳修斯将“等价值”定义为熵。<font color = 'red'><s>也就是在</s></font><font color = 'blue'>基于这个理论，</font>同一年，第二定律最著名的版本在4月24日苏黎世哲学学会的一次演讲中被<font color = 'red'><s>宣读</s></font><font color = 'blue'>提出</font>，在演讲的最后克劳修斯总结道:<br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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宇宙的熵趋于最大<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. [[universe]], as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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这句话是第二定律最著名的<font color = 'red'><s>措辞</s></font><font color = 'blue'>陈述</font>。由于其语言松散<font color = 'blue'>模糊</font>，如“宇宙universe”，<font color = 'red'><s>与缺乏具体的条件相同</s></font><font color = 'blue'>和缺乏具体的条件</font>，如“开放open”，“封闭closed”，或“孤立isolated”，许多人认为这一简单的陈述意味着热力学第二定律几乎适用于每一个可以想象的<font color = 'red'><s>学科</s></font><font color = 'blue'>主题</font>。这不是真的；这句话只是一个更广泛和更精确的描述的简化版本。<br />
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In terms of time variation, the mathematical statement of the second law for an [[isolated system]] undergoing an arbitrary transformation is:<br />
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In terms of time variation, the mathematical statement of the second law for an isolated system undergoing an arbitrary transformation is:<br />
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就时间变化而言，<font color = 'blue'>一个</font>经历任意变换的孤立系统<font color = 'blue'>的</font>第二定律的数学表述是:<br />
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: <math>\frac{dS}{dt} \ge 0</math><br />
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<math>\frac{dS}{dt} \ge 0</math><br />
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<math>\frac{dS}{dt} \ge 0</math><br />
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where<br />
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where<br />
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这里''S'' 是系统的熵<br />
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: ''S'' is the entropy of the system and<br />
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S is the entropy of the system and<br />
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: ''t'' is [[time]].<br />
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t is time.<br />
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T是时间<br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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!-- 可逆过程需要与周围环境保持平衡。这对于一个孤立的系统是不可能的。因此，关于可逆过程的讨论已经转移到对封闭系统的分析----<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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平衡后用等号。另一种表述孤立系统第二定律的方法是：<br />
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: <math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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with <math> \dot S_{i}</math> the sum of the rate of [[entropy production]] by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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with <math> \dot S_{i}</math> the sum of the rate of entropy production by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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用<math> \dot S_{i}</math>表示系统内所有进程<font color = 'red'><s>产生熵</s></font>'''<font color = '#ff8000'>熵产生Entropy Production</font>'''的速率之和。这个公式的优点是它显示了熵产生的<font color = 'red'><s>影响</s></font>'''<font color = 'blue'>效果</font>。熵产生率是一个非常重要的概念，因为它决定（或限制）热机的效率。乘以环境温度<math>T_{a}</math>，<font color = 'red'><s>得到</s></font>'''<font color = 'blue'>它给出</font>所谓的耗散能<math> P_{diss}=T_{a}\dot S_{i}</math>。<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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封闭系统的第二定律(允许热<s>量</s>交换和移动边界，但不允许物质交换)的表达式是:<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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Here<br />
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Here<br />
这里<br />
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: <math>\dot Q</math> is the heat flow into the system<br />
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<math>\dot Q</math> is the heat flow into the system<br />
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<math>\dot Q</math> 是进入系统的热流。<br />
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: <math>T</math> is the temperature at the point where the heat enters the system.<br />
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<math>T</math> is the temperature at the point where the heat enters the system.<br />
<math>T</math> 是热<s>量</s>进入系统时的温度。<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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<font color = 'red'><s>等式符号在只有可逆过程在系统内发生的情况下成立。</s></font><font color = 'blue'>只有在系统内发生可逆过程的情况下等号才成立</font>如果发生不可逆过程<font color = 'red'><s>(</s></font><font color = 'blue'>（</font>在实际操作系统中就是这种情况）,<font color = 'red'><s>符号保持不变</s></font><font color = 'blue'>则“＞”成立</font>。如果系统有多处供热，必须求相应项的代数和。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 因为是中文环境所以统一用中文标点<br />
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For open systems (also allowing exchange of matter):<br />
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For open systems (also allowing exchange of matter):<br />
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对于开放系统(也允许物质交换) :<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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这里<math>\dot S</math> 是进入系统的熵流，与进入系统的物质流有关。它不应该与熵的时间导数混淆。如果物质在几个地方被供给，需要取这些贡献的代数和。<br />
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==Statistical mechanics==<br />
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统计力学<br />
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[[Statistical mechanics]] gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a [[microstate (statistical mechanics)|microstate]] of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/{{radic|''N''}} where ''N'' is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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Statistical mechanics gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a microstate of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/{{radic|''N''}} where N is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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<font color = 'blue'>统计力学</font>通过假设物质是由不断运动的原子和分子组成的，<font color = 'blue'>来</font><font color = 'red'><s>统计力学</s></font>对第二定律给出了解释。系统中每个粒子的一组特定的位置和速度称为系统的微观状态，由于系统的不断运动，系统不断地改变其微观状态。统计力学假设，在平衡状态下，系统<font color = 'red'><s>可能</s></font>处于<s>的</s>每个微观状态<font color = 'red'><s>发生</s></font>的可能性是相等的<font color = 'red'><s>，当这个假设被提出时，它直接导致了第二定律必须在统计意义上成立的结论。</s></font><font color = 'blue'>这个假设的提出直接导致第二定律必须子啊统计学意义上成立</font>也就是说，第二定律<font color = 'red'><s>平均成立，其中 N 是系统中粒子数的1 / 数量级的统计变化。</s></font><font color = 'blue'>在平均意义上成立，其中统计学变异取决于数量级1/{{radic|''N''}}，其中 N 是系统中粒子数</font>。在日常(宏观)情况下，违反第二定律的概率几乎为零。然而，对于粒子数量很少的系统，热力学参数，包括熵，可能显示出与第二定律预测<font color = 'red'><s>的</s></font><font color = 'blue'>结果的</font>显著的统计偏差。经典热力学理论<font color = 'red'><s>没有</s></font><font color = 'blue'>不</font>处理这些统计变量。<br />
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（by Jie XIANG) 在通过假定一种物质由不断运动的原子和分子组成的前提下，统计力学对热力学第二定律进行了解释。系统中每个粒子其特定的位置和速度组成为系统的微观状态，由于（原子和分子的）不断运动，系统也持续改变其微观状态。统计力学假设，在系统力平衡的情况下，其微观状态其实是等可能发生的，进而能够得出结论：热力学第二定律必须在统计学意义上成立。也就是说，第二定律将保持均值不变， 其统计变化为1 /√N，N为系统中的粒子数。在日常(宏观)情况下，违反热力学第二定律的概率几乎为零，但是，对于粒子数量非常少的系统，其热力学参数（包括熵）可能会显示出与第二定律所预测的明显统计偏差。然而经典热力学理论并不处理这些统计变量。<br />
<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 为什么会有两个？迷茫。<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）是不是两个译者翻译的？如果第二段翻译不用，可以括号或删除线标一下<br />
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==Derivation from statistical mechanics==<br />
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从统计力学导出<br />
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{{Further information|H-theorem}}<br />
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The first mechanical argument of the [[Kinetic theory of gases]] that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to [[James Clerk Maxwell]] in 1860;<ref>{{Cite journal | last1 = Gyenis | first1 = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G }}</ref> [[Ludwig Boltzmann]] with his [[H-theorem]] of 1872 also argued that due to collisions gases should over time tend toward the [[Maxwell–Boltzmann distribution]].<br />
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The first mechanical argument of the Kinetic theory of gases that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to James Clerk Maxwell in 1860; Ludwig Boltzmann with his H-theorem of 1872 also argued that due to collisions gases should over time tend toward the Maxwell–Boltzmann distribution.<br />
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'''<font color="#ff8000">气体动力学Kinetic theory of gases</font>'''理论的第一个力学论证由'''<font color="#ff8000">麦克斯韦James Clerk Maxwell</font>'''在1860年给出，指出分子碰撞引起温度均衡<font color = 'blue'>化</font>，因此整体趋向于'''<font color="#ff8000">平衡 Equilibrium </font>'''； '''<font color="#ff8000">玻尔兹曼Ludwig Boltzmann</font>'''在1872年提出的'''<font color="#ff8000"> H 定理H-theorem</font>'''也认为，气体由于碰撞应该随着时间的推移趋向于'''<font color="#ff8000">麦克斯韦-波兹曼分布Maxwell–Boltzmann distribution</font>'''。<br />
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Due to [[Loschmidt's paradox]], derivations of the Second Law have to make an assumption regarding the past, namely that the system is [[Correlation and dependence|uncorrelated]] at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a [[boundary condition]], and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the [[Big Bang]]), though [[Boltzmann brain|other scenarios]] have also been suggested.<ref name="Hawking AOT">{{cite journal|last=Hawking|first=SW|title=Arrow of time in cosmology|journal=Phys. Rev. D|year=1985|volume=32|issue=10|pages=2489–2495|doi=10.1103/PhysRevD.32.2489|pmid=9956019|bibcode = 1985PhRvD..32.2489H }}</ref><ref>{{cite book | last = Greene | first = Brian | authorlink = Brian Greene | title = The Fabric of the Cosmos | url = https://archive.org/details/fabricofcosmossp00gree | url-access = registration | publisher = Alfred A. Knopf | year = 2004 | page = [https://archive.org/details/fabricofcosmossp00gree/page/171 171] | isbn = 978-0-375-41288-2}}</ref><ref name=Lebowitz>{{cite journal|last=Lebowitz|first=Joel L.|title= Boltzmann's Entropy and Time's Arrow|journal=Physics Today|date=September 1993|volume=46|issue=9|pages=32–38|url=http://users.df.uba.ar/ariel/materias/FT3_2008_1C/papers_pdf/lebowitz_370.pdf|accessdate=2013-02-22|doi=10.1063/1.881363|bibcode = 1993PhT....46i..32L }}</ref><br />
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Due to Loschmidt's paradox, derivations of the Second Law have to make an assumption regarding the past, namely that the system is uncorrelated at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a boundary condition, and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the Big Bang), though other scenarios have also been suggested.<br />
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由于'''<font color="#ff8000">洛施密特悖论Loschmidt's paradox</font>'''，第二定律的导出必须对过去做出一个假设，即系统在过去的某个时刻是不相关的；这样的假设允许进行简单的概率处理。这个假设通常被认为是一个'''<font color="#ff8000">边界条件boundary condition</font>'''，因此热力学第二定律最终是过去某个地方的初始条件的结果，可能是在宇宙的开始('''<font color="#ff8000">大爆炸 the Big Bang</font>''') ，<font color = 'blue'>尽管</font>也有人提出了其他<font color = 'red'><s>假设</s></font><font color = 'blue'>场景</font>。<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the [[Statistical mechanics#Fundamental postulate|fundamental postulate]], also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy.{{citation needed|date=August 2012}} The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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<font color = 'red'><s>考虑到</s></font><font color = 'blue'>基于</font>这些假设，在统计力学中，第二定律不是一个假设，而是'''统计力学基本假设Statistical mechanics#Fundamental postulate|fundamental postulate'''的一个结果，也被称为'''<font color="#ff8000">等先验概率假设 equal prior probability postulate</font>'''。这个基本假设表明，只要一个人清楚地知道，简单的概率论证只适用于未来，而对于过去，有辅助的信息来源告诉我们，它是低熵的。<font color = 'red'><s>热力学第二定律的第一部分指出，热孤立系统的熵只能增加。</s></font>如果我们把熵的概念限制在热平衡系统中，那么热力学第二定律的第一部分<font color = 'blue'>即热孤立系统的熵只能增加，</font>是等先验概率假设的一个显然结果。处于热平衡状态的孤立系统<font color = 'red'><s>包含</s></font><font color = 'blue'>且具有</font>能量<math>E</math>的熵表示为：<br />
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: <math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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<math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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其中<math>\Omega\left(E\right)</math> 是处于 <math>E</math>和<math>E +\delta E</math这个小区间内的量子态数目。这里的 <math>\delta E</math> 是一个宏观上很小的固定能量区间。严格地说，这意味着熵取决于对<math>\delta E</math>的选择。然而在热力学极限下（例如无穷大系统的极限），狭义的熵(单位体积或单位质量的熵)不依赖于 <math>\delta E</math>。<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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假设我们有一个孤立系统，其宏观状态由许多变量<font color = 'red'><s>描述</s></font><font color = 'blue'>指定</font>。这些宏观变量可以是总体积、活塞在系统中的位置等。从而<math>\Omega</math>将取决于这些变量的值。如果某个变量不是固定的（我们不会在某个位置夹住活塞) ，那么因为在平衡状态下<font color = 'red'><s>所有可到达状态的等几率到达的，</s></font><font color = 'blue'>到达所有可到达状态的可能性是相等的，</font>平衡状态下的自由变量会使 <math>\Omega</math> 最大，因为这是平衡状态下最可能的情况。<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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如果该变量最初固定到某个值然后释放，当达到新的平衡时，变量将自我调整使得 <math>\Omega</math> 最大，这意味着熵将增加或保持不变（如果变量初始固定的值恰好是平衡值）。<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's [[H-theorem]], however, proves that the quantity {{math|''H''}} increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the quantity increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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假设我们初始位于一个平衡状态，突然移除了对一个变量的约束。我们做完这件事的时候，可达到的微观状态的数为<math>\Omega</math>，但是系统还没有达到平衡，所以系统处于某些可达到的状态的实际概率还不等于先验概率 <math>1/\Omega</math>。我们已经知道，最终的平衡状态相对于之前的平衡状态，熵会增加或者保持不变。然而，玻耳兹曼的'''<font color="#ff8000">H定理H-theorem</font>'''证明系统在不处于平衡态的期间，那个量作为时间的函数单调增加。<br />
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===Derivation of the entropy change for reversible processes===<br />
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可逆过程熵变的推导<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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热力学第二定律的第二部分指出，经历可逆过程的系统的熵变为:<br />
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: <math>dS =\frac{\delta Q}{T}</math><br />
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<math>dS =\frac{\delta Q}{T}</math><br />
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where the temperature is defined as:<br />
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where the temperature is defined as:<br />
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其中温度定义为:<br />
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:<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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See [[Microcanonical ensemble|here]] for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the [[adiabatic theorem]] of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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请参阅'''此处Microcanonical ensemble|here'''查看该定义的正当性。假设系统有一些可以改变的外部参数 x。一般来说，系统的能量本征态将依赖于 x。根据量子力学的'''<font color="#ff8000">绝热定理adiabatic theorem</font>'''，在系统哈密顿量无限缓慢变化的极限下，系统将保持在相同的'''<font color = '#ff8000'>能量本征态 energy eigenstate</font>'''，因此系统的能量会随着其所在能量本征态的能量变化而变化。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 能量本征态 不懂翻译是否正确<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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对于外部变量 x 可以定义'''<font color="#ff8000">广义力generalized force</font>''' X ，因此如果 x 增加 dx，那么<math>X dx</math> 就是系统所做的功。例如，如果 x 是体积，那么 X 就是压强。一个已知处于能量本征态<math>E_{r}</math>的系统的广义力为：<br />
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: <math>X = -\frac{dE_{r}}{dx}</math><br />
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<math>X = -\frac{dE_{r}}{dx}</math><br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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由于系统可以在<math>\delta E</math> 区间内的任意能量本征态内，我们将系统的广义力定义为上述表达式的期望值:<br />
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: <math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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<math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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为了计算平均值，我们按照能量本征态来划分<math>\Omega\left(E\right)</math>，通过计算有多少能量本征态的<math>\frac{dE_{r}}{dx}</math>值在区间在<math>Y</math> 到 <math>Y + \delta Y</math>之间。把这个数字叫做为 <math>\Omega_{Y}\left(E\right)</math>，我们有:<br />
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: <math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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<math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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The average defining the generalized force can now be written:<br />
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The average defining the generalized force can now be written:<br />
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根据平均值定义的广义力现在可以写成:<br />
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: <math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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<math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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我们可以把它和由恒定能量 E 下的 x 导出来的熵联系起来。假定我们把 x 改变至 x + dx。然后因为能量本征态依赖于 x， <math>\Omega\left(E\right)</math> 将会改变，这导致能量本征态进入或超出<math>E</math> 和<math>E+\delta E</math> 之间的范围。让我们再次关注<math>\frac{dE_{r}}{dx}</math> 处于 <math>Y</math> 和 <math>Y + \delta Y</math> 之间的能量本征态。由于这些能量本征态的能量增加了 Y dx，所有这些在 E-Y dx 到 E 之间能量本征态<font color = 'blue'>都</font>从 E 以下移动到 E 以上。 因此有<br />
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: <math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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<math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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<font color = 'red'><s>这么多</s></font><font color = 'blue'>这些</font>的能量本征态。如果<math>Y dx\leq\delta E</math>，<font color = 'blue'>则</font>所有这些能量本征态将移动到 <math>E</math> 到 <math>E+\delta E</math>的范围内，使得<math>\Omega</math>增加。从<math>E+\delta E</math>以下移动到<math>E+\delta E</math>以上的能量本征态数目为 <math>N_{Y}\left(E+\delta E\right)</math>。它们的差<br />
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: <math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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<math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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因此是 <math>\Omega</math>增长的净贡献。请注意如果 Y dx 大于 <math>\delta E</math>，那么将会有能量本征态从 E 以下移动到<math>E+\delta E</math>以上。它们在<math>N_{Y}\left(E\right)</math> 和 <math>N_{Y}\left(E+\delta E\right)</math>中都有计数，因此上述表达式在这种情况下也是有效的。<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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将上面的表达式表示为对 E 的导数，并且对 Y 求和得到表达式:<br />
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: <math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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<math>\Omega</math>的对数关于x的导数由以下方式给出:<br />
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: <math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
<br />
The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
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第一项是集约型的，<font color = 'red'><s>例如</s></font><font color = 'blue'>即</font>它不能根据系统大小进行缩放。相反，<font color = 'red'><s>最后一项的规模与逆系统的规模一样</s></font><font color = 'blue'>最后一项跟随逆系统的大小而缩放</font>，因此将在热力学极限中消失。因此我们发现:<br />
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<br />
<br />
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: <math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
<br />
<math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
<br />
<br />
<br />
<br />
<br />
<br />
Combining this with<br />
<br />
Combining this with<br />
<br />
将这个和下面式子结合<br />
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<br />
<br />
<br />
<br />
: <math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
<br />
<math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
<br />
<br />
<br />
<br />
<br />
Gives:<br />
<br />
Gives:<br />
<br />
给出：<br />
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<br />
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: <math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
<br />
<math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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===Derivation for systems described by the canonical ensemble===<br />
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正则系综系统的推导<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the [[canonical ensemble]]:<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:<br />
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如果一个系统与某个温度为T的热浴热接触，那么在平衡状态下，关于能量本征值的概率分布值由'''<font color = '#ff8000'>正则系综canonical ensemble</font>'''给出:<br />
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: <math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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<math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the [[Partition function (statistical mechanics)|partition function]]. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. <font color = 'red'>We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. </font>It follows from the general formula for the entropy:<br />
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这里 Z 是一个使所有概率之和<font color = 'red'><s>归一化</s></font><font color = 'blue'>正态化到 1 </font>的因子，这个函数被称为'''<font color = '#ff8000'>配分函数Partition function (statistical mechanics)|partition function</font>'''。现在我们考虑对温度和能级所依赖的外部参数的无限小的可逆改变。它遵循熵的一般公式:<br />
<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 能级依赖温度吗？即“现在我们考虑……的可逆改变”当中，“on which the energy levels depend”修饰（1）in the external parameters，还是（2）in the temperature and in the external parameters<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）修饰in the external parameters<br />
<br />
<br />
<br />
: <math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
<br />
<math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
<br />
<br />
<br />
<br />
<br />
<br />
that<br />
<br />
that<br />
<br />
以及<br />
<br />
<br />
<br />
<br />
<br />
: <math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
<br />
<math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
<br />
<br />
<br />
<br />
<br />
Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
<br />
把正则系综的数学形式 <math>P_{j}</math> 插入，可以得到:<br />
<br />
<br />
<br />
<br />
<br />
: <math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
<br />
<math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
<br />
==Living organisms==<br />
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==Living organisms==<br />
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生命体<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as [[What is Life?|Erwin Schrödinger]], [[Léon Brillouin]]<ref>Léon Brillouin [https://books.google.com/books?hl=en&lr=&id=tPXVbiw_1P0C ''Science and Information Theory''] (Academic Press, 1962) (Dover, 2004)</ref> and [[Life and Energy|Isaac Asimov]]. It is also the topic of current{{When|date=July 2019}} research.{{Citation needed|date=July 2019}}<br />
<br />
There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as Erwin Schrödinger, Léon Brillouin and Isaac Asimov. It is also the topic of current research.<br />
<br />
表述热力学主要有两种方式：(a)描述从一种热力学平衡状态到另一种状态的路径，(b)通过<font color = 'red'><s>使系统保持不变，同时增加周围环境的总熵</s></font><font color = 'blue'>系统保持不变但周围环境的总熵增加</font>的循环过程。这两种方法有助于理解生命过程。这个话题大大超出了本文的范围，但是已经有一些研究者考虑过这个问题，比如'''<font color = '#ff8000'>薛定谔Erwin Schrödinger</font>'''、'''<font color = '#ff8000'>布里渊Léon Brillouin</font>'''和'''<font color = '#ff8000'>阿西莫夫Isaac Asimov</font>'''。这也是当前研究的主题。<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of [[metabolism]], give out breakdown products and heat. Plants [[Photosynthesis|take in radiative energy]] from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
<br />
To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of metabolism, give out breakdown products and heat. Plants take in radiative energy from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
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从近似的角度来看，生命体可以被认为是(b)的一个例子。近似地，一只动物的身体状态每天循环，使得它几乎没有什么变化。动物吸收食物、水和氧气，经过'''<font color = '#ff8000'>新陈代谢metabolism</font>'''，输出分解的产物和热<s>量</s>。植物吸收来自太阳的辐射能量，这可以认为是热<s>量</s>，以及二氧化碳和水，然后它们释放氧气。它们就是这样生长的。<font color = 'blue'>它们</font>最终会死亡，尸体腐烂，大部分重新变成二氧化碳和水。这可以看作是一个循环过程。总的来说，阳光来自一个高温的源——太阳，它的能量被传递到一个较低的温度源，<font color = 'red'><s>例如</s></font><font color = 'blue'>即</font>向太空辐射。这个过程使得植物周围环境的熵增加。因此从循环过程的角度来看，动物和植物服从热力学第二定律。简单的热机效率概念很难适用于这个问题，因为它们假定系统是封闭的。<br />
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From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to [[Non-equilibrium thermodynamics|unsettled current debate or research]]. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to unsettled current debate or research. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
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从(a)的热力学观点考虑，即考虑从一个平衡状态到另一个平衡状态的路径，只会产生一个粗略近似的图像，因为生命体从来不会处于热力学平衡状态。生物体经常必须被认为是开放的系统，因为它们吸收营养物质并排出废弃物。开放系统热力学目前通常从一个热力学平衡状态到另一个状态的路径的角度来考虑，或者是考虑局部热力学平衡近似下的流。生命体的问题可以通过假定一个流不变的稳态<font color = 'blue'>的近似</font>来进一步简化。对于这样的近似，<font color = 'red'><s>熵产生的一般原理在'''目前有争论和研究的不确定性Non-equilibrium thermodynamics|unsettled current debate or research'''</s></font><font color = 'blue'>目前有很多关于熵产生的一般原理的未解决的争论或研究</font>。尽管如此，从热力学第二定律的这个角度出发衍生出来的想法对生物是有启发意义的。<br />
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==Gravitational systems==<br />
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==Gravitational systems==<br />
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重力系统<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive [[heat capacity]], meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it.<ref name="Grandy 151">Grandy, W.T. (Jr) (2008), p. 151.</ref> Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
<br />
In systems that do not require for their descriptions the general theory of relativity, bodies always have positive heat capacity, meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it. Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
<br />
<font color = 'red'><s>在不需要描述相对论</s></font><font color = 'blue'>在不需要其对广义相对论的解释的</font>的系统中，物体始终具有正热容，这意味着温度随能量而升高。因此，当能量从高温物体流向低温物体时，发射源温度降低而冷源温度升高；因此，温差会随着时间的流逝而减小。对于引力很重要并且<font color = 'red'><s>基于相对论的一般</s></font><font color = 'blue'>需要广义相对论的</font>系统而言，情况并非总是如此。这样的系统可以自发地<font color = 'red'><s>转变为质量和能量的不均匀分布</s></font><font color = 'blue'>向质量和能量不均匀的分布转变</font>。这适用于大规模的宇宙，因此可能很难或不可能对它应用第二定律。除此之外，广义相对论描述的系统的热力学超出了本文的范围。<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
非平衡状态<br />
{{main article|Non-equilibrium thermodynamics}}<br />
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The theory of classical or [[thermodynamic equilibrium|equilibrium thermodynamics]] is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<ref name="Grandy 151"/><ref>Callen, H.B. (1960/1985), p. 15.</ref><br />
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The theory of classical or equilibrium thermodynamics is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<br />
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经典热力学或平衡热力学理论是理想化的。其主要假定或假设（<font color = 'red'><s>往往无明确说明</s></font><font color = 'blue'>通常甚至没有明确地陈述</font>），是基于系统处在自身内部的热力学平衡状态。通常，在自然界可能发现的，在给定时间包含物理系统的空间区域，其热力学平衡状态并不是能用最严格的术语来表示。用较宽泛的术语来说，整个宇宙中没有任何东西是或曾经是真正处于精确的热力学平衡状态中。<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of [[thermodynamic equilibrium]]. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable [[thermal fluctuations|fluctuations]]. There is an exception, the case of [[critical point (thermodynamics)|critical states]], which exhibit to the naked eye the phenomenon of [[critical opalescence]]. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of thermodynamic equilibrium. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable fluctuations. There is an exception, the case of critical states, which exhibit to the naked eye the phenomenon of critical opalescence. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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为了进行物理分析，通常<font color = 'red'><s>假设热力学平衡就足以方便可行</s></font><font color = 'blue'>做一个热力学平衡的假设就足够方便可行</font>。 这样的假设可能依赖于反复试验和试错来证明其合理性。如果假设是合理的，那么它通常会非常有价值，因为它使热力学理论成为可能。平衡假设的要素是，观察到的系统会无限期地保持不变，并且因为系统中有很多粒子，以至于完全可以忽略其微粒<font color = 'red'><s>性质</s></font><font color = 'blue'>的本质</font>。通常在这种平衡假设下，<font color = 'red'><s>没有</s></font><font color = 'blue'>不存在一个</font>宏观上可检测到的波动。临界状态是一个例外，<font color = 'red'><s>它以肉眼显示出</s></font><font color = 'blue'>肉眼可以观察到</font>临界'''<font color = '#ff8000'>乳光现象opalescence</font>'''。对于临界状态的实验室研究，需要非常长的观察时间。<br />
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In all cases, the assumption of [[thermodynamic equilibrium]], once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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In all cases, the assumption of thermodynamic equilibrium, once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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在所有情况下，一旦<font color = 'red'><s>完成</s></font><font color = 'blue'>做出</font>了热力学平衡的假设，则<font color = 'blue'>作为一个结果</font>意味着假定<font color = 'blue'>中的</font>能改变系统熵的<font color = 'red'><s>“fluctuation”（待定的波动因子）</s></font><font color = 'blue'>候选波动</font>不存在。<br />
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'''<font color = '#32CD32'>则作为一个结果意味着假定中的能改变系统熵的候选波动不存在。implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.</font>'''<br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<ref>Lieb, E.H., Yngvason, J. (2003), p. 190.</ref><br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<br />
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物理系统很容易发生内部宏观变化，其速度之快足以使熵的恒定性假设失效。或物理系统中的粒子太少，以至于微粒的本质在可观测的波动中显而易见。那么热力学平衡的假设就会被放弃。对于非平衡态，熵没有绝对的通俗定义。<br />
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There are intermediate cases, in which the assumption of local [[thermodynamic equilibrium]] is a very good approximation,<ref>Gyarmati, I. (1967/1970), pp. 4-14.</ref><ref>Glansdorff, P., Prigogine, I. (1971).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003).</ref> but strictly speaking it is still an approximation, not theoretically ideal.<br />
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There are intermediate cases, in which the assumption of local thermodynamic equilibrium is a very good approximation, but strictly speaking it is still an approximation, not theoretically ideal.<br />
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还有一些中间情况，其中局部热力学平衡的假设是一个非常好的近似<font color = 'red'><s>值</s></font>，但严格地说它仍然是一个近似值，不是理论上的理想值。<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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对于一般的非平衡状态，考虑其他量的统计力学定义可能是有用的，这些量可以方便地称为“熵”，但是它们不应该与第二定律正确定义的熵混淆<font color = 'blue'>或合并</font>。这些其他量确实属于统计力学，而不属于第二定律的主要领域——热力学。<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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宏观上可观察到的波动的物理学超出了本文的范围。<br />
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==Arrow of time==<br />
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==Arrow of time==<br />
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时间之箭<br />
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{{Further information|Entropy (arrow of time)}}Thermodynamic Asymmetry in Time<br />
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Thermodynamic Asymmetry in Time<br />
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时间中的热力学不对称性<br />
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{{See also|Arrow of time}}<br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely [[CPT symmetry]], since the second law applies statistically, it is hypothesized, on time-asymmetric ''boundary conditions''.<ref>{{cite web|first=Craig|last=Callender|url=https://plato.stanford.edu/archives/fall2011/entries/time-thermo/|title=Thermodynamic Asymmetry in Time|encyclopedia=Stanford Encyclopedia of Philosophy|date=29 July 2011}}</ref><br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with symmetries observed in the fundamental laws of physics (particularly CPT symmetry) since the second law applies statistically on time-asymmetric boundary conditions.[78] The second law has been related to the difference between moving forwards and backwards in time, or to the principle that cause precedes effect (the causal arrow of time, or causality).[79]<br />
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热力学第二定律是<font color = 'red'><s>与时间方向逆转不对称的物理定律</s></font><font color = 'blue'>在时间方向上不对称的物理定律</font>。这与在物理学基本定律中观察到的对称性概念（特别是CPT对称性）<s>并</s>没有冲突，因为第二定律在统计上适用于时间非对称边界条件。第二定律与时间向前和向后移动之间的差异有关，或者与<font color = 'red'><s>导致先于效果</s></font><font color = 'blue'>原因先于结果</font>的原理（时间的因果关系或因果关系）有关。<br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect ([[Arrow of time#The causal arrow of time|the causal arrow of time]]).<ref>{{cite book | first = J.J.| last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1|display-authors=etal}} chapter 6</ref><br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect (the causal arrow of time).<br />
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第二定律被提出来<font color = 'red'><s>部分</s></font><font color = 'blue'>侧面</font>解释时间向前和向后移动的区别，例如为什么原因先于结果(因果关系的时间箭头)。<br />
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==Irreversibility==<br />
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==Irreversibility==<br />
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不可逆性<br />
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Irreversibility in [[thermodynamic process]]es is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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Irreversibility in thermodynamic processes is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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热力学过程中的不可逆性是热力学作用不对称特性的结果，而不是物体内部任何不可逆的微观性质<font color = 'blue'>的结果</font>。热力学作用是对参与体施加的宏观外部干预，并非来自其内部属性。由于没有认识到这一点，产生了被称为“悖论”的说法。<br />
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===Loschmidt's paradox===<br />
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<font color = 'red'><s>洛斯密特悖论 </s></font><font color = 'blue'>洛施密特悖论</font><br />
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{{main article|Loschmidt's paradox}}<br />
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[[Loschmidt's paradox]], also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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'''<font color="#ff8000">洛施密特悖论 Loschmidt's paradox</font>'''，也被称为'''<font color="#ff8000">可逆性悖论reversibility paradox</font>'''，<br />
<font color = 'red'><s>其目的在于不可能从描述宏观系统微观演化的时间对称动力学中推导出不可逆过程。</s></font><font color = 'blue'><br />
它反驳说，从描述宏观系统微观演化的时间对称动力学中，不可能推导出不可逆过程。</font><br />
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In the opinion of [[Erwin Schrödinger|Schrödinger]], "It is now quite obvious in what manner you have to reformulate the law of entropy{{snd}}or for that matter, all other irreversible statements{{snd}}so that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other."<ref>[[Erwin Schrödinger|Schrödinger, E.]] (1950), p.&nbsp;192.</ref> The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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In the opinion of Schrödinger, "It is now quite obvious in what manner you have to reformulate the law of entropyor for that matter, all other irreversible statementsso that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other." The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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薛定谔的认为：“现在很明显，您必须以何种方式重新制定熵定律（亦或者所有其他不可逆陈述）以使它们能够从可逆模型中得出。你不能只谈论一个孤立的系统，而应该至少谈论两个。你可以暂时认为它们与世界其它地方是孤立的，但并不总是相互孤立的。”这两个系统被隔离内壁彼此分开，直到被热力学（由热力学定律设想出来的）作用而拆除。热力学作用是从外部施加的，不受支配系统组成部分的可逆微观动力学定律的制约。这是不可逆转的原因。本文中<font color = 'red'><s>的法律声明</s></font><font color = 'blue'>第二定律的陈述</font>符合薛定谔的的建议。 因果关系在逻辑上先于第二定律，而不是由第二定律推导而来。<br />
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===Poincaré recurrence theorem===<br />
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庞加莱始态复现定理<br />
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{{Main article|Poincaré recurrence theorem}}<br />
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The [[Poincaré recurrence theorem]] considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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The Poincaré recurrence theorem considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics.Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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庞加莱始态复现定理<font color = 'red'><s>（又叫递归定理）</s></font>考虑了孤立物理系统的理论微观描述。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）又叫递归定理吗？存疑<br />
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经过热力学作用去除隔离内壁之后，便可以认为其是热力学系统的模型。在足够长的时间后，系统将返回到非常接近初始状态的微观定义状态。庞加莱复现时间是指回归前经过的时间长度。它极其漫长，可能比宇宙的寿命还要长，并且<font color = 'red'><s>敏感地依赖于被热力学作用拆除的墙体的几何形状</s></font><font color = 'blue'>对那个被热力学作用拆除的内壁的几何形状敏感</font>。复现定理可能被认为是明显与热力学第二定律相矛盾的。但是，更显而易见的是，它只是通过移除两个系统之间的内壁而形成的隔离系统中热力学平衡的微观模型。<font color = 'red'><s>对于典型的热力学系统而言，重复时间是如此之长（比宇宙的寿命长很多倍），以至于在所有的实际目的中，人们都无法观察到这种重复。</s></font><font color = 'blue'>对所有典型的热力学系统的实际目的，人们无法观察到如此之长的复现时间很长（比宇宙的寿命长很多倍）。</font>尽管如此，还是有人会想像一个机会可以等待庞加莱复现的出现，然后重新插入被热力学作用去除的内壁。<br />
然后很明显，不可逆性的出现是由于<font color = 'red'><s>庞加莱递归的完全不可预测性</s></font><font color = 'blue'>庞加莱复现完全不可预测的特性</font>，因为仅仅给出了初始状态<font color = 'red'><s>遵守热力学平衡的条件之一</s></font><font color = 'blue'>是热力学平衡之一</font>，就像宏观热力学的情况一样。<br />
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'''<font color="#32CD32">然后很明显，不可逆性的出现是由于庞加莱复现完全不可预测的特性，因为仅仅给出了初始状态是热力学平衡之一，就像宏观热力学的情况一样。It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics.</font>''' （这句话重复上面的，是为了标注疑难句的同时保留审校痕迹。生产修改后可以只保留这句，删掉上面重复部分）<br />
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即使<font color = 'red'><s>可以等待，也没有实际可操作性</s></font><font color = 'blue'>一个人可以等那么长时间，他也没有实际操作的可能性</font>来选择合适的时间重新插入内壁。庞加莱始态复现定理为洛施密特悖论提供了一个解决方案。如果一个孤立的热力学系统能以多倍于平均庞加莱复现时间的长度下<font color = 'red'><s>进行</s></font><font color = 'blue'>被</font>监控，则该系统的热力学行为在时间反转下将变得恒定。<br />
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[[File:James-clerk-maxwell3.jpg|thumb|upright|James Clerk Maxwell]]<br />
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James Clerk Maxwell<br />
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詹姆斯·克拉克·麦克斯韦<br />
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===Maxwell's demon===<br />
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麦克斯韦<font color = 'red'><s>的恶魔</s></font><font color = 'blue'>妖</font><br />
{{unreferenced|section|date=August 2018}}<br />
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{{main article|Maxwell's demon}}<br />
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[[James Clerk Maxwell]] imagined one container divided into two parts, ''A'' and ''B''. Both parts are filled with the same [[gas]] at equal temperatures and placed next to each other, separated by a wall. Observing the [[molecule]]s on both sides, an imaginary [[demon]] guards a microscopic trapdoor in the wall. When a faster-than-average molecule from ''A'' flies towards the trapdoor, the demon opens it, and the molecule will fly from ''A'' to ''B''. The average [[speed]] of the molecules in ''B'' will have increased while in ''A'' they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in ''A'' and increases in ''B'', contrary to the second law of thermodynamics.<br />
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James Clerk Maxwell imagined one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other, separated by a wall. Observing the molecules on both sides, an imaginary demon guards a microscopic trapdoor in the wall. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics.<br />
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'''麦克斯韦 James Clerk Maxwell'''设想一个容器分为A和B两部分。两部分均在相同温度下充满相同的气体，并彼此相邻由内壁隔开。<font color = 'red'><s>观察到两侧的分子，一个虚构的恶魔在墙上守着一个微观活板门</s></font><font color = 'blue'>一个虚构的妖精看守着一个内壁上的微型陷门，并观察者内壁两边的分子</font>。当来自A的速度快于平均水平的分子飞向<font color = 'red'><s>活板</s></font>门时，<font color = 'red'><s>恶魔将其打开</s></font><font color = 'blue'>妖精将门打开</font>，该分子将从A飞向B。B中分子的平均速度将增加，而A中分子的平均速度将<font color = 'red'><s>平均</s></font>降低。由于平均分子速度与温度相对应，因此与热力学第二定律相反，温度在A中降低，在B中升高。<br />
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One response to this question was suggested in 1929 by [[Leó Szilárd]] and later by [[Léon Brillouin]]. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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One response to this question was suggested in 1929 by Leó Szilárd and later by Léon Brillouin. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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1929年，Leó Szilárd和后来的 Léon Brillouin 对这个问题提出了一个答案。Szilárd指出，现实生活中的麦克斯韦<font color = 'red'><s>恶魔</s></font><font color = 'blue'>妖</font>需要一些测量分子速度的方法，而获取信息<font color = 'blue'>的过程</font>需要消耗<font color = 'red'><s>大量</s></font>能量。<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between ''A'' and ''B''. It is therefore performing [[thermodynamic operation]]s on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between A and B. It is therefore performing thermodynamic operations on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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麦克斯韦的<font color = 'red'><s>“恶魔”不断地</s></font><font color = 'blue'>“妖”重复地</font>改变 A 和 B 之间<font color = 'red'><s>墙体的渗透性</s></font><font color = 'blue'>壁的热透性</font>。因此，它是在微观尺度上执行热力学操作，而不仅仅是观察普通的自发或自然的宏观热力学过程。<br />
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==Quotations==<br />
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==Quotations==<br />
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语录<br />
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{{quote|The law that entropy always increases holds, I think, the supreme position among the [[Laws of science|laws of Nature]]. If someone points out to you that your pet theory of the [[universe]] is in disagreement with [[Maxwell's equations]] – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.|Sir [[Arthur Stanley Eddington]], ''The Nature of the Physical World'' (1927)}}<br />
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The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.<br />
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— Sir Arthur Stanley Eddington, The Nature of the Physical World (1927)<br />
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我认为，熵总是增加的定律在自然定律中占有至高无上的地位。如果有人指出您<font color = 'red'><s>的宇宙宠物理论</s></font><font color = 'blue'>杜撰的宇宙理论</font>与麦克斯韦方程不符，那么麦克斯韦方程就遇到麻烦了。如果发现<font color = 'blue'>您的理论</font>与观察不符，那么，可能是这些实验学家搞砸了。但是，如果发现您的理论违反了热力学第二定律，那么就没有希望了。除了在深深的耻辱中崩溃，别无他法。<br />
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—亚瑟·斯坦利·爱丁顿爵士，《自然世界》（1927年）<br />
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{{quote|There have been nearly as many formulations of the second law as there have been discussions of it.|Philosopher / Physicist [[Percy Williams Bridgman|P.W. Bridgman]], (1941)}}<br />
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There have been nearly as many formulations of the second law as there have been discussions of it.<br />
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— Philosopher / Physicist P.W. Bridgman, (1941)<br />
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关于第二定律的<font color = 'red'><s>阐述</s></font><font color = 'blue'>公式</font>几乎与关于第二定律的讨论一样多。<br />
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—哲学家/物理学家 布里奇曼（1941）<br />
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{{quote|Clausius is the author of the sibyllic utterance, "The energy of the universe is constant; the entropy of the universe tends to a maximum." The objectives of continuum thermomechanics stop far short of explaining the "universe", but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size.|[[Clifford Truesdell|Truesdell, C.]], Muncaster, R. G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 17.}}<br />
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Clausius is the author of the sibyllic utterance, "The energy of the universe is constant; the entropy of the universe tends to a maximum." The objectives of continuum thermomechanics stop far short of explaining the "universe", but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size.<br />
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— Truesdell, C., Muncaster, R. G. (1980). Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics, Academic Press, New York, ISBN 0-12-701350-4, p. 17.<br />
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克劳修斯（Sir Clausius）是西伯利亚文“宇宙的能量是恒定的；宇宙的熵趋于最大。”的作者。<font color = 'red'><s>不间断热力学的</s></font><font color = 'blue'>连续热力学的目标</font>远远不能解释“宇宙”，但是在该理论内，我们可以很容易地以某种类似于克劳修斯的方式得出明确的陈述，只是它仅仅指代一个简单的物体：一个有限大小的孤立物体。<br />
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— Truesdell, C., Muncaster, R. G. (1980). Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics, Academic Press, New York, ISBN 0-12-701350-4, p. 17. 特雷斯德尔，C.，蒙卡斯特，R.G.（1980）。麦克斯韦的单原子气体动力学理论的基本原理，被作为“理性力学”的一个分支，纽约学术出版社，ISBN 0-12-701350-4，第1页. 17。<br />
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==See also==<br />
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参见<br />
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*[[Clausius–Duhem inequality]]<br />
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*[[Fluctuation theorem]]<br />
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*[[Heat death of the universe]]<br />
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*[[History of thermodynamics]]<br />
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*[[Jarzynski equality]]<br />
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*[[Laws of thermodynamics]]<br />
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*[[Maximum entropy thermodynamics]]<br />
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*[[Quantum thermodynamics]]<br />
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*[[Reflections on the Motive Power of Fire]]<br />
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<br />
*[[Relativistic heat conduction]]<br />
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*[[Thermal diode]]<br />
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*[[Thermodynamic equilibrium]]<br />
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<br />
{{colend}}<br />
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<br />
==References==<br />
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==References==<br />
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参考资料<br />
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{{reflist|20em}}<br />
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===Sources===<br />
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===Sources===<br />
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资料来源<br />
<br />
{{refbegin|50em}}<br />
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<br />
<br />
*Adkins, C.J. (1968/1983). ''Equilibrium Thermodynamics'', (1st edition 1968), third edition 1983, Cambridge University Press, Cambridge UK, {{ISBN|0-521-25445-0}}.<br />
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<br />
*[[Peter Atkins|Atkins, P.W.]], de Paula, J. (2006). ''Atkins' Physical Chemistry'', eighth edition, W.H. Freeman, New York, {{ISBN|978-0-7167-8759-4}}.<br />
<br />
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*Attard, P. (2012). ''Non-equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications'', Oxford University Press, Oxford UK, {{ISBN|978-0-19-966276-0}}.<br />
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<br />
*Baierlein, R. (1999). ''Thermal Physics'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-59082-5}}.<br />
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*Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics, New York, {{ISBN|0-88318-797-3}}.<br />
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*{{Cite book|title=Concepts in thermal physics|last=Blundell|first=Stephen J.|last2=Blundell|authorlink1=Stephen Blundell|first2=Katherine M.|authorlink2=Katherine Blundell|publisher=[[Oxford University Press]]|year=2010|isbn=9780199562107|edition=2nd|location=Oxford|pages=|oclc=607907330|doi=10.1093/acprof:oso/9780199562091.001.0001 |url=http://cds.cern.ch/record/1235139}}<br />
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*[[Ludwig Boltzmann|Boltzmann, L.]] (1896/1964). ''Lectures on Gas Theory'', translated by S.G. Brush, University of California Press, Berkeley.<br />
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<br />
*Borgnakke, C., Sonntag., R.E. (2009). ''Fundamentals of Thermodynamics'', seventh edition, Wiley, {{ISBN|978-0-470-04192-5}}.<br />
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*Buchdahl, H.A. (1966). ''The Concepts of Classical Thermodynamics'', Cambridge University Press, Cambridge UK.<br />
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<br />
*[[Percy Williams Bridgman|Bridgman, P.W.]] (1943). ''The Nature of Thermodynamics'', Harvard University Press, Cambridge MA.<br />
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*[[Herbert Callen|Callen, H.B.]] (1960/1985). ''Thermodynamics and an Introduction to Thermostatistics'', (1st edition 1960) 2nd edition 1985, Wiley, New York, {{ISBN|0-471-86256-8}}.<br />
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*{{cite journal|author=C. Carathéodory|author1-link=Constantin Carathéodory<br />
<br />
<br />
<br />
|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
<br />
|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
<br />
|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
<br />
|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
<br />
|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
<br />
2010年3月15日 | 第355-386页 | http://gdz.sub.uni-goettingen.de/index.php?id=11&ppn=ppn235181684_0067&dmdid=dmdlog_0033&l=1<br />
<br />
|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). ''The Second Law of Thermodynamics'', Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
<br />
|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
<br />
|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können.(p. 363) | doi 10.1007 / bf01450409}.译文可以在这里找到[ http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf ]。同时，在 Kestin，j。(1976).热力学第二定律，Dowden，Hutchinson & Ross，Stroudsburg PA. 。<br />
<br />
*[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986). [http://www.worldcat.org/title/reflections-on-the-motive-power-of-fire-a-critical-edition-with-the-surviving-scientific-manuscripts-translated-and-edited-by-fox-robert/oclc/812944517&referer=brief_results ''Reflections on the motive power of fire''], Manchester University Press, Manchester UK, {{ISBN|0-7190-1741-6}}. [https://archive.org/stream/reflectionsonmot00carnrich#page/n7/mode/2up Also here.]<br />
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*[[Sydney Chapman (mathematician)|Chapman, S.]], [[Thomas George Cowling|Cowling, T.G.]] (1939/1970). ''The Mathematical Theory of Non-uniform gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases'', third edition 1970, Cambridge University Press, London.<br />
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*{{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Ueber Die Bewegende Kraft Der Wärme Und Die Gesetze, Welche Sich Daraus Für Die Wärmelehre Selbst Ableiten Lassen|journal=Annalen der Physik|year=1850|volume=79|issue=4|pages=368–397, 500–524|url=http://gallica.bnf.fr/ark:/12148/bpt6k15164w/f518.image|accessdate=26 June 2012|doi=10.1002/andp.18501550403|bibcode = 1850AnP...155..500C |hdl=2027/uc1.$b242250|hdl-access=free}} Translated into English: {{cite journal|last=Clausius|first=R.|title=On the Moving Force of Heat, and the Laws regarding the Nature of Heat itself which are deducible therefrom|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1851| volume=2|series=4th|issue=VIII|pages=1–21; 102–119 |url=https://archive.org/stream/londonedinburghd02lond#page/1/mode/1up|accessdate=26 June 2012<br />
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|doi=10.1080/14786445108646819}}<br />
<br />
|doi=10.1080/14786445108646819}}<br />
<br />
10.1080 / 14786445108646819}<br />
<br />
* {{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie|year=1854|journal=Annalen der Physik|volume=xciii|issue=12|pages=481–506|url=http://zfbb.thulb.uni-jena.de/servlets/MCRFileNodeServlet/jportal_derivate_00140956/18541691202_ftp.pdf|accessdate=24 March 2014|doi=10.1002/andp.18541691202<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
<br />
|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
<br />
| bibcode 1854AnP... 169. 1856年7月 | 第2卷 | 第4页 | 第86页 | https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=242014年3月<br />
<br />
}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
<br />
}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
<br />
热力学理论及其在蒸汽机和物体物理性质中的应用 | 1867年 | 出版商 John van Voorst | 位置: 伦敦 | https://archive.org/details/mechanicaltheor04claugoog|quote=editions:pwr_sbkwa8ic。 2012年6月19日<br />
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}}<br />
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}}<br />
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}}<br />
<br />
*Denbigh, K. (1954/1981). ''The Principles of Chemical Equilibrium. With Applications in Chemistry and Chemical Engineering'', fourth edition, Cambridge University Press, Cambridge UK, {{ISBN|0-521-23682-7}}.<br />
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*Eu, B.C. (2002). ''Generalized Thermodynamics. The Thermodynamics of Irreversible Processes and Generalized Hydrodynamics'', Kluwer Academic Publishers, Dordrecht, {{ISBN|1-4020-0788-4}}.<br />
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*[[Josiah Willard Gibbs|Gibbs, J.W.]] (1876/1878). On the equilibrium of heterogeneous substances, ''Trans. Conn. Acad.'', '''3''': 108-248, 343-524, reprinted in ''The Collected Works of J. Willard Gibbs, Ph.D, LL. D.'', edited by W.R. Longley, R.G. Van Name, Longmans, Green & Co., New York, 1928, volume 1, pp.&nbsp;55–353.<br />
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*Griem, H.R. (2005). ''Principles of Plasma Spectroscopy (Cambridge Monographs on Plasma Physics)'', Cambridge University Press, New York {{ISBN|0-521-61941-6}}.<br />
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*Glansdorff, P., Prigogine, I. (1971). ''Thermodynamic Theory of Structure, Stability, and Fluctuations'', Wiley-Interscience, London, 1971, {{ISBN|0-471-30280-5}}.<br />
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*Grandy, W.T., Jr (2008). [http://global.oup.com/academic/product/entropy-and-the-time-evolution-of-macroscopic-systems-9780199546176?cc=au&lang=en& ''Entropy and the Time Evolution of Macroscopic Systems'']. Oxford University Press. {{ISBN|978-0-19-954617-6}}.<br />
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*Greven, A., Keller, G., Warnecke (editors) (2003). ''Entropy'', Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[Edward A. Guggenheim|Guggenheim, E.A.]] (1949). 'Statistical basis of thermodynamics', ''Research'', '''2''': 450–454.<br />
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*[[Edward A. Guggenheim|Guggenheim, E.A.]] (1967). ''Thermodynamics. An Advanced Treatment for Chemists and Physicists'', fifth revised edition, North Holland, Amsterdam.<br />
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*Gyarmati, I. (1967/1970) ''Non-equilibrium Thermodynamics. Field Theory and Variational Principles'', translated by E. Gyarmati and W.F. Heinz, Springer, New York.<br />
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*[[Charles Kittel|Kittel, C.]], [[Herbert Kroemer|Kroemer, H.]] (1969/1980). ''Thermal Physics'', second edition, Freeman, San Francisco CA, {{ISBN|0-7167-1088-9}}.<br />
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*Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998). ''Modern Thermodynamics: From Heat Engines to Dissipative Structures'', John Wiley & Sons, Chichester, {{ISBN|0-471-97393-9}}.<br />
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*Lebon, G., Jou, D., Casas-Vázquez, J. (2008). ''Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers'', Springer-Verlag, Berlin, {{ISBN|978-3-540-74252-4}}.<br />
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*{{cite journal|ref=harv|title=The Physics and Mathematics of the Second Law of Thermodynamics|last1=Lieb |first1=E. H. |last2=Yngvason |first2=J.|journal=Physics Reports|volume=310|issue=1|pages=1–96 |year=1999|doi=10.1016/S0370-1573(98)00082-9|arxiv = cond-mat/9708200 |bibcode = 1999PhR...310....1L}}<br />
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*Lieb, E.H., Yngvason, J. (2003). The Entropy of Classical Thermodynamics, pp.&nbsp;147–195, Chapter 8 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*{{cite book|ref=harv|first=F.|last=Mandl|title=Statistical physics|edition=second|year=1988|publisher=[[Wiley & Sons]]|isbn=978-0-471-91533-1}}<br />
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*{{cite journal | last1 = Maxwell | first1 = J.C. | authorlink = James Clerk Maxwell | year = 1867 | title = On the dynamical theory of gases | url = | journal = Phil. Trans. R. Soc. Lond. | volume = 157 | issue = | pages = 49–88 | doi = 10.1098/rstl.1867.0004 }}<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985). ''Thermodynamics'', Pitman, London, {{ISBN|0-273-08577-8}}.<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003). Entropy in Nonequilibrium, pp.&nbsp;79–109, Chapter 5 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*Münster, A. (1970), ''Classical Thermodynamics'', translated by E.S. Halberstadt, Wiley–Interscience, London, {{ISBN|0-471-62430-6}}.<br />
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*[[Brian Pippard|Pippard, A.B.]] (1957/1966). ''Elements of Classical Thermodynamics for Advanced Students of Physics'', original publication 1957, reprint 1966, Cambridge University Press, Cambridge UK.<br />
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*[[Max Planck|Planck, M.]] (1897/1903). [https://archive.org/stream/treatiseonthermo00planrich#page/100/mode/2up ''Treatise on Thermodynamics'', translated by A. Ogg, Longmans Green, London, p. 100.]<br />
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*[[Max Planck|Planck. M.]] (1914). [https://archive.org/details/theoryofheatradi00planrich ''The Theory of Heat Radiation''], a translation by Masius, M. of the second German edition, P. Blakiston's Son & Co., Philadelphia.<br />
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*[[Max Planck|Planck, M.]] (1926). Über die Begründung des zweiten Hauptsatzes der Thermodynamik, ''Sitzungsberichte der Preussischen Akademie der Wissenschaften: Physikalisch-mathematische Klasse'': 453–463.<br />
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* Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769-781.<br />
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* {{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}<br />
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*Quinn, T.J. (1983). ''Temperature'', Academic Press, London, {{ISBN|0-12-569680-9}}.<br />
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*{{cite book|ref=harv|first=Y.V.C.|last=Rao|title=An Introduction to thermodynamics|url=https://books.google.com/books?id=iYWiCXziWsEC&pg=PA213|date=2004|publisher=Universities Press|isbn=978-81-7371-461-0|page=213}}<br />
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*Roberts, J.K., Miller, A.R. (1928/1960). ''Heat and Thermodynamics'', (first edition 1928), fifth edition, Blackie & Son Limited, Glasgow.<br />
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*[[Erwin Schrödinger|Schrödinger, E.]] (1950). Irreversibility, ''Proc. R. Ir. Acad.'', '''A53''': 189–195.<br />
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*[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966). ''Elements of Thermodynamics'', Addison-Wesley Publishing, Reading MA.<br />
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*{{cite journal|ref=harv|last=Thomson|first=W.|author-link=William Thomson, 1st Baron Kelvin|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Transactions of the Royal Society of Edinburgh|year=1851|volume=XX|issue=part II|pages=261–268; 289–298|url=https://www.biodiversitylibrary.org/item/126047#page/295/mode/1up}} Also published in {{cite journal|last=Thomson|first=W.|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Philos. Mag. |date=December 1852 |volume=IV |series=4 |issue=22 |page=13|url=https://archive.org/stream/londonedinburghp04maga#page/12/mode/2up |accessdate=25 June 2012}}<br />
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*[[William Thomson, 1st Baron Kelvin|Thomson, W.]] (1852). ''On the universal tendency in nature to the dissipation of mechanical energy'' Philosophical Magazine, Ser. 4, p.&nbsp;304.<br />
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*[[László Tisza|Tisza, L.]] (1966). ''Generalized Thermodynamics'', M.I.T Press, Cambridge MA.<br />
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*[[Clifford Truesdell|Truesdell, C.]] (1980). ''The Tragicomical History of Thermodynamics 1822–1854'', Springer, New York, {{ISBN|0-387-90403-4}}.<br />
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*Uffink, J. (2001). Bluff your way in the second law of thermodynamics, ''Stud. Hist. Phil. Mod. Phys.'', '''32'''(3): 305–394.<br />
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*Uffink, J. (2003). Irreversibility and the Second Law of Thermodynamics, Chapter 7 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003), Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963). ''Lectures in Statistical Mechanics'', American Mathematical Society, Providence RI.<br />
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*[[Mark Zemansky|Zemansky, M.W.]] (1968). ''Heat and Thermodynamics. An Intermediate Textbook'', fifth edition, McGraw-Hill Book Company, New York.<br />
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==Further reading==<br />
<br />
==Further reading==<br />
<br />
进一步阅读<br />
<br />
*Goldstein, Martin, and Inge F., 1993. ''The Refrigerator and the Universe''. Harvard Univ. Press. Chpts. 4–9 contain an introduction to the Second Law, one a bit less technical than this entry. {{ISBN|978-0-674-75324-2}}<br />
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*Leff, Harvey S., and Rex, Andrew F. (eds.) 2003. ''Maxwell's Demon 2 : Entropy, classical and quantum information, computing''. Bristol UK; Philadelphia PA: [[Institute of Physics]]. {{ISBN|978-0-585-49237-7}}<br />
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<br />
*{{Cite book | first = J.J. | last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1}}(technical).<br />
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*{{cite book |title=Reflections on the Motive Power of Heat and on Machines Fitted to Develop That Power |last=Carnot |first=Sadi |editor=[[Robert Henry Thurston|Thurston, Robert Henry]] |year=1890 |publisher=J. Wiley & Sons |location=New York |isbn= |pages= }} ([https://books.google.com/books?id=tgdJAAAAIAAJ full text of 1897 ed.]) ([http://www.history.rochester.edu/steam/carnot/1943/ html])<br />
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*Stephen Jay Kline (1999). ''The Low-Down on Entropy and Interpretive Thermodynamics'', La Cañada, CA: DCW Industries. {{ISBN|1-928729-01-0}}.<br />
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*{{cite book | last1 = Kostic | first1 = M | year = 2011 | title = Revisiting The Second Law of Energy Degradation and Entropy Generation: From Sadi Carnot's Ingenious Reasoning to Holistic Generalization | journal = AIP Conf. Proc. | volume = 1411 | issue = 1| pages = 327–350 | doi = 10.1063/1.3665247 | isbn = 978-0-7354-0985-9 | bibcode = 2011AIPC.1411..327K | series = AIP Conference Proceedings | citeseerx = 10.1.1.405.1945 }} also at [https://web.archive.org/web/20130420222450/http://www.kostic.niu.edu/2ndLaw/Revisiting%20The%20Second%20Law%20of%20Energy%20Degradation%20and%20Entropy%20Generation%20-%20From%20Carnot%20to%20Holistic%20Generalization-4.pdf].<br />
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==External links==<br />
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==External links==<br />
<br />
外部链接<br />
<br />
*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics]" – by Lawrence Sklar.<br />
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*[http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node30.html ''Second law of thermodynamics''] in the MIT Course [http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/notes.html ''Unified Thermodynamics and Propulsion''] from Prof. Z. S. Spakovszky<br />
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*[[E.T. Jaynes]], 1988, "[http://bayes.wustl.edu/etj/articles/ccarnot.pdf The evolution of Carnot's principle]," in G. J. Erickson and C. R. Smith (eds.)''Maximum-Entropy and Bayesian Methods in Science and Engineering, Vol'' 1: p.&nbsp;267.<br />
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*[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf Caratheodory, C., "Examination of the foundations of thermodynamics," trans. by D. H. Delphenich]<br />
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* [https://www.bbc.co.uk/programmes/p004y2bm The Second Law of Thermodynamics], BBC Radio 4 discussion with John Gribbin, Peter Atkins & Monica Grady (''In Our Time'', Dec. 16, 2004)<br />
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* [http://mdpi.org/entropy/papers/e6010001.pdf [[Entropy (journal)]], 2004]<br />
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* [https://www.journals.uchicago.edu/doi/abs/10.1086/663835 The Journal of the International Society for the History of Philosophy of Science, 2012]<br />
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{{DEFAULTSORT:Second Law Of Thermodynamics}}<br />
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[[Category:Concepts in physics]]<br />
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Category:Concepts in physics<br />
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分类: 物理概念<br />
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[[Category:Laws of thermodynamics|2]]<br />
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2<br />
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[[Category:Non-equilibrium thermodynamics]]<br />
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Category:Non-equilibrium thermodynamics<br />
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类别: 非平衡态热力学<br />
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[[Category:Philosophy of thermal and statistical physics]]<br />
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Category:Philosophy of thermal and statistical physics<br />
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类别: 热力学和统计物理学哲学<br />
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<small>This page was moved from [[wikipedia:en:Second law of thermodynamics]]. Its edit history can be viewed at [[热力学第二定律/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E7%83%AD%E5%8A%9B%E5%AD%A6%E7%AC%AC%E4%BA%8C%E5%AE%9A%E5%BE%8B&diff=12748热力学第二定律2020-08-16T13:20:06Z<p>Dorr：/* Principle of Carathéodory 卡拉西奥多里原理 */</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Law of physics}}<br />
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{{Thermodynamics|laws}}<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 全文审校中，heat一般统一译为“热”，caloric 或者 caloric of heat 译为“热量”。这一点是否准确，有待各位同学和专家的批评指正<br />
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The '''second law of thermodynamics''' states that the total [[entropy]] of an [[isolated system]] can never decrease over time, and is constant if and only if all processes are reversible.<ref>http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node38.html#SECTION05224000000000000000</ref> Isolated systems spontaneously evolve towards [[thermodynamic equilibrium]], the state with maximum entropy. <br />
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The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, and is constant if and only if all processes are reversible. Isolated systems spontaneously evolve towards thermodynamic equilibrium, the state with maximum entropy. <br />
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<font color="#ff8000">'''热力学第二定律 Second Law Of Thermodynamics'''</font>指出，<font color="#ff8000">'''孤立系统Isolated System'''</font>的总熵永远不会随着时间而减少，且当且仅当所有过程都是可逆时，总熵才恒定。孤立系统自发地到达到热力学平衡状态，此时为熵最大的状态。<br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in [[thermodynamic equilibrium]], or is undergoing a (fictive) [[Reversible process (thermodynamics)|reversible process]]. In all processes that occur, including [[spontaneous process]]es,<ref>Atkins and de Paula, p.78</ref> the total entropy of the system and its surroundings increases and the process is [[Irreversible process|irreversible in the thermodynamic sense]]. The increase in entropy accounts for the irreversibility of natural processes, and the [[Arrow of time|asymmetry between future and past]].<ref>{{cite book|last=Zohuri|first=Bahman|title=Dimensional Analysis Beyond the Pi Theorem|url=https://books.google.com/books?id=pRVuDQAAQBAJ|year=2016|publisher=Springer|isbn=978-3-319-45726-0|page=[https://books.google.com.ph/books?id=pRVuDQAAQBAJ&pg=PA111&dq=%22increase+in+entropy+accounts+for+the+irreversibility+of+natural+processes+and+the+asymmetry+between+future+and+past.%22&hl=en&sa=X&ved=0ahUKEwj6spb61tbaAhUFS7wKHfftDtIQ6AEIKDAA#v=onepage&q=%22increase%20in%20entropy%20accounts%20for%20the%20irreversibility%20of%20natural%20processes%20and%20the%20asymmetry%20between%20future%20and%20past.%22 111]}}</ref><br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in thermodynamic equilibrium, or is undergoing a (fictive) reversible process. In all processes that occur, including spontaneous processes, the total entropy of the system and its surroundings increases and the process is irreversible in the thermodynamic sense. The increase in entropy accounts for the irreversibility of natural processes, and the asymmetry between future and past.<br />
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系统及其周围环境的总熵在理想情况下可以保持不变，在这种情况下，系统处于热力学平衡状态，或者正在经历一个(<font color = 'red'><s>虚拟的</s></font><font color = 'blue'>假想的</font>)可逆过程。<font color = 'red'><s>在所有发生的</s></font><font color = 'blue'>所有</font>过程中，包括<font color="#ff8000">'''自发过程 Spontaneous Processes'''</font>，系统及其周围环境的总熵增加，这一过程在热力学意义上是不可逆的。熵的增加解释了自然过程的不可逆性，以及未来和过去之间的不对称性。<br />
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Historically, the second law was an [[Empirical evidence|empirical finding]] that was accepted as an axiom of [[thermodynamics|thermodynamic theory]]. [[Statistical mechanics]], classical or [[quantum thermodynamics|quantum]], explains the microscopic origin of the law.<br />
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Historically, the second law was an empirical finding that was accepted as an axiom of thermodynamic theory. Statistical mechanics, classical or quantum, explains the microscopic origin of the law.<br />
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从历史上看，第二定律是一个经验性的发现，它被认为是热力学理论中的公理。经典统计力学、量子统计力学都可以解释这个定律的微观起源。<br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist [[Nicolas Léonard Sadi Carnot|Sadi Carnot]], who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as [[Carnot's theorem (thermodynamics)|Carnot's rule]] or limit.<ref>Jaffe, R.L., Taylor, W. (2018). ''The Physics of Energy'', Cambridge University Press, Cambridge UK, pages 150, 151, 259, 772, 743.</ref><br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist Sadi Carnot, who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as Carnot's rule or limit.<br />
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<font color = 'blue'>热力学第二定律可以使用多种方法表述。它的第一个公式归功于法国科学家<font color="#ff8000">'''卡诺Sadi Carnot'''</font>，卡诺在1824年证明了在热机中将热转化为功的效率有一个上限。第二定律的这个方面也被称为<font color="#ff8000">'''卡诺规则Carnot's Rule'''</font>或<font color="#ff8000">'''卡诺限制Carnot's Limit'''</font>。</font><br />
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'''热力学第二定律 Second Law of Thermodynamics'''是热力学的四条基本定律之一，它表述了热力学过程的不可逆性——孤立系统自发地朝着热力学平衡方向（即最大熵状态）演化，另一种表述为：第二类永动机永不可能实现。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）这里是不是少了一段原文？<br />
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==Introduction引言==<br />
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[[File:Heat flow hot to cold.png|thumb|upright|Heat flow from hot water to cold water. 热总是从热水流向冷水]]<br />
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The [[first law of thermodynamics]] provides the definition of the [[internal energy]] of a [[thermodynamic system]], and expresses the law of [[conservation of energy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 40–41.</ref><ref>Munster A. (1970), pp. 8–9, 50–51.</ref> The second law is concerned with the direction of natural processes.<ref>{{harvnb|Mandl|1988}}</ref> It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such [[Phenomenon|phenomena]] are accounted for in terms of [[entropy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 79–107.</ref><ref>Bailyn, M. (1994), Section 71, pp. 113–154.</ref> If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, {{math|''S''}}, increases.<br />
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The first law of thermodynamics provides the definition of the internal energy of a thermodynamic system, and expresses the law of conservation of energy. The second law is concerned with the direction of natural processes. It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such phenomena are accounted for in terms of entropy. If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, S, increases.<br />
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'''热力学第一定律 First Law Of Thermodynamics''' <font color = 'red'><s>对</s></font> <font color = 'blue'>定义了</font>热力学系统所涉及到的'''内能 Internal Energy'''<font color = 'red'><s>进行了定义</s></font> ，并体现了能量守恒定律。热力学第二定律与'''自然过程Natural Processes'''的方向有关。它<font color = 'red'><s>设定</s></font> <font color = 'blue'>断言</font>自然过程只在一种意义上进行，且<font color = 'red'><s>是</s></font> 不可逆<font color = 'red'><s>的</s></font> 。例如，当有了传导和辐射的（传播）路径时，热<s>量</s>总是自发地从一个较热的物体流向一个较冷的物体。这种现象可以用'''熵Entropy'''来解释。<font color = 'red'><s>若一个孤立系统最初在内部不可渗透的薄膜维持内部的热力学平衡，通过一些操作使得薄膜具有渗透性，则该系统可自发地演变，最终达到一个新的内部热力学平衡，其总熵增加。</s></font><br />
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<font color = '#32CD32'>若一个'''孤立系统Isolated System'''最初在具有隔热内壁的系统内维持热力学平衡，通过一些操作使内壁透热，则该系统可自发地演变，最终达到一个新的内部热力学平衡，且其总熵{{math|''S''}}增加。If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, S, increases.</font><br />
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--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]） more permeable不知道是否需要译为更具有 impermeable wall是不可渗透 如果要使用比较级 是否应该再有一个 permeable<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） internal partitioning impermeable walls翻译为隔热的内壁，从而permeable指的是透热的。我觉得这里的渗透性应该是说系统可以和外界进行能量/热的交换。？不知道是否正确。<br />
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In a fictive reversible process, an infinitesimal increment in the entropy ({{math|d''S''}}) of a system is defined to result from an infinitesimal transfer of heat ({{math|δ''Q''}}) to a [[closed system]] (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature ({{math|''T''}}) of the system in equilibrium and the surroundings which supply the heat:<ref>Bailyn, M. (1994), p. 120.</ref><br />
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In a fictive reversible process, an infinitesimal increment in the entropy () of a system is defined to result from an infinitesimal transfer of heat () to a closed system (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature () of the system in equilibrium and the surroundings which supply the heat:<br />
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在<font color = 'red'><s>设置的虚拟</s></font> <font color = 'blue'>假想的</font>可逆过程中，系统的熵的无穷小增量<math>dS</math>被定义为由无穷小的热<s>量</s><math>δQ</math>传递到一个封闭系统(允许能量进入或出去，但不允许物质传递<font color = 'blue'>的系统</font>)除以<font color = 'blue'>该</font>平衡系统和提供热<s>量</s>的环境的共同温度 <math>T</math>:<br />
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: <math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process封闭系统中理想状态下的可逆过程).}</math><br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 这里的公式和备注貌似有误，公式应为 <math>\mathrm dS = \frac{\delta Q}{T}</math>，备注应为“closed system, idealized fictive reversible process”封闭系统，理想的可逆过程。或者？<br />
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Different notations are used for infinitesimal amounts of heat ({{mvar|δ}}) and infinitesimal amounts of entropy ({{mvar|d}}) because entropy is a [[function of state]], while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the [[Clausius Theorem|inequality]] <ref name="MortimerBook"> Mortimer, R. G. ''Physical Chemistry''. 3rd ed., p. 120, Academic Press, 2008. </ref><ref name="FermiBook"> Fermi, E. ''Thermodynamics'', footnote on p. 48, Dover Publications,1956 (still in print). </ref> <br />
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Different notations are used for infinitesimal amounts of heat () and infinitesimal amounts of entropy () because entropy is a function of state, while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the inequality <br />
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用不同的符号''δ''和''d''表示无穷小量的热<s>量</s>和无穷小量的熵，因为熵是状态函数，而热<s>量</s>和功一样<font color = 'blue'>则</font>不是状态函数。<font color = 'red'><s>第二定律要求系统熵的增量满足不等式，</s></font>对于实际上可能存在的不与环境发生<font color = 'red'><s>质量</s></font><font color = 'blue'>物质</font>交换的无穷小过程，<font color = 'blue'>第二定律要求</font>系统熵增量满足不等式：<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 根据上下文（讲到封闭系统），此处疑“mass”译为物质更佳<br />
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: <math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process 封闭系统中，实际上可能发生的可逆过程).}</math><br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature ({{math|''T''}}) and the temperature of the surroundings ({{math|''T''}}<sub>''surr''</sub>).<br />
<ref name=":0">Adkins, C.J. (1968/1983), p. 75.</ref><ref name="Munster 45"/> <br />
Note that the equality still applies for pure heat flow,<br />
<ref name="Schmidt-Rohr 14"> Schmidt-Rohr, K. (2014). "Expansion Work without the External Pressure, and Thermodynamics in Terms of Quasistatic Irreversible Processes" ''J. Chem. Educ.'' '''91''': 402-409. https://dx.doi.org/10.1021/ed3008704 </ref><br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature () and the temperature of the surroundings (<sub>surr</sub>).<br />
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<font color = 'blue'>这是因为</font>这种情况下的一般过程可能包括周围环境对系统所做的功，<font color = 'red'><s>这是因为</s></font><font color = 'blue'>这个功</font>在系统内部会产生摩擦或粘滞效应，<font color = 'blue'>或是因为</font>此时<font color = 'red'><s>是由于</s></font><font color = 'blue'>一个</font>化学反应可能正在<font color = 'blue'>系统内部</font>进行，或<font color = 'blue'>因为</font>热传递实际上是不可逆地发生，<font color = 'red'><s>通过</s></font>系统温度<math>T</math>和周围环境温度<math>T_surr</math>之间<font color = 'red'><s>存在差异而进行驱动</s></font><font color = 'blue'>的差异驱动了热传递的发生</font>。<br />
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）该句有些不太理解<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 我觉得应该是“This is because a general process……，because a chemical reaction……，or because heat transfer……surroundings (<sub>surr</sub>).”的并列结构。<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）我的理解，第一个because是对上文的解释；后两个because是对“功在系统内部产生……效应”的解释；后两个为并列，不与第一个并列，因此“粘滞效应”后面的because前不用加“或是”<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）and the temperature of the surroundings (<sub>surr</sub>)后面缺少半句原文：“Note that the equality still applies for pure heat flow,”<br />
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<font color='blue'>注意这个等式也适用于'''纯热流Pure Heat Flow''',</font><br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change 实际上可能的,不改变成分的准静态不可逆性).}</math><br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry.<ref name="Oxtoby8th">Oxtoby, D. W; Gillis, H.P., [[Laurie Butler|Butler, L. J.]] (2015).''Principles of Modern Chemistry'', Brooks Cole. p. 617. {{ISBN|978-1305079113}}</ref> <ref name="MortimerBook"></ref> Introducing a set of internal variables <math>\xi</math> to describe the deviation of a thermodynamic system in physical equilibrium (with the required well-defined uniform pressure ''P'' and temperature ''T'')<ref name="Schmidt-Rohr 14"></ref> from the chemical equilibrium state, one can record the equality<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry.[16] [11] Introducing a set of internal variables {\displaystyle \xi }\xi to describe the deviation of a thermodynamic system in physical equilibrium (with the required well-defined uniform pressure P and temperature T)[15] from the chemical equilibrium state, one can record the equality<br />
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<font color='blue'>这种方法</font>是通过测量'''热容曲线 Heat Capacity Curves'''和相变中熵的变化，来准确测定纯物质的绝对熵的基础，比如'''量热法calorimetry'''。<br />
为了描述一个热力学系统在物理平衡状态下(要求有明确定义的等压P和等温T)偏离化学平衡状态，引入一组内部变量<math>x_i</math>,可以用该等式<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process 封闭系统，实际可能的准静态不可逆过程).}</math><br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of ''the second principle of thermodynamics'' – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<ref> Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769–781.</ref><ref>{{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}</ref><br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of the second principle of thermodynamics – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<br />
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第二项代表内部变量的功，这些内部变量<font color='red'><s>可以</s></font><font color= 'blue'>可能</font>受到外部影响的干扰，但是系统不能通过内部变量做任何正功。这种说法介绍了热力学系统在时间上<font color='red'><s>不可能逆转演化的事实</s></font>演化不可能逆转的性质<font color= 'blue'></font>，并且<font color='red'><s>可以被认为是热力学第二原理的另外一种相当于熵原理的表述。</s></font><font color= 'blue'>该公式可以被看作使用'''熵Entropy'''的热力学第二定律等价表述</font><br />
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The [[zeroth law of thermodynamics]] in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body.<ref name=dugdale>{{cite book|author=J. S. Dugdale|title=Entropy and its Physical Meaning|publisher=Taylor & Francis|year=1996|isbn=978-0-7484-0569-5|page=13|quote=This law is the basis of temperature.}}</ref> For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows{{how?|date=August 2018}} a distinguished temperature scale, which defines an absolute, [[thermodynamic temperature]], independent of the properties of any particular reference thermometric body.<ref>[[Mark Zemansky|Zemansky, M.W.]] (1968), pp. 207–209.</ref><ref>Quinn, T.J. (1983), p. 8.</ref><br />
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The zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body. For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows a distinguished temperature scale, which defines an absolute, thermodynamic temperature, independent of the properties of any particular reference thermometric body.<br />
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热力学第零定律是指如果两个热力学系统<font color='red'><s>中的每一个</s></font>都与第三个热力学系统处于热平衡(温度相同)<font color= 'blue'>状态</font>，则它们彼此也必定处于热平衡<font color= 'blue'>状态</font>。热力学第零定律在<font color='red'><s>它</s></font>这个<font color= 'blue'></font><font color='red'><s>通常的</s></font>简短叙述中让人们认识到热平衡关系中的两个物体具有相同的温度，特别是当一个被测物体与一个参考测温物体具有相同的温度时，对于<font color='red'><s>一个与另一个</s></font>两个<font color= 'blue'></font>处于热平衡<font color= 'blue'>状态</font>的物体，有无限多的'''<font color = '#ff8000'>经验温标empirical temperature scales</font>'''，这通常<font color= 'blue'>分别</font>取决于特定参考温度<font color='red'><s>标度</s></font><font color= 'blue'>体</font>的性质。热力学第二定律允许区分'''<font color = '#ff8000'>温度标度temperature scale</font>'''，它定义了一个绝对的热力学温度，与任何特定的参考温度体的性质无关。<br />
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--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）补充热力学第零定律<br />
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==Various statements of the law热力学第二定律的不同表述==<br />
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The second law of thermodynamics may be expressed in many specific ways,<ref name=MIT>{{cite web|title=Concept and Statements of the Second Law|url=http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node37.html|accessdate=2010-10-07 |publisher=web.mit.edu}}</ref> the most prominent classical statements{{sfnp|Lieb|Yngvason|1999}} being the statement by [[Rudolf Clausius]] (1854), the statement by [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] (1851), and the statement in axiomatic thermodynamics by [[Constantin Carathéodory]] (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.{{sfnp|Rao|2004|p=213}}<br />
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The second law of thermodynamics may be expressed in many specific ways, the most prominent classical statements being the statement by Rudolf Clausius (1854), the statement by Lord Kelvin (1851), and the statement in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.<br />
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热力学第二定律可以用许多特定的方式来表达，最突出的经典陈述是 '''克劳修斯 Rudolf Clausius''' (1854)<font color = 'red'><s>陈述</s></font><font color = 'blue'>表述</font>，'''开尔文 Kelvin''' (1851)<font color = 'red'><s>陈述</s></font><font color = 'blue'>表述</font>，以及康斯坦丁·卡拉西奥多里 Constantin Carathéodory(1909)在'''<font color = '#ff8000'>公理化热力学axiomatic thermodynamics</font>'''中的<font color = 'red'><s>陈述</s></font><font color = 'blue'>表述</font>。这些<font color = 'red'><s>陈述</s></font><font color = 'blue'>表述</font>用一般的物理术语来描述定律，引用某些过程的不可能性。克劳修斯和开尔文表述被证明是等价的。<br />
--[[用户:厚朴|厚朴]]（[[用户讨论:厚朴|讨论]]）statement的表述需要统一，同样的statement by Rudolf Clausius这里需要统一开尔文 克劳修斯 Kelvin Clausius<br />
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===Carnot's principle卡诺原理===<br />
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The historical origin<ref>[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986).</ref> of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a [[Carnot heat engine]], fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the [[caloric theory]] of heat was seriously considered, before the recognition of the [[first law of thermodynamics]], and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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The historical origin of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a Carnot heat engine, fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the caloric theory of heat was seriously considered, before the recognition of the first law of thermodynamics, and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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热力学第二定律的历史起源是'''卡诺原理Carnot's principle'''。<br />
它指的是'''<font color = '#ff8000'>卡诺热机Carnot heat engine</font>'''的一个循环，<font color = 'red'><s>它</s></font><font color = 'blue'>卡诺热机</font>以'''<font color = '#ff8000'>准静态quasi-static</font>'''的极限慢速运转，因此热<s>量</s>和功在子系统之间进行传递，子系统总是处于它们自己内部的热力学平衡状态。<br />
卡诺热机是研究热机效率的工程师特别感兴趣的理想装置。<br />
当卡诺发现卡诺原理时，'''<font color = '#ff8000'>热量理论caloric theory of heat</font>'''还没有得到重视，热力学第一定律还没有得到承认，熵的概念还没有数学表达。<br />
根据第一定律的解释，它在物理上等同于热力学第二定律，并沿用至今。<br />
在热力学第一定律被发现之前，卡诺最初的论点是从热量理论的观点出发的。<br />
下面是他书中的一些例子：<br />
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::...''wherever there exists a difference of temperature, motive power can be produced.''<ref>Carnot, S. (1824/1986), p. 51.</ref><br />
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...wherever there exists a difference of temperature, motive power can be produced.<br />
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……只要有温差，就能产生动力。<br />
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::The production of motive power is then due in steam engines not to an actual consumption of caloric, but ''to its transportation from a warm body to a cold body ...''<ref>Carnot, S. (1824/1986), p. 46.</ref><br />
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The production of motive power is then due in steam engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body ...<br />
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动力的产生不是由于蒸汽机实际消耗的热量，而是由于它从一个较热的物体<font color = 'red'><s>运输</s></font><font color = 'blue'>转移</font>到一个较冷的物体……。<br />
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::''The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.''<ref>Carnot, S. (1824/1986), p. 68.</ref><br />
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The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.<br />
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热<s>量</s>的原动力与实现热的媒介无关，<font color = 'red'><s>热量的数量</s></font><font color = 'blue'>热的量</font>完全取决于两个物体之间的温度，最后是热量的传递。<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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用现代术语<font color = 'red'><s>来说</s></font>，卡诺原理可能更为准确:<br />
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::The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<ref>[[Clifford Truesdell|Truesdell, C.]] (1980), Chapter 5.</ref><ref>Adkins, C.J. (1968/1983), pp. 56–58.</ref><ref>Münster, A. (1970), p. 11.</ref><ref>Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998), pp.67–75.</ref><ref>Lebon, G., Jou, D., Casas-Vázquez, J. (2008), p. 10.</ref><ref>Eu, B.C. (2002), pp. 32–35.</ref><br />
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The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<br />
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准静态卡诺循环或可逆卡诺循环的效率只取决于两种热源的温度，而且无论工作物质是什么，效率是相同的。<font color = 'red'><s>只有</s></font><font color = 'blue'>使用</font>两个热源与工作物质交换热源（一个高温热源温度<math>T_1</math>和一个低温热源温度<math>T_2</math>)的卡诺热机是最有效的热机。<br />
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'''<font color="#32CD32">使用两个热源与工作物质交换热源（一个高温热源温度<math>T_1</math>和一个低温热源温度<math>T_2</math>)的卡诺热机是最有效的热机。 A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.</font>''' <br />
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--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）最后一句进行了补充 意译<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 我没太看懂卡诺表述，所以不能确定这里的补充是否正确。可以再看一下“与工作物质交换热源”部分的表述是否准确<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）如果不确定，可以考虑将这句话标注为疑难句；或直接按原文直译<br />
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===Clausius statement克劳修斯表述===<br />
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The German scientist [[Rudolf Clausius]] laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work.{{sfnp|Clausius|1850}} His formulation of the second law, which was published in German in 1854, is known as the ''Clausius statement'':<br />
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The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work. His formulation of the second law, which was published in German in 1854, is known as the Clausius statement:<br />
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1850年，德国科学家'''鲁道夫·克劳修斯 Rudolf Clausius''' 通过研究热传递和功之间的关系，为热力学第二定律<font color = 'red'><s>实验室</s></font>奠定了基础。他在1854年用德语发表的论文中所提及的热力学第二定律定义被称为克劳修斯表述:<br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.{{sfnp|Clausius|1854|p=86}}</blockquote><br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.</blockquote><br />
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不可能把热<s>量</s>从低温物体传递到高温物体而不产生其他影响。 <br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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克劳修斯的表述使用了“'''<font color = '#ff8000'>热通道Passage Of Heat</font>'''”的概念。<font color = 'red'><s>正如通常在热力学的讨论，这意味着“能量作为热的净转移” ，而不是另一种形式上的"分摊转账"或其他。</s></font><font color = 'blue'><br />
在热力学的讨论中，通常这意味着“能量作为热的形式的净转移” ，而不是指其他方式的转移。</font><br />
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）不太理解分摊转账contributory transfers <br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 我觉得应该是说，energy as '''heat''' 而不是 as其他方式的转移<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）我认为这里不要把contributory transfer看作一个专有名词去理解，从contributory的本意“A contributory factor of a problem or accident is one of the things which caused it to exist or happen.”来理解即可，感觉嘉树理解的是对的。<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of [[refrigeration]], for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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如果不对系统<font color = 'red'><s>进行外部功</s></font><font color = 'blue'>外部做功</font>，热<font color = 'red'><s>量就</s></font>不能自发地从冷区流向热区，这一点从制冷的普通经验中也可以看出。在冰箱中，<font color = 'red'><s>热量从冷到热，但只有在外部媒介——制冷系统的强制作用下才会发生变化。</s></font><font color = 'blue'>只有在外部媒介也就是制冷系统的强制作用下热才会从冷区流到热区。</font><br />
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===Kelvin statements开尔文表述===<br />
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[[William Thomson, 1st Baron Kelvin|Lord Kelvin]] expressed the second law in several wordings.<br />
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Lord Kelvin expressed the second law in several wordings.<br />
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'''<font color = '#ff8000'>开尔文勋爵 Lord Kelvin</font>''' <font color = 'red'><s>用几个字表达</s></font><font color = 'blue'>表述</font>了热力学第二定律。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 感觉”用几个字“怪怪的，所以删掉了<br />
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::It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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<font color = 'red'><s>无法从单一热源取热使其完全转化为有用功而不对环境产生影响。</s></font><font color = 'blue'>不可能从单一热源取热使之完全转换为有用的功而不产生其他影响。</font><br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 改的内容和原文内容不太对应。这个摘自百度百科，另参见 严子浚(1994). 对热力学第二定律开尔文表述的一点讨论, 大学物理: 13(2).<br />
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::It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.{{sfnp|Thomson|1851}}<br />
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It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.<br />
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<font color = 'red'><s>通过无生命物质的作用，不可能通过将物质的任何部分冷却到低于周围物体最冷的温度来产生机械效应。</s></font><font color = 'blue'>不可能通过无生命物质的作用，将物质的任何部分冷却到低于周围物体最低的温度并产生'''<font color = '#ff8000'>机械效应 Mechanical Effect</font>'''。</font><br />
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--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）无生命物质可以意译为机器吗<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 我不太懂这个，不过感觉最好不要吧。<br />
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===Equivalence of the Clausius and the Kelvin statements克劳修斯和开尔文表述的等价性===<br />
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[[Image:Deriving Kelvin Statement from Clausius Statement.svg|thumb|Derive Kelvin Statement from Clausius Statement从克劳修斯表述推导出开尔文表述]]<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed [[Carnot engine]] as shown by the figure. The [[Heat engine#Efficiency|efficiency]] of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the [[first law of thermodynamics]], as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the figure. The efficiency of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat from <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the first law of thermodynamics, as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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假设有一个热机违反了开尔文定理: 也就是说，这个热机以循环的方式吸收热<s>量</s>并将其<font color = 'blue'>完全</font>转化为功，而<font color = 'blue'>且</font>不产生任何影响。现<font color = 'blue'>在将其</font>与反向卡诺<font color = 'red'><s>发动</s></font>机<font color = 'red'><s>相对应</s></font><font color = 'blue'>比较</font>，如图所示。<br />
普通热机的效率为η，反向热机的效率为1/η。这对<font color = 'red'><s>热机的联合作用为</s></font><font color = 'blue'>联合热机的净效应和唯一效应是</font><math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> 将热<s>量</s>从较冷热源<font color = 'blue'>转移</font>到较热热源，这违反了克劳修斯表述。（这是能量守恒定律的结果，因为系统的总能量保持不变<math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>，所以<math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math>。）因此，违反开尔文表述意味着违反克劳修斯表述，即克劳修斯表述暗示了开尔文表述。我们可以用类似的方式证明开尔文表述暗示了克劳修斯表述，因此两者是等价的。<br />
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===Planck's proposition普朗克<font color = 'red'><s>假设</s></font><font color = 'blue'>命题</font>===<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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普朗克提出了直接来自经验的如下命题。这有时被认为是他对热力学第二定律的表述，但他认为这是热力学第二定律推导的起点。<br />
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::It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 86.</ref><ref>Roberts, J.K., Miller, A.R. (1928/1960), p. 319.</ref><br />
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It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<br />
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<font color = 'red'><s>建造一台发动机是不可能的，这台发动机将在一个完整的循环中工作，</s></font><font color = 'blue'>不可能建造一台发动机，使其在一个完整的循环中工作，</font>并且除了提高重量和冷却热储以外不会产生任何效果。<br />
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===Relation between Kelvin's statement and Planck's proposition 开尔文表述与普朗克命题的关系===<br />
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It is almost customary in textbooks to speak of the "[[Kelvin-Planck statement]]" of the law, as for example in the text by [[Dirk ter Haar|ter Haar]] and [[Harald Wergeland|Wergeland]].<ref>[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966), p. 17.</ref><br />
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It is almost customary in textbooks to speak of the "Kelvin-Planck statement" of the law, as for example in the text by ter Haar and Wergeland.<br />
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教科书中几乎总是用“'''<font color = '#ff8000'>开尔文-普朗克表述 Kelvin-Planck Statement</font>'''”来称呼该定律，例如'''<font color = '#ff8000'>德克·特哈尔 Diek ter Haar</font>''' 和'''<font color = '#ff8000'>哈拉尔德·沃格兰 Harald Wergeland</font>''' 就是这样表述的。<br />
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The '''Kelvin–Planck statement''' (or the ''heat engine statement'') of the second law of thermodynamics states that <br />
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The Kelvin–Planck statement (or the heat engine statement) of the second law of thermodynamics states that <br />
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热力学第二定律的开尔文-普朗克表述（或称“'''<font color = '#ff8000'>热机表述 Heat Engine Statement</font>'''”）指出<br />
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::It is impossible to devise a [[thermodynamic cycle|cyclically]] operating device, the sole effect of which is to absorb energy in the form of heat from a single [[heat reservoir|thermal reservoir]] and to deliver an equivalent amount of [[Work (physics)|work]].<ref name="Rao">{{cite book|last=Rao|first=Y. V. C.|title=Chemical Engineering Thermodynamics|publisher=Universities Press|isbn=978-81-7371-048-3|page=158|year=1997}}</ref><br />
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It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.<br />
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设计一种唯一效果是从单一热源吸收热并提供等量的功的循环运行装置是不可能的。<br />
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===Planck's statement 普朗克表述===<br />
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Planck stated the second law as follows.<br />
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Planck stated the second law as follows.<br />
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普朗克表述第二定律如下。<br />
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::Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<ref name="Planck 100">[[Max Planck|Planck, M.]] (1897/1903), p. 100.</ref><ref name="Planck 463">[[Max Planck|Planck, M.]] (1926), p. 463, translation by Uffink, J. (2003), p. 131.</ref><ref name="Roberts & Miller 382">Roberts, J.K., Miller, A.R. (1928/1960), p. 382. This source is partly verbatim from Planck's statement, but does not cite Planck. This source calls the statement the principle of the increase of entropy.</ref><br />
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Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<br />
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自然界中发生的任一过程都是沿参与其中的所有物体的熵之和增加的方向进行的。在极限情况中，即对于可逆过程，熵的总和保持不变。<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for ''irreversible phenomena''.<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for irreversible phenomena.<br />
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与普朗克表述非常相似的是'''<font color = '#ff8000'>乌伦贝克 Uhlenbeck</font>'''和'''<font color = '#ff8000'>福特 Ford</font>'''关于不可逆现象的表述。<br />
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::... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<ref>[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963), p. 16.</ref><br />
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... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<br />
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……在从一个平衡态到另一个平衡态的不可逆或自发的变化中（例如，当两个物体 A 和 B 接触时的温度平衡过程），熵总是增加。<br />
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===Principle of Carathéodory 卡拉西奥多里原理===<br />
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<!-- [[Caratheodory's principle]] redirects here --><br />
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—— caratheodory 的原理在这里重新定向——<br />
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[[Constantin Carathéodory]] formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<ref>[[Constantin Carathéodory|Carathéodory, C.]] (1909).</ref><br />
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Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<br />
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'''<font color = '#ff8000'>康斯坦丁·卡拉西奥多里 Constantin Carathéodory</font>'''在纯数学公理的基础上进行了热力学'''<font color = '#ff8000'>公理化formulated</font>'''阐明。他对第二定律的陈述被称为'''<font color = '#ff8000'>卡拉西奥多里原理 Principle of Carathéodory</font>'''，可以这样表述:<br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.<ref>Buchdahl, H.A. (1966), p. 68.</ref></blockquote><br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.</blockquote><br />
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在绝热封闭系统的任意状态 S 附近，总有从 S 出发不可达的状态。<br />
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With this formulation, he described the concept of [[adiabatic accessibility]] for the first time and provided the foundation for a new subfield of classical thermodynamics, often called [[Ruppeiner geometry|geometrical thermodynamics]]. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic [[process function]], in other words, <math>\delta Q=TdS</math>.<ref name="Sychev1991">{{cite book |last=Sychev |first=V. V. |title=The Differential Equations of Thermodynamics |year=1991 |publisher=Taylor & Francis |isbn=978-1-56032-121-7}}</ref> {{clarify|date=February 2014}}<br />
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With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, <math>\delta Q=TdS</math>. <br />
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通过这个阐明，他首次描述了'''<font color = '#ff8000'>绝热可达性 Adiabatic Accessibility</font>'''的概念，并为经典热力学的一个新的子领域，即通常所说的'''<font color = '#ff8000'>几何热力学 Geometrical Thermodynamics</font>'''<font color = 'red'><s>奠定了基础</s></font>。由卡拉西奥多里原理可以推出，作为热的能量的准静态转移是一个'''<font color="#ff8000"> 完整的过程函数holonomic process function</font>'''即<math>\delta Q=TdS</math>。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）准静态转移的热量值是一个可积过程函数 存疑<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）本句的修改供讨论。可积函数的英文应该是Integrable function。但是什么是完整的函数存疑，是连续的函数？<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）这里建议标注一下'''<font color="#ff8000"> 完整的过程函数 holonomic process function</font>'''<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）奠定了基础为何要删<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.<ref name="Munster 45">Münster, A. (1970), p. 45.</ref>{{sfnp|Lieb|Yngvason|1999|p=49}}<ref name="Planck 1926">[[Max Planck|Planck, M.]] (1926).</ref><ref>Buchdahl, H.A. (1966), p. 69.</ref> {{clarify|date=February 2014}}<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium. <br />
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尽管在教科书几乎惯称卡拉西奥多里原理也是第二定律的一种表述，并认为其与克劳修斯表述或开尔文-普朗克表述等价，但事实并非如此。为了得到第二定律的所有内容，需要对卡拉西奥多里原理补充普朗克表述，即等量功总是增加一个最初处于自身内部热力学平衡的封闭系统的内部能量。<br />
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===Planck's principle 普朗克原理===<br />
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In 1926, [[Max Planck]] wrote an important paper on the basics of thermodynamics.<ref name="Planck 1926"/><ref>Uffink, J. (2003), pp. 129–132.</ref> He indicated the principle<br />
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In 1926, Max Planck wrote an important paper on the basics of thermodynamics. He indicated the principle<br />
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1926年，'''<font color = '#ff8000'>马克斯·普朗克 Max Planck</font>'''写了一篇关于热力学基础的重要论文。他指出了以下原理<br />
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::The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<ref name="Munster 45"/>{{sfnp |Lieb|Yngvason|1999|p=49}}<br />
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The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<br />
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一个封闭系统的内部能量因绝热过程增加，在整个过程中，系统的体积保持不变。<br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work."<ref>[[Clifford Truesdell|Truesdell, C.]], Muncaster, R.G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 15.</ref> Planck wrote: "The production of heat by friction is irreversible."<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 81.</ref><ref>[[Max Planck|Planck, M.]] (1926), p. 457, Wikipedia editor's translation.</ref><br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work." Planck wrote: "The production of heat by friction is irreversible."<br />
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这个公式没有提到热，没有提到温度，甚至没有提到熵，也不一定隐含地依赖于这些概念，但它暗示了第二定律的内容。一个密切相关的表述为，“摩擦力从来不做正功。”普朗克写道: “摩擦生热是不可逆的。”<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above.<ref>Lieb, E.H., Yngvason, J. (2003), p. 149.</ref> It is relevant that for a system at constant volume and [[Mole (unit)|mole numbers]], the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above. It is relevant that for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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这个普朗克定理没有提到熵，是用物理术语来表述的。它与上面给出的开尔文表述密切相关。相关的是，对于恒定体积和摩尔数的系统，熵是内能的单调函数。然而，普朗克的这个原理实际上并不是普朗克对第二定律的首选表述（见前面小节），而是依赖于熵的概念。<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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Borgnakke 和 Sonntag 提出的以下表述在某种意义上是对普朗克原理的补充。他们没有将其作为第二定律的完整表述:<br />
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::... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<ref>Borgnakke, C., Sonntag., R.E. (2009), p. 304.</ref><br />
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... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<br />
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……只有一种方法可以减少（封闭）系统的熵，将热从系统中转移出去。<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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与普朗克之前提出的原理不同，这一原理明确地用熵的变化来表示。从系统中去除物质也可以减少系统的熵。<br />
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===Statement for a system that has a known expression of its internal energy as a function of its extensive state variables 一个其内能有已知表达式（其扩展状态变量的函数）的系统的表述===<br />
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The second law has been shown to be equivalent to the [[internal energy]] ''U'' being a weakly [[convex function]], when written as a function of extensive properties (mass, volume, entropy, ...).<ref>{{cite book |last1=van Gool |first1=W. |last2=Bruggink |first2=J.J.C. (Eds) |url= |title=Energy and time in the economic and physical sciences |publisher=North-Holland |year=1985 |pages=41–56 |quote= |isbn=978-0-444-87748-2}}</ref><ref>{{Cite journal | last1 = Grubbström | first1 = Robert W. | doi = 10.1016/j.apenergy.2007.01.003 | title = An Attempt to Introduce Dynamics Into Generalised Exergy Considerations| journal = Applied Energy| volume = 84| issue = 7–8 | pages = 701–718 | year = 2007}}</ref> {{clarify|date=February 2014}}<br />
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The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...). <br />
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'''<font color="#32CD32">写成广泛性质（质量，体积，熵……）的函数时，第二定律被证明等价于弱凸函数内能 U。The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...).</font>'''<br />
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==Corollaries 推论==<br />
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===Perpetual motion of the second kind 第二类永动机===<br />
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{{main article|Perpetual motion}}<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of [[first law of thermodynamics]] by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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在第二定律建立之前，许多志在发明永动机的人试图通过提取环境的巨大内能作为机器动力，以突破'''热力学第一定律 First Law of Thermodynamics''' 的限制。这种机器被称为“'''第二类永动机 Perpetual Motion Machine of the Second Kind'''”。<font color = 'blue'>热力学</font>第二定律表明这种机器是不可能的。<br />
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===Carnot theorem 卡诺定理===<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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'''卡诺定理 Carnot's Theorem'''（1824）是一条限制任何可能的热机的最大效率的原理。效率完全取决于热源中热和冷之间的温差。 卡诺定理指出:<br />
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*All irreversible heat engines between two heat reservoirs are less efficient than a [[Carnot engine]] operating between the same reservoirs.<br />
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*All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs.<br />
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* 所有<font color = 'blue'>在两个热源之间工作的</font>不可逆热机的效率低于在<font color = 'red'><s>两相同</s></font><font color = 'blue'>同一个</font>热源之间工作的'''<font color = '#ff8000'>卡诺热机 Carnot Engine</font>'''。<br />
* 所有<font color = 'blue'>在两个热源之间工作的</font>可逆热机的效率与在<font color = 'red'><s>两相同</s></font><font color = 'blue'>同一个</font>热源之间工作的卡诺机相等。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）改成同一个正确吗<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as [[thermodynamic reversibility]]. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the [[Carnot cycle]]'s reversibility and was condemned to be less efficient.<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the Carnot cycle's reversibility and was condemned to be less efficient.<br />
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在他的理想模型中，热转化为功的过程可以通过逆转循环的运动而恢复，这个概念后来被称为'''<font color = '#ff8000'>热力学可逆性 Thermodynamic Reversibility</font>'''。然而，卡诺进一步假定，一些热<s>量</s>损失了，并没有转化为机械功。因此，没有一个真实的热机能够实现'''<font color = '#ff8000'>卡诺循环 Carnot Cycle</font>'''的可逆性，并且被认为效率较低。<br />
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Though formulated in terms of caloric (see the obsolete [[caloric theory]]), rather than [[entropy]], this was an early insight into the second law.<br />
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Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law.<br />
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该理论尽管是用热<s>量</s>表述的（见被取代的'''<font color = '#ff8000'>热质说</font>'''），而不是熵，但是它是对第二定律的早期认识。<br />
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===Clausius inequality 克劳修斯不等式===<br />
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The [[Clausius theorem]] (1854) states that in a cyclic process<br />
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The Clausius theorem (1854) states that in a cyclic process<br />
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'''<font color = '#ff8000'>克劳修斯定理 Clausius Theorem</font>'''（1854）指出，在一个循环的过程中<br />
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: <math>\oint \frac{\delta Q}{T} \leq 0.</math><br />
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The equality holds in the reversible case<ref>[http://scienceworld.wolfram.com/physics/ClausiusTheorem.html ''Clausius theorem''] at [[Wolfram Research]]</ref> and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function [[entropy]]. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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The equality holds in the reversible case and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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等号在可逆情况下成立，严格不等号在不可逆情况下成立。可逆情况下引入状态函数熵。'''<font color="#32CD32">这是因为在循环过程中，状态功能的变化相对于状态功能为零。This is because in cyclic processes the variation of a state function is zero from state functionality.</font>'''<br />
<br />
--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）末句存疑<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 我也没看懂，按照字面意思修改了一下<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）感觉修改后的句子还是怪怪的，这里标成疑难句吧 '''<font color="#32CD32">这是因为在循环过程中，状态功能的变化相对于状态功能为零。This is because in cyclic processes the variation of a state function is zero from state functionality.</font>'''<br />
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===Thermodynamic temperature 热力学温度===<br />
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{{main article|Thermodynamic temperature}}<br />
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For an arbitrary heat engine, the efficiency is:<br />
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For an arbitrary heat engine, the efficiency is:<br />
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对于任意热机，效率为:<br />
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: <math>\eta = \frac {W_n}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H} \qquad (1)</math><br />
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where ''W''<sub>n</sub> is for the net work done per cycle. Thus the efficiency depends only on q<sub>C</sub>/q<sub>H</sub>.<br />
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其中 ''W''<sub>n</sub> 表示每个循环所做的净功。因此效率只取决于 q<sub>C</sub>/q<sub>H</sub>。<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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卡诺定理指出，所有在相同热源间运行的可逆<font color = 'red'><s>发动机</s></font><font color = 'blue'>热机</font>的效率是相同的。<br />
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Thus, any reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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Thus, any reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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因此，任何在温度''T''<sub>1</sub>和''T''<sub>2</sub>之间运行的可逆热机必须具有相同的效率，也就是说，效率只是温度的函数:<br />
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<math>\frac{q_C}{q_H} = f(T_H,T_C)\qquad (2).</math><br />
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In addition, a reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between ''T''<sub>1</sub> and another (intermediate) temperature ''T''<sub>2</sub>, and the second between ''T''<sub>2</sub> and''T''<sub>3</sub>. This can only be the case if<br />
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In addition, a reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between T<sub>1</sub> and another (intermediate) temperature T<sub>2</sub>, and the second between T<sub>2</sub> andT<sub>3</sub>. This can only be the case if<br />
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另外，在温度 ''T''<sub>1</sub> 和 ''T''<sub>3</sub> 之间工作的可逆热机必须具有与由分别在温度''T''<sub>1</sub> 和（中间）温度 ''T''<sub>2</sub>之间、在 ''T''<sub>2</sub> 和 ''T''<sub>3</sub> 之间的两个循环组成的系统效率相同。只有下式成立才会出现这种情况:<br />
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: <math>f(T_1,T_3) = \frac{q_3}{q_1} = \frac{q_2 q_3} {q_1 q_2} = f(T_1,T_2)f(T_2,T_3).</math><br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the [[triple point]] of water. Then for any ''T''<sub>2</sub> and ''T''<sub>3</sub>,<br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the triple point of water. Then for any T<sub>2</sub> and T<sub>3</sub>,<br />
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现在考虑如下情形，<math>T_1</math> 是一个固定的参考温度: 水的<font color="#ff8000">'''<font color = '#ff8000'>三相点 Triple Point</font>'''</font>的温度。则对于任意 T<sub>2</sub> 和 T<sub>3</sub>,<br />
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: <math>f(T_2,T_3) = \frac{f(T_1,T_3)}{f(T_1,T_2)} = \frac{273.16 \cdot f(T_1,T_3)}{273.16 \cdot f(T_1,T_2)}.</math><br />
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Therefore, if thermodynamic temperature is defined by<br />
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Therefore, if thermodynamic temperature is defined by<br />
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因此，如果热力学温度被定义为<br />
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: <math>T = 273.16 \cdot f(T_1,T) \,</math><br />
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then the function ''f'', viewed as a function of thermodynamic temperature, is simply<br />
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then the function f, viewed as a function of thermodynamic temperature, is simply<br />
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那么函数 ''f'' 作为热力学温度的函数，为<br />
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: <math>f(T_2,T_3) = \frac{T_3}{T_2},</math><br />
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and the reference temperature ''T''<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be used{{snd}}the choice here corresponds to the [[Kelvin]] scale.)<br />
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and the reference temperature T<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be used the choice here corresponds to the Kelvin scale.)<br />
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参考温度 ''T''<sub>1</sub> 的值为273.16。（任何参考温度和任何正值均可用——此处的选择对应开尔文标度。）<br />
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===Entropy 熵===<br />
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{{main article|Entropy (classical thermodynamics)}}<br />
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According to the [[Clausius theorem|Clausius equality]], for a reversible process<br />
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According to the Clausius equality, for a reversible process<br />
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根据克劳修斯等式，对可逆过程有<br />
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: <math>\oint \frac{\delta Q}{T}=0</math><br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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这意味着线积分 <math>\int_L \frac{\delta Q}{T}</math> 对于可逆过程是路径无关的。<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer<ref name="Schmidt-Rohr 14" /> satisfies<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer satisfies<br />
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所以我们可以定义一个叫做熵的状态函数 S，对于可逆过程或者纯热传递满足<br />
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: <math>dS = \frac{\delta Q}{T} \!</math><br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the [[third law of thermodynamics]], which states that ''S'' = 0 at [[absolute zero]] for perfect crystals.<br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the third law of thermodynamics, which states that S = 0 at absolute zero for perfect crystals.<br />
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据此，只有对上述公式进行积分，才能得到熵的差值。为了获得绝对值，我们需要<font color="#ff8000">'''热力学第三定律 Third Law of Thermodynamics'''</font>，它指出<font color="#ff8000">'''绝对零度 Absolute Zero'''</font>下完美晶体的 ''S'' = 0。<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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对于任意不可逆过程，由于熵是一个状态函数，我们总是可以将初始状态和最终状态与一个虚拟的可逆过程联系起来，并在这条路径上积分以计算熵的差值。<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the [[Clausius inequality]] on this loop,<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the Clausius inequality on this loop,<br />
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现在把可逆过程逆过来，将其与上述不可逆过程结合。把克劳修斯不等式应用到这个循环,<br />
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: <math>-\Delta S+\int\frac{\delta Q}{T}=\oint\frac{\delta Q}{T}< 0</math><br />
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Thus,<br />
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Thus,<br />
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故,<br />
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: <math>\Delta S \ge \int \frac{\delta Q}{T} \,\!</math><br />
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where the equality holds if the transformation is reversible.<br />
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where the equality holds if the transformation is reversible.<br />
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如果变换可逆，等号成立。<br />
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Notice that if the process is an [[adiabatic process]], then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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Notice that if the process is an adiabatic process, then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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注意，若该过程是一个<font color="#ff8000">'''绝热过程 Adiabatic Process'''</font>，则<math>\delta Q=0</math>，故<math>\Delta S\ge 0</math>。<br />
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===Energy, available useful work===<br />
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===Energy, available useful work===<br />
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能量，<font color = 'red'><s>有用的工作</s></font>'''<font color = '#32CD32'>可用的有用工作available useful work</font>'''<br />
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{{See also|Exergy}}<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an ''unlimited'' heat reservoir at temperature ''T<sub>R</sub>'' and pressure ''P<sub>R</sub>'' {{snd}}so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain ''T<sub>R</sub>''; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain ''P<sub>R</sub>''.<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an unlimited heat reservoir at temperature T<sub>R</sub> and pressure P<sub>R</sub> so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain T<sub>R</sub>; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain P<sub>R</sub>.<br />
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考虑将第二定律应用于孤立系统(称为整体系统或宇宙)<font color = 'red'><s>的情形，</s></font>是一个重要<font color = 'red'><s>的、</s></font><font color = 'blue'>而</font>具有启发性的理想情形。该系统由两部分组成：<font color = 'red'>感兴趣的<s></s></font>研究关心的<font color = 'blue'></font>子系统和子系统的周围环境。因为<font color = 'red'><s>环境被想象的</s></font><font color = 'blue'>想象的环境</font>非常<font color = 'red'><s>巨</s></font>大，<font color = 'blue'>以至于</font>它们可以被视为一个温度为 T<sub>R</sub> 且压力为 P<sub>R</sub> 的无限<font color = 'red'><s>蓄热器</s></font><font color = 'blue'>热源</font>，<font color = 'blue'>因此</font>无论有多少热<font color = 'red'><s>量</s></font>被转移到（或来自）子系统，周围的温度将保持T<sub>R</sub>；无论子系统的体积膨胀（或收缩）有多大，周围环境的压力都将保持不变。<br />
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Whatever changes to ''dS'' and ''dS<sub>R</sub>'' occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy ''S<sub>tot</sub>'' of the isolated total system must not decrease:<br />
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Whatever changes to dS and dS<sub>R</sub> occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy S<sub>tot</sub> of the isolated total system must not decrease:<br />
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无论子系统和周围环境<font color = 'red'><s>的 ''dS''和''dS<sub>R</sub>'' 发生</s></font><font color = 'blue'>单独地发生什么熵的变化 ''dS''和dS<sub>R</sub></font>，根据第二定律，孤立总体系统的熵S<sub>tot</sub>不能减小。<br />
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: <math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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<math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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<math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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According to the [[first law of thermodynamics]], the change ''dU'' in the internal energy of the sub-system is the sum of the heat ''δq'' added to the sub-system, ''less'' any work ''δw'' done ''by'' the sub-system, ''plus'' any net chemical energy entering the sub-system ''d ∑μ<sub>iR</sub>N<sub>i</sub>'', so that:<br />
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According to the first law of thermodynamics, the change dU in the internal energy of the sub-system is the sum of the heat δq added to the sub-system, less any work δw done by the sub-system, plus any net chemical energy entering the sub-system d ∑μ<sub>iR</sub>N<sub>i</sub>, so that:<br />
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根据<font color = 'red'><s>能量守恒定律</s></font><font color = 'blue'>热力学第一定律</font>，子系统<font color = 'red'><s>内部能量</s></font><font color = 'blue'>内能</font>的变化 dU 是<font color = 'red'><s>子系统内部能量 q 的总和</s></font>加在子系统上的热δq的和<font color = 'blue'></font>，减去子系统所做的任何功w，再加上进入子系统的任何净化学能 d ∑μ<sub>iR</sub>N<sub>i</sub>，因此<br />
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: <math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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<math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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<math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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where μ<sub>iR</sub> are the [[chemical potential]]s of chemical species in the external surroundings.<br />
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where μ<sub>iR</sub> are the chemical potentials of chemical species in the external surroundings.<br />
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其中μ<sub>iR</sub>是外部环境中<font color = 'red'><s>化学物类</s></font>'''<font color = '#ff8000'>化学形态 chemical species</font>'''的化学势。<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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现在热<font color = 'red'><s>量</s></font>离开<font color = 'red'><s>储存器</s></font><font color = 'blue'>热源</font>进入子系统是<br />
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: <math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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<math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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数学△ qtr (- dS r) le tr dS / math<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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在这个过程中，首先使用了经典热力学中熵的定义(在统计热力学中，熵变、温度和吸收热量之间的关系'''<font color = '#32CD32'>可以将其推导出来 can be derived</font>''') ，然后从上面的公式可以推导出第二定律的不等式。<br />
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It therefore follows that any net work ''δw'' done by the sub-system must obey<br />
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It therefore follows that any net work δw done by the sub-system must obey<br />
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因此，子系统所做的任何净功δw必须服从<br />
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: <math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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<math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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数学增量 w le-dU + tds + sum mu { iR } dN i，/ math<br />
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It is useful to separate the work ''δw'' done by the subsystem into the ''useful'' work ''δw<sub>u</sub>'' that can be done ''by'' the sub-system, over and beyond the work ''p<sub>R</sub> dV'' done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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It is useful to separate the work δw done by the subsystem into the useful work δw<sub>u</sub> that can be done by the sub-system, over and beyond the work p<sub>R</sub> dV done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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将子系统所做的功δw划分为子系统可以完成的有用功δw<sub>u</sub> ，除了子系统在周围外部压力下膨胀所做的功p<sub>R</sub> dV外，<font color = 'red'><s>还可以</s></font>给出以下可用功（<font color = 'red'><s>放射本能</s></font>'''<font color = 'ff8000'>有用能 exergy</font>'''）关系式:<br />
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: <math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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<math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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<math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the ''availability'' or ''[[exergy]]'' ''E'' of the subsystem,<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the availability or exergy E of the subsystem,<br />
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为方便起见，可以把右边定义为热力学势的精确导数，称为子系统的'''<font color = 'ff8000'>可用性 Availability </font>'''或<font color = 'red'><s>放射本能E</s></font>'''<font color = 'ff8000'>有用能 exergy</font>'''<br />
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: <math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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<math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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<math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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<font color="#32CD32">因此，第二定律意味着，对于任何可以简单地被视为分为一个子系统和一个与之接触的无限温度和压力<font color = 'red'><s>储存器</s></font> '''<font color="#ff8000">热源 Reservoir</font>''' 的过程，The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,</font><br />
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: <math> dE + \delta w_u \le 0 \, </math><br />
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<math> dE + \delta w_u \le 0 \, </math><br />
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<math> dE + \delta w_u \le 0 \, </math><br />
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i.e. the change in the subsystem's exergy plus the useful work done ''by'' the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done ''on'' the system) must be less than or equal to zero.<br />
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i.e. the change in the subsystem's exergy plus the useful work done by the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done on the system) must be less than or equal to zero.<br />
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也就是。子系统<font color = 'red'><s>(火用)</s></font><font color = 'blue'>有用能</font>的变化加上子系统所做的有用功(或者，子系统<font color = 'red'><s>(火用)</s></font><font color = 'blue'>有用能</font>的变化<font color = 'red'><s>不包括任何功，除了压力贮存器所做的功之外，在系统中所做的功</s></font><font color = 'blue'>减去除了'''<font color = '#ff8000'>压力热源</font>'''外任何对系统做的功，</font>)必须小于或等于零。<br />
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In sum, if a proper ''infinite-reservoir-like'' reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in ''E'' for an irreversible process and no change for a reversible process.<br />
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In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in E for an irreversible process and no change for a reversible process.<br />
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总之，如果选择一个合适的类似于无限库的参考状态作为现实世界中的系统环境，<br />
<font color = 'red'><s>那么第二定律预测的不可逆性的 e 值会减少，而可逆过程的 e 值不会变化。</s></font><font color = 'blue'>则第二定律预测不可逆过程的''E''值减少，可逆过程的''E''值不变。</font><br />
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: <math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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<math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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<math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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This expression together with the associated reference state permits a [[design engineer]] working at the macroscopic scale (above the [[thermodynamic limit]]) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (''Also, see [[process engineer]]''). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (''See [[Exergy efficiency|second law efficiency]]''.)<br />
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This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (Also, see process engineer). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (See second law efficiency.)<br />
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这个表达式和相关的参考状态允许<font color = 'red'><s>在宏观尺度（高于热力学极限）下工作的使用第二定律</s></font><font color = 'blue'>设计工程师在宏观尺度（高于热力学极限）下使用第二定律</font>，而无需直接测量或考虑整个孤立系统中的熵变。（另见'''<font color = '#ff8000'>工艺工程师process engineer</font>'''）。<font color = 'red'><s>考虑到这些变化，假设</s></font><font color = 'blue'>这些变化已经在假设中被考虑到，该假设认为</font>所考虑的系统可以在不改变参考状态的情况下与参考状态达到平衡。将其与可逆理想<font color = 'blue'>状态</font>进行比较，还可以找到一个过程或过程集合的效率（见'''<font color = '#ff8000'>第二定律效率second law efficiency</font>'''）<br />
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This approach to the Second Law is widely utilized in [[engineering]] practice, [[environmental accounting]], [[systems ecology]], and other disciplines.<br />
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This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines.<br />
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第二定律的这种方法被广泛应用于'''<font color = '#ff8000'>工程实践engineering practice</font>'''、'''<font color = '#ff8000'>环境会计environmental accounting</font>'''、'''<font color = '#ff8000'>系统生态学systems ecology</font>'''等其他学科。<br />
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===The second law in chemical thermodynamics===<br />
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化学热力学的第二定律<br />
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For a [[spontaneous process|spontaneous chemical process]] in a closed system at constant temperature and pressure without non-''PV'' work, the Clausius inequality Δ''S > Q/T<sub>surr</sub>'' transforms into a condition for the change in [[Gibbs free energy]] <br />
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For a spontaneous chemical process in a closed system at constant temperature and pressure without non-PV work, the Clausius inequality ΔS > Q/T<sub>surr</sub> transforms into a condition for the change in Gibbs free energy <br />
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对于一个<font color = 'blue'>恒温恒压</font>封闭系统中的自发化学过程，在没有 non-PV 功的情况下，<font color = 'red'><s>在恒定温度和压力下，将</s></font>克劳修斯不等式Δ''S > Q/T<sub>surr</sub>'' <font color = 'red'><s>转化为吉布斯自由能变化的条件</s></font><br />
<font color = 'blue'>由于'''<font color = '#ff8000'>吉布斯自由能Gibbs free energy</font>'''的变化而转化为：</font><br />
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: <math>\Delta G < 0 </math><br />
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<math>\Delta G < 0 </math><br />
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<math>\Delta G < 0 </math><br />
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or d''G'' < 0. For a similar process at constant temperature and volume, the change in [[Helmholtz free energy]] must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products.<ref name="Oxtoby8th"/> <ref name="MortimerBook"></ref> The chemical equilibrium condition at constant ''T'' and ''p'' without electrical work is d''G'' = 0.<br />
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or dG < 0. For a similar process at constant temperature and volume, the change in Helmholtz free energy must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products. The chemical equilibrium condition at constant T and p without electrical work is dG = 0.<br />
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或者 dG < 0。对于一个相似的<font color = 'blue'>恒温恒压</font>过程<font color = 'red'><s>，在恒定的温度和体积下</s></font>，'''<font color = '#ff8000'>亥姆霍兹自由能 Helmholtz free energy </font>'''的变化一定是负的， <math>\Delta A < 0 </math>。因此，<font color = 'blue'>一个负的</font>自由能(G 或 A)变化<font color = 'red'><s>的负值</s></font>是过程自发的必要条件。这是热力学第二定律在化学中最有用的形式，其中自由能的变化可以通过<font color = 'red'><s>列表的生成焓</s></font>'''<font color = '#ff8000'>表列生成焓 Tabulated Enthalpies of Formation</font>'''和反应物及产物的标准摩尔熵来计算。在<font color = 'red'><s>T 和 P </s></font><font color = 'blue'>温度和压力</font>不变的情况下，化学平衡条件是 dG = 0。<br />
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-- [[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） tabulated enthalpies of formation 表列生成焓？？<br />
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==History==<br />
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历史<br />
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{{See also|History of entropy}}<br />
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[[File:Sadi Carnot.jpeg|thumb|upright|Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique]].]]<br />
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Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique.]]<br />
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尼古拉·莱昂纳尔·萨迪·卡诺穿着传统的巴黎综合理工学院制服<br />
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The first theory of the conversion of heat into mechanical work is due to [[Nicolas Léonard Sadi Carnot]] in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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The first theory of the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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<font color = 'blue'>卡诺在1824年提出热</font><font color = 'red'><s>首先提出类热</s></font>转化为机械功的<font color = 'blue'>第一个</font>理论。他<font color = 'blue'>是第一个</font>正确认识到了转换效率取决于发动机和环境之间的温差<font color = 'blue'>的人</font>。<br />
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Recognizing the significance of [[James Prescott Joule]]'s work on the conservation of energy, [[Rudolf Clausius]] was the first to formulate the second law during 1850, in this form: heat does not flow ''spontaneously'' from cold to hot bodies. While common knowledge now, this was contrary to the [[caloric theory]] of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law during 1850, in this form: heat does not flow spontaneously from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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'''<font color="#ff8000">克劳修斯Rudolf Clausius </font>'''认识到'''<font color="#ff8000">焦耳James Prescott Joule</font>'''在能量守恒方面工作的重要性后，在1850年<font color = 'red'><s>第一个提出了第二定律</s></font><font color = 'blue'>提出了第二定律的第一个公式，在这个公式中</font>: 热不会自发地从冷物体流向热物体。虽然<font color = 'red'><s>现在的常识是这样的</s></font><font color = 'blue'>现在这是常识</font>，但是这与当时流行的热理论相反，当时的热理论认为热是一种流体。从这些他推断出了'''<font color="#ff8000"> 萨迪卡诺定律the principle of Sadi Carnot</font>'''和熵的定义(1865年)。<br />
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Established during the 19th century, the [[Kelvin-Planck statement|Kelvin-Planck statement of the Second Law]] says, "It is impossible for any device that operates on a [[cyclic process|cycle]] to receive heat from a single [[heat reservoir|reservoir]] and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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Established during the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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<font color = 'red'><s>19世纪提出的开尔文-普朗克第二定律(Kelvin-Planck)表示：“任何循环运行的设备都不可能从单个蓄热体接收热<s>量</s>并产生净功。”这被证明相当于克劳修斯的陈述。</s></font><font color = 'blue'>19世纪提出的开尔文-普朗克第二陈述(Kelvin-Planck)表示：“任何循环运行的设备都不可能从单个热源接收热并产生净功。”这被证明与克劳修斯的陈述等价。</font><br />
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The [[ergodic hypothesis]] is also important for the [[Boltzmann]] approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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'''<font color="#ff8000">遍历假设ergodic hypothesis</font>'''对'''<font color="#ff8000">玻尔兹曼方法Boltzmann approach</font>'''也很重要。<font color = 'blue'>遍历假设认为</font>在很长一段时间内，在具有相同能量的微观态相空间的某些区域所花费的时间与这个区域的体积成正比，即在很长一段时间内，所有可访问的微观状态<font color = 'blue'>出现/成立</font>的可能性都是一样的。<font color = 'red'><s>同样的，</s></font><font color = 'blue'>等价于说，</font>它表明时间平均值和统计集合的平均值是相同的。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 觉得加上出现/成立好一点？<br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a [[Macroscopic bodies|macroscopic system]]. This doctrine is obsolescent.<ref>Denbigh, K.G., Denbigh, J.S. (1985). ''Entropy in Relation to Incomplete Knowledge'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-25677-1}}, pp. 43–44.</ref><ref>Grandy, W.T., Jr (2008). ''Entropy and the Time Evolution of Macroscopic Systems'', Oxford University Press, Oxford, {{ISBN|978-0-19-954617-6}}, pp. 55–58.</ref><ref name=Lambert>[http://entropysite.oxy.edu Entropy Sites — A Guide] Content selected by [[Frank L. Lambert]]</ref><br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a macroscopic system. This doctrine is obsolescent.<br />
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克劳修斯还提出了一种传统的学说，他认为熵可以被理解为宏观系统中的'''<font color = '#ff8000'>分子“无序”molecular 'disorder'</font>'''，但这种学说已经过时了。<br />
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===Account given by Clausius===<br />
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作者: 克劳修斯<br />
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[[File:Clausius-1.jpg|thumb|upright|Rudolf Clausius]]<br />
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Rudolf Clausius<br />
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鲁道夫 · 克劳修斯<br />
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In 1856, the German physicist [[Rudolf Clausius]] stated what he called the "second fundamental theorem in the [[mechanical theory of heat]]" in the following form:{{sfnp|Clausius|1867}}<br />
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In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:<br />
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1856年，德国物理学家'''<font color = '#ff8000'>鲁道夫 • 克劳修斯Rudolf Clausius </font>'''阐述了他所谓的“热力学理论中的第二个基本定理second fundamental theorem in the mechanical theory of heat” ，其形式如下:<br />
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: <math>\int \frac{\delta Q}{T} = -N</math><br />
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<math>\int \frac{\delta Q}{T} = -N</math><br />
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<math>\int \frac{\delta Q}{T} = -N</math><br />
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where ''Q'' is heat, ''T'' is temperature and ''N'' is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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其中 Q 是热，T 是温度，N 是一个循环过程中所有<font color = 'red'><s>未补偿</s></font><font color = 'blue'>非补偿</font>的相变的“等价值”。后来在1865年，克劳修斯将“等价值”定义为熵。<font color = 'red'><s>也就是在</s></font><font color = 'blue'>基于这个理论，</font>同一年，第二定律最著名的版本在4月24日苏黎世哲学学会的一次演讲中被<font color = 'red'><s>宣读</s></font><font color = 'blue'>提出</font>，在演讲的最后克劳修斯总结道:<br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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宇宙的熵趋于最大<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. [[universe]], as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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这句话是第二定律最著名的<font color = 'red'><s>措辞</s></font><font color = 'blue'>陈述</font>。由于其语言松散<font color = 'blue'>模糊</font>，如“宇宙universe”，<font color = 'red'><s>与缺乏具体的条件相同</s></font><font color = 'blue'>和缺乏具体的条件</font>，如“开放open”，“封闭closed”，或“孤立isolated”，许多人认为这一简单的陈述意味着热力学第二定律几乎适用于每一个可以想象的<font color = 'red'><s>学科</s></font><font color = 'blue'>主题</font>。这不是真的；这句话只是一个更广泛和更精确的描述的简化版本。<br />
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In terms of time variation, the mathematical statement of the second law for an [[isolated system]] undergoing an arbitrary transformation is:<br />
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In terms of time variation, the mathematical statement of the second law for an isolated system undergoing an arbitrary transformation is:<br />
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就时间变化而言，<font color = 'blue'>一个</font>经历任意变换的孤立系统<font color = 'blue'>的</font>第二定律的数学表述是:<br />
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: <math>\frac{dS}{dt} \ge 0</math><br />
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<math>\frac{dS}{dt} \ge 0</math><br />
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<math>\frac{dS}{dt} \ge 0</math><br />
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where<br />
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where<br />
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这里''S'' 是系统的熵<br />
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: ''S'' is the entropy of the system and<br />
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S is the entropy of the system and<br />
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: ''t'' is [[time]].<br />
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t is time.<br />
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T是时间<br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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!-- 可逆过程需要与周围环境保持平衡。这对于一个孤立的系统是不可能的。因此，关于可逆过程的讨论已经转移到对封闭系统的分析----<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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平衡后用等号。另一种表述孤立系统第二定律的方法是：<br />
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: <math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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with <math> \dot S_{i}</math> the sum of the rate of [[entropy production]] by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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with <math> \dot S_{i}</math> the sum of the rate of entropy production by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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用<math> \dot S_{i}</math>表示系统内所有进程<font color = 'red'><s>产生熵</s></font>'''<font color = '#ff8000'>熵产生Entropy Production</font>'''的速率之和。这个公式的优点是它显示了熵产生的<font color = 'red'><s>影响</s></font>'''<font color = 'blue'>效果</font>。熵产生率是一个非常重要的概念，因为它决定（或限制）热机的效率。乘以环境温度<math>T_{a}</math>，<font color = 'red'><s>得到</s></font>'''<font color = 'blue'>它给出</font>所谓的耗散能<math> P_{diss}=T_{a}\dot S_{i}</math>。<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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封闭系统的第二定律(允许热<s>量</s>交换和移动边界，但不允许物质交换)的表达式是:<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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Here<br />
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: <math>\dot Q</math> is the heat flow into the system<br />
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<math>\dot Q</math> is the heat flow into the system<br />
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<math>\dot Q</math> 是进入系统的热流。<br />
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: <math>T</math> is the temperature at the point where the heat enters the system.<br />
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<math>T</math> is the temperature at the point where the heat enters the system.<br />
<math>T</math> 是热<s>量</s>进入系统时的温度。<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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<font color = 'red'><s>等式符号在只有可逆过程在系统内发生的情况下成立。</s></font><font color = 'blue'>只有在系统内发生可逆过程的情况下等号才成立</font>如果发生不可逆过程<font color = 'red'><s>(</s></font><font color = 'blue'>（</font>在实际操作系统中就是这种情况）,<font color = 'red'><s>符号保持不变</s></font><font color = 'blue'>则“＞”成立</font>。如果系统有多处供热，必须求相应项的代数和。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 因为是中文环境所以统一用中文标点<br />
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For open systems (also allowing exchange of matter):<br />
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For open systems (also allowing exchange of matter):<br />
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对于开放系统(也允许物质交换) :<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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这里<math>\dot S</math> 是进入系统的熵流，与进入系统的物质流有关。它不应该与熵的时间导数混淆。如果物质在几个地方被供给，需要取这些贡献的代数和。<br />
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==Statistical mechanics==<br />
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统计力学<br />
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[[Statistical mechanics]] gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a [[microstate (statistical mechanics)|microstate]] of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/{{radic|''N''}} where ''N'' is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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Statistical mechanics gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a microstate of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/{{radic|''N''}} where N is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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<font color = 'blue'>统计力学</font>通过假设物质是由不断运动的原子和分子组成的，<font color = 'blue'>来</font><font color = 'red'><s>统计力学</s></font>对第二定律给出了解释。系统中每个粒子的一组特定的位置和速度称为系统的微观状态，由于系统的不断运动，系统不断地改变其微观状态。统计力学假设，在平衡状态下，系统<font color = 'red'><s>可能</s></font>处于<s>的</s>每个微观状态<font color = 'red'><s>发生</s></font>的可能性是相等的<font color = 'red'><s>，当这个假设被提出时，它直接导致了第二定律必须在统计意义上成立的结论。</s></font><font color = 'blue'>这个假设的提出直接导致第二定律必须子啊统计学意义上成立</font>也就是说，第二定律<font color = 'red'><s>平均成立，其中 N 是系统中粒子数的1 / 数量级的统计变化。</s></font><font color = 'blue'>在平均意义上成立，其中统计学变异取决于数量级1/{{radic|''N''}}，其中 N 是系统中粒子数</font>。在日常(宏观)情况下，违反第二定律的概率几乎为零。然而，对于粒子数量很少的系统，热力学参数，包括熵，可能显示出与第二定律预测<font color = 'red'><s>的</s></font><font color = 'blue'>结果的</font>显著的统计偏差。经典热力学理论<font color = 'red'><s>没有</s></font><font color = 'blue'>不</font>处理这些统计变量。<br />
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（by Jie XIANG) 在通过假定一种物质由不断运动的原子和分子组成的前提下，统计力学对热力学第二定律进行了解释。系统中每个粒子其特定的位置和速度组成为系统的微观状态，由于（原子和分子的）不断运动，系统也持续改变其微观状态。统计力学假设，在系统力平衡的情况下，其微观状态其实是等可能发生的，进而能够得出结论：热力学第二定律必须在统计学意义上成立。也就是说，第二定律将保持均值不变， 其统计变化为1 /√N，N为系统中的粒子数。在日常(宏观)情况下，违反热力学第二定律的概率几乎为零，但是，对于粒子数量非常少的系统，其热力学参数（包括熵）可能会显示出与第二定律所预测的明显统计偏差。然而经典热力学理论并不处理这些统计变量。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 为什么会有两个？迷茫。<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）是不是两个译者翻译的？如果第二段翻译不用，可以括号或删除线标一下<br />
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==Derivation from statistical mechanics==<br />
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从统计力学导出<br />
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{{Further information|H-theorem}}<br />
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The first mechanical argument of the [[Kinetic theory of gases]] that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to [[James Clerk Maxwell]] in 1860;<ref>{{Cite journal | last1 = Gyenis | first1 = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G }}</ref> [[Ludwig Boltzmann]] with his [[H-theorem]] of 1872 also argued that due to collisions gases should over time tend toward the [[Maxwell–Boltzmann distribution]].<br />
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The first mechanical argument of the Kinetic theory of gases that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to James Clerk Maxwell in 1860; Ludwig Boltzmann with his H-theorem of 1872 also argued that due to collisions gases should over time tend toward the Maxwell–Boltzmann distribution.<br />
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'''<font color="#ff8000">气体动力学Kinetic theory of gases</font>'''理论的第一个力学论证由'''<font color="#ff8000">麦克斯韦James Clerk Maxwell</font>'''在1860年给出，指出分子碰撞引起温度均衡<font color = 'blue'>化</font>，因此整体趋向于'''<font color="#ff8000">平衡 Equilibrium </font>'''； '''<font color="#ff8000">玻尔兹曼Ludwig Boltzmann</font>'''在1872年提出的'''<font color="#ff8000"> H 定理H-theorem</font>'''也认为，气体由于碰撞应该随着时间的推移趋向于'''<font color="#ff8000">麦克斯韦-波兹曼分布Maxwell–Boltzmann distribution</font>'''。<br />
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Due to [[Loschmidt's paradox]], derivations of the Second Law have to make an assumption regarding the past, namely that the system is [[Correlation and dependence|uncorrelated]] at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a [[boundary condition]], and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the [[Big Bang]]), though [[Boltzmann brain|other scenarios]] have also been suggested.<ref name="Hawking AOT">{{cite journal|last=Hawking|first=SW|title=Arrow of time in cosmology|journal=Phys. Rev. D|year=1985|volume=32|issue=10|pages=2489–2495|doi=10.1103/PhysRevD.32.2489|pmid=9956019|bibcode = 1985PhRvD..32.2489H }}</ref><ref>{{cite book | last = Greene | first = Brian | authorlink = Brian Greene | title = The Fabric of the Cosmos | url = https://archive.org/details/fabricofcosmossp00gree | url-access = registration | publisher = Alfred A. Knopf | year = 2004 | page = [https://archive.org/details/fabricofcosmossp00gree/page/171 171] | isbn = 978-0-375-41288-2}}</ref><ref name=Lebowitz>{{cite journal|last=Lebowitz|first=Joel L.|title= Boltzmann's Entropy and Time's Arrow|journal=Physics Today|date=September 1993|volume=46|issue=9|pages=32–38|url=http://users.df.uba.ar/ariel/materias/FT3_2008_1C/papers_pdf/lebowitz_370.pdf|accessdate=2013-02-22|doi=10.1063/1.881363|bibcode = 1993PhT....46i..32L }}</ref><br />
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Due to Loschmidt's paradox, derivations of the Second Law have to make an assumption regarding the past, namely that the system is uncorrelated at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a boundary condition, and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the Big Bang), though other scenarios have also been suggested.<br />
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由于'''<font color="#ff8000">洛施密特悖论Loschmidt's paradox</font>'''，第二定律的导出必须对过去做出一个假设，即系统在过去的某个时刻是不相关的；这样的假设允许进行简单的概率处理。这个假设通常被认为是一个'''<font color="#ff8000">边界条件boundary condition</font>'''，因此热力学第二定律最终是过去某个地方的初始条件的结果，可能是在宇宙的开始('''<font color="#ff8000">大爆炸 the Big Bang</font>''') ，<font color = 'blue'>尽管</font>也有人提出了其他<font color = 'red'><s>假设</s></font><font color = 'blue'>场景</font>。<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the [[Statistical mechanics#Fundamental postulate|fundamental postulate]], also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy.{{citation needed|date=August 2012}} The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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<font color = 'red'><s>考虑到</s></font><font color = 'blue'>基于</font>这些假设，在统计力学中，第二定律不是一个假设，而是'''统计力学基本假设Statistical mechanics#Fundamental postulate|fundamental postulate'''的一个结果，也被称为'''<font color="#ff8000">等先验概率假设 equal prior probability postulate</font>'''。这个基本假设表明，只要一个人清楚地知道，简单的概率论证只适用于未来，而对于过去，有辅助的信息来源告诉我们，它是低熵的。<font color = 'red'><s>热力学第二定律的第一部分指出，热孤立系统的熵只能增加。</s></font>如果我们把熵的概念限制在热平衡系统中，那么热力学第二定律的第一部分<font color = 'blue'>即热孤立系统的熵只能增加，</font>是等先验概率假设的一个显然结果。处于热平衡状态的孤立系统<font color = 'red'><s>包含</s></font><font color = 'blue'>且具有</font>能量<math>E</math>的熵表示为：<br />
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: <math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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<math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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其中<math>\Omega\left(E\right)</math> 是处于 <math>E</math>和<math>E +\delta E</math这个小区间内的量子态数目。这里的 <math>\delta E</math> 是一个宏观上很小的固定能量区间。严格地说，这意味着熵取决于对<math>\delta E</math>的选择。然而在热力学极限下（例如无穷大系统的极限），狭义的熵(单位体积或单位质量的熵)不依赖于 <math>\delta E</math>。<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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假设我们有一个孤立系统，其宏观状态由许多变量<font color = 'red'><s>描述</s></font><font color = 'blue'>指定</font>。这些宏观变量可以是总体积、活塞在系统中的位置等。从而<math>\Omega</math>将取决于这些变量的值。如果某个变量不是固定的（我们不会在某个位置夹住活塞) ，那么因为在平衡状态下<font color = 'red'><s>所有可到达状态的等几率到达的，</s></font><font color = 'blue'>到达所有可到达状态的可能性是相等的，</font>平衡状态下的自由变量会使 <math>\Omega</math> 最大，因为这是平衡状态下最可能的情况。<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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如果该变量最初固定到某个值然后释放，当达到新的平衡时，变量将自我调整使得 <math>\Omega</math> 最大，这意味着熵将增加或保持不变（如果变量初始固定的值恰好是平衡值）。<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's [[H-theorem]], however, proves that the quantity {{math|''H''}} increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the quantity increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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假设我们初始位于一个平衡状态，突然移除了对一个变量的约束。我们做完这件事的时候，可达到的微观状态的数为<math>\Omega</math>，但是系统还没有达到平衡，所以系统处于某些可达到的状态的实际概率还不等于先验概率 <math>1/\Omega</math>。我们已经知道，最终的平衡状态相对于之前的平衡状态，熵会增加或者保持不变。然而，玻耳兹曼的'''<font color="#ff8000">H定理H-theorem</font>'''证明系统在不处于平衡态的期间，那个量作为时间的函数单调增加。<br />
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===Derivation of the entropy change for reversible processes===<br />
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可逆过程熵变的推导<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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热力学第二定律的第二部分指出，经历可逆过程的系统的熵变为:<br />
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: <math>dS =\frac{\delta Q}{T}</math><br />
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<math>dS =\frac{\delta Q}{T}</math><br />
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where the temperature is defined as:<br />
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where the temperature is defined as:<br />
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其中温度定义为:<br />
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:<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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See [[Microcanonical ensemble|here]] for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the [[adiabatic theorem]] of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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请参阅'''此处Microcanonical ensemble|here'''查看该定义的正当性。假设系统有一些可以改变的外部参数 x。一般来说，系统的能量本征态将依赖于 x。根据量子力学的'''<font color="#ff8000">绝热定理adiabatic theorem</font>'''，在系统哈密顿量无限缓慢变化的极限下，系统将保持在相同的'''<font color = '#ff8000'>能量本征态 energy eigenstate</font>'''，因此系统的能量会随着其所在能量本征态的能量变化而变化。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 能量本征态 不懂翻译是否正确<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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对于外部变量 x 可以定义'''<font color="#ff8000">广义力generalized force</font>''' X ，因此如果 x 增加 dx，那么<math>X dx</math> 就是系统所做的功。例如，如果 x 是体积，那么 X 就是压强。一个已知处于能量本征态<math>E_{r}</math>的系统的广义力为：<br />
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: <math>X = -\frac{dE_{r}}{dx}</math><br />
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<math>X = -\frac{dE_{r}}{dx}</math><br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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由于系统可以在<math>\delta E</math> 区间内的任意能量本征态内，我们将系统的广义力定义为上述表达式的期望值:<br />
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: <math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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<math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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为了计算平均值，我们按照能量本征态来划分<math>\Omega\left(E\right)</math>，通过计算有多少能量本征态的<math>\frac{dE_{r}}{dx}</math>值在区间在<math>Y</math> 到 <math>Y + \delta Y</math>之间。把这个数字叫做为 <math>\Omega_{Y}\left(E\right)</math>，我们有:<br />
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: <math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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<math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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The average defining the generalized force can now be written:<br />
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The average defining the generalized force can now be written:<br />
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根据平均值定义的广义力现在可以写成:<br />
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: <math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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<math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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我们可以把它和由恒定能量 E 下的 x 导出来的熵联系起来。假定我们把 x 改变至 x + dx。然后因为能量本征态依赖于 x， <math>\Omega\left(E\right)</math> 将会改变，这导致能量本征态进入或超出<math>E</math> 和<math>E+\delta E</math> 之间的范围。让我们再次关注<math>\frac{dE_{r}}{dx}</math> 处于 <math>Y</math> 和 <math>Y + \delta Y</math> 之间的能量本征态。由于这些能量本征态的能量增加了 Y dx，所有这些在 E-Y dx 到 E 之间能量本征态<font color = 'blue'>都</font>从 E 以下移动到 E 以上。 因此有<br />
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: <math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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<math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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<font color = 'red'><s>这么多</s></font><font color = 'blue'>这些</font>的能量本征态。如果<math>Y dx\leq\delta E</math>，<font color = 'blue'>则</font>所有这些能量本征态将移动到 <math>E</math> 到 <math>E+\delta E</math>的范围内，使得<math>\Omega</math>增加。从<math>E+\delta E</math>以下移动到<math>E+\delta E</math>以上的能量本征态数目为 <math>N_{Y}\left(E+\delta E\right)</math>。它们的差<br />
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: <math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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<math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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因此是 <math>\Omega</math>增长的净贡献。请注意如果 Y dx 大于 <math>\delta E</math>，那么将会有能量本征态从 E 以下移动到<math>E+\delta E</math>以上。它们在<math>N_{Y}\left(E\right)</math> 和 <math>N_{Y}\left(E+\delta E\right)</math>中都有计数，因此上述表达式在这种情况下也是有效的。<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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将上面的表达式表示为对 E 的导数，并且对 Y 求和得到表达式:<br />
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: <math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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<math>\Omega</math>的对数关于x的导数由以下方式给出:<br />
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: <math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
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第一项是集约型的，<font color = 'red'><s>例如</s></font><font color = 'blue'>即</font>它不能根据系统大小进行缩放。相反，<font color = 'red'><s>最后一项的规模与逆系统的规模一样</s></font><font color = 'blue'>最后一项跟随逆系统的大小而缩放</font>，因此将在热力学极限中消失。因此我们发现:<br />
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: <math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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<math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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Combining this with<br />
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Combining this with<br />
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将这个和下面式子结合<br />
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: <math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
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<math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
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Gives:<br />
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Gives:<br />
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给出：<br />
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: <math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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<math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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===Derivation for systems described by the canonical ensemble===<br />
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正则系综系统的推导<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the [[canonical ensemble]]:<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:<br />
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如果一个系统与某个温度为T的热浴热接触，那么在平衡状态下，关于能量本征值的概率分布值由'''<font color = '#ff8000'>正则系综canonical ensemble</font>'''给出:<br />
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: <math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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<math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the [[Partition function (statistical mechanics)|partition function]]. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. <font color = 'red'>We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. </font>It follows from the general formula for the entropy:<br />
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这里 Z 是一个使所有概率之和<font color = 'red'><s>归一化</s></font><font color = 'blue'>正态化到 1 </font>的因子，这个函数被称为'''<font color = '#ff8000'>配分函数Partition function (statistical mechanics)|partition function</font>'''。现在我们考虑对温度和能级所依赖的外部参数的无限小的可逆改变。它遵循熵的一般公式:<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 能级依赖温度吗？即“现在我们考虑……的可逆改变”当中，“on which the energy levels depend”修饰（1）in the external parameters，还是（2）in the temperature and in the external parameters<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）修饰in the external parameters<br />
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: <math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
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<math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
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that<br />
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that<br />
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以及<br />
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: <math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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<math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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把正则系综的数学形式 <math>P_{j}</math> 插入，可以得到:<br />
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: <math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
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<math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
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==Living organisms==<br />
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==Living organisms==<br />
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生命体<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as [[What is Life?|Erwin Schrödinger]], [[Léon Brillouin]]<ref>Léon Brillouin [https://books.google.com/books?hl=en&lr=&id=tPXVbiw_1P0C ''Science and Information Theory''] (Academic Press, 1962) (Dover, 2004)</ref> and [[Life and Energy|Isaac Asimov]]. It is also the topic of current{{When|date=July 2019}} research.{{Citation needed|date=July 2019}}<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as Erwin Schrödinger, Léon Brillouin and Isaac Asimov. It is also the topic of current research.<br />
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表述热力学主要有两种方式：(a)描述从一种热力学平衡状态到另一种状态的路径，(b)通过<font color = 'red'><s>使系统保持不变，同时增加周围环境的总熵</s></font><font color = 'blue'>系统保持不变但周围环境的总熵增加</font>的循环过程。这两种方法有助于理解生命过程。这个话题大大超出了本文的范围，但是已经有一些研究者考虑过这个问题，比如'''<font color = '#ff8000'>薛定谔Erwin Schrödinger</font>'''、'''<font color = '#ff8000'>布里渊Léon Brillouin</font>'''和'''<font color = '#ff8000'>阿西莫夫Isaac Asimov</font>'''。这也是当前研究的主题。<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of [[metabolism]], give out breakdown products and heat. Plants [[Photosynthesis|take in radiative energy]] from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of metabolism, give out breakdown products and heat. Plants take in radiative energy from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
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从近似的角度来看，生命体可以被认为是(b)的一个例子。近似地，一只动物的身体状态每天循环，使得它几乎没有什么变化。动物吸收食物、水和氧气，经过'''<font color = '#ff8000'>新陈代谢metabolism</font>'''，输出分解的产物和热<s>量</s>。植物吸收来自太阳的辐射能量，这可以认为是热<s>量</s>，以及二氧化碳和水，然后它们释放氧气。它们就是这样生长的。<font color = 'blue'>它们</font>最终会死亡，尸体腐烂，大部分重新变成二氧化碳和水。这可以看作是一个循环过程。总的来说，阳光来自一个高温的源——太阳，它的能量被传递到一个较低的温度源，<font color = 'red'><s>例如</s></font><font color = 'blue'>即</font>向太空辐射。这个过程使得植物周围环境的熵增加。因此从循环过程的角度来看，动物和植物服从热力学第二定律。简单的热机效率概念很难适用于这个问题，因为它们假定系统是封闭的。<br />
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From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to [[Non-equilibrium thermodynamics|unsettled current debate or research]]. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to unsettled current debate or research. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
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从(a)的热力学观点考虑，即考虑从一个平衡状态到另一个平衡状态的路径，只会产生一个粗略近似的图像，因为生命体从来不会处于热力学平衡状态。生物体经常必须被认为是开放的系统，因为它们吸收营养物质并排出废弃物。开放系统热力学目前通常从一个热力学平衡状态到另一个状态的路径的角度来考虑，或者是考虑局部热力学平衡近似下的流。生命体的问题可以通过假定一个流不变的稳态<font color = 'blue'>的近似</font>来进一步简化。对于这样的近似，<font color = 'red'><s>熵产生的一般原理在'''目前有争论和研究的不确定性Non-equilibrium thermodynamics|unsettled current debate or research'''</s></font><font color = 'blue'>目前有很多关于熵产生的一般原理的未解决的争论或研究</font>。尽管如此，从热力学第二定律的这个角度出发衍生出来的想法对生物是有启发意义的。<br />
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==Gravitational systems==<br />
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==Gravitational systems==<br />
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重力系统<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive [[heat capacity]], meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it.<ref name="Grandy 151">Grandy, W.T. (Jr) (2008), p. 151.</ref> Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive heat capacity, meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it. Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
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<font color = 'red'><s>在不需要描述相对论</s></font><font color = 'blue'>在不需要其对广义相对论的解释的</font>的系统中，物体始终具有正热容，这意味着温度随能量而升高。因此，当能量从高温物体流向低温物体时，发射源温度降低而冷源温度升高；因此，温差会随着时间的流逝而减小。对于引力很重要并且<font color = 'red'><s>基于相对论的一般</s></font><font color = 'blue'>需要广义相对论的</font>系统而言，情况并非总是如此。这样的系统可以自发地<font color = 'red'><s>转变为质量和能量的不均匀分布</s></font><font color = 'blue'>向质量和能量不均匀的分布转变</font>。这适用于大规模的宇宙，因此可能很难或不可能对它应用第二定律。除此之外，广义相对论描述的系统的热力学超出了本文的范围。<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
非平衡状态<br />
{{main article|Non-equilibrium thermodynamics}}<br />
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The theory of classical or [[thermodynamic equilibrium|equilibrium thermodynamics]] is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<ref name="Grandy 151"/><ref>Callen, H.B. (1960/1985), p. 15.</ref><br />
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The theory of classical or equilibrium thermodynamics is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<br />
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经典热力学或平衡热力学理论是理想化的。其主要假定或假设（<font color = 'red'><s>往往无明确说明</s></font><font color = 'blue'>通常甚至没有明确地陈述</font>），是基于系统处在自身内部的热力学平衡状态。通常，在自然界可能发现的，在给定时间包含物理系统的空间区域，其热力学平衡状态并不是能用最严格的术语来表示。用较宽泛的术语来说，整个宇宙中没有任何东西是或曾经是真正处于精确的热力学平衡状态中。<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of [[thermodynamic equilibrium]]. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable [[thermal fluctuations|fluctuations]]. There is an exception, the case of [[critical point (thermodynamics)|critical states]], which exhibit to the naked eye the phenomenon of [[critical opalescence]]. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of thermodynamic equilibrium. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable fluctuations. There is an exception, the case of critical states, which exhibit to the naked eye the phenomenon of critical opalescence. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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为了进行物理分析，通常<font color = 'red'><s>假设热力学平衡就足以方便可行</s></font><font color = 'blue'>做一个热力学平衡的假设就足够方便可行</font>。 这样的假设可能依赖于反复试验和试错来证明其合理性。如果假设是合理的，那么它通常会非常有价值，因为它使热力学理论成为可能。平衡假设的要素是，观察到的系统会无限期地保持不变，并且因为系统中有很多粒子，以至于完全可以忽略其微粒<font color = 'red'><s>性质</s></font><font color = 'blue'>的本质</font>。通常在这种平衡假设下，<font color = 'red'><s>没有</s></font><font color = 'blue'>不存在一个</font>宏观上可检测到的波动。临界状态是一个例外，<font color = 'red'><s>它以肉眼显示出</s></font><font color = 'blue'>肉眼可以观察到</font>临界'''<font color = '#ff8000'>乳光现象opalescence</font>'''。对于临界状态的实验室研究，需要非常长的观察时间。<br />
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In all cases, the assumption of [[thermodynamic equilibrium]], once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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In all cases, the assumption of thermodynamic equilibrium, once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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在所有情况下，一旦<font color = 'red'><s>完成</s></font><font color = 'blue'>做出</font>了热力学平衡的假设，则<font color = 'blue'>作为一个结果</font>意味着假定<font color = 'blue'>中的</font>能改变系统熵的<font color = 'red'><s>“fluctuation”（待定的波动因子）</s></font><font color = 'blue'>候选波动</font>不存在。<br />
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'''<font color = '#32CD32'>则作为一个结果意味着假定中的能改变系统熵的候选波动不存在。implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.</font>'''<br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<ref>Lieb, E.H., Yngvason, J. (2003), p. 190.</ref><br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<br />
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物理系统很容易发生内部宏观变化，其速度之快足以使熵的恒定性假设失效。或物理系统中的粒子太少，以至于微粒的本质在可观测的波动中显而易见。那么热力学平衡的假设就会被放弃。对于非平衡态，熵没有绝对的通俗定义。<br />
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There are intermediate cases, in which the assumption of local [[thermodynamic equilibrium]] is a very good approximation,<ref>Gyarmati, I. (1967/1970), pp. 4-14.</ref><ref>Glansdorff, P., Prigogine, I. (1971).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003).</ref> but strictly speaking it is still an approximation, not theoretically ideal.<br />
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There are intermediate cases, in which the assumption of local thermodynamic equilibrium is a very good approximation, but strictly speaking it is still an approximation, not theoretically ideal.<br />
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还有一些中间情况，其中局部热力学平衡的假设是一个非常好的近似<font color = 'red'><s>值</s></font>，但严格地说它仍然是一个近似值，不是理论上的理想值。<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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对于一般的非平衡状态，考虑其他量的统计力学定义可能是有用的，这些量可以方便地称为“熵”，但是它们不应该与第二定律正确定义的熵混淆<font color = 'blue'>或合并</font>。这些其他量确实属于统计力学，而不属于第二定律的主要领域——热力学。<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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宏观上可观察到的波动的物理学超出了本文的范围。<br />
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==Arrow of time==<br />
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==Arrow of time==<br />
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时间之箭<br />
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{{Further information|Entropy (arrow of time)}}Thermodynamic Asymmetry in Time<br />
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Thermodynamic Asymmetry in Time<br />
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时间中的热力学不对称性<br />
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{{See also|Arrow of time}}<br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely [[CPT symmetry]], since the second law applies statistically, it is hypothesized, on time-asymmetric ''boundary conditions''.<ref>{{cite web|first=Craig|last=Callender|url=https://plato.stanford.edu/archives/fall2011/entries/time-thermo/|title=Thermodynamic Asymmetry in Time|encyclopedia=Stanford Encyclopedia of Philosophy|date=29 July 2011}}</ref><br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with symmetries observed in the fundamental laws of physics (particularly CPT symmetry) since the second law applies statistically on time-asymmetric boundary conditions.[78] The second law has been related to the difference between moving forwards and backwards in time, or to the principle that cause precedes effect (the causal arrow of time, or causality).[79]<br />
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热力学第二定律是<font color = 'red'><s>与时间方向逆转不对称的物理定律</s></font><font color = 'blue'>在时间方向上不对称的物理定律</font>。这与在物理学基本定律中观察到的对称性概念（特别是CPT对称性）<s>并</s>没有冲突，因为第二定律在统计上适用于时间非对称边界条件。第二定律与时间向前和向后移动之间的差异有关，或者与<font color = 'red'><s>导致先于效果</s></font><font color = 'blue'>原因先于结果</font>的原理（时间的因果关系或因果关系）有关。<br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect ([[Arrow of time#The causal arrow of time|the causal arrow of time]]).<ref>{{cite book | first = J.J.| last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1|display-authors=etal}} chapter 6</ref><br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect (the causal arrow of time).<br />
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第二定律被提出来<font color = 'red'><s>部分</s></font><font color = 'blue'>侧面</font>解释时间向前和向后移动的区别，例如为什么原因先于结果(因果关系的时间箭头)。<br />
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==Irreversibility==<br />
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==Irreversibility==<br />
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不可逆性<br />
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Irreversibility in [[thermodynamic process]]es is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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Irreversibility in thermodynamic processes is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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热力学过程中的不可逆性是热力学作用不对称特性的结果，而不是物体内部任何不可逆的微观性质<font color = 'blue'>的结果</font>。热力学作用是对参与体施加的宏观外部干预，并非来自其内部属性。由于没有认识到这一点，产生了被称为“悖论”的说法。<br />
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===Loschmidt's paradox===<br />
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<font color = 'red'><s>洛斯密特悖论 </s></font><font color = 'blue'>洛施密特悖论</font><br />
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{{main article|Loschmidt's paradox}}<br />
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[[Loschmidt's paradox]], also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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'''<font color="#ff8000">洛施密特悖论 Loschmidt's paradox</font>'''，也被称为'''<font color="#ff8000">可逆性悖论reversibility paradox</font>'''，<br />
<font color = 'red'><s>其目的在于不可能从描述宏观系统微观演化的时间对称动力学中推导出不可逆过程。</s></font><font color = 'blue'><br />
它反驳说，从描述宏观系统微观演化的时间对称动力学中，不可能推导出不可逆过程。</font><br />
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In the opinion of [[Erwin Schrödinger|Schrödinger]], "It is now quite obvious in what manner you have to reformulate the law of entropy{{snd}}or for that matter, all other irreversible statements{{snd}}so that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other."<ref>[[Erwin Schrödinger|Schrödinger, E.]] (1950), p.&nbsp;192.</ref> The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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In the opinion of Schrödinger, "It is now quite obvious in what manner you have to reformulate the law of entropyor for that matter, all other irreversible statementsso that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other." The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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薛定谔的认为：“现在很明显，您必须以何种方式重新制定熵定律（亦或者所有其他不可逆陈述）以使它们能够从可逆模型中得出。你不能只谈论一个孤立的系统，而应该至少谈论两个。你可以暂时认为它们与世界其它地方是孤立的，但并不总是相互孤立的。”这两个系统被隔离内壁彼此分开，直到被热力学（由热力学定律设想出来的）作用而拆除。热力学作用是从外部施加的，不受支配系统组成部分的可逆微观动力学定律的制约。这是不可逆转的原因。本文中<font color = 'red'><s>的法律声明</s></font><font color = 'blue'>第二定律的陈述</font>符合薛定谔的的建议。 因果关系在逻辑上先于第二定律，而不是由第二定律推导而来。<br />
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===Poincaré recurrence theorem===<br />
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庞加莱始态复现定理<br />
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{{Main article|Poincaré recurrence theorem}}<br />
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The [[Poincaré recurrence theorem]] considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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The Poincaré recurrence theorem considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics.Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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庞加莱始态复现定理<font color = 'red'><s>（又叫递归定理）</s></font>考虑了孤立物理系统的理论微观描述。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）又叫递归定理吗？存疑<br />
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经过热力学作用去除隔离内壁之后，便可以认为其是热力学系统的模型。在足够长的时间后，系统将返回到非常接近初始状态的微观定义状态。庞加莱复现时间是指回归前经过的时间长度。它极其漫长，可能比宇宙的寿命还要长，并且<font color = 'red'><s>敏感地依赖于被热力学作用拆除的墙体的几何形状</s></font><font color = 'blue'>对那个被热力学作用拆除的内壁的几何形状敏感</font>。复现定理可能被认为是明显与热力学第二定律相矛盾的。但是，更显而易见的是，它只是通过移除两个系统之间的内壁而形成的隔离系统中热力学平衡的微观模型。<font color = 'red'><s>对于典型的热力学系统而言，重复时间是如此之长（比宇宙的寿命长很多倍），以至于在所有的实际目的中，人们都无法观察到这种重复。</s></font><font color = 'blue'>对所有典型的热力学系统的实际目的，人们无法观察到如此之长的复现时间很长（比宇宙的寿命长很多倍）。</font>尽管如此，还是有人会想像一个机会可以等待庞加莱复现的出现，然后重新插入被热力学作用去除的内壁。<br />
然后很明显，不可逆性的出现是由于<font color = 'red'><s>庞加莱递归的完全不可预测性</s></font><font color = 'blue'>庞加莱复现完全不可预测的特性</font>，因为仅仅给出了初始状态<font color = 'red'><s>遵守热力学平衡的条件之一</s></font><font color = 'blue'>是热力学平衡之一</font>，就像宏观热力学的情况一样。<br />
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'''<font color="#32CD32">然后很明显，不可逆性的出现是由于庞加莱复现完全不可预测的特性，因为仅仅给出了初始状态是热力学平衡之一，就像宏观热力学的情况一样。It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics.</font>''' （这句话重复上面的，是为了标注疑难句的同时保留审校痕迹。生产修改后可以只保留这句，删掉上面重复部分）<br />
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即使<font color = 'red'><s>可以等待，也没有实际可操作性</s></font><font color = 'blue'>一个人可以等那么长时间，他也没有实际操作的可能性</font>来选择合适的时间重新插入内壁。庞加莱始态复现定理为洛施密特悖论提供了一个解决方案。如果一个孤立的热力学系统能以多倍于平均庞加莱复现时间的长度下<font color = 'red'><s>进行</s></font><font color = 'blue'>被</font>监控，则该系统的热力学行为在时间反转下将变得恒定。<br />
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[[File:James-clerk-maxwell3.jpg|thumb|upright|James Clerk Maxwell]]<br />
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James Clerk Maxwell<br />
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詹姆斯·克拉克·麦克斯韦<br />
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===Maxwell's demon===<br />
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麦克斯韦<font color = 'red'><s>的恶魔</s></font><font color = 'blue'>妖</font><br />
{{unreferenced|section|date=August 2018}}<br />
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{{main article|Maxwell's demon}}<br />
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[[James Clerk Maxwell]] imagined one container divided into two parts, ''A'' and ''B''. Both parts are filled with the same [[gas]] at equal temperatures and placed next to each other, separated by a wall. Observing the [[molecule]]s on both sides, an imaginary [[demon]] guards a microscopic trapdoor in the wall. When a faster-than-average molecule from ''A'' flies towards the trapdoor, the demon opens it, and the molecule will fly from ''A'' to ''B''. The average [[speed]] of the molecules in ''B'' will have increased while in ''A'' they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in ''A'' and increases in ''B'', contrary to the second law of thermodynamics.<br />
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James Clerk Maxwell imagined one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other, separated by a wall. Observing the molecules on both sides, an imaginary demon guards a microscopic trapdoor in the wall. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics.<br />
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'''麦克斯韦 James Clerk Maxwell'''设想一个容器分为A和B两部分。两部分均在相同温度下充满相同的气体，并彼此相邻由内壁隔开。<font color = 'red'><s>观察到两侧的分子，一个虚构的恶魔在墙上守着一个微观活板门</s></font><font color = 'blue'>一个虚构的妖精看守着一个内壁上的微型陷门，并观察者内壁两边的分子</font>。当来自A的速度快于平均水平的分子飞向<font color = 'red'><s>活板</s></font>门时，<font color = 'red'><s>恶魔将其打开</s></font><font color = 'blue'>妖精将门打开</font>，该分子将从A飞向B。B中分子的平均速度将增加，而A中分子的平均速度将<font color = 'red'><s>平均</s></font>降低。由于平均分子速度与温度相对应，因此与热力学第二定律相反，温度在A中降低，在B中升高。<br />
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One response to this question was suggested in 1929 by [[Leó Szilárd]] and later by [[Léon Brillouin]]. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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One response to this question was suggested in 1929 by Leó Szilárd and later by Léon Brillouin. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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1929年，Leó Szilárd和后来的 Léon Brillouin 对这个问题提出了一个答案。Szilárd指出，现实生活中的麦克斯韦<font color = 'red'><s>恶魔</s></font><font color = 'blue'>妖</font>需要一些测量分子速度的方法，而获取信息<font color = 'blue'>的过程</font>需要消耗<font color = 'red'><s>大量</s></font>能量。<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between ''A'' and ''B''. It is therefore performing [[thermodynamic operation]]s on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between A and B. It is therefore performing thermodynamic operations on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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麦克斯韦的<font color = 'red'><s>“恶魔”不断地</s></font><font color = 'blue'>“妖”重复地</font>改变 A 和 B 之间<font color = 'red'><s>墙体的渗透性</s></font><font color = 'blue'>壁的热透性</font>。因此，它是在微观尺度上执行热力学操作，而不仅仅是观察普通的自发或自然的宏观热力学过程。<br />
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==Quotations==<br />
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==Quotations==<br />
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语录<br />
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{{quote|The law that entropy always increases holds, I think, the supreme position among the [[Laws of science|laws of Nature]]. If someone points out to you that your pet theory of the [[universe]] is in disagreement with [[Maxwell's equations]] – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.|Sir [[Arthur Stanley Eddington]], ''The Nature of the Physical World'' (1927)}}<br />
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The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.<br />
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— Sir Arthur Stanley Eddington, The Nature of the Physical World (1927)<br />
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我认为，熵总是增加的定律在自然定律中占有至高无上的地位。如果有人指出您<font color = 'red'><s>的宇宙宠物理论</s></font><font color = 'blue'>杜撰的宇宙理论</font>与麦克斯韦方程不符，那么麦克斯韦方程就遇到麻烦了。如果发现<font color = 'blue'>您的理论</font>与观察不符，那么，可能是这些实验学家搞砸了。但是，如果发现您的理论违反了热力学第二定律，那么就没有希望了。除了在深深的耻辱中崩溃，别无他法。<br />
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—亚瑟·斯坦利·爱丁顿爵士，《自然世界》（1927年）<br />
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{{quote|There have been nearly as many formulations of the second law as there have been discussions of it.|Philosopher / Physicist [[Percy Williams Bridgman|P.W. Bridgman]], (1941)}}<br />
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There have been nearly as many formulations of the second law as there have been discussions of it.<br />
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— Philosopher / Physicist P.W. Bridgman, (1941)<br />
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关于第二定律的<font color = 'red'><s>阐述</s></font><font color = 'blue'>公式</font>几乎与关于第二定律的讨论一样多。<br />
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—哲学家/物理学家 布里奇曼（1941）<br />
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{{quote|Clausius is the author of the sibyllic utterance, "The energy of the universe is constant; the entropy of the universe tends to a maximum." The objectives of continuum thermomechanics stop far short of explaining the "universe", but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size.|[[Clifford Truesdell|Truesdell, C.]], Muncaster, R. G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 17.}}<br />
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Clausius is the author of the sibyllic utterance, "The energy of the universe is constant; the entropy of the universe tends to a maximum." The objectives of continuum thermomechanics stop far short of explaining the "universe", but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size.<br />
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— Truesdell, C., Muncaster, R. G. (1980). Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics, Academic Press, New York, ISBN 0-12-701350-4, p. 17.<br />
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克劳修斯（Sir Clausius）是西伯利亚文“宇宙的能量是恒定的；宇宙的熵趋于最大。”的作者。<font color = 'red'><s>不间断热力学的</s></font><font color = 'blue'>连续热力学的目标</font>远远不能解释“宇宙”，但是在该理论内，我们可以很容易地以某种类似于克劳修斯的方式得出明确的陈述，只是它仅仅指代一个简单的物体：一个有限大小的孤立物体。<br />
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— Truesdell, C., Muncaster, R. G. (1980). Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics, Academic Press, New York, ISBN 0-12-701350-4, p. 17. 特雷斯德尔，C.，蒙卡斯特，R.G.（1980）。麦克斯韦的单原子气体动力学理论的基本原理，被作为“理性力学”的一个分支，纽约学术出版社，ISBN 0-12-701350-4，第1页. 17。<br />
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==See also==<br />
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==See also==<br />
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参见<br />
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*[[Clausius–Duhem inequality]]<br />
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*[[Fluctuation theorem]]<br />
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*[[Heat death of the universe]]<br />
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*[[History of thermodynamics]]<br />
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*[[Jarzynski equality]]<br />
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*[[Laws of thermodynamics]]<br />
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*[[Maximum entropy thermodynamics]]<br />
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*[[Quantum thermodynamics]]<br />
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*[[Reflections on the Motive Power of Fire]]<br />
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*[[Relativistic heat conduction]]<br />
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*[[Thermal diode]]<br />
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*[[Thermodynamic equilibrium]]<br />
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==References==<br />
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==References==<br />
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参考资料<br />
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===Sources===<br />
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===Sources===<br />
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资料来源<br />
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*Adkins, C.J. (1968/1983). ''Equilibrium Thermodynamics'', (1st edition 1968), third edition 1983, Cambridge University Press, Cambridge UK, {{ISBN|0-521-25445-0}}.<br />
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*[[Peter Atkins|Atkins, P.W.]], de Paula, J. (2006). ''Atkins' Physical Chemistry'', eighth edition, W.H. Freeman, New York, {{ISBN|978-0-7167-8759-4}}.<br />
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*Attard, P. (2012). ''Non-equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications'', Oxford University Press, Oxford UK, {{ISBN|978-0-19-966276-0}}.<br />
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*Baierlein, R. (1999). ''Thermal Physics'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-59082-5}}.<br />
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*Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics, New York, {{ISBN|0-88318-797-3}}.<br />
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*{{Cite book|title=Concepts in thermal physics|last=Blundell|first=Stephen J.|last2=Blundell|authorlink1=Stephen Blundell|first2=Katherine M.|authorlink2=Katherine Blundell|publisher=[[Oxford University Press]]|year=2010|isbn=9780199562107|edition=2nd|location=Oxford|pages=|oclc=607907330|doi=10.1093/acprof:oso/9780199562091.001.0001 |url=http://cds.cern.ch/record/1235139}}<br />
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*[[Ludwig Boltzmann|Boltzmann, L.]] (1896/1964). ''Lectures on Gas Theory'', translated by S.G. Brush, University of California Press, Berkeley.<br />
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*Borgnakke, C., Sonntag., R.E. (2009). ''Fundamentals of Thermodynamics'', seventh edition, Wiley, {{ISBN|978-0-470-04192-5}}.<br />
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*Buchdahl, H.A. (1966). ''The Concepts of Classical Thermodynamics'', Cambridge University Press, Cambridge UK.<br />
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*[[Percy Williams Bridgman|Bridgman, P.W.]] (1943). ''The Nature of Thermodynamics'', Harvard University Press, Cambridge MA.<br />
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*[[Herbert Callen|Callen, H.B.]] (1960/1985). ''Thermodynamics and an Introduction to Thermostatistics'', (1st edition 1960) 2nd edition 1985, Wiley, New York, {{ISBN|0-471-86256-8}}.<br />
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*{{cite journal|author=C. Carathéodory|author1-link=Constantin Carathéodory<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
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|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
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2010年3月15日 | 第355-386页 | http://gdz.sub.uni-goettingen.de/index.php?id=11&ppn=ppn235181684_0067&dmdid=dmdlog_0033&l=1<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). ''The Second Law of Thermodynamics'', Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können.(p. 363) | doi 10.1007 / bf01450409}.译文可以在这里找到[ http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf ]。同时，在 Kestin，j。(1976).热力学第二定律，Dowden，Hutchinson & Ross，Stroudsburg PA. 。<br />
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*[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986). [http://www.worldcat.org/title/reflections-on-the-motive-power-of-fire-a-critical-edition-with-the-surviving-scientific-manuscripts-translated-and-edited-by-fox-robert/oclc/812944517&referer=brief_results ''Reflections on the motive power of fire''], Manchester University Press, Manchester UK, {{ISBN|0-7190-1741-6}}. [https://archive.org/stream/reflectionsonmot00carnrich#page/n7/mode/2up Also here.]<br />
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*[[Sydney Chapman (mathematician)|Chapman, S.]], [[Thomas George Cowling|Cowling, T.G.]] (1939/1970). ''The Mathematical Theory of Non-uniform gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases'', third edition 1970, Cambridge University Press, London.<br />
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*{{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Ueber Die Bewegende Kraft Der Wärme Und Die Gesetze, Welche Sich Daraus Für Die Wärmelehre Selbst Ableiten Lassen|journal=Annalen der Physik|year=1850|volume=79|issue=4|pages=368–397, 500–524|url=http://gallica.bnf.fr/ark:/12148/bpt6k15164w/f518.image|accessdate=26 June 2012|doi=10.1002/andp.18501550403|bibcode = 1850AnP...155..500C |hdl=2027/uc1.$b242250|hdl-access=free}} Translated into English: {{cite journal|last=Clausius|first=R.|title=On the Moving Force of Heat, and the Laws regarding the Nature of Heat itself which are deducible therefrom|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1851| volume=2|series=4th|issue=VIII|pages=1–21; 102–119 |url=https://archive.org/stream/londonedinburghd02lond#page/1/mode/1up|accessdate=26 June 2012<br />
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|doi=10.1080/14786445108646819}}<br />
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|doi=10.1080/14786445108646819}}<br />
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10.1080 / 14786445108646819}<br />
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* {{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie|year=1854|journal=Annalen der Physik|volume=xciii|issue=12|pages=481–506|url=http://zfbb.thulb.uni-jena.de/servlets/MCRFileNodeServlet/jportal_derivate_00140956/18541691202_ftp.pdf|accessdate=24 March 2014|doi=10.1002/andp.18541691202<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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| bibcode 1854AnP... 169. 1856年7月 | 第2卷 | 第4页 | 第86页 | https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=242014年3月<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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热力学理论及其在蒸汽机和物体物理性质中的应用 | 1867年 | 出版商 John van Voorst | 位置: 伦敦 | https://archive.org/details/mechanicaltheor04claugoog|quote=editions:pwr_sbkwa8ic。 2012年6月19日<br />
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*Denbigh, K. (1954/1981). ''The Principles of Chemical Equilibrium. With Applications in Chemistry and Chemical Engineering'', fourth edition, Cambridge University Press, Cambridge UK, {{ISBN|0-521-23682-7}}.<br />
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*Eu, B.C. (2002). ''Generalized Thermodynamics. The Thermodynamics of Irreversible Processes and Generalized Hydrodynamics'', Kluwer Academic Publishers, Dordrecht, {{ISBN|1-4020-0788-4}}.<br />
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*[[Josiah Willard Gibbs|Gibbs, J.W.]] (1876/1878). On the equilibrium of heterogeneous substances, ''Trans. Conn. Acad.'', '''3''': 108-248, 343-524, reprinted in ''The Collected Works of J. Willard Gibbs, Ph.D, LL. D.'', edited by W.R. Longley, R.G. Van Name, Longmans, Green & Co., New York, 1928, volume 1, pp.&nbsp;55–353.<br />
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*Griem, H.R. (2005). ''Principles of Plasma Spectroscopy (Cambridge Monographs on Plasma Physics)'', Cambridge University Press, New York {{ISBN|0-521-61941-6}}.<br />
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*Glansdorff, P., Prigogine, I. (1971). ''Thermodynamic Theory of Structure, Stability, and Fluctuations'', Wiley-Interscience, London, 1971, {{ISBN|0-471-30280-5}}.<br />
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*Grandy, W.T., Jr (2008). [http://global.oup.com/academic/product/entropy-and-the-time-evolution-of-macroscopic-systems-9780199546176?cc=au&lang=en& ''Entropy and the Time Evolution of Macroscopic Systems'']. Oxford University Press. {{ISBN|978-0-19-954617-6}}.<br />
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*Greven, A., Keller, G., Warnecke (editors) (2003). ''Entropy'', Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[Edward A. Guggenheim|Guggenheim, E.A.]] (1949). 'Statistical basis of thermodynamics', ''Research'', '''2''': 450–454.<br />
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*[[Edward A. Guggenheim|Guggenheim, E.A.]] (1967). ''Thermodynamics. An Advanced Treatment for Chemists and Physicists'', fifth revised edition, North Holland, Amsterdam.<br />
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*Gyarmati, I. (1967/1970) ''Non-equilibrium Thermodynamics. Field Theory and Variational Principles'', translated by E. Gyarmati and W.F. Heinz, Springer, New York.<br />
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*[[Charles Kittel|Kittel, C.]], [[Herbert Kroemer|Kroemer, H.]] (1969/1980). ''Thermal Physics'', second edition, Freeman, San Francisco CA, {{ISBN|0-7167-1088-9}}.<br />
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*Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998). ''Modern Thermodynamics: From Heat Engines to Dissipative Structures'', John Wiley & Sons, Chichester, {{ISBN|0-471-97393-9}}.<br />
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*Lebon, G., Jou, D., Casas-Vázquez, J. (2008). ''Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers'', Springer-Verlag, Berlin, {{ISBN|978-3-540-74252-4}}.<br />
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*{{cite journal|ref=harv|title=The Physics and Mathematics of the Second Law of Thermodynamics|last1=Lieb |first1=E. H. |last2=Yngvason |first2=J.|journal=Physics Reports|volume=310|issue=1|pages=1–96 |year=1999|doi=10.1016/S0370-1573(98)00082-9|arxiv = cond-mat/9708200 |bibcode = 1999PhR...310....1L}}<br />
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*Lieb, E.H., Yngvason, J. (2003). The Entropy of Classical Thermodynamics, pp.&nbsp;147–195, Chapter 8 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*{{cite book|ref=harv|first=F.|last=Mandl|title=Statistical physics|edition=second|year=1988|publisher=[[Wiley & Sons]]|isbn=978-0-471-91533-1}}<br />
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*{{cite journal | last1 = Maxwell | first1 = J.C. | authorlink = James Clerk Maxwell | year = 1867 | title = On the dynamical theory of gases | url = | journal = Phil. Trans. R. Soc. Lond. | volume = 157 | issue = | pages = 49–88 | doi = 10.1098/rstl.1867.0004 }}<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985). ''Thermodynamics'', Pitman, London, {{ISBN|0-273-08577-8}}.<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003). Entropy in Nonequilibrium, pp.&nbsp;79–109, Chapter 5 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*Münster, A. (1970), ''Classical Thermodynamics'', translated by E.S. Halberstadt, Wiley–Interscience, London, {{ISBN|0-471-62430-6}}.<br />
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*[[Brian Pippard|Pippard, A.B.]] (1957/1966). ''Elements of Classical Thermodynamics for Advanced Students of Physics'', original publication 1957, reprint 1966, Cambridge University Press, Cambridge UK.<br />
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*[[Max Planck|Planck, M.]] (1897/1903). [https://archive.org/stream/treatiseonthermo00planrich#page/100/mode/2up ''Treatise on Thermodynamics'', translated by A. Ogg, Longmans Green, London, p. 100.]<br />
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*[[Max Planck|Planck. M.]] (1914). [https://archive.org/details/theoryofheatradi00planrich ''The Theory of Heat Radiation''], a translation by Masius, M. of the second German edition, P. Blakiston's Son & Co., Philadelphia.<br />
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*[[Max Planck|Planck, M.]] (1926). Über die Begründung des zweiten Hauptsatzes der Thermodynamik, ''Sitzungsberichte der Preussischen Akademie der Wissenschaften: Physikalisch-mathematische Klasse'': 453–463.<br />
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* Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769-781.<br />
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* {{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}<br />
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*Quinn, T.J. (1983). ''Temperature'', Academic Press, London, {{ISBN|0-12-569680-9}}.<br />
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*{{cite book|ref=harv|first=Y.V.C.|last=Rao|title=An Introduction to thermodynamics|url=https://books.google.com/books?id=iYWiCXziWsEC&pg=PA213|date=2004|publisher=Universities Press|isbn=978-81-7371-461-0|page=213}}<br />
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*Roberts, J.K., Miller, A.R. (1928/1960). ''Heat and Thermodynamics'', (first edition 1928), fifth edition, Blackie & Son Limited, Glasgow.<br />
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*[[Erwin Schrödinger|Schrödinger, E.]] (1950). Irreversibility, ''Proc. R. Ir. Acad.'', '''A53''': 189–195.<br />
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*[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966). ''Elements of Thermodynamics'', Addison-Wesley Publishing, Reading MA.<br />
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*{{cite journal|ref=harv|last=Thomson|first=W.|author-link=William Thomson, 1st Baron Kelvin|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Transactions of the Royal Society of Edinburgh|year=1851|volume=XX|issue=part II|pages=261–268; 289–298|url=https://www.biodiversitylibrary.org/item/126047#page/295/mode/1up}} Also published in {{cite journal|last=Thomson|first=W.|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Philos. Mag. |date=December 1852 |volume=IV |series=4 |issue=22 |page=13|url=https://archive.org/stream/londonedinburghp04maga#page/12/mode/2up |accessdate=25 June 2012}}<br />
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*[[William Thomson, 1st Baron Kelvin|Thomson, W.]] (1852). ''On the universal tendency in nature to the dissipation of mechanical energy'' Philosophical Magazine, Ser. 4, p.&nbsp;304.<br />
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*[[László Tisza|Tisza, L.]] (1966). ''Generalized Thermodynamics'', M.I.T Press, Cambridge MA.<br />
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*[[Clifford Truesdell|Truesdell, C.]] (1980). ''The Tragicomical History of Thermodynamics 1822–1854'', Springer, New York, {{ISBN|0-387-90403-4}}.<br />
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*Uffink, J. (2001). Bluff your way in the second law of thermodynamics, ''Stud. Hist. Phil. Mod. Phys.'', '''32'''(3): 305–394.<br />
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*Uffink, J. (2003). Irreversibility and the Second Law of Thermodynamics, Chapter 7 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003), Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963). ''Lectures in Statistical Mechanics'', American Mathematical Society, Providence RI.<br />
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*[[Mark Zemansky|Zemansky, M.W.]] (1968). ''Heat and Thermodynamics. An Intermediate Textbook'', fifth edition, McGraw-Hill Book Company, New York.<br />
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==Further reading==<br />
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==Further reading==<br />
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进一步阅读<br />
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*Goldstein, Martin, and Inge F., 1993. ''The Refrigerator and the Universe''. Harvard Univ. Press. Chpts. 4–9 contain an introduction to the Second Law, one a bit less technical than this entry. {{ISBN|978-0-674-75324-2}}<br />
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*Leff, Harvey S., and Rex, Andrew F. (eds.) 2003. ''Maxwell's Demon 2 : Entropy, classical and quantum information, computing''. Bristol UK; Philadelphia PA: [[Institute of Physics]]. {{ISBN|978-0-585-49237-7}}<br />
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*{{Cite book | first = J.J. | last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1}}(technical).<br />
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*{{cite book |title=Reflections on the Motive Power of Heat and on Machines Fitted to Develop That Power |last=Carnot |first=Sadi |editor=[[Robert Henry Thurston|Thurston, Robert Henry]] |year=1890 |publisher=J. Wiley & Sons |location=New York |isbn= |pages= }} ([https://books.google.com/books?id=tgdJAAAAIAAJ full text of 1897 ed.]) ([http://www.history.rochester.edu/steam/carnot/1943/ html])<br />
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*Stephen Jay Kline (1999). ''The Low-Down on Entropy and Interpretive Thermodynamics'', La Cañada, CA: DCW Industries. {{ISBN|1-928729-01-0}}.<br />
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*{{cite book | last1 = Kostic | first1 = M | year = 2011 | title = Revisiting The Second Law of Energy Degradation and Entropy Generation: From Sadi Carnot's Ingenious Reasoning to Holistic Generalization | journal = AIP Conf. Proc. | volume = 1411 | issue = 1| pages = 327–350 | doi = 10.1063/1.3665247 | isbn = 978-0-7354-0985-9 | bibcode = 2011AIPC.1411..327K | series = AIP Conference Proceedings | citeseerx = 10.1.1.405.1945 }} also at [https://web.archive.org/web/20130420222450/http://www.kostic.niu.edu/2ndLaw/Revisiting%20The%20Second%20Law%20of%20Energy%20Degradation%20and%20Entropy%20Generation%20-%20From%20Carnot%20to%20Holistic%20Generalization-4.pdf].<br />
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==External links==<br />
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==External links==<br />
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外部链接<br />
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*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics]" – by Lawrence Sklar.<br />
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*[http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node30.html ''Second law of thermodynamics''] in the MIT Course [http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/notes.html ''Unified Thermodynamics and Propulsion''] from Prof. Z. S. Spakovszky<br />
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*[[E.T. Jaynes]], 1988, "[http://bayes.wustl.edu/etj/articles/ccarnot.pdf The evolution of Carnot's principle]," in G. J. Erickson and C. R. Smith (eds.)''Maximum-Entropy and Bayesian Methods in Science and Engineering, Vol'' 1: p.&nbsp;267.<br />
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*[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf Caratheodory, C., "Examination of the foundations of thermodynamics," trans. by D. H. Delphenich]<br />
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* [https://www.bbc.co.uk/programmes/p004y2bm The Second Law of Thermodynamics], BBC Radio 4 discussion with John Gribbin, Peter Atkins & Monica Grady (''In Our Time'', Dec. 16, 2004)<br />
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* [http://mdpi.org/entropy/papers/e6010001.pdf [[Entropy (journal)]], 2004]<br />
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* [https://www.journals.uchicago.edu/doi/abs/10.1086/663835 The Journal of the International Society for the History of Philosophy of Science, 2012]<br />
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{{DEFAULTSORT:Second Law Of Thermodynamics}}<br />
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[[Category:Concepts in physics]]<br />
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Category:Concepts in physics<br />
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分类: 物理概念<br />
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[[Category:Laws of thermodynamics|2]]<br />
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[[Category:Non-equilibrium thermodynamics]]<br />
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Category:Non-equilibrium thermodynamics<br />
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类别: 非平衡态热力学<br />
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[[Category:Philosophy of thermal and statistical physics]]<br />
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Category:Philosophy of thermal and statistical physics<br />
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类别: 热力学和统计物理学哲学<br />
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<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Second law of thermodynamics]]. Its edit history can be viewed at [[热力学第二定律/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E7%83%AD%E5%8A%9B%E5%AD%A6%E7%AC%AC%E4%BA%8C%E5%AE%9A%E5%BE%8B&diff=12744热力学第二定律2020-08-16T13:18:00Z<p>Dorr：</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Law of physics}}<br />
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{{Thermodynamics|laws}}<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 全文审校中，heat一般统一译为“热”，caloric 或者 caloric of heat 译为“热量”。这一点是否准确，有待各位同学和专家的批评指正<br />
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The '''second law of thermodynamics''' states that the total [[entropy]] of an [[isolated system]] can never decrease over time, and is constant if and only if all processes are reversible.<ref>http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node38.html#SECTION05224000000000000000</ref> Isolated systems spontaneously evolve towards [[thermodynamic equilibrium]], the state with maximum entropy. <br />
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The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, and is constant if and only if all processes are reversible. Isolated systems spontaneously evolve towards thermodynamic equilibrium, the state with maximum entropy. <br />
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<font color="#ff8000">'''热力学第二定律 Second Law Of Thermodynamics'''</font>指出，<font color="#ff8000">'''孤立系统Isolated System'''</font>的总熵永远不会随着时间而减少，且当且仅当所有过程都是可逆时，总熵才恒定。孤立系统自发地到达到热力学平衡状态，此时为熵最大的状态。<br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in [[thermodynamic equilibrium]], or is undergoing a (fictive) [[Reversible process (thermodynamics)|reversible process]]. In all processes that occur, including [[spontaneous process]]es,<ref>Atkins and de Paula, p.78</ref> the total entropy of the system and its surroundings increases and the process is [[Irreversible process|irreversible in the thermodynamic sense]]. The increase in entropy accounts for the irreversibility of natural processes, and the [[Arrow of time|asymmetry between future and past]].<ref>{{cite book|last=Zohuri|first=Bahman|title=Dimensional Analysis Beyond the Pi Theorem|url=https://books.google.com/books?id=pRVuDQAAQBAJ|year=2016|publisher=Springer|isbn=978-3-319-45726-0|page=[https://books.google.com.ph/books?id=pRVuDQAAQBAJ&pg=PA111&dq=%22increase+in+entropy+accounts+for+the+irreversibility+of+natural+processes+and+the+asymmetry+between+future+and+past.%22&hl=en&sa=X&ved=0ahUKEwj6spb61tbaAhUFS7wKHfftDtIQ6AEIKDAA#v=onepage&q=%22increase%20in%20entropy%20accounts%20for%20the%20irreversibility%20of%20natural%20processes%20and%20the%20asymmetry%20between%20future%20and%20past.%22 111]}}</ref><br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in thermodynamic equilibrium, or is undergoing a (fictive) reversible process. In all processes that occur, including spontaneous processes, the total entropy of the system and its surroundings increases and the process is irreversible in the thermodynamic sense. The increase in entropy accounts for the irreversibility of natural processes, and the asymmetry between future and past.<br />
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系统及其周围环境的总熵在理想情况下可以保持不变，在这种情况下，系统处于热力学平衡状态，或者正在经历一个(<font color = 'red'><s>虚拟的</s></font><font color = 'blue'>假想的</font>)可逆过程。<font color = 'red'><s>在所有发生的</s></font><font color = 'blue'>所有</font>过程中，包括<font color="#ff8000">'''自发过程 Spontaneous Processes'''</font>，系统及其周围环境的总熵增加，这一过程在热力学意义上是不可逆的。熵的增加解释了自然过程的不可逆性，以及未来和过去之间的不对称性。<br />
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Historically, the second law was an [[Empirical evidence|empirical finding]] that was accepted as an axiom of [[thermodynamics|thermodynamic theory]]. [[Statistical mechanics]], classical or [[quantum thermodynamics|quantum]], explains the microscopic origin of the law.<br />
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Historically, the second law was an empirical finding that was accepted as an axiom of thermodynamic theory. Statistical mechanics, classical or quantum, explains the microscopic origin of the law.<br />
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从历史上看，第二定律是一个经验性的发现，它被认为是热力学理论中的公理。经典统计力学、量子统计力学都可以解释这个定律的微观起源。<br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist [[Nicolas Léonard Sadi Carnot|Sadi Carnot]], who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as [[Carnot's theorem (thermodynamics)|Carnot's rule]] or limit.<ref>Jaffe, R.L., Taylor, W. (2018). ''The Physics of Energy'', Cambridge University Press, Cambridge UK, pages 150, 151, 259, 772, 743.</ref><br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist Sadi Carnot, who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as Carnot's rule or limit.<br />
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<font color = 'blue'>热力学第二定律可以使用多种方法表述。它的第一个公式归功于法国科学家<font color="#ff8000">'''卡诺Sadi Carnot'''</font>，卡诺在1824年证明了在热机中将热转化为功的效率有一个上限。第二定律的这个方面也被称为<font color="#ff8000">'''卡诺规则Carnot's Rule'''</font>或<font color="#ff8000">'''卡诺限制Carnot's Limit'''</font>。</font><br />
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'''热力学第二定律 Second Law of Thermodynamics'''是热力学的四条基本定律之一，它表述了热力学过程的不可逆性——孤立系统自发地朝着热力学平衡方向（即最大熵状态）演化，另一种表述为：第二类永动机永不可能实现。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）这里是不是少了一段原文？<br />
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==Introduction引言==<br />
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[[File:Heat flow hot to cold.png|thumb|upright|Heat flow from hot water to cold water. 热总是从热水流向冷水]]<br />
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The [[first law of thermodynamics]] provides the definition of the [[internal energy]] of a [[thermodynamic system]], and expresses the law of [[conservation of energy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 40–41.</ref><ref>Munster A. (1970), pp. 8–9, 50–51.</ref> The second law is concerned with the direction of natural processes.<ref>{{harvnb|Mandl|1988}}</ref> It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such [[Phenomenon|phenomena]] are accounted for in terms of [[entropy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 79–107.</ref><ref>Bailyn, M. (1994), Section 71, pp. 113–154.</ref> If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, {{math|''S''}}, increases.<br />
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The first law of thermodynamics provides the definition of the internal energy of a thermodynamic system, and expresses the law of conservation of energy. The second law is concerned with the direction of natural processes. It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such phenomena are accounted for in terms of entropy. If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, S, increases.<br />
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'''热力学第一定律 First Law Of Thermodynamics''' <font color = 'red'><s>对</s></font> <font color = 'blue'>定义了</font>热力学系统所涉及到的'''内能 Internal Energy'''<font color = 'red'><s>进行了定义</s></font> ，并体现了能量守恒定律。热力学第二定律与'''自然过程Natural Processes'''的方向有关。它<font color = 'red'><s>设定</s></font> <font color = 'blue'>断言</font>自然过程只在一种意义上进行，且<font color = 'red'><s>是</s></font> 不可逆<font color = 'red'><s>的</s></font> 。例如，当有了传导和辐射的（传播）路径时，热<s>量</s>总是自发地从一个较热的物体流向一个较冷的物体。这种现象可以用'''熵Entropy'''来解释。<font color = 'red'><s>若一个孤立系统最初在内部不可渗透的薄膜维持内部的热力学平衡，通过一些操作使得薄膜具有渗透性，则该系统可自发地演变，最终达到一个新的内部热力学平衡，其总熵增加。</s></font><br />
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<font color = '#32CD32'>若一个'''孤立系统Isolated System'''最初在具有隔热内壁的系统内维持热力学平衡，通过一些操作使内壁透热，则该系统可自发地演变，最终达到一个新的内部热力学平衡，且其总熵{{math|''S''}}增加。If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, S, increases.</font><br />
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--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]） more permeable不知道是否需要译为更具有 impermeable wall是不可渗透 如果要使用比较级 是否应该再有一个 permeable<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） internal partitioning impermeable walls翻译为隔热的内壁，从而permeable指的是透热的。我觉得这里的渗透性应该是说系统可以和外界进行能量/热的交换。？不知道是否正确。<br />
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In a fictive reversible process, an infinitesimal increment in the entropy ({{math|d''S''}}) of a system is defined to result from an infinitesimal transfer of heat ({{math|δ''Q''}}) to a [[closed system]] (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature ({{math|''T''}}) of the system in equilibrium and the surroundings which supply the heat:<ref>Bailyn, M. (1994), p. 120.</ref><br />
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In a fictive reversible process, an infinitesimal increment in the entropy () of a system is defined to result from an infinitesimal transfer of heat () to a closed system (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature () of the system in equilibrium and the surroundings which supply the heat:<br />
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在<font color = 'red'><s>设置的虚拟</s></font> <font color = 'blue'>假想的</font>可逆过程中，系统的熵的无穷小增量<math>dS</math>被定义为由无穷小的热<s>量</s><math>δQ</math>传递到一个封闭系统(允许能量进入或出去，但不允许物质传递<font color = 'blue'>的系统</font>)除以<font color = 'blue'>该</font>平衡系统和提供热<s>量</s>的环境的共同温度 <math>T</math>:<br />
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: <math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process封闭系统中理想状态下的可逆过程).}</math><br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 这里的公式和备注貌似有误，公式应为 <math>\mathrm dS = \frac{\delta Q}{T}</math>，备注应为“closed system, idealized fictive reversible process”封闭系统，理想的可逆过程。或者？<br />
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Different notations are used for infinitesimal amounts of heat ({{mvar|δ}}) and infinitesimal amounts of entropy ({{mvar|d}}) because entropy is a [[function of state]], while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the [[Clausius Theorem|inequality]] <ref name="MortimerBook"> Mortimer, R. G. ''Physical Chemistry''. 3rd ed., p. 120, Academic Press, 2008. </ref><ref name="FermiBook"> Fermi, E. ''Thermodynamics'', footnote on p. 48, Dover Publications,1956 (still in print). </ref> <br />
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Different notations are used for infinitesimal amounts of heat () and infinitesimal amounts of entropy () because entropy is a function of state, while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the inequality <br />
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用不同的符号''δ''和''d''表示无穷小量的热<s>量</s>和无穷小量的熵，因为熵是状态函数，而热<s>量</s>和功一样<font color = 'blue'>则</font>不是状态函数。<font color = 'red'><s>第二定律要求系统熵的增量满足不等式，</s></font>对于实际上可能存在的不与环境发生<font color = 'red'><s>质量</s></font><font color = 'blue'>物质</font>交换的无穷小过程，<font color = 'blue'>第二定律要求</font>系统熵增量满足不等式：<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 根据上下文（讲到封闭系统），此处疑“mass”译为物质更佳<br />
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: <math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process 封闭系统中，实际上可能发生的可逆过程).}</math><br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature ({{math|''T''}}) and the temperature of the surroundings ({{math|''T''}}<sub>''surr''</sub>).<br />
<ref name=":0">Adkins, C.J. (1968/1983), p. 75.</ref><ref name="Munster 45"/> <br />
Note that the equality still applies for pure heat flow,<br />
<ref name="Schmidt-Rohr 14"> Schmidt-Rohr, K. (2014). "Expansion Work without the External Pressure, and Thermodynamics in Terms of Quasistatic Irreversible Processes" ''J. Chem. Educ.'' '''91''': 402-409. https://dx.doi.org/10.1021/ed3008704 </ref><br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature () and the temperature of the surroundings (<sub>surr</sub>).<br />
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<font color = 'blue'>这是因为</font>这种情况下的一般过程可能包括周围环境对系统所做的功，<font color = 'red'><s>这是因为</s></font><font color = 'blue'>这个功</font>在系统内部会产生摩擦或粘滞效应，<font color = 'blue'>或是因为</font>此时<font color = 'red'><s>是由于</s></font><font color = 'blue'>一个</font>化学反应可能正在<font color = 'blue'>系统内部</font>进行，或<font color = 'blue'>因为</font>热传递实际上是不可逆地发生，<font color = 'red'><s>通过</s></font>系统温度<math>T</math>和周围环境温度<math>T_surr</math>之间<font color = 'red'><s>存在差异而进行驱动</s></font><font color = 'blue'>的差异驱动了热传递的发生</font>。<br />
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）该句有些不太理解<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 我觉得应该是“This is because a general process……，because a chemical reaction……，or because heat transfer……surroundings (<sub>surr</sub>).”的并列结构。<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）我的理解，第一个because是对上文的解释；后两个because是对“功在系统内部产生……效应”的解释；后两个为并列，不与第一个并列，因此“粘滞效应”后面的because前不用加“或是”<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）and the temperature of the surroundings (<sub>surr</sub>)后面缺少半句原文：“Note that the equality still applies for pure heat flow,”<br />
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<font color='blue'>注意这个等式也适用于'''纯热流Pure Heat Flow''',</font><br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change 实际上可能的,不改变成分的准静态不可逆性).}</math><br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry.<ref name="Oxtoby8th">Oxtoby, D. W; Gillis, H.P., [[Laurie Butler|Butler, L. J.]] (2015).''Principles of Modern Chemistry'', Brooks Cole. p. 617. {{ISBN|978-1305079113}}</ref> <ref name="MortimerBook"></ref> Introducing a set of internal variables <math>\xi</math> to describe the deviation of a thermodynamic system in physical equilibrium (with the required well-defined uniform pressure ''P'' and temperature ''T'')<ref name="Schmidt-Rohr 14"></ref> from the chemical equilibrium state, one can record the equality<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry.[16] [11] Introducing a set of internal variables {\displaystyle \xi }\xi to describe the deviation of a thermodynamic system in physical equilibrium (with the required well-defined uniform pressure P and temperature T)[15] from the chemical equilibrium state, one can record the equality<br />
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<font color='blue'>这种方法</font>是通过测量'''热容曲线 Heat Capacity Curves'''和相变中熵的变化，来准确测定纯物质的绝对熵的基础，比如'''量热法calorimetry'''。<br />
为了描述一个热力学系统在物理平衡状态下(要求有明确定义的等压P和等温T)偏离化学平衡状态，引入一组内部变量<math>x_i</math>,可以用该等式<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process 封闭系统，实际可能的准静态不可逆过程).}</math><br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of ''the second principle of thermodynamics'' – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<ref> Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769–781.</ref><ref>{{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}</ref><br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of the second principle of thermodynamics – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<br />
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第二项代表内部变量的功，这些内部变量<font color='red'><s>可以</s></font><font color= 'blue'>可能</font>受到外部影响的干扰，但是系统不能通过内部变量做任何正功。这种说法介绍了热力学系统在时间上<font color='red'><s>不可能逆转演化的事实</s></font>演化不可能逆转的性质<font color= 'blue'></font>，并且<font color='red'><s>可以被认为是热力学第二原理的另外一种相当于熵原理的表述。</s></font><font color= 'blue'>该公式可以被看作使用'''熵Entropy'''的热力学第二定律等价表述</font><br />
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The [[zeroth law of thermodynamics]] in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body.<ref name=dugdale>{{cite book|author=J. S. Dugdale|title=Entropy and its Physical Meaning|publisher=Taylor & Francis|year=1996|isbn=978-0-7484-0569-5|page=13|quote=This law is the basis of temperature.}}</ref> For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows{{how?|date=August 2018}} a distinguished temperature scale, which defines an absolute, [[thermodynamic temperature]], independent of the properties of any particular reference thermometric body.<ref>[[Mark Zemansky|Zemansky, M.W.]] (1968), pp. 207–209.</ref><ref>Quinn, T.J. (1983), p. 8.</ref><br />
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The zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body. For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows a distinguished temperature scale, which defines an absolute, thermodynamic temperature, independent of the properties of any particular reference thermometric body.<br />
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热力学第零定律是指如果两个热力学系统<font color='red'><s>中的每一个</s></font>都与第三个热力学系统处于热平衡(温度相同)<font color= 'blue'>状态</font>，则它们彼此也必定处于热平衡<font color= 'blue'>状态</font>。热力学第零定律在<font color='red'><s>它</s></font>这个<font color= 'blue'></font><font color='red'><s>通常的</s></font>简短叙述中让人们认识到热平衡关系中的两个物体具有相同的温度，特别是当一个被测物体与一个参考测温物体具有相同的温度时，对于<font color='red'><s>一个与另一个</s></font>两个<font color= 'blue'></font>处于热平衡<font color= 'blue'>状态</font>的物体，有无限多的'''<font color = '#ff8000'>经验温标empirical temperature scales</font>'''，这通常<font color= 'blue'>分别</font>取决于特定参考温度<font color='red'><s>标度</s></font><font color= 'blue'>体</font>的性质。热力学第二定律允许区分'''<font color = '#ff8000'>温度标度temperature scale</font>'''，它定义了一个绝对的热力学温度，与任何特定的参考温度体的性质无关。<br />
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--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）补充热力学第零定律<br />
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==Various statements of the law热力学第二定律的不同表述==<br />
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The second law of thermodynamics may be expressed in many specific ways,<ref name=MIT>{{cite web|title=Concept and Statements of the Second Law|url=http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node37.html|accessdate=2010-10-07 |publisher=web.mit.edu}}</ref> the most prominent classical statements{{sfnp|Lieb|Yngvason|1999}} being the statement by [[Rudolf Clausius]] (1854), the statement by [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] (1851), and the statement in axiomatic thermodynamics by [[Constantin Carathéodory]] (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.{{sfnp|Rao|2004|p=213}}<br />
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The second law of thermodynamics may be expressed in many specific ways, the most prominent classical statements being the statement by Rudolf Clausius (1854), the statement by Lord Kelvin (1851), and the statement in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.<br />
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热力学第二定律可以用许多特定的方式来表达，最突出的经典陈述是 '''克劳修斯 Rudolf Clausius''' (1854)<font color = 'red'><s>陈述</s></font><font color = 'blue'>表述</font>，'''开尔文 Kelvin''' (1851)<font color = 'red'><s>陈述</s></font><font color = 'blue'>表述</font>，以及康斯坦丁·卡拉西奥多里 Constantin Carathéodory(1909)在'''<font color = '#ff8000'>公理化热力学axiomatic thermodynamics</font>'''中的<font color = 'red'><s>陈述</s></font><font color = 'blue'>表述</font>。这些<font color = 'red'><s>陈述</s></font><font color = 'blue'>表述</font>用一般的物理术语来描述定律，引用某些过程的不可能性。克劳修斯和开尔文表述被证明是等价的。<br />
--[[用户:厚朴|厚朴]]（[[用户讨论:厚朴|讨论]]）statement的表述需要统一，同样的statement by Rudolf Clausius这里需要统一开尔文 克劳修斯 Kelvin Clausius<br />
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===Carnot's principle卡诺原理===<br />
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The historical origin<ref>[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986).</ref> of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a [[Carnot heat engine]], fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the [[caloric theory]] of heat was seriously considered, before the recognition of the [[first law of thermodynamics]], and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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The historical origin of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a Carnot heat engine, fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the caloric theory of heat was seriously considered, before the recognition of the first law of thermodynamics, and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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热力学第二定律的历史起源是'''卡诺原理Carnot's principle'''。<br />
它指的是'''<font color = '#ff8000'>卡诺热机Carnot heat engine</font>'''的一个循环，<font color = 'red'><s>它</s></font><font color = 'blue'>卡诺热机</font>以'''<font color = '#ff8000'>准静态quasi-static</font>'''的极限慢速运转，因此热<s>量</s>和功在子系统之间进行传递，子系统总是处于它们自己内部的热力学平衡状态。<br />
卡诺热机是研究热机效率的工程师特别感兴趣的理想装置。<br />
当卡诺发现卡诺原理时，'''<font color = '#ff8000'>热量理论caloric theory of heat</font>'''还没有得到重视，热力学第一定律还没有得到承认，熵的概念还没有数学表达。<br />
根据第一定律的解释，它在物理上等同于热力学第二定律，并沿用至今。<br />
在热力学第一定律被发现之前，卡诺最初的论点是从热量理论的观点出发的。<br />
下面是他书中的一些例子：<br />
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::...''wherever there exists a difference of temperature, motive power can be produced.''<ref>Carnot, S. (1824/1986), p. 51.</ref><br />
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...wherever there exists a difference of temperature, motive power can be produced.<br />
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……只要有温差，就能产生动力。<br />
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::The production of motive power is then due in steam engines not to an actual consumption of caloric, but ''to its transportation from a warm body to a cold body ...''<ref>Carnot, S. (1824/1986), p. 46.</ref><br />
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The production of motive power is then due in steam engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body ...<br />
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动力的产生不是由于蒸汽机实际消耗的热量，而是由于它从一个较热的物体<font color = 'red'><s>运输</s></font><font color = 'blue'>转移</font>到一个较冷的物体……。<br />
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::''The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.''<ref>Carnot, S. (1824/1986), p. 68.</ref><br />
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The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.<br />
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热<s>量</s>的原动力与实现热的媒介无关，<font color = 'red'><s>热量的数量</s></font><font color = 'blue'>热的量</font>完全取决于两个物体之间的温度，最后是热量的传递。<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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用现代术语<font color = 'red'><s>来说</s></font>，卡诺原理可能更为准确:<br />
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::The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<ref>[[Clifford Truesdell|Truesdell, C.]] (1980), Chapter 5.</ref><ref>Adkins, C.J. (1968/1983), pp. 56–58.</ref><ref>Münster, A. (1970), p. 11.</ref><ref>Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998), pp.67–75.</ref><ref>Lebon, G., Jou, D., Casas-Vázquez, J. (2008), p. 10.</ref><ref>Eu, B.C. (2002), pp. 32–35.</ref><br />
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The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<br />
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准静态卡诺循环或可逆卡诺循环的效率只取决于两种热源的温度，而且无论工作物质是什么，效率是相同的。<font color = 'red'><s>只有</s></font><font color = 'blue'>使用</font>两个热源与工作物质交换热源（一个高温热源温度<math>T_1</math>和一个低温热源温度<math>T_2</math>)的卡诺热机是最有效的热机。<br />
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'''<font color="#32CD32">使用两个热源与工作物质交换热源（一个高温热源温度<math>T_1</math>和一个低温热源温度<math>T_2</math>)的卡诺热机是最有效的热机。 A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.</font>''' <br />
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--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）最后一句进行了补充 意译<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 我没太看懂卡诺表述，所以不能确定这里的补充是否正确。可以再看一下“与工作物质交换热源”部分的表述是否准确<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）如果不确定，可以考虑将这句话标注为疑难句；或直接按原文直译<br />
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===Clausius statement克劳修斯表述===<br />
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The German scientist [[Rudolf Clausius]] laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work.{{sfnp|Clausius|1850}} His formulation of the second law, which was published in German in 1854, is known as the ''Clausius statement'':<br />
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The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work. His formulation of the second law, which was published in German in 1854, is known as the Clausius statement:<br />
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1850年，德国科学家'''鲁道夫·克劳修斯 Rudolf Clausius''' 通过研究热传递和功之间的关系，为热力学第二定律<font color = 'red'><s>实验室</s></font>奠定了基础。他在1854年用德语发表的论文中所提及的热力学第二定律定义被称为克劳修斯表述:<br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.{{sfnp|Clausius|1854|p=86}}</blockquote><br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.</blockquote><br />
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不可能把热<s>量</s>从低温物体传递到高温物体而不产生其他影响。 <br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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克劳修斯的表述使用了“'''<font color = '#ff8000'>热通道Passage Of Heat</font>'''”的概念。<font color = 'red'><s>正如通常在热力学的讨论，这意味着“能量作为热的净转移” ，而不是另一种形式上的"分摊转账"或其他。</s></font><font color = 'blue'><br />
在热力学的讨论中，通常这意味着“能量作为热的形式的净转移” ，而不是指其他方式的转移。</font><br />
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）不太理解分摊转账contributory transfers <br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 我觉得应该是说，energy as '''heat''' 而不是 as其他方式的转移<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）我认为这里不要把contributory transfer看作一个专有名词去理解，从contributory的本意“A contributory factor of a problem or accident is one of the things which caused it to exist or happen.”来理解即可，感觉嘉树理解的是对的。<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of [[refrigeration]], for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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如果不对系统<font color = 'red'><s>进行外部功</s></font><font color = 'blue'>外部做功</font>，热<font color = 'red'><s>量就</s></font>不能自发地从冷区流向热区，这一点从制冷的普通经验中也可以看出。在冰箱中，<font color = 'red'><s>热量从冷到热，但只有在外部媒介——制冷系统的强制作用下才会发生变化。</s></font><font color = 'blue'>只有在外部媒介也就是制冷系统的强制作用下热才会从冷区流到热区。</font><br />
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===Kelvin statements开尔文表述===<br />
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[[William Thomson, 1st Baron Kelvin|Lord Kelvin]] expressed the second law in several wordings.<br />
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Lord Kelvin expressed the second law in several wordings.<br />
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'''<font color = '#ff8000'>开尔文勋爵 Lord Kelvin</font>''' <font color = 'red'><s>用几个字表达</s></font><font color = 'blue'>表述</font>了热力学第二定律。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 感觉”用几个字“怪怪的，所以删掉了<br />
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::It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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<font color = 'red'><s>无法从单一热源取热使其完全转化为有用功而不对环境产生影响。</s></font><font color = 'blue'>不可能从单一热源取热使之完全转换为有用的功而不产生其他影响。</font><br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 改的内容和原文内容不太对应。这个摘自百度百科，另参见 严子浚(1994). 对热力学第二定律开尔文表述的一点讨论, 大学物理: 13(2).<br />
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::It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.{{sfnp|Thomson|1851}}<br />
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It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.<br />
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<font color = 'red'><s>通过无生命物质的作用，不可能通过将物质的任何部分冷却到低于周围物体最冷的温度来产生机械效应。</s></font><font color = 'blue'>不可能通过无生命物质的作用，将物质的任何部分冷却到低于周围物体最低的温度并产生'''<font color = '#ff8000'>机械效应 Mechanical Effect</font>'''。</font><br />
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--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）无生命物质可以意译为机器吗<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 我不太懂这个，不过感觉最好不要吧。<br />
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===Equivalence of the Clausius and the Kelvin statements克劳修斯和开尔文表述的等价性===<br />
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[[Image:Deriving Kelvin Statement from Clausius Statement.svg|thumb|Derive Kelvin Statement from Clausius Statement从克劳修斯表述推导出开尔文表述]]<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed [[Carnot engine]] as shown by the figure. The [[Heat engine#Efficiency|efficiency]] of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the [[first law of thermodynamics]], as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the figure. The efficiency of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat from <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the first law of thermodynamics, as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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假设有一个热机违反了开尔文定理: 也就是说，这个热机以循环的方式吸收热<s>量</s>并将其<font color = 'blue'>完全</font>转化为功，而<font color = 'blue'>且</font>不产生任何影响。现<font color = 'blue'>在将其</font>与反向卡诺<font color = 'red'><s>发动</s></font>机<font color = 'red'><s>相对应</s></font><font color = 'blue'>比较</font>，如图所示。<br />
普通热机的效率为η，反向热机的效率为1/η。这对<font color = 'red'><s>热机的联合作用为</s></font><font color = 'blue'>联合热机的净效应和唯一效应是</font><math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> 将热<s>量</s>从较冷热源<font color = 'blue'>转移</font>到较热热源，这违反了克劳修斯表述。（这是能量守恒定律的结果，因为系统的总能量保持不变<math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>，所以<math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math>。）因此，违反开尔文表述意味着违反克劳修斯表述，即克劳修斯表述暗示了开尔文表述。我们可以用类似的方式证明开尔文表述暗示了克劳修斯表述，因此两者是等价的。<br />
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===Planck's proposition普朗克<font color = 'red'><s>假设</s></font><font color = 'blue'>命题</font>===<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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普朗克提出了直接来自经验的如下命题。这有时被认为是他对热力学第二定律的表述，但他认为这是热力学第二定律推导的起点。<br />
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::It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 86.</ref><ref>Roberts, J.K., Miller, A.R. (1928/1960), p. 319.</ref><br />
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It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<br />
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<font color = 'red'><s>建造一台发动机是不可能的，这台发动机将在一个完整的循环中工作，</s></font><font color = 'blue'>不可能建造一台发动机，使其在一个完整的循环中工作，</font>并且除了提高重量和冷却热储以外不会产生任何效果。<br />
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===Relation between Kelvin's statement and Planck's proposition 开尔文表述与普朗克命题的关系===<br />
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It is almost customary in textbooks to speak of the "[[Kelvin-Planck statement]]" of the law, as for example in the text by [[Dirk ter Haar|ter Haar]] and [[Harald Wergeland|Wergeland]].<ref>[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966), p. 17.</ref><br />
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It is almost customary in textbooks to speak of the "Kelvin-Planck statement" of the law, as for example in the text by ter Haar and Wergeland.<br />
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教科书中几乎总是用“'''<font color = '#ff8000'>开尔文-普朗克表述 Kelvin-Planck Statement</font>'''”来称呼该定律，例如'''<font color = '#ff8000'>德克·特哈尔 Diek ter Haar</font>''' 和'''<font color = '#ff8000'>哈拉尔德·沃格兰 Harald Wergeland</font>''' 就是这样表述的。<br />
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The '''Kelvin–Planck statement''' (or the ''heat engine statement'') of the second law of thermodynamics states that <br />
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The Kelvin–Planck statement (or the heat engine statement) of the second law of thermodynamics states that <br />
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热力学第二定律的开尔文-普朗克表述（或称“'''<font color = '#ff8000'>热机表述 Heat Engine Statement</font>'''”）指出<br />
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::It is impossible to devise a [[thermodynamic cycle|cyclically]] operating device, the sole effect of which is to absorb energy in the form of heat from a single [[heat reservoir|thermal reservoir]] and to deliver an equivalent amount of [[Work (physics)|work]].<ref name="Rao">{{cite book|last=Rao|first=Y. V. C.|title=Chemical Engineering Thermodynamics|publisher=Universities Press|isbn=978-81-7371-048-3|page=158|year=1997}}</ref><br />
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It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.<br />
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设计一种唯一效果是从单一热源吸收热并提供等量的功的循环运行装置是不可能的。<br />
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===Planck's statement 普朗克表述===<br />
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Planck stated the second law as follows.<br />
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Planck stated the second law as follows.<br />
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普朗克表述第二定律如下。<br />
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::Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<ref name="Planck 100">[[Max Planck|Planck, M.]] (1897/1903), p. 100.</ref><ref name="Planck 463">[[Max Planck|Planck, M.]] (1926), p. 463, translation by Uffink, J. (2003), p. 131.</ref><ref name="Roberts & Miller 382">Roberts, J.K., Miller, A.R. (1928/1960), p. 382. This source is partly verbatim from Planck's statement, but does not cite Planck. This source calls the statement the principle of the increase of entropy.</ref><br />
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Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<br />
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自然界中发生的任一过程都是沿参与其中的所有物体的熵之和增加的方向进行的。在极限情况中，即对于可逆过程，熵的总和保持不变。<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for ''irreversible phenomena''.<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for irreversible phenomena.<br />
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与普朗克表述非常相似的是'''<font color = '#ff8000'>乌伦贝克 Uhlenbeck</font>'''和'''<font color = '#ff8000'>福特 Ford</font>'''关于不可逆现象的表述。<br />
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::... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<ref>[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963), p. 16.</ref><br />
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... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<br />
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……在从一个平衡态到另一个平衡态的不可逆或自发的变化中（例如，当两个物体 A 和 B 接触时的温度平衡过程），熵总是增加。<br />
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===Principle of Carathéodory 卡拉西奥多里原理===<br />
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<!-- [[Caratheodory's principle]] redirects here --><br />
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—— caratheodory 的原理在这里重新定向——<br />
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[[Constantin Carathéodory]] formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<ref>[[Constantin Carathéodory|Carathéodory, C.]] (1909).</ref><br />
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Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<br />
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'''<font color = '#ff8000'>康斯坦丁·卡拉西奥多里 Constantin Carathéodory</font>'''在纯数学公理的基础上进行了热力学<font color = '#ff8000'>公理化formulated阐明。他对第二定律的陈述被称为'''<font color = '#ff8000'>卡拉西奥多里原理 Principle of Carathéodory</font>'''，可以这样表述:<br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.<ref>Buchdahl, H.A. (1966), p. 68.</ref></blockquote><br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.</blockquote><br />
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在绝热封闭系统的任意状态 S 附近，总有从 S 出发不可达的状态。<br />
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With this formulation, he described the concept of [[adiabatic accessibility]] for the first time and provided the foundation for a new subfield of classical thermodynamics, often called [[Ruppeiner geometry|geometrical thermodynamics]]. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic [[process function]], in other words, <math>\delta Q=TdS</math>.<ref name="Sychev1991">{{cite book |last=Sychev |first=V. V. |title=The Differential Equations of Thermodynamics |year=1991 |publisher=Taylor & Francis |isbn=978-1-56032-121-7}}</ref> {{clarify|date=February 2014}}<br />
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With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, <math>\delta Q=TdS</math>. <br />
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通过这个阐明，他首次描述了'''<font color = '#ff8000'>绝热可达性 Adiabatic Accessibility</font>'''的概念，并为经典热力学的一个新的子领域，即通常所说的'''<font color = '#ff8000'>几何热力学 Geometrical Thermodynamics</font>'''<font color = 'red'><s>奠定了基础</s></font>。由卡拉西奥多里原理可以推出，作为热的能量的准静态转移是一个'''<font color="#ff8000"> 完整的过程函数holonomic process function</font>'''即<math>\delta Q=TdS</math>。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）准静态转移的热量值是一个可积过程函数 存疑<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）本句的修改供讨论。可积函数的英文应该是Integrable function。但是什么是完整的函数存疑，是连续的函数？<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）这里建议标注一下'''<font color="#ff8000"> 完整的过程函数 holonomic process function</font>'''<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）奠定了基础为何要删<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.<ref name="Munster 45">Münster, A. (1970), p. 45.</ref>{{sfnp|Lieb|Yngvason|1999|p=49}}<ref name="Planck 1926">[[Max Planck|Planck, M.]] (1926).</ref><ref>Buchdahl, H.A. (1966), p. 69.</ref> {{clarify|date=February 2014}}<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium. <br />
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尽管在教科书几乎惯称卡拉西奥多里原理也是第二定律的一种表述，并认为其与克劳修斯表述或开尔文-普朗克表述等价，但事实并非如此。为了得到第二定律的所有内容，需要对卡拉西奥多里原理补充普朗克表述，即等量功总是增加一个最初处于自身内部热力学平衡的封闭系统的内部能量。<br />
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===Planck's principle 普朗克原理===<br />
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In 1926, [[Max Planck]] wrote an important paper on the basics of thermodynamics.<ref name="Planck 1926"/><ref>Uffink, J. (2003), pp. 129–132.</ref> He indicated the principle<br />
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In 1926, Max Planck wrote an important paper on the basics of thermodynamics. He indicated the principle<br />
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1926年，'''<font color = '#ff8000'>马克斯·普朗克 Max Planck</font>'''写了一篇关于热力学基础的重要论文。他指出了以下原理<br />
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::The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<ref name="Munster 45"/>{{sfnp |Lieb|Yngvason|1999|p=49}}<br />
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The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<br />
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一个封闭系统的内部能量因绝热过程增加，在整个过程中，系统的体积保持不变。<br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work."<ref>[[Clifford Truesdell|Truesdell, C.]], Muncaster, R.G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 15.</ref> Planck wrote: "The production of heat by friction is irreversible."<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 81.</ref><ref>[[Max Planck|Planck, M.]] (1926), p. 457, Wikipedia editor's translation.</ref><br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work." Planck wrote: "The production of heat by friction is irreversible."<br />
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这个公式没有提到热，没有提到温度，甚至没有提到熵，也不一定隐含地依赖于这些概念，但它暗示了第二定律的内容。一个密切相关的表述为，“摩擦力从来不做正功。”普朗克写道: “摩擦生热是不可逆的。”<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above.<ref>Lieb, E.H., Yngvason, J. (2003), p. 149.</ref> It is relevant that for a system at constant volume and [[Mole (unit)|mole numbers]], the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above. It is relevant that for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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这个普朗克定理没有提到熵，是用物理术语来表述的。它与上面给出的开尔文表述密切相关。相关的是，对于恒定体积和摩尔数的系统，熵是内能的单调函数。然而，普朗克的这个原理实际上并不是普朗克对第二定律的首选表述（见前面小节），而是依赖于熵的概念。<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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Borgnakke 和 Sonntag 提出的以下表述在某种意义上是对普朗克原理的补充。他们没有将其作为第二定律的完整表述:<br />
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::... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<ref>Borgnakke, C., Sonntag., R.E. (2009), p. 304.</ref><br />
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... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<br />
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……只有一种方法可以减少（封闭）系统的熵，将热从系统中转移出去。<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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与普朗克之前提出的原理不同，这一原理明确地用熵的变化来表示。从系统中去除物质也可以减少系统的熵。<br />
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===Statement for a system that has a known expression of its internal energy as a function of its extensive state variables 一个其内能有已知表达式（其扩展状态变量的函数）的系统的表述===<br />
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The second law has been shown to be equivalent to the [[internal energy]] ''U'' being a weakly [[convex function]], when written as a function of extensive properties (mass, volume, entropy, ...).<ref>{{cite book |last1=van Gool |first1=W. |last2=Bruggink |first2=J.J.C. (Eds) |url= |title=Energy and time in the economic and physical sciences |publisher=North-Holland |year=1985 |pages=41–56 |quote= |isbn=978-0-444-87748-2}}</ref><ref>{{Cite journal | last1 = Grubbström | first1 = Robert W. | doi = 10.1016/j.apenergy.2007.01.003 | title = An Attempt to Introduce Dynamics Into Generalised Exergy Considerations| journal = Applied Energy| volume = 84| issue = 7–8 | pages = 701–718 | year = 2007}}</ref> {{clarify|date=February 2014}}<br />
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The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...). <br />
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'''<font color="#32CD32">写成广泛性质（质量，体积，熵……）的函数时，第二定律被证明等价于弱凸函数内能 U。The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...).</font>'''<br />
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==Corollaries 推论==<br />
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===Perpetual motion of the second kind 第二类永动机===<br />
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{{main article|Perpetual motion}}<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of [[first law of thermodynamics]] by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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在第二定律建立之前，许多志在发明永动机的人试图通过提取环境的巨大内能作为机器动力，以突破'''热力学第一定律 First Law of Thermodynamics''' 的限制。这种机器被称为“'''第二类永动机 Perpetual Motion Machine of the Second Kind'''”。<font color = 'blue'>热力学</font>第二定律表明这种机器是不可能的。<br />
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===Carnot theorem 卡诺定理===<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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'''卡诺定理 Carnot's Theorem'''（1824）是一条限制任何可能的热机的最大效率的原理。效率完全取决于热源中热和冷之间的温差。 卡诺定理指出:<br />
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*All irreversible heat engines between two heat reservoirs are less efficient than a [[Carnot engine]] operating between the same reservoirs.<br />
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*All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs.<br />
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* 所有<font color = 'blue'>在两个热源之间工作的</font>不可逆热机的效率低于在<font color = 'red'><s>两相同</s></font><font color = 'blue'>同一个</font>热源之间工作的'''<font color = '#ff8000'>卡诺热机 Carnot Engine</font>'''。<br />
* 所有<font color = 'blue'>在两个热源之间工作的</font>可逆热机的效率与在<font color = 'red'><s>两相同</s></font><font color = 'blue'>同一个</font>热源之间工作的卡诺机相等。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）改成同一个正确吗<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as [[thermodynamic reversibility]]. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the [[Carnot cycle]]'s reversibility and was condemned to be less efficient.<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the Carnot cycle's reversibility and was condemned to be less efficient.<br />
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在他的理想模型中，热转化为功的过程可以通过逆转循环的运动而恢复，这个概念后来被称为'''<font color = '#ff8000'>热力学可逆性 Thermodynamic Reversibility</font>'''。然而，卡诺进一步假定，一些热<s>量</s>损失了，并没有转化为机械功。因此，没有一个真实的热机能够实现'''<font color = '#ff8000'>卡诺循环 Carnot Cycle</font>'''的可逆性，并且被认为效率较低。<br />
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Though formulated in terms of caloric (see the obsolete [[caloric theory]]), rather than [[entropy]], this was an early insight into the second law.<br />
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Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law.<br />
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该理论尽管是用热<s>量</s>表述的（见被取代的'''<font color = '#ff8000'>热质说</font>'''），而不是熵，但是它是对第二定律的早期认识。<br />
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===Clausius inequality 克劳修斯不等式===<br />
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The [[Clausius theorem]] (1854) states that in a cyclic process<br />
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The Clausius theorem (1854) states that in a cyclic process<br />
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'''<font color = '#ff8000'>克劳修斯定理 Clausius Theorem</font>'''（1854）指出，在一个循环的过程中<br />
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: <math>\oint \frac{\delta Q}{T} \leq 0.</math><br />
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The equality holds in the reversible case<ref>[http://scienceworld.wolfram.com/physics/ClausiusTheorem.html ''Clausius theorem''] at [[Wolfram Research]]</ref> and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function [[entropy]]. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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The equality holds in the reversible case and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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等号在可逆情况下成立，严格不等号在不可逆情况下成立。可逆情况下引入状态函数熵。'''<font color="#32CD32">这是因为在循环过程中，状态功能的变化相对于状态功能为零。This is because in cyclic processes the variation of a state function is zero from state functionality.</font>'''<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）末句存疑<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 我也没看懂，按照字面意思修改了一下<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）感觉修改后的句子还是怪怪的，这里标成疑难句吧 '''<font color="#32CD32">这是因为在循环过程中，状态功能的变化相对于状态功能为零。This is because in cyclic processes the variation of a state function is zero from state functionality.</font>'''<br />
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===Thermodynamic temperature 热力学温度===<br />
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{{main article|Thermodynamic temperature}}<br />
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For an arbitrary heat engine, the efficiency is:<br />
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For an arbitrary heat engine, the efficiency is:<br />
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对于任意热机，效率为:<br />
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: <math>\eta = \frac {W_n}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H} \qquad (1)</math><br />
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where ''W''<sub>n</sub> is for the net work done per cycle. Thus the efficiency depends only on q<sub>C</sub>/q<sub>H</sub>.<br />
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其中 ''W''<sub>n</sub> 表示每个循环所做的净功。因此效率只取决于 q<sub>C</sub>/q<sub>H</sub>。<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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卡诺定理指出，所有在相同热源间运行的可逆<font color = 'red'><s>发动机</s></font><font color = 'blue'>热机</font>的效率是相同的。<br />
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Thus, any reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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Thus, any reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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因此，任何在温度''T''<sub>1</sub>和''T''<sub>2</sub>之间运行的可逆热机必须具有相同的效率，也就是说，效率只是温度的函数:<br />
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<math>\frac{q_C}{q_H} = f(T_H,T_C)\qquad (2).</math><br />
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In addition, a reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between ''T''<sub>1</sub> and another (intermediate) temperature ''T''<sub>2</sub>, and the second between ''T''<sub>2</sub> and''T''<sub>3</sub>. This can only be the case if<br />
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In addition, a reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between T<sub>1</sub> and another (intermediate) temperature T<sub>2</sub>, and the second between T<sub>2</sub> andT<sub>3</sub>. This can only be the case if<br />
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另外，在温度 ''T''<sub>1</sub> 和 ''T''<sub>3</sub> 之间工作的可逆热机必须具有与由分别在温度''T''<sub>1</sub> 和（中间）温度 ''T''<sub>2</sub>之间、在 ''T''<sub>2</sub> 和 ''T''<sub>3</sub> 之间的两个循环组成的系统效率相同。只有下式成立才会出现这种情况:<br />
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: <math>f(T_1,T_3) = \frac{q_3}{q_1} = \frac{q_2 q_3} {q_1 q_2} = f(T_1,T_2)f(T_2,T_3).</math><br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the [[triple point]] of water. Then for any ''T''<sub>2</sub> and ''T''<sub>3</sub>,<br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the triple point of water. Then for any T<sub>2</sub> and T<sub>3</sub>,<br />
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现在考虑如下情形，<math>T_1</math> 是一个固定的参考温度: 水的<font color="#ff8000">'''<font color = '#ff8000'>三相点 Triple Point</font>'''</font>的温度。则对于任意 T<sub>2</sub> 和 T<sub>3</sub>,<br />
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: <math>f(T_2,T_3) = \frac{f(T_1,T_3)}{f(T_1,T_2)} = \frac{273.16 \cdot f(T_1,T_3)}{273.16 \cdot f(T_1,T_2)}.</math><br />
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Therefore, if thermodynamic temperature is defined by<br />
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Therefore, if thermodynamic temperature is defined by<br />
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因此，如果热力学温度被定义为<br />
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: <math>T = 273.16 \cdot f(T_1,T) \,</math><br />
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then the function ''f'', viewed as a function of thermodynamic temperature, is simply<br />
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then the function f, viewed as a function of thermodynamic temperature, is simply<br />
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那么函数 ''f'' 作为热力学温度的函数，为<br />
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: <math>f(T_2,T_3) = \frac{T_3}{T_2},</math><br />
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and the reference temperature ''T''<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be used{{snd}}the choice here corresponds to the [[Kelvin]] scale.)<br />
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and the reference temperature T<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be used the choice here corresponds to the Kelvin scale.)<br />
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参考温度 ''T''<sub>1</sub> 的值为273.16。（任何参考温度和任何正值均可用——此处的选择对应开尔文标度。）<br />
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===Entropy 熵===<br />
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{{main article|Entropy (classical thermodynamics)}}<br />
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According to the [[Clausius theorem|Clausius equality]], for a reversible process<br />
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According to the Clausius equality, for a reversible process<br />
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根据克劳修斯等式，对可逆过程有<br />
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: <math>\oint \frac{\delta Q}{T}=0</math><br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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这意味着线积分 <math>\int_L \frac{\delta Q}{T}</math> 对于可逆过程是路径无关的。<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer<ref name="Schmidt-Rohr 14" /> satisfies<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer satisfies<br />
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所以我们可以定义一个叫做熵的状态函数 S，对于可逆过程或者纯热传递满足<br />
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: <math>dS = \frac{\delta Q}{T} \!</math><br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the [[third law of thermodynamics]], which states that ''S'' = 0 at [[absolute zero]] for perfect crystals.<br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the third law of thermodynamics, which states that S = 0 at absolute zero for perfect crystals.<br />
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据此，只有对上述公式进行积分，才能得到熵的差值。为了获得绝对值，我们需要<font color="#ff8000">'''热力学第三定律 Third Law of Thermodynamics'''</font>，它指出<font color="#ff8000">'''绝对零度 Absolute Zero'''</font>下完美晶体的 ''S'' = 0。<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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对于任意不可逆过程，由于熵是一个状态函数，我们总是可以将初始状态和最终状态与一个虚拟的可逆过程联系起来，并在这条路径上积分以计算熵的差值。<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the [[Clausius inequality]] on this loop,<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the Clausius inequality on this loop,<br />
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现在把可逆过程逆过来，将其与上述不可逆过程结合。把克劳修斯不等式应用到这个循环,<br />
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: <math>-\Delta S+\int\frac{\delta Q}{T}=\oint\frac{\delta Q}{T}< 0</math><br />
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Thus,<br />
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Thus,<br />
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故,<br />
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: <math>\Delta S \ge \int \frac{\delta Q}{T} \,\!</math><br />
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where the equality holds if the transformation is reversible.<br />
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where the equality holds if the transformation is reversible.<br />
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如果变换可逆，等号成立。<br />
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Notice that if the process is an [[adiabatic process]], then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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Notice that if the process is an adiabatic process, then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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注意，若该过程是一个<font color="#ff8000">'''绝热过程 Adiabatic Process'''</font>，则<math>\delta Q=0</math>，故<math>\Delta S\ge 0</math>。<br />
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===Energy, available useful work===<br />
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===Energy, available useful work===<br />
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能量，<font color = 'red'><s>有用的工作</s></font>'''<font color = '#32CD32'>可用的有用工作available useful work</font>'''<br />
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{{See also|Exergy}}<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an ''unlimited'' heat reservoir at temperature ''T<sub>R</sub>'' and pressure ''P<sub>R</sub>'' {{snd}}so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain ''T<sub>R</sub>''; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain ''P<sub>R</sub>''.<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an unlimited heat reservoir at temperature T<sub>R</sub> and pressure P<sub>R</sub> so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain T<sub>R</sub>; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain P<sub>R</sub>.<br />
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考虑将第二定律应用于孤立系统(称为整体系统或宇宙)<font color = 'red'><s>的情形，</s></font>是一个重要<font color = 'red'><s>的、</s></font><font color = 'blue'>而</font>具有启发性的理想情形。该系统由两部分组成：<font color = 'red'>感兴趣的<s></s></font>研究关心的<font color = 'blue'></font>子系统和子系统的周围环境。因为<font color = 'red'><s>环境被想象的</s></font><font color = 'blue'>想象的环境</font>非常<font color = 'red'><s>巨</s></font>大，<font color = 'blue'>以至于</font>它们可以被视为一个温度为 T<sub>R</sub> 且压力为 P<sub>R</sub> 的无限<font color = 'red'><s>蓄热器</s></font><font color = 'blue'>热源</font>，<font color = 'blue'>因此</font>无论有多少热<font color = 'red'><s>量</s></font>被转移到（或来自）子系统，周围的温度将保持T<sub>R</sub>；无论子系统的体积膨胀（或收缩）有多大，周围环境的压力都将保持不变。<br />
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Whatever changes to ''dS'' and ''dS<sub>R</sub>'' occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy ''S<sub>tot</sub>'' of the isolated total system must not decrease:<br />
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Whatever changes to dS and dS<sub>R</sub> occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy S<sub>tot</sub> of the isolated total system must not decrease:<br />
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无论子系统和周围环境<font color = 'red'><s>的 ''dS''和''dS<sub>R</sub>'' 发生</s></font><font color = 'blue'>单独地发生什么熵的变化 ''dS''和dS<sub>R</sub></font>，根据第二定律，孤立总体系统的熵S<sub>tot</sub>不能减小。<br />
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: <math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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<math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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<math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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According to the [[first law of thermodynamics]], the change ''dU'' in the internal energy of the sub-system is the sum of the heat ''δq'' added to the sub-system, ''less'' any work ''δw'' done ''by'' the sub-system, ''plus'' any net chemical energy entering the sub-system ''d ∑μ<sub>iR</sub>N<sub>i</sub>'', so that:<br />
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According to the first law of thermodynamics, the change dU in the internal energy of the sub-system is the sum of the heat δq added to the sub-system, less any work δw done by the sub-system, plus any net chemical energy entering the sub-system d ∑μ<sub>iR</sub>N<sub>i</sub>, so that:<br />
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根据<font color = 'red'><s>能量守恒定律</s></font><font color = 'blue'>热力学第一定律</font>，子系统<font color = 'red'><s>内部能量</s></font><font color = 'blue'>内能</font>的变化 dU 是<font color = 'red'><s>子系统内部能量 q 的总和</s></font>加在子系统上的热δq的和<font color = 'blue'></font>，减去子系统所做的任何功w，再加上进入子系统的任何净化学能 d ∑μ<sub>iR</sub>N<sub>i</sub>，因此<br />
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: <math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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<math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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<math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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where μ<sub>iR</sub> are the [[chemical potential]]s of chemical species in the external surroundings.<br />
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where μ<sub>iR</sub> are the chemical potentials of chemical species in the external surroundings.<br />
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其中μ<sub>iR</sub>是外部环境中<font color = 'red'><s>化学物类</s></font>'''<font color = '#ff8000'>化学形态 chemical species</font>'''的化学势。<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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现在热<font color = 'red'><s>量</s></font>离开<font color = 'red'><s>储存器</s></font><font color = 'blue'>热源</font>进入子系统是<br />
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: <math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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<math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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数学△ qtr (- dS r) le tr dS / math<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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在这个过程中，首先使用了经典热力学中熵的定义(在统计热力学中，熵变、温度和吸收热量之间的关系'''<font color = '#32CD32'>可以将其推导出来 can be derived</font>''') ，然后从上面的公式可以推导出第二定律的不等式。<br />
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It therefore follows that any net work ''δw'' done by the sub-system must obey<br />
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It therefore follows that any net work δw done by the sub-system must obey<br />
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因此，子系统所做的任何净功δw必须服从<br />
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: <math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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<math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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数学增量 w le-dU + tds + sum mu { iR } dN i，/ math<br />
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It is useful to separate the work ''δw'' done by the subsystem into the ''useful'' work ''δw<sub>u</sub>'' that can be done ''by'' the sub-system, over and beyond the work ''p<sub>R</sub> dV'' done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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It is useful to separate the work δw done by the subsystem into the useful work δw<sub>u</sub> that can be done by the sub-system, over and beyond the work p<sub>R</sub> dV done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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将子系统所做的功δw划分为子系统可以完成的有用功δw<sub>u</sub> ，除了子系统在周围外部压力下膨胀所做的功p<sub>R</sub> dV外，<font color = 'red'><s>还可以</s></font>给出以下可用功（<font color = 'red'><s>放射本能</s></font>'''<font color = 'ff8000'>有用能 exergy</font>'''）关系式:<br />
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: <math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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<math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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<math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the ''availability'' or ''[[exergy]]'' ''E'' of the subsystem,<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the availability or exergy E of the subsystem,<br />
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为方便起见，可以把右边定义为热力学势的精确导数，称为子系统的'''<font color = 'ff8000'>可用性 Availability </font>'''或<font color = 'red'><s>放射本能E</s></font>'''<font color = 'ff8000'>有用能 exergy</font>'''<br />
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: <math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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<math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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<math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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<font color="#32CD32">因此，第二定律意味着，对于任何可以简单地被视为分为一个子系统和一个与之接触的无限温度和压力<font color = 'red'><s>储存器</s></font> '''<font color="#ff8000">热源 Reservoir</font>''' 的过程，The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,</font><br />
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: <math> dE + \delta w_u \le 0 \, </math><br />
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<math> dE + \delta w_u \le 0 \, </math><br />
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<math> dE + \delta w_u \le 0 \, </math><br />
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i.e. the change in the subsystem's exergy plus the useful work done ''by'' the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done ''on'' the system) must be less than or equal to zero.<br />
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i.e. the change in the subsystem's exergy plus the useful work done by the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done on the system) must be less than or equal to zero.<br />
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也就是。子系统<font color = 'red'><s>(火用)</s></font><font color = 'blue'>有用能</font>的变化加上子系统所做的有用功(或者，子系统<font color = 'red'><s>(火用)</s></font><font color = 'blue'>有用能</font>的变化<font color = 'red'><s>不包括任何功，除了压力贮存器所做的功之外，在系统中所做的功</s></font><font color = 'blue'>减去除了'''<font color = '#ff8000'>压力热源</font>'''外任何对系统做的功，</font>)必须小于或等于零。<br />
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In sum, if a proper ''infinite-reservoir-like'' reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in ''E'' for an irreversible process and no change for a reversible process.<br />
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In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in E for an irreversible process and no change for a reversible process.<br />
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总之，如果选择一个合适的类似于无限库的参考状态作为现实世界中的系统环境，<br />
<font color = 'red'><s>那么第二定律预测的不可逆性的 e 值会减少，而可逆过程的 e 值不会变化。</s></font><font color = 'blue'>则第二定律预测不可逆过程的''E''值减少，可逆过程的''E''值不变。</font><br />
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: <math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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<math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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<math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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This expression together with the associated reference state permits a [[design engineer]] working at the macroscopic scale (above the [[thermodynamic limit]]) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (''Also, see [[process engineer]]''). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (''See [[Exergy efficiency|second law efficiency]]''.)<br />
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This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (Also, see process engineer). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (See second law efficiency.)<br />
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这个表达式和相关的参考状态允许<font color = 'red'><s>在宏观尺度（高于热力学极限）下工作的使用第二定律</s></font><font color = 'blue'>设计工程师在宏观尺度（高于热力学极限）下使用第二定律</font>，而无需直接测量或考虑整个孤立系统中的熵变。（另见'''<font color = '#ff8000'>工艺工程师process engineer</font>'''）。<font color = 'red'><s>考虑到这些变化，假设</s></font><font color = 'blue'>这些变化已经在假设中被考虑到，该假设认为</font>所考虑的系统可以在不改变参考状态的情况下与参考状态达到平衡。将其与可逆理想<font color = 'blue'>状态</font>进行比较，还可以找到一个过程或过程集合的效率（见'''<font color = '#ff8000'>第二定律效率second law efficiency</font>'''）<br />
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This approach to the Second Law is widely utilized in [[engineering]] practice, [[environmental accounting]], [[systems ecology]], and other disciplines.<br />
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This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines.<br />
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第二定律的这种方法被广泛应用于'''<font color = '#ff8000'>工程实践engineering practice</font>'''、'''<font color = '#ff8000'>环境会计environmental accounting</font>'''、'''<font color = '#ff8000'>系统生态学systems ecology</font>'''等其他学科。<br />
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===The second law in chemical thermodynamics===<br />
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化学热力学的第二定律<br />
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For a [[spontaneous process|spontaneous chemical process]] in a closed system at constant temperature and pressure without non-''PV'' work, the Clausius inequality Δ''S > Q/T<sub>surr</sub>'' transforms into a condition for the change in [[Gibbs free energy]] <br />
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For a spontaneous chemical process in a closed system at constant temperature and pressure without non-PV work, the Clausius inequality ΔS > Q/T<sub>surr</sub> transforms into a condition for the change in Gibbs free energy <br />
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对于一个<font color = 'blue'>恒温恒压</font>封闭系统中的自发化学过程，在没有 non-PV 功的情况下，<font color = 'red'><s>在恒定温度和压力下，将</s></font>克劳修斯不等式Δ''S > Q/T<sub>surr</sub>'' <font color = 'red'><s>转化为吉布斯自由能变化的条件</s></font><br />
<font color = 'blue'>由于'''<font color = '#ff8000'>吉布斯自由能Gibbs free energy</font>'''的变化而转化为：</font><br />
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: <math>\Delta G < 0 </math><br />
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<math>\Delta G < 0 </math><br />
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<math>\Delta G < 0 </math><br />
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or d''G'' < 0. For a similar process at constant temperature and volume, the change in [[Helmholtz free energy]] must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products.<ref name="Oxtoby8th"/> <ref name="MortimerBook"></ref> The chemical equilibrium condition at constant ''T'' and ''p'' without electrical work is d''G'' = 0.<br />
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or dG < 0. For a similar process at constant temperature and volume, the change in Helmholtz free energy must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products. The chemical equilibrium condition at constant T and p without electrical work is dG = 0.<br />
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或者 dG < 0。对于一个相似的<font color = 'blue'>恒温恒压</font>过程<font color = 'red'><s>，在恒定的温度和体积下</s></font>，'''<font color = '#ff8000'>亥姆霍兹自由能 Helmholtz free energy </font>'''的变化一定是负的， <math>\Delta A < 0 </math>。因此，<font color = 'blue'>一个负的</font>自由能(G 或 A)变化<font color = 'red'><s>的负值</s></font>是过程自发的必要条件。这是热力学第二定律在化学中最有用的形式，其中自由能的变化可以通过<font color = 'red'><s>列表的生成焓</s></font>'''<font color = '#ff8000'>表列生成焓 Tabulated Enthalpies of Formation</font>'''和反应物及产物的标准摩尔熵来计算。在<font color = 'red'><s>T 和 P </s></font><font color = 'blue'>温度和压力</font>不变的情况下，化学平衡条件是 dG = 0。<br />
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-- [[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） tabulated enthalpies of formation 表列生成焓？？<br />
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==History==<br />
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历史<br />
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{{See also|History of entropy}}<br />
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[[File:Sadi Carnot.jpeg|thumb|upright|Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique]].]]<br />
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Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique.]]<br />
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尼古拉·莱昂纳尔·萨迪·卡诺穿着传统的巴黎综合理工学院制服<br />
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The first theory of the conversion of heat into mechanical work is due to [[Nicolas Léonard Sadi Carnot]] in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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The first theory of the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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<font color = 'blue'>卡诺在1824年提出热</font><font color = 'red'><s>首先提出类热</s></font>转化为机械功的<font color = 'blue'>第一个</font>理论。他<font color = 'blue'>是第一个</font>正确认识到了转换效率取决于发动机和环境之间的温差<font color = 'blue'>的人</font>。<br />
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Recognizing the significance of [[James Prescott Joule]]'s work on the conservation of energy, [[Rudolf Clausius]] was the first to formulate the second law during 1850, in this form: heat does not flow ''spontaneously'' from cold to hot bodies. While common knowledge now, this was contrary to the [[caloric theory]] of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law during 1850, in this form: heat does not flow spontaneously from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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'''<font color="#ff8000">克劳修斯Rudolf Clausius </font>'''认识到'''<font color="#ff8000">焦耳James Prescott Joule</font>'''在能量守恒方面工作的重要性后，在1850年<font color = 'red'><s>第一个提出了第二定律</s></font><font color = 'blue'>提出了第二定律的第一个公式，在这个公式中</font>: 热不会自发地从冷物体流向热物体。虽然<font color = 'red'><s>现在的常识是这样的</s></font><font color = 'blue'>现在这是常识</font>，但是这与当时流行的热理论相反，当时的热理论认为热是一种流体。从这些他推断出了'''<font color="#ff8000"> 萨迪卡诺定律the principle of Sadi Carnot</font>'''和熵的定义(1865年)。<br />
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Established during the 19th century, the [[Kelvin-Planck statement|Kelvin-Planck statement of the Second Law]] says, "It is impossible for any device that operates on a [[cyclic process|cycle]] to receive heat from a single [[heat reservoir|reservoir]] and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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Established during the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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<font color = 'red'><s>19世纪提出的开尔文-普朗克第二定律(Kelvin-Planck)表示：“任何循环运行的设备都不可能从单个蓄热体接收热<s>量</s>并产生净功。”这被证明相当于克劳修斯的陈述。</s></font><font color = 'blue'>19世纪提出的开尔文-普朗克第二陈述(Kelvin-Planck)表示：“任何循环运行的设备都不可能从单个热源接收热并产生净功。”这被证明与克劳修斯的陈述等价。</font><br />
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The [[ergodic hypothesis]] is also important for the [[Boltzmann]] approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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'''<font color="#ff8000">遍历假设ergodic hypothesis</font>'''对'''<font color="#ff8000">玻尔兹曼方法Boltzmann approach</font>'''也很重要。<font color = 'blue'>遍历假设认为</font>在很长一段时间内，在具有相同能量的微观态相空间的某些区域所花费的时间与这个区域的体积成正比，即在很长一段时间内，所有可访问的微观状态<font color = 'blue'>出现/成立</font>的可能性都是一样的。<font color = 'red'><s>同样的，</s></font><font color = 'blue'>等价于说，</font>它表明时间平均值和统计集合的平均值是相同的。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 觉得加上出现/成立好一点？<br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a [[Macroscopic bodies|macroscopic system]]. This doctrine is obsolescent.<ref>Denbigh, K.G., Denbigh, J.S. (1985). ''Entropy in Relation to Incomplete Knowledge'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-25677-1}}, pp. 43–44.</ref><ref>Grandy, W.T., Jr (2008). ''Entropy and the Time Evolution of Macroscopic Systems'', Oxford University Press, Oxford, {{ISBN|978-0-19-954617-6}}, pp. 55–58.</ref><ref name=Lambert>[http://entropysite.oxy.edu Entropy Sites — A Guide] Content selected by [[Frank L. Lambert]]</ref><br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a macroscopic system. This doctrine is obsolescent.<br />
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克劳修斯还提出了一种传统的学说，他认为熵可以被理解为宏观系统中的'''<font color = '#ff8000'>分子“无序”molecular 'disorder'</font>'''，但这种学说已经过时了。<br />
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===Account given by Clausius===<br />
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作者: 克劳修斯<br />
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[[File:Clausius-1.jpg|thumb|upright|Rudolf Clausius]]<br />
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Rudolf Clausius<br />
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鲁道夫 · 克劳修斯<br />
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In 1856, the German physicist [[Rudolf Clausius]] stated what he called the "second fundamental theorem in the [[mechanical theory of heat]]" in the following form:{{sfnp|Clausius|1867}}<br />
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In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:<br />
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1856年，德国物理学家'''<font color = '#ff8000'>鲁道夫 • 克劳修斯Rudolf Clausius </font>'''阐述了他所谓的“热力学理论中的第二个基本定理second fundamental theorem in the mechanical theory of heat” ，其形式如下:<br />
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: <math>\int \frac{\delta Q}{T} = -N</math><br />
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<math>\int \frac{\delta Q}{T} = -N</math><br />
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<math>\int \frac{\delta Q}{T} = -N</math><br />
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where ''Q'' is heat, ''T'' is temperature and ''N'' is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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其中 Q 是热，T 是温度，N 是一个循环过程中所有<font color = 'red'><s>未补偿</s></font><font color = 'blue'>非补偿</font>的相变的“等价值”。后来在1865年，克劳修斯将“等价值”定义为熵。<font color = 'red'><s>也就是在</s></font><font color = 'blue'>基于这个理论，</font>同一年，第二定律最著名的版本在4月24日苏黎世哲学学会的一次演讲中被<font color = 'red'><s>宣读</s></font><font color = 'blue'>提出</font>，在演讲的最后克劳修斯总结道:<br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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宇宙的熵趋于最大<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. [[universe]], as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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这句话是第二定律最著名的<font color = 'red'><s>措辞</s></font><font color = 'blue'>陈述</font>。由于其语言松散<font color = 'blue'>模糊</font>，如“宇宙universe”，<font color = 'red'><s>与缺乏具体的条件相同</s></font><font color = 'blue'>和缺乏具体的条件</font>，如“开放open”，“封闭closed”，或“孤立isolated”，许多人认为这一简单的陈述意味着热力学第二定律几乎适用于每一个可以想象的<font color = 'red'><s>学科</s></font><font color = 'blue'>主题</font>。这不是真的；这句话只是一个更广泛和更精确的描述的简化版本。<br />
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In terms of time variation, the mathematical statement of the second law for an [[isolated system]] undergoing an arbitrary transformation is:<br />
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In terms of time variation, the mathematical statement of the second law for an isolated system undergoing an arbitrary transformation is:<br />
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就时间变化而言，<font color = 'blue'>一个</font>经历任意变换的孤立系统<font color = 'blue'>的</font>第二定律的数学表述是:<br />
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: <math>\frac{dS}{dt} \ge 0</math><br />
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<math>\frac{dS}{dt} \ge 0</math><br />
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<math>\frac{dS}{dt} \ge 0</math><br />
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where<br />
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where<br />
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这里''S'' 是系统的熵<br />
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: ''S'' is the entropy of the system and<br />
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S is the entropy of the system and<br />
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: ''t'' is [[time]].<br />
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t is time.<br />
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T是时间<br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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!-- 可逆过程需要与周围环境保持平衡。这对于一个孤立的系统是不可能的。因此，关于可逆过程的讨论已经转移到对封闭系统的分析----<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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平衡后用等号。另一种表述孤立系统第二定律的方法是：<br />
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: <math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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with <math> \dot S_{i}</math> the sum of the rate of [[entropy production]] by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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with <math> \dot S_{i}</math> the sum of the rate of entropy production by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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用<math> \dot S_{i}</math>表示系统内所有进程<font color = 'red'><s>产生熵</s></font>'''<font color = '#ff8000'>熵产生Entropy Production</font>'''的速率之和。这个公式的优点是它显示了熵产生的<font color = 'red'><s>影响</s></font>'''<font color = 'blue'>效果</font>。熵产生率是一个非常重要的概念，因为它决定（或限制）热机的效率。乘以环境温度<math>T_{a}</math>，<font color = 'red'><s>得到</s></font>'''<font color = 'blue'>它给出</font>所谓的耗散能<math> P_{diss}=T_{a}\dot S_{i}</math>。<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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封闭系统的第二定律(允许热<s>量</s>交换和移动边界，但不允许物质交换)的表达式是:<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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Here<br />
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Here<br />
这里<br />
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: <math>\dot Q</math> is the heat flow into the system<br />
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<math>\dot Q</math> is the heat flow into the system<br />
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<math>\dot Q</math> 是进入系统的热流。<br />
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: <math>T</math> is the temperature at the point where the heat enters the system.<br />
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<math>T</math> is the temperature at the point where the heat enters the system.<br />
<math>T</math> 是热<s>量</s>进入系统时的温度。<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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<font color = 'red'><s>等式符号在只有可逆过程在系统内发生的情况下成立。</s></font><font color = 'blue'>只有在系统内发生可逆过程的情况下等号才成立</font>如果发生不可逆过程<font color = 'red'><s>(</s></font><font color = 'blue'>（</font>在实际操作系统中就是这种情况）,<font color = 'red'><s>符号保持不变</s></font><font color = 'blue'>则“＞”成立</font>。如果系统有多处供热，必须求相应项的代数和。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 因为是中文环境所以统一用中文标点<br />
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For open systems (also allowing exchange of matter):<br />
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For open systems (also allowing exchange of matter):<br />
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对于开放系统(也允许物质交换) :<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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这里<math>\dot S</math> 是进入系统的熵流，与进入系统的物质流有关。它不应该与熵的时间导数混淆。如果物质在几个地方被供给，需要取这些贡献的代数和。<br />
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==Statistical mechanics==<br />
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统计力学<br />
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[[Statistical mechanics]] gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a [[microstate (statistical mechanics)|microstate]] of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/{{radic|''N''}} where ''N'' is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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Statistical mechanics gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a microstate of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/{{radic|''N''}} where N is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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<font color = 'blue'>统计力学</font>通过假设物质是由不断运动的原子和分子组成的，<font color = 'blue'>来</font><font color = 'red'><s>统计力学</s></font>对第二定律给出了解释。系统中每个粒子的一组特定的位置和速度称为系统的微观状态，由于系统的不断运动，系统不断地改变其微观状态。统计力学假设，在平衡状态下，系统<font color = 'red'><s>可能</s></font>处于<s>的</s>每个微观状态<font color = 'red'><s>发生</s></font>的可能性是相等的<font color = 'red'><s>，当这个假设被提出时，它直接导致了第二定律必须在统计意义上成立的结论。</s></font><font color = 'blue'>这个假设的提出直接导致第二定律必须子啊统计学意义上成立</font>也就是说，第二定律<font color = 'red'><s>平均成立，其中 N 是系统中粒子数的1 / 数量级的统计变化。</s></font><font color = 'blue'>在平均意义上成立，其中统计学变异取决于数量级1/{{radic|''N''}}，其中 N 是系统中粒子数</font>。在日常(宏观)情况下，违反第二定律的概率几乎为零。然而，对于粒子数量很少的系统，热力学参数，包括熵，可能显示出与第二定律预测<font color = 'red'><s>的</s></font><font color = 'blue'>结果的</font>显著的统计偏差。经典热力学理论<font color = 'red'><s>没有</s></font><font color = 'blue'>不</font>处理这些统计变量。<br />
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（by Jie XIANG) 在通过假定一种物质由不断运动的原子和分子组成的前提下，统计力学对热力学第二定律进行了解释。系统中每个粒子其特定的位置和速度组成为系统的微观状态，由于（原子和分子的）不断运动，系统也持续改变其微观状态。统计力学假设，在系统力平衡的情况下，其微观状态其实是等可能发生的，进而能够得出结论：热力学第二定律必须在统计学意义上成立。也就是说，第二定律将保持均值不变， 其统计变化为1 /√N，N为系统中的粒子数。在日常(宏观)情况下，违反热力学第二定律的概率几乎为零，但是，对于粒子数量非常少的系统，其热力学参数（包括熵）可能会显示出与第二定律所预测的明显统计偏差。然而经典热力学理论并不处理这些统计变量。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 为什么会有两个？迷茫。<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）是不是两个译者翻译的？如果第二段翻译不用，可以括号或删除线标一下<br />
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==Derivation from statistical mechanics==<br />
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从统计力学导出<br />
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{{Further information|H-theorem}}<br />
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The first mechanical argument of the [[Kinetic theory of gases]] that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to [[James Clerk Maxwell]] in 1860;<ref>{{Cite journal | last1 = Gyenis | first1 = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G }}</ref> [[Ludwig Boltzmann]] with his [[H-theorem]] of 1872 also argued that due to collisions gases should over time tend toward the [[Maxwell–Boltzmann distribution]].<br />
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The first mechanical argument of the Kinetic theory of gases that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to James Clerk Maxwell in 1860; Ludwig Boltzmann with his H-theorem of 1872 also argued that due to collisions gases should over time tend toward the Maxwell–Boltzmann distribution.<br />
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'''<font color="#ff8000">气体动力学Kinetic theory of gases</font>'''理论的第一个力学论证由'''<font color="#ff8000">麦克斯韦James Clerk Maxwell</font>'''在1860年给出，指出分子碰撞引起温度均衡<font color = 'blue'>化</font>，因此整体趋向于'''<font color="#ff8000">平衡 Equilibrium </font>'''； '''<font color="#ff8000">玻尔兹曼Ludwig Boltzmann</font>'''在1872年提出的'''<font color="#ff8000"> H 定理H-theorem</font>'''也认为，气体由于碰撞应该随着时间的推移趋向于'''<font color="#ff8000">麦克斯韦-波兹曼分布Maxwell–Boltzmann distribution</font>'''。<br />
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Due to [[Loschmidt's paradox]], derivations of the Second Law have to make an assumption regarding the past, namely that the system is [[Correlation and dependence|uncorrelated]] at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a [[boundary condition]], and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the [[Big Bang]]), though [[Boltzmann brain|other scenarios]] have also been suggested.<ref name="Hawking AOT">{{cite journal|last=Hawking|first=SW|title=Arrow of time in cosmology|journal=Phys. Rev. D|year=1985|volume=32|issue=10|pages=2489–2495|doi=10.1103/PhysRevD.32.2489|pmid=9956019|bibcode = 1985PhRvD..32.2489H }}</ref><ref>{{cite book | last = Greene | first = Brian | authorlink = Brian Greene | title = The Fabric of the Cosmos | url = https://archive.org/details/fabricofcosmossp00gree | url-access = registration | publisher = Alfred A. Knopf | year = 2004 | page = [https://archive.org/details/fabricofcosmossp00gree/page/171 171] | isbn = 978-0-375-41288-2}}</ref><ref name=Lebowitz>{{cite journal|last=Lebowitz|first=Joel L.|title= Boltzmann's Entropy and Time's Arrow|journal=Physics Today|date=September 1993|volume=46|issue=9|pages=32–38|url=http://users.df.uba.ar/ariel/materias/FT3_2008_1C/papers_pdf/lebowitz_370.pdf|accessdate=2013-02-22|doi=10.1063/1.881363|bibcode = 1993PhT....46i..32L }}</ref><br />
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Due to Loschmidt's paradox, derivations of the Second Law have to make an assumption regarding the past, namely that the system is uncorrelated at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a boundary condition, and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the Big Bang), though other scenarios have also been suggested.<br />
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由于'''<font color="#ff8000">洛施密特悖论Loschmidt's paradox</font>'''，第二定律的导出必须对过去做出一个假设，即系统在过去的某个时刻是不相关的；这样的假设允许进行简单的概率处理。这个假设通常被认为是一个'''<font color="#ff8000">边界条件boundary condition</font>'''，因此热力学第二定律最终是过去某个地方的初始条件的结果，可能是在宇宙的开始('''<font color="#ff8000">大爆炸 the Big Bang</font>''') ，<font color = 'blue'>尽管</font>也有人提出了其他<font color = 'red'><s>假设</s></font><font color = 'blue'>场景</font>。<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the [[Statistical mechanics#Fundamental postulate|fundamental postulate]], also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy.{{citation needed|date=August 2012}} The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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<font color = 'red'><s>考虑到</s></font><font color = 'blue'>基于</font>这些假设，在统计力学中，第二定律不是一个假设，而是'''统计力学基本假设Statistical mechanics#Fundamental postulate|fundamental postulate'''的一个结果，也被称为'''<font color="#ff8000">等先验概率假设 equal prior probability postulate</font>'''。这个基本假设表明，只要一个人清楚地知道，简单的概率论证只适用于未来，而对于过去，有辅助的信息来源告诉我们，它是低熵的。<font color = 'red'><s>热力学第二定律的第一部分指出，热孤立系统的熵只能增加。</s></font>如果我们把熵的概念限制在热平衡系统中，那么热力学第二定律的第一部分<font color = 'blue'>即热孤立系统的熵只能增加，</font>是等先验概率假设的一个显然结果。处于热平衡状态的孤立系统<font color = 'red'><s>包含</s></font><font color = 'blue'>且具有</font>能量<math>E</math>的熵表示为：<br />
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: <math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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<math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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其中<math>\Omega\left(E\right)</math> 是处于 <math>E</math>和<math>E +\delta E</math这个小区间内的量子态数目。这里的 <math>\delta E</math> 是一个宏观上很小的固定能量区间。严格地说，这意味着熵取决于对<math>\delta E</math>的选择。然而在热力学极限下（例如无穷大系统的极限），狭义的熵(单位体积或单位质量的熵)不依赖于 <math>\delta E</math>。<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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假设我们有一个孤立系统，其宏观状态由许多变量<font color = 'red'><s>描述</s></font><font color = 'blue'>指定</font>。这些宏观变量可以是总体积、活塞在系统中的位置等。从而<math>\Omega</math>将取决于这些变量的值。如果某个变量不是固定的（我们不会在某个位置夹住活塞) ，那么因为在平衡状态下<font color = 'red'><s>所有可到达状态的等几率到达的，</s></font><font color = 'blue'>到达所有可到达状态的可能性是相等的，</font>平衡状态下的自由变量会使 <math>\Omega</math> 最大，因为这是平衡状态下最可能的情况。<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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如果该变量最初固定到某个值然后释放，当达到新的平衡时，变量将自我调整使得 <math>\Omega</math> 最大，这意味着熵将增加或保持不变（如果变量初始固定的值恰好是平衡值）。<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's [[H-theorem]], however, proves that the quantity {{math|''H''}} increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the quantity increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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假设我们初始位于一个平衡状态，突然移除了对一个变量的约束。我们做完这件事的时候，可达到的微观状态的数为<math>\Omega</math>，但是系统还没有达到平衡，所以系统处于某些可达到的状态的实际概率还不等于先验概率 <math>1/\Omega</math>。我们已经知道，最终的平衡状态相对于之前的平衡状态，熵会增加或者保持不变。然而，玻耳兹曼的'''<font color="#ff8000">H定理H-theorem</font>'''证明系统在不处于平衡态的期间，那个量作为时间的函数单调增加。<br />
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===Derivation of the entropy change for reversible processes===<br />
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可逆过程熵变的推导<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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热力学第二定律的第二部分指出，经历可逆过程的系统的熵变为:<br />
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: <math>dS =\frac{\delta Q}{T}</math><br />
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<math>dS =\frac{\delta Q}{T}</math><br />
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where the temperature is defined as:<br />
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where the temperature is defined as:<br />
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其中温度定义为:<br />
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:<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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See [[Microcanonical ensemble|here]] for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the [[adiabatic theorem]] of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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请参阅'''此处Microcanonical ensemble|here'''查看该定义的正当性。假设系统有一些可以改变的外部参数 x。一般来说，系统的能量本征态将依赖于 x。根据量子力学的'''<font color="#ff8000">绝热定理adiabatic theorem</font>'''，在系统哈密顿量无限缓慢变化的极限下，系统将保持在相同的'''<font color = '#ff8000'>能量本征态 energy eigenstate</font>'''，因此系统的能量会随着其所在能量本征态的能量变化而变化。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 能量本征态 不懂翻译是否正确<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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对于外部变量 x 可以定义'''<font color="#ff8000">广义力generalized force</font>''' X ，因此如果 x 增加 dx，那么<math>X dx</math> 就是系统所做的功。例如，如果 x 是体积，那么 X 就是压强。一个已知处于能量本征态<math>E_{r}</math>的系统的广义力为：<br />
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: <math>X = -\frac{dE_{r}}{dx}</math><br />
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<math>X = -\frac{dE_{r}}{dx}</math><br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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由于系统可以在<math>\delta E</math> 区间内的任意能量本征态内，我们将系统的广义力定义为上述表达式的期望值:<br />
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: <math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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<math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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为了计算平均值，我们按照能量本征态来划分<math>\Omega\left(E\right)</math>，通过计算有多少能量本征态的<math>\frac{dE_{r}}{dx}</math>值在区间在<math>Y</math> 到 <math>Y + \delta Y</math>之间。把这个数字叫做为 <math>\Omega_{Y}\left(E\right)</math>，我们有:<br />
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: <math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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<math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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The average defining the generalized force can now be written:<br />
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The average defining the generalized force can now be written:<br />
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根据平均值定义的广义力现在可以写成:<br />
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: <math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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<math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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我们可以把它和由恒定能量 E 下的 x 导出来的熵联系起来。假定我们把 x 改变至 x + dx。然后因为能量本征态依赖于 x， <math>\Omega\left(E\right)</math> 将会改变，这导致能量本征态进入或超出<math>E</math> 和<math>E+\delta E</math> 之间的范围。让我们再次关注<math>\frac{dE_{r}}{dx}</math> 处于 <math>Y</math> 和 <math>Y + \delta Y</math> 之间的能量本征态。由于这些能量本征态的能量增加了 Y dx，所有这些在 E-Y dx 到 E 之间能量本征态<font color = 'blue'>都</font>从 E 以下移动到 E 以上。 因此有<br />
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: <math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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<math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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<font color = 'red'><s>这么多</s></font><font color = 'blue'>这些</font>的能量本征态。如果<math>Y dx\leq\delta E</math>，<font color = 'blue'>则</font>所有这些能量本征态将移动到 <math>E</math> 到 <math>E+\delta E</math>的范围内，使得<math>\Omega</math>增加。从<math>E+\delta E</math>以下移动到<math>E+\delta E</math>以上的能量本征态数目为 <math>N_{Y}\left(E+\delta E\right)</math>。它们的差<br />
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: <math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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<math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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因此是 <math>\Omega</math>增长的净贡献。请注意如果 Y dx 大于 <math>\delta E</math>，那么将会有能量本征态从 E 以下移动到<math>E+\delta E</math>以上。它们在<math>N_{Y}\left(E\right)</math> 和 <math>N_{Y}\left(E+\delta E\right)</math>中都有计数，因此上述表达式在这种情况下也是有效的。<br />
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<br />
<br />
Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
<br />
Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
<br />
将上面的表达式表示为对 E 的导数，并且对 Y 求和得到表达式:<br />
<br />
<br />
<br />
<br />
<br />
: <math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
<br />
<math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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<br />
<br />
<br />
<br />
<br />
The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
<br />
The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
<br />
<math>\Omega</math>的对数关于x的导数由以下方式给出:<br />
<br />
<br />
<br />
<br />
<br />
: <math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
<br />
<math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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<br />
<br />
<br />
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<br />
The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
<br />
The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
<br />
第一项是集约型的，<font color = 'red'><s>例如</s></font><font color = 'blue'>即</font>它不能根据系统大小进行缩放。相反，<font color = 'red'><s>最后一项的规模与逆系统的规模一样</s></font><font color = 'blue'>最后一项跟随逆系统的大小而缩放</font>，因此将在热力学极限中消失。因此我们发现:<br />
<br />
<br />
<br />
<br />
<br />
: <math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
<br />
<math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
<br />
<br />
<br />
<br />
<br />
<br />
Combining this with<br />
<br />
Combining this with<br />
<br />
将这个和下面式子结合<br />
<br />
<br />
<br />
<br />
<br />
: <math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
<br />
<math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
<br />
<br />
<br />
<br />
<br />
Gives:<br />
<br />
Gives:<br />
<br />
给出：<br />
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<br />
<br />
<br />
<br />
: <math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
<br />
<math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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===Derivation for systems described by the canonical ensemble===<br />
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正则系综系统的推导<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the [[canonical ensemble]]:<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:<br />
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如果一个系统与某个温度为T的热浴热接触，那么在平衡状态下，关于能量本征值的概率分布值由'''<font color = '#ff8000'>正则系综canonical ensemble</font>'''给出:<br />
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: <math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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<math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the [[Partition function (statistical mechanics)|partition function]]. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
<br />
Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. <font color = 'red'>We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. </font>It follows from the general formula for the entropy:<br />
<br />
这里 Z 是一个使所有概率之和<font color = 'red'><s>归一化</s></font><font color = 'blue'>正态化到 1 </font>的因子，这个函数被称为'''<font color = '#ff8000'>配分函数Partition function (statistical mechanics)|partition function</font>'''。现在我们考虑对温度和能级所依赖的外部参数的无限小的可逆改变。它遵循熵的一般公式:<br />
<br />
--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 能级依赖温度吗？即“现在我们考虑……的可逆改变”当中，“on which the energy levels depend”修饰（1）in the external parameters，还是（2）in the temperature and in the external parameters<br />
--[[用户:小头盔|小头盔]]（[[用户讨论:小头盔|讨论]]）修饰in the external parameters<br />
<br />
<br />
<br />
: <math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
<br />
<math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
<br />
<br />
<br />
<br />
<br />
<br />
that<br />
<br />
that<br />
<br />
以及<br />
<br />
<br />
<br />
<br />
<br />
: <math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
<br />
<math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
<br />
<br />
<br />
<br />
<br />
Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
<br />
Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
<br />
把正则系综的数学形式 <math>P_{j}</math> 插入，可以得到:<br />
<br />
<br />
<br />
<br />
<br />
: <math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
<br />
<math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
<br />
==Living organisms==<br />
<br />
==Living organisms==<br />
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生命体<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as [[What is Life?|Erwin Schrödinger]], [[Léon Brillouin]]<ref>Léon Brillouin [https://books.google.com/books?hl=en&lr=&id=tPXVbiw_1P0C ''Science and Information Theory''] (Academic Press, 1962) (Dover, 2004)</ref> and [[Life and Energy|Isaac Asimov]]. It is also the topic of current{{When|date=July 2019}} research.{{Citation needed|date=July 2019}}<br />
<br />
There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as Erwin Schrödinger, Léon Brillouin and Isaac Asimov. It is also the topic of current research.<br />
<br />
表述热力学主要有两种方式：(a)描述从一种热力学平衡状态到另一种状态的路径，(b)通过<font color = 'red'><s>使系统保持不变，同时增加周围环境的总熵</s></font><font color = 'blue'>系统保持不变但周围环境的总熵增加</font>的循环过程。这两种方法有助于理解生命过程。这个话题大大超出了本文的范围，但是已经有一些研究者考虑过这个问题，比如'''<font color = '#ff8000'>薛定谔Erwin Schrödinger</font>'''、'''<font color = '#ff8000'>布里渊Léon Brillouin</font>'''和'''<font color = '#ff8000'>阿西莫夫Isaac Asimov</font>'''。这也是当前研究的主题。<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of [[metabolism]], give out breakdown products and heat. Plants [[Photosynthesis|take in radiative energy]] from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
<br />
To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of metabolism, give out breakdown products and heat. Plants take in radiative energy from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
<br />
从近似的角度来看，生命体可以被认为是(b)的一个例子。近似地，一只动物的身体状态每天循环，使得它几乎没有什么变化。动物吸收食物、水和氧气，经过'''<font color = '#ff8000'>新陈代谢metabolism</font>'''，输出分解的产物和热<s>量</s>。植物吸收来自太阳的辐射能量，这可以认为是热<s>量</s>，以及二氧化碳和水，然后它们释放氧气。它们就是这样生长的。<font color = 'blue'>它们</font>最终会死亡，尸体腐烂，大部分重新变成二氧化碳和水。这可以看作是一个循环过程。总的来说，阳光来自一个高温的源——太阳，它的能量被传递到一个较低的温度源，<font color = 'red'><s>例如</s></font><font color = 'blue'>即</font>向太空辐射。这个过程使得植物周围环境的熵增加。因此从循环过程的角度来看，动物和植物服从热力学第二定律。简单的热机效率概念很难适用于这个问题，因为它们假定系统是封闭的。<br />
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From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to [[Non-equilibrium thermodynamics|unsettled current debate or research]]. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to unsettled current debate or research. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
<br />
从(a)的热力学观点考虑，即考虑从一个平衡状态到另一个平衡状态的路径，只会产生一个粗略近似的图像，因为生命体从来不会处于热力学平衡状态。生物体经常必须被认为是开放的系统，因为它们吸收营养物质并排出废弃物。开放系统热力学目前通常从一个热力学平衡状态到另一个状态的路径的角度来考虑，或者是考虑局部热力学平衡近似下的流。生命体的问题可以通过假定一个流不变的稳态<font color = 'blue'>的近似</font>来进一步简化。对于这样的近似，<font color = 'red'><s>熵产生的一般原理在'''目前有争论和研究的不确定性Non-equilibrium thermodynamics|unsettled current debate or research'''</s></font><font color = 'blue'>目前有很多关于熵产生的一般原理的未解决的争论或研究</font>。尽管如此，从热力学第二定律的这个角度出发衍生出来的想法对生物是有启发意义的。<br />
<br />
==Gravitational systems==<br />
<br />
==Gravitational systems==<br />
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重力系统<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive [[heat capacity]], meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it.<ref name="Grandy 151">Grandy, W.T. (Jr) (2008), p. 151.</ref> Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
<br />
In systems that do not require for their descriptions the general theory of relativity, bodies always have positive heat capacity, meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it. Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
<br />
<font color = 'red'><s>在不需要描述相对论</s></font><font color = 'blue'>在不需要其对广义相对论的解释的</font>的系统中，物体始终具有正热容，这意味着温度随能量而升高。因此，当能量从高温物体流向低温物体时，发射源温度降低而冷源温度升高；因此，温差会随着时间的流逝而减小。对于引力很重要并且<font color = 'red'><s>基于相对论的一般</s></font><font color = 'blue'>需要广义相对论的</font>系统而言，情况并非总是如此。这样的系统可以自发地<font color = 'red'><s>转变为质量和能量的不均匀分布</s></font><font color = 'blue'>向质量和能量不均匀的分布转变</font>。这适用于大规模的宇宙，因此可能很难或不可能对它应用第二定律。除此之外，广义相对论描述的系统的热力学超出了本文的范围。<br />
<br />
==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
非平衡状态<br />
{{main article|Non-equilibrium thermodynamics}}<br />
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The theory of classical or [[thermodynamic equilibrium|equilibrium thermodynamics]] is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<ref name="Grandy 151"/><ref>Callen, H.B. (1960/1985), p. 15.</ref><br />
<br />
The theory of classical or equilibrium thermodynamics is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<br />
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经典热力学或平衡热力学理论是理想化的。其主要假定或假设（<font color = 'red'><s>往往无明确说明</s></font><font color = 'blue'>通常甚至没有明确地陈述</font>），是基于系统处在自身内部的热力学平衡状态。通常，在自然界可能发现的，在给定时间包含物理系统的空间区域，其热力学平衡状态并不是能用最严格的术语来表示。用较宽泛的术语来说，整个宇宙中没有任何东西是或曾经是真正处于精确的热力学平衡状态中。<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of [[thermodynamic equilibrium]]. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable [[thermal fluctuations|fluctuations]]. There is an exception, the case of [[critical point (thermodynamics)|critical states]], which exhibit to the naked eye the phenomenon of [[critical opalescence]]. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
<br />
For purposes of physical analysis, it is often enough convenient to make an assumption of thermodynamic equilibrium. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable fluctuations. There is an exception, the case of critical states, which exhibit to the naked eye the phenomenon of critical opalescence. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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为了进行物理分析，通常<font color = 'red'><s>假设热力学平衡就足以方便可行</s></font><font color = 'blue'>做一个热力学平衡的假设就足够方便可行</font>。 这样的假设可能依赖于反复试验和试错来证明其合理性。如果假设是合理的，那么它通常会非常有价值，因为它使热力学理论成为可能。平衡假设的要素是，观察到的系统会无限期地保持不变，并且因为系统中有很多粒子，以至于完全可以忽略其微粒<font color = 'red'><s>性质</s></font><font color = 'blue'>的本质</font>。通常在这种平衡假设下，<font color = 'red'><s>没有</s></font><font color = 'blue'>不存在一个</font>宏观上可检测到的波动。临界状态是一个例外，<font color = 'red'><s>它以肉眼显示出</s></font><font color = 'blue'>肉眼可以观察到</font>临界'''<font color = '#ff8000'>乳光现象opalescence</font>'''。对于临界状态的实验室研究，需要非常长的观察时间。<br />
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In all cases, the assumption of [[thermodynamic equilibrium]], once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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In all cases, the assumption of thermodynamic equilibrium, once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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在所有情况下，一旦<font color = 'red'><s>完成</s></font><font color = 'blue'>做出</font>了热力学平衡的假设，则<font color = 'blue'>作为一个结果</font>意味着假定<font color = 'blue'>中的</font>能改变系统熵的<font color = 'red'><s>“fluctuation”（待定的波动因子）</s></font><font color = 'blue'>候选波动</font>不存在。<br />
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'''<font color = '#32CD32'>则作为一个结果意味着假定中的能改变系统熵的候选波动不存在。implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.</font>'''<br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<ref>Lieb, E.H., Yngvason, J. (2003), p. 190.</ref><br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<br />
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物理系统很容易发生内部宏观变化，其速度之快足以使熵的恒定性假设失效。或物理系统中的粒子太少，以至于微粒的本质在可观测的波动中显而易见。那么热力学平衡的假设就会被放弃。对于非平衡态，熵没有绝对的通俗定义。<br />
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There are intermediate cases, in which the assumption of local [[thermodynamic equilibrium]] is a very good approximation,<ref>Gyarmati, I. (1967/1970), pp. 4-14.</ref><ref>Glansdorff, P., Prigogine, I. (1971).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003).</ref> but strictly speaking it is still an approximation, not theoretically ideal.<br />
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There are intermediate cases, in which the assumption of local thermodynamic equilibrium is a very good approximation, but strictly speaking it is still an approximation, not theoretically ideal.<br />
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还有一些中间情况，其中局部热力学平衡的假设是一个非常好的近似<font color = 'red'><s>值</s></font>，但严格地说它仍然是一个近似值，不是理论上的理想值。<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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对于一般的非平衡状态，考虑其他量的统计力学定义可能是有用的，这些量可以方便地称为“熵”，但是它们不应该与第二定律正确定义的熵混淆<font color = 'blue'>或合并</font>。这些其他量确实属于统计力学，而不属于第二定律的主要领域——热力学。<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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宏观上可观察到的波动的物理学超出了本文的范围。<br />
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==Arrow of time==<br />
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==Arrow of time==<br />
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时间之箭<br />
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{{Further information|Entropy (arrow of time)}}Thermodynamic Asymmetry in Time<br />
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Thermodynamic Asymmetry in Time<br />
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时间中的热力学不对称性<br />
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{{See also|Arrow of time}}<br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely [[CPT symmetry]], since the second law applies statistically, it is hypothesized, on time-asymmetric ''boundary conditions''.<ref>{{cite web|first=Craig|last=Callender|url=https://plato.stanford.edu/archives/fall2011/entries/time-thermo/|title=Thermodynamic Asymmetry in Time|encyclopedia=Stanford Encyclopedia of Philosophy|date=29 July 2011}}</ref><br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with symmetries observed in the fundamental laws of physics (particularly CPT symmetry) since the second law applies statistically on time-asymmetric boundary conditions.[78] The second law has been related to the difference between moving forwards and backwards in time, or to the principle that cause precedes effect (the causal arrow of time, or causality).[79]<br />
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热力学第二定律是<font color = 'red'><s>与时间方向逆转不对称的物理定律</s></font><font color = 'blue'>在时间方向上不对称的物理定律</font>。这与在物理学基本定律中观察到的对称性概念（特别是CPT对称性）<s>并</s>没有冲突，因为第二定律在统计上适用于时间非对称边界条件。第二定律与时间向前和向后移动之间的差异有关，或者与<font color = 'red'><s>导致先于效果</s></font><font color = 'blue'>原因先于结果</font>的原理（时间的因果关系或因果关系）有关。<br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect ([[Arrow of time#The causal arrow of time|the causal arrow of time]]).<ref>{{cite book | first = J.J.| last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1|display-authors=etal}} chapter 6</ref><br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect (the causal arrow of time).<br />
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第二定律被提出来<font color = 'red'><s>部分</s></font><font color = 'blue'>侧面</font>解释时间向前和向后移动的区别，例如为什么原因先于结果(因果关系的时间箭头)。<br />
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==Irreversibility==<br />
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==Irreversibility==<br />
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不可逆性<br />
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Irreversibility in [[thermodynamic process]]es is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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Irreversibility in thermodynamic processes is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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热力学过程中的不可逆性是热力学作用不对称特性的结果，而不是物体内部任何不可逆的微观性质<font color = 'blue'>的结果</font>。热力学作用是对参与体施加的宏观外部干预，并非来自其内部属性。由于没有认识到这一点，产生了被称为“悖论”的说法。<br />
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===Loschmidt's paradox===<br />
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<font color = 'red'><s>洛斯密特悖论 </s></font><font color = 'blue'>洛施密特悖论</font><br />
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{{main article|Loschmidt's paradox}}<br />
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[[Loschmidt's paradox]], also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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'''<font color="#ff8000">洛施密特悖论 Loschmidt's paradox</font>'''，也被称为'''<font color="#ff8000">可逆性悖论reversibility paradox</font>'''，<br />
<font color = 'red'><s>其目的在于不可能从描述宏观系统微观演化的时间对称动力学中推导出不可逆过程。</s></font><font color = 'blue'><br />
它反驳说，从描述宏观系统微观演化的时间对称动力学中，不可能推导出不可逆过程。</font><br />
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In the opinion of [[Erwin Schrödinger|Schrödinger]], "It is now quite obvious in what manner you have to reformulate the law of entropy{{snd}}or for that matter, all other irreversible statements{{snd}}so that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other."<ref>[[Erwin Schrödinger|Schrödinger, E.]] (1950), p.&nbsp;192.</ref> The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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In the opinion of Schrödinger, "It is now quite obvious in what manner you have to reformulate the law of entropyor for that matter, all other irreversible statementsso that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other." The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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薛定谔的认为：“现在很明显，您必须以何种方式重新制定熵定律（亦或者所有其他不可逆陈述）以使它们能够从可逆模型中得出。你不能只谈论一个孤立的系统，而应该至少谈论两个。你可以暂时认为它们与世界其它地方是孤立的，但并不总是相互孤立的。”这两个系统被隔离内壁彼此分开，直到被热力学（由热力学定律设想出来的）作用而拆除。热力学作用是从外部施加的，不受支配系统组成部分的可逆微观动力学定律的制约。这是不可逆转的原因。本文中<font color = 'red'><s>的法律声明</s></font><font color = 'blue'>第二定律的陈述</font>符合薛定谔的的建议。 因果关系在逻辑上先于第二定律，而不是由第二定律推导而来。<br />
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===Poincaré recurrence theorem===<br />
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庞加莱始态复现定理<br />
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{{Main article|Poincaré recurrence theorem}}<br />
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The [[Poincaré recurrence theorem]] considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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The Poincaré recurrence theorem considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics.Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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庞加莱始态复现定理<font color = 'red'><s>（又叫递归定理）</s></font>考虑了孤立物理系统的理论微观描述。<br />
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--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）又叫递归定理吗？存疑<br />
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经过热力学作用去除隔离内壁之后，便可以认为其是热力学系统的模型。在足够长的时间后，系统将返回到非常接近初始状态的微观定义状态。庞加莱复现时间是指回归前经过的时间长度。它极其漫长，可能比宇宙的寿命还要长，并且<font color = 'red'><s>敏感地依赖于被热力学作用拆除的墙体的几何形状</s></font><font color = 'blue'>对那个被热力学作用拆除的内壁的几何形状敏感</font>。复现定理可能被认为是明显与热力学第二定律相矛盾的。但是，更显而易见的是，它只是通过移除两个系统之间的内壁而形成的隔离系统中热力学平衡的微观模型。<font color = 'red'><s>对于典型的热力学系统而言，重复时间是如此之长（比宇宙的寿命长很多倍），以至于在所有的实际目的中，人们都无法观察到这种重复。</s></font><font color = 'blue'>对所有典型的热力学系统的实际目的，人们无法观察到如此之长的复现时间很长（比宇宙的寿命长很多倍）。</font>尽管如此，还是有人会想像一个机会可以等待庞加莱复现的出现，然后重新插入被热力学作用去除的内壁。<br />
然后很明显，不可逆性的出现是由于<font color = 'red'><s>庞加莱递归的完全不可预测性</s></font><font color = 'blue'>庞加莱复现完全不可预测的特性</font>，因为仅仅给出了初始状态<font color = 'red'><s>遵守热力学平衡的条件之一</s></font><font color = 'blue'>是热力学平衡之一</font>，就像宏观热力学的情况一样。<br />
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'''<font color="#32CD32">然后很明显，不可逆性的出现是由于庞加莱复现完全不可预测的特性，因为仅仅给出了初始状态是热力学平衡之一，就像宏观热力学的情况一样。It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics.</font>''' （这句话重复上面的，是为了标注疑难句的同时保留审校痕迹。生产修改后可以只保留这句，删掉上面重复部分）<br />
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即使<font color = 'red'><s>可以等待，也没有实际可操作性</s></font><font color = 'blue'>一个人可以等那么长时间，他也没有实际操作的可能性</font>来选择合适的时间重新插入内壁。庞加莱始态复现定理为洛施密特悖论提供了一个解决方案。如果一个孤立的热力学系统能以多倍于平均庞加莱复现时间的长度下<font color = 'red'><s>进行</s></font><font color = 'blue'>被</font>监控，则该系统的热力学行为在时间反转下将变得恒定。<br />
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[[File:James-clerk-maxwell3.jpg|thumb|upright|James Clerk Maxwell]]<br />
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James Clerk Maxwell<br />
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詹姆斯·克拉克·麦克斯韦<br />
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===Maxwell's demon===<br />
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麦克斯韦<font color = 'red'><s>的恶魔</s></font><font color = 'blue'>妖</font><br />
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{{main article|Maxwell's demon}}<br />
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[[James Clerk Maxwell]] imagined one container divided into two parts, ''A'' and ''B''. Both parts are filled with the same [[gas]] at equal temperatures and placed next to each other, separated by a wall. Observing the [[molecule]]s on both sides, an imaginary [[demon]] guards a microscopic trapdoor in the wall. When a faster-than-average molecule from ''A'' flies towards the trapdoor, the demon opens it, and the molecule will fly from ''A'' to ''B''. The average [[speed]] of the molecules in ''B'' will have increased while in ''A'' they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in ''A'' and increases in ''B'', contrary to the second law of thermodynamics.<br />
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James Clerk Maxwell imagined one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other, separated by a wall. Observing the molecules on both sides, an imaginary demon guards a microscopic trapdoor in the wall. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics.<br />
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'''麦克斯韦 James Clerk Maxwell'''设想一个容器分为A和B两部分。两部分均在相同温度下充满相同的气体，并彼此相邻由内壁隔开。<font color = 'red'><s>观察到两侧的分子，一个虚构的恶魔在墙上守着一个微观活板门</s></font><font color = 'blue'>一个虚构的妖精看守着一个内壁上的微型陷门，并观察者内壁两边的分子</font>。当来自A的速度快于平均水平的分子飞向<font color = 'red'><s>活板</s></font>门时，<font color = 'red'><s>恶魔将其打开</s></font><font color = 'blue'>妖精将门打开</font>，该分子将从A飞向B。B中分子的平均速度将增加，而A中分子的平均速度将<font color = 'red'><s>平均</s></font>降低。由于平均分子速度与温度相对应，因此与热力学第二定律相反，温度在A中降低，在B中升高。<br />
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One response to this question was suggested in 1929 by [[Leó Szilárd]] and later by [[Léon Brillouin]]. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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One response to this question was suggested in 1929 by Leó Szilárd and later by Léon Brillouin. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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1929年，Leó Szilárd和后来的 Léon Brillouin 对这个问题提出了一个答案。Szilárd指出，现实生活中的麦克斯韦<font color = 'red'><s>恶魔</s></font><font color = 'blue'>妖</font>需要一些测量分子速度的方法，而获取信息<font color = 'blue'>的过程</font>需要消耗<font color = 'red'><s>大量</s></font>能量。<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between ''A'' and ''B''. It is therefore performing [[thermodynamic operation]]s on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between A and B. It is therefore performing thermodynamic operations on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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麦克斯韦的<font color = 'red'><s>“恶魔”不断地</s></font><font color = 'blue'>“妖”重复地</font>改变 A 和 B 之间<font color = 'red'><s>墙体的渗透性</s></font><font color = 'blue'>壁的热透性</font>。因此，它是在微观尺度上执行热力学操作，而不仅仅是观察普通的自发或自然的宏观热力学过程。<br />
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==Quotations==<br />
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==Quotations==<br />
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语录<br />
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{{quote|The law that entropy always increases holds, I think, the supreme position among the [[Laws of science|laws of Nature]]. If someone points out to you that your pet theory of the [[universe]] is in disagreement with [[Maxwell's equations]] – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.|Sir [[Arthur Stanley Eddington]], ''The Nature of the Physical World'' (1927)}}<br />
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The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.<br />
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— Sir Arthur Stanley Eddington, The Nature of the Physical World (1927)<br />
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我认为，熵总是增加的定律在自然定律中占有至高无上的地位。如果有人指出您<font color = 'red'><s>的宇宙宠物理论</s></font><font color = 'blue'>杜撰的宇宙理论</font>与麦克斯韦方程不符，那么麦克斯韦方程就遇到麻烦了。如果发现<font color = 'blue'>您的理论</font>与观察不符，那么，可能是这些实验学家搞砸了。但是，如果发现您的理论违反了热力学第二定律，那么就没有希望了。除了在深深的耻辱中崩溃，别无他法。<br />
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—亚瑟·斯坦利·爱丁顿爵士，《自然世界》（1927年）<br />
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{{quote|There have been nearly as many formulations of the second law as there have been discussions of it.|Philosopher / Physicist [[Percy Williams Bridgman|P.W. Bridgman]], (1941)}}<br />
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There have been nearly as many formulations of the second law as there have been discussions of it.<br />
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— Philosopher / Physicist P.W. Bridgman, (1941)<br />
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关于第二定律的<font color = 'red'><s>阐述</s></font><font color = 'blue'>公式</font>几乎与关于第二定律的讨论一样多。<br />
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—哲学家/物理学家 布里奇曼（1941）<br />
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{{quote|Clausius is the author of the sibyllic utterance, "The energy of the universe is constant; the entropy of the universe tends to a maximum." The objectives of continuum thermomechanics stop far short of explaining the "universe", but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size.|[[Clifford Truesdell|Truesdell, C.]], Muncaster, R. G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 17.}}<br />
<br />
Clausius is the author of the sibyllic utterance, "The energy of the universe is constant; the entropy of the universe tends to a maximum." The objectives of continuum thermomechanics stop far short of explaining the "universe", but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size.<br />
<br />
— Truesdell, C., Muncaster, R. G. (1980). Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics, Academic Press, New York, ISBN 0-12-701350-4, p. 17.<br />
<br />
克劳修斯（Sir Clausius）是西伯利亚文“宇宙的能量是恒定的；宇宙的熵趋于最大。”的作者。<font color = 'red'><s>不间断热力学的</s></font><font color = 'blue'>连续热力学的目标</font>远远不能解释“宇宙”，但是在该理论内，我们可以很容易地以某种类似于克劳修斯的方式得出明确的陈述，只是它仅仅指代一个简单的物体：一个有限大小的孤立物体。<br />
<br />
— Truesdell, C., Muncaster, R. G. (1980). Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics, Academic Press, New York, ISBN 0-12-701350-4, p. 17. 特雷斯德尔，C.，蒙卡斯特，R.G.（1980）。麦克斯韦的单原子气体动力学理论的基本原理，被作为“理性力学”的一个分支，纽约学术出版社，ISBN 0-12-701350-4，第1页. 17。<br />
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==See also==<br />
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==See also==<br />
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参见<br />
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{{colbegin|colwidth=20em}}<br />
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<br />
<br />
*[[Clausius–Duhem inequality]]<br />
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*[[Fluctuation theorem]]<br />
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*[[Heat death of the universe]]<br />
<br />
<br />
<br />
*[[History of thermodynamics]]<br />
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<br />
<br />
*[[Jarzynski equality]]<br />
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<br />
<br />
*[[Laws of thermodynamics]]<br />
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<br />
<br />
*[[Maximum entropy thermodynamics]]<br />
<br />
<br />
<br />
*[[Quantum thermodynamics]]<br />
<br />
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*[[Reflections on the Motive Power of Fire]]<br />
<br />
<br />
<br />
*[[Relativistic heat conduction]]<br />
<br />
<br />
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*[[Thermal diode]]<br />
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*[[Thermodynamic equilibrium]]<br />
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{{colend}}<br />
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<br />
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==References==<br />
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==References==<br />
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参考资料<br />
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{{reflist|20em}}<br />
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===Sources===<br />
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===Sources===<br />
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资料来源<br />
<br />
{{refbegin|50em}}<br />
<br />
<br />
<br />
*Adkins, C.J. (1968/1983). ''Equilibrium Thermodynamics'', (1st edition 1968), third edition 1983, Cambridge University Press, Cambridge UK, {{ISBN|0-521-25445-0}}.<br />
<br />
<br />
<br />
*[[Peter Atkins|Atkins, P.W.]], de Paula, J. (2006). ''Atkins' Physical Chemistry'', eighth edition, W.H. Freeman, New York, {{ISBN|978-0-7167-8759-4}}.<br />
<br />
<br />
<br />
*Attard, P. (2012). ''Non-equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications'', Oxford University Press, Oxford UK, {{ISBN|978-0-19-966276-0}}.<br />
<br />
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*Baierlein, R. (1999). ''Thermal Physics'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-59082-5}}.<br />
<br />
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*Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics, New York, {{ISBN|0-88318-797-3}}.<br />
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*{{Cite book|title=Concepts in thermal physics|last=Blundell|first=Stephen J.|last2=Blundell|authorlink1=Stephen Blundell|first2=Katherine M.|authorlink2=Katherine Blundell|publisher=[[Oxford University Press]]|year=2010|isbn=9780199562107|edition=2nd|location=Oxford|pages=|oclc=607907330|doi=10.1093/acprof:oso/9780199562091.001.0001 |url=http://cds.cern.ch/record/1235139}}<br />
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*[[Ludwig Boltzmann|Boltzmann, L.]] (1896/1964). ''Lectures on Gas Theory'', translated by S.G. Brush, University of California Press, Berkeley.<br />
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*Borgnakke, C., Sonntag., R.E. (2009). ''Fundamentals of Thermodynamics'', seventh edition, Wiley, {{ISBN|978-0-470-04192-5}}.<br />
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*Buchdahl, H.A. (1966). ''The Concepts of Classical Thermodynamics'', Cambridge University Press, Cambridge UK.<br />
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*[[Percy Williams Bridgman|Bridgman, P.W.]] (1943). ''The Nature of Thermodynamics'', Harvard University Press, Cambridge MA.<br />
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*[[Herbert Callen|Callen, H.B.]] (1960/1985). ''Thermodynamics and an Introduction to Thermostatistics'', (1st edition 1960) 2nd edition 1985, Wiley, New York, {{ISBN|0-471-86256-8}}.<br />
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*{{cite journal|author=C. Carathéodory|author1-link=Constantin Carathéodory<br />
<br />
<br />
<br />
|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
<br />
|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
<br />
|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
<br />
|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
<br />
|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
<br />
2010年3月15日 | 第355-386页 | http://gdz.sub.uni-goettingen.de/index.php?id=11&ppn=ppn235181684_0067&dmdid=dmdlog_0033&l=1<br />
<br />
|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). ''The Second Law of Thermodynamics'', Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
<br />
|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
<br />
|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können.(p. 363) | doi 10.1007 / bf01450409}.译文可以在这里找到[ http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf ]。同时，在 Kestin，j。(1976).热力学第二定律，Dowden，Hutchinson & Ross，Stroudsburg PA. 。<br />
<br />
*[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986). [http://www.worldcat.org/title/reflections-on-the-motive-power-of-fire-a-critical-edition-with-the-surviving-scientific-manuscripts-translated-and-edited-by-fox-robert/oclc/812944517&referer=brief_results ''Reflections on the motive power of fire''], Manchester University Press, Manchester UK, {{ISBN|0-7190-1741-6}}. [https://archive.org/stream/reflectionsonmot00carnrich#page/n7/mode/2up Also here.]<br />
<br />
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*[[Sydney Chapman (mathematician)|Chapman, S.]], [[Thomas George Cowling|Cowling, T.G.]] (1939/1970). ''The Mathematical Theory of Non-uniform gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases'', third edition 1970, Cambridge University Press, London.<br />
<br />
<br />
<br />
*{{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Ueber Die Bewegende Kraft Der Wärme Und Die Gesetze, Welche Sich Daraus Für Die Wärmelehre Selbst Ableiten Lassen|journal=Annalen der Physik|year=1850|volume=79|issue=4|pages=368–397, 500–524|url=http://gallica.bnf.fr/ark:/12148/bpt6k15164w/f518.image|accessdate=26 June 2012|doi=10.1002/andp.18501550403|bibcode = 1850AnP...155..500C |hdl=2027/uc1.$b242250|hdl-access=free}} Translated into English: {{cite journal|last=Clausius|first=R.|title=On the Moving Force of Heat, and the Laws regarding the Nature of Heat itself which are deducible therefrom|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1851| volume=2|series=4th|issue=VIII|pages=1–21; 102–119 |url=https://archive.org/stream/londonedinburghd02lond#page/1/mode/1up|accessdate=26 June 2012<br />
<br />
<br />
<br />
|doi=10.1080/14786445108646819}}<br />
<br />
|doi=10.1080/14786445108646819}}<br />
<br />
10.1080 / 14786445108646819}<br />
<br />
* {{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie|year=1854|journal=Annalen der Physik|volume=xciii|issue=12|pages=481–506|url=http://zfbb.thulb.uni-jena.de/servlets/MCRFileNodeServlet/jportal_derivate_00140956/18541691202_ftp.pdf|accessdate=24 March 2014|doi=10.1002/andp.18541691202<br />
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<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
<br />
|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
<br />
| bibcode 1854AnP... 169. 1856年7月 | 第2卷 | 第4页 | 第86页 | https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=242014年3月<br />
<br />
}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
<br />
}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
<br />
热力学理论及其在蒸汽机和物体物理性质中的应用 | 1867年 | 出版商 John van Voorst | 位置: 伦敦 | https://archive.org/details/mechanicaltheor04claugoog|quote=editions:pwr_sbkwa8ic。 2012年6月19日<br />
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}}<br />
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}}<br />
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}}<br />
<br />
*Denbigh, K. (1954/1981). ''The Principles of Chemical Equilibrium. With Applications in Chemistry and Chemical Engineering'', fourth edition, Cambridge University Press, Cambridge UK, {{ISBN|0-521-23682-7}}.<br />
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*Eu, B.C. (2002). ''Generalized Thermodynamics. The Thermodynamics of Irreversible Processes and Generalized Hydrodynamics'', Kluwer Academic Publishers, Dordrecht, {{ISBN|1-4020-0788-4}}.<br />
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*[[Josiah Willard Gibbs|Gibbs, J.W.]] (1876/1878). On the equilibrium of heterogeneous substances, ''Trans. Conn. Acad.'', '''3''': 108-248, 343-524, reprinted in ''The Collected Works of J. Willard Gibbs, Ph.D, LL. D.'', edited by W.R. Longley, R.G. Van Name, Longmans, Green & Co., New York, 1928, volume 1, pp.&nbsp;55–353.<br />
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*Griem, H.R. (2005). ''Principles of Plasma Spectroscopy (Cambridge Monographs on Plasma Physics)'', Cambridge University Press, New York {{ISBN|0-521-61941-6}}.<br />
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*Glansdorff, P., Prigogine, I. (1971). ''Thermodynamic Theory of Structure, Stability, and Fluctuations'', Wiley-Interscience, London, 1971, {{ISBN|0-471-30280-5}}.<br />
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*Grandy, W.T., Jr (2008). [http://global.oup.com/academic/product/entropy-and-the-time-evolution-of-macroscopic-systems-9780199546176?cc=au&lang=en& ''Entropy and the Time Evolution of Macroscopic Systems'']. Oxford University Press. {{ISBN|978-0-19-954617-6}}.<br />
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*Greven, A., Keller, G., Warnecke (editors) (2003). ''Entropy'', Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[Edward A. Guggenheim|Guggenheim, E.A.]] (1949). 'Statistical basis of thermodynamics', ''Research'', '''2''': 450–454.<br />
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*[[Edward A. Guggenheim|Guggenheim, E.A.]] (1967). ''Thermodynamics. An Advanced Treatment for Chemists and Physicists'', fifth revised edition, North Holland, Amsterdam.<br />
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*Gyarmati, I. (1967/1970) ''Non-equilibrium Thermodynamics. Field Theory and Variational Principles'', translated by E. Gyarmati and W.F. Heinz, Springer, New York.<br />
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*[[Charles Kittel|Kittel, C.]], [[Herbert Kroemer|Kroemer, H.]] (1969/1980). ''Thermal Physics'', second edition, Freeman, San Francisco CA, {{ISBN|0-7167-1088-9}}.<br />
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*Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998). ''Modern Thermodynamics: From Heat Engines to Dissipative Structures'', John Wiley & Sons, Chichester, {{ISBN|0-471-97393-9}}.<br />
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*Lebon, G., Jou, D., Casas-Vázquez, J. (2008). ''Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers'', Springer-Verlag, Berlin, {{ISBN|978-3-540-74252-4}}.<br />
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*{{cite journal|ref=harv|title=The Physics and Mathematics of the Second Law of Thermodynamics|last1=Lieb |first1=E. H. |last2=Yngvason |first2=J.|journal=Physics Reports|volume=310|issue=1|pages=1–96 |year=1999|doi=10.1016/S0370-1573(98)00082-9|arxiv = cond-mat/9708200 |bibcode = 1999PhR...310....1L}}<br />
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*Lieb, E.H., Yngvason, J. (2003). The Entropy of Classical Thermodynamics, pp.&nbsp;147–195, Chapter 8 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*{{cite book|ref=harv|first=F.|last=Mandl|title=Statistical physics|edition=second|year=1988|publisher=[[Wiley & Sons]]|isbn=978-0-471-91533-1}}<br />
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*{{cite journal | last1 = Maxwell | first1 = J.C. | authorlink = James Clerk Maxwell | year = 1867 | title = On the dynamical theory of gases | url = | journal = Phil. Trans. R. Soc. Lond. | volume = 157 | issue = | pages = 49–88 | doi = 10.1098/rstl.1867.0004 }}<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985). ''Thermodynamics'', Pitman, London, {{ISBN|0-273-08577-8}}.<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003). Entropy in Nonequilibrium, pp.&nbsp;79–109, Chapter 5 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*Münster, A. (1970), ''Classical Thermodynamics'', translated by E.S. Halberstadt, Wiley–Interscience, London, {{ISBN|0-471-62430-6}}.<br />
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*[[Brian Pippard|Pippard, A.B.]] (1957/1966). ''Elements of Classical Thermodynamics for Advanced Students of Physics'', original publication 1957, reprint 1966, Cambridge University Press, Cambridge UK.<br />
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*[[Max Planck|Planck, M.]] (1897/1903). [https://archive.org/stream/treatiseonthermo00planrich#page/100/mode/2up ''Treatise on Thermodynamics'', translated by A. Ogg, Longmans Green, London, p. 100.]<br />
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*[[Max Planck|Planck. M.]] (1914). [https://archive.org/details/theoryofheatradi00planrich ''The Theory of Heat Radiation''], a translation by Masius, M. of the second German edition, P. Blakiston's Son & Co., Philadelphia.<br />
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*[[Max Planck|Planck, M.]] (1926). Über die Begründung des zweiten Hauptsatzes der Thermodynamik, ''Sitzungsberichte der Preussischen Akademie der Wissenschaften: Physikalisch-mathematische Klasse'': 453–463.<br />
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* Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769-781.<br />
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* {{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}<br />
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*Quinn, T.J. (1983). ''Temperature'', Academic Press, London, {{ISBN|0-12-569680-9}}.<br />
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*{{cite book|ref=harv|first=Y.V.C.|last=Rao|title=An Introduction to thermodynamics|url=https://books.google.com/books?id=iYWiCXziWsEC&pg=PA213|date=2004|publisher=Universities Press|isbn=978-81-7371-461-0|page=213}}<br />
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*Roberts, J.K., Miller, A.R. (1928/1960). ''Heat and Thermodynamics'', (first edition 1928), fifth edition, Blackie & Son Limited, Glasgow.<br />
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*[[Erwin Schrödinger|Schrödinger, E.]] (1950). Irreversibility, ''Proc. R. Ir. Acad.'', '''A53''': 189–195.<br />
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*[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966). ''Elements of Thermodynamics'', Addison-Wesley Publishing, Reading MA.<br />
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*{{cite journal|ref=harv|last=Thomson|first=W.|author-link=William Thomson, 1st Baron Kelvin|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Transactions of the Royal Society of Edinburgh|year=1851|volume=XX|issue=part II|pages=261–268; 289–298|url=https://www.biodiversitylibrary.org/item/126047#page/295/mode/1up}} Also published in {{cite journal|last=Thomson|first=W.|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Philos. Mag. |date=December 1852 |volume=IV |series=4 |issue=22 |page=13|url=https://archive.org/stream/londonedinburghp04maga#page/12/mode/2up |accessdate=25 June 2012}}<br />
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*[[William Thomson, 1st Baron Kelvin|Thomson, W.]] (1852). ''On the universal tendency in nature to the dissipation of mechanical energy'' Philosophical Magazine, Ser. 4, p.&nbsp;304.<br />
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*[[László Tisza|Tisza, L.]] (1966). ''Generalized Thermodynamics'', M.I.T Press, Cambridge MA.<br />
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*[[Clifford Truesdell|Truesdell, C.]] (1980). ''The Tragicomical History of Thermodynamics 1822–1854'', Springer, New York, {{ISBN|0-387-90403-4}}.<br />
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*Uffink, J. (2001). Bluff your way in the second law of thermodynamics, ''Stud. Hist. Phil. Mod. Phys.'', '''32'''(3): 305–394.<br />
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*Uffink, J. (2003). Irreversibility and the Second Law of Thermodynamics, Chapter 7 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003), Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963). ''Lectures in Statistical Mechanics'', American Mathematical Society, Providence RI.<br />
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*[[Mark Zemansky|Zemansky, M.W.]] (1968). ''Heat and Thermodynamics. An Intermediate Textbook'', fifth edition, McGraw-Hill Book Company, New York.<br />
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==Further reading==<br />
<br />
==Further reading==<br />
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进一步阅读<br />
<br />
*Goldstein, Martin, and Inge F., 1993. ''The Refrigerator and the Universe''. Harvard Univ. Press. Chpts. 4–9 contain an introduction to the Second Law, one a bit less technical than this entry. {{ISBN|978-0-674-75324-2}}<br />
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*Leff, Harvey S., and Rex, Andrew F. (eds.) 2003. ''Maxwell's Demon 2 : Entropy, classical and quantum information, computing''. Bristol UK; Philadelphia PA: [[Institute of Physics]]. {{ISBN|978-0-585-49237-7}}<br />
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*{{Cite book | first = J.J. | last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1}}(technical).<br />
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*{{cite book |title=Reflections on the Motive Power of Heat and on Machines Fitted to Develop That Power |last=Carnot |first=Sadi |editor=[[Robert Henry Thurston|Thurston, Robert Henry]] |year=1890 |publisher=J. Wiley & Sons |location=New York |isbn= |pages= }} ([https://books.google.com/books?id=tgdJAAAAIAAJ full text of 1897 ed.]) ([http://www.history.rochester.edu/steam/carnot/1943/ html])<br />
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*Stephen Jay Kline (1999). ''The Low-Down on Entropy and Interpretive Thermodynamics'', La Cañada, CA: DCW Industries. {{ISBN|1-928729-01-0}}.<br />
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*{{cite book | last1 = Kostic | first1 = M | year = 2011 | title = Revisiting The Second Law of Energy Degradation and Entropy Generation: From Sadi Carnot's Ingenious Reasoning to Holistic Generalization | journal = AIP Conf. Proc. | volume = 1411 | issue = 1| pages = 327–350 | doi = 10.1063/1.3665247 | isbn = 978-0-7354-0985-9 | bibcode = 2011AIPC.1411..327K | series = AIP Conference Proceedings | citeseerx = 10.1.1.405.1945 }} also at [https://web.archive.org/web/20130420222450/http://www.kostic.niu.edu/2ndLaw/Revisiting%20The%20Second%20Law%20of%20Energy%20Degradation%20and%20Entropy%20Generation%20-%20From%20Carnot%20to%20Holistic%20Generalization-4.pdf].<br />
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==External links==<br />
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*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics]" – by Lawrence Sklar.<br />
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*[http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node30.html ''Second law of thermodynamics''] in the MIT Course [http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/notes.html ''Unified Thermodynamics and Propulsion''] from Prof. Z. S. Spakovszky<br />
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*[[E.T. Jaynes]], 1988, "[http://bayes.wustl.edu/etj/articles/ccarnot.pdf The evolution of Carnot's principle]," in G. J. Erickson and C. R. Smith (eds.)''Maximum-Entropy and Bayesian Methods in Science and Engineering, Vol'' 1: p.&nbsp;267.<br />
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*[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf Caratheodory, C., "Examination of the foundations of thermodynamics," trans. by D. H. Delphenich]<br />
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* [https://www.bbc.co.uk/programmes/p004y2bm The Second Law of Thermodynamics], BBC Radio 4 discussion with John Gribbin, Peter Atkins & Monica Grady (''In Our Time'', Dec. 16, 2004)<br />
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* [http://mdpi.org/entropy/papers/e6010001.pdf [[Entropy (journal)]], 2004]<br />
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* [https://www.journals.uchicago.edu/doi/abs/10.1086/663835 The Journal of the International Society for the History of Philosophy of Science, 2012]<br />
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类别: 热力学和统计物理学哲学<br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E7%A4%BE%E4%BC%9A%E7%BD%91%E7%BB%9C&diff=12522社会网络2020-08-14T04:14:59Z<p>Dorr：</p>
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此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Social structure made up of a set of social actors}}<br />
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{{about|the theoretical concept as used in the social and behavioral sciences|social networking sites|Social networking service|the 2010 movie|The Social Network|other uses|Social network (disambiguation)}}<br />
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{{Network Science}}<br />
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A '''social network ''' is a [[social structure]] made up of a set of [[social]] actors (such as individuals or organizations), sets of [[Dyad (sociology)|dyadic]] ties, and other [[Social relation|social interactions]] between actors. The social network perspective provides a set of methods for analyzing the structure of whole social entities as well as a variety of theories explaining the patterns observed in these structures.<ref name="WF94CH1">{{cite book|last1=Wasserman |first1=Stanley | author-link1= Stanley Wasserman|last2=Faust |first2=Katherine |year=1994 |title=Social Network Analysis: Methods and Applications |isbn=9780521387071 |chapter=Social Network Analysis in the Social and Behavioral Sciences |pages=1–27 |publisher=Cambridge University Press}}</ref> The study of these structures uses [[social network analysis]] to identify local and global patterns, locate influential entities, and examine network dynamics.<br />
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A social network is a social structure made up of a set of social actors (such as individuals or organizations), sets of dyadic ties, and other social interactions between actors. The social network perspective provides a set of methods for analyzing the structure of whole social entities as well as a variety of theories explaining the patterns observed in these structures. The study of these structures uses social network analysis to identify local and global patterns, locate influential entities, and examine network dynamics.<br />
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'''<font color="#ff8000">社会网络 Social Network</font>'''是一种由一组社会行为者（个人或组织）、一组二元关系以及行为者之间的其他社会互动组成的社会结构。社会网络视角为分析整个社会实体的结构提供了一套方法，也为解释在这些结构中观察到的模式提供了各种理论。对这些结构的研究使用'''<font color="#ff8000">社会网络分析 Social Network Analysis</font>'''以确定本地及全球模式，定位有影响力的实体，并审查网络动态。<br />
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Social networks and the analysis of them is an inherently [[Interdisciplinarity|interdisciplinary]] academic field which emerged from [[social psychology]], [[sociology]], [[statistics]], and [[graph theory]]. [[Georg Simmel]] authored early structural theories in sociology emphasizing the dynamics of triads and "web of group affiliations".<ref>{{cite book|last1=Scott |first1=W. Richard |last2=Davis |first2=Gerald F. |author-link1= William Richard Scott|author-link2= Gerald F. Davis|title=Organizations and Organizing |chapter=Networks In and Around Organizations |year=2003 |isbn=978-0-13-195893-7 |publisher=Pearson Prentice Hall}}</ref> [[Jacob Moreno]] is credited with developing the first [[sociogram]]s in the 1930s to study interpersonal relationships. These approaches were mathematically formalized in the 1950s and theories and methods of social networks became pervasive in the [[Social science|social and behavioral sciences]] by the 1980s.<ref name="WF94CH1"/><ref name="Freeman History">{{cite book|last=Freeman |first=Linton |year=2004 |publisher=Empirical Press |isbn=978-1-59457-714-7 |title=The Development of Social Network Analysis: A Study in the Sociology of Science}}</ref> [[Social network analysis]] is now one of the major paradigms in contemporary sociology, and is also employed in a number of other social and formal sciences. Together with other [[complex network]]s, it forms part of the nascent field of [[network science]].<ref>{{cite journal|journal=Science |year=2009 |volume=323 |number=5916 |pages=892–895 |doi=10.1126/science.1165821 |pmid=19213908 |title=Network Analysis in the Social Sciences |first1=Stephen P. |last1=Borgatti |first2=Ajay |last2=Mehra |first3=Daniel J. |last3=Brass |first4=Giuseppe |last4=Labianca|bibcode=2009Sci...323..892B |citeseerx=10.1.1.536.5568 }}</ref><ref name="EK">{{cite book|title=Networks, Crowds, and Markets: Reasoning about a Highly Connected World |first1=David |last1=Easley |first2=Jon |last2=Kleinberg |publisher=Cambridge University Press |year=2010 |chapter=Overview |pages=1–20 |isbn=978-0-521-19533-1}}</ref><br />
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Social networks and the analysis of them is an inherently interdisciplinary academic field which emerged from social psychology, sociology, statistics, and graph theory. Georg Simmel authored early structural theories in sociology emphasizing the dynamics of triads and "web of group affiliations". Jacob Moreno is credited with developing the first sociograms in the 1930s to study interpersonal relationships. These approaches were mathematically formalized in the 1950s and theories and methods of social networks became pervasive in the social and behavioral sciences by the 1980s. Social network analysis is now one of the major paradigms in contemporary sociology, and is also employed in a number of other social and formal sciences. Together with other complex networks, it forms part of the nascent field of network science.<br />
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社会网络及其分析是社会心理学、社会学、统计学和图论中内在的跨学科学术领域。'''格奥尔格·齐美尔 Georg Simmel'''是早期社会学结构主义理论的作者，他强调三合会和“群体关系网”的动态性。'''雅各布·莫雷诺 Jacob Moreno'''被认为是在20世纪30年代发展了第一份'''<font color="#ff8000">社交关系图 Sociogram</font>'''以研究人际关系的人。这些方法在20世纪50年代得到数学化，到20世纪80年代，社会网络的理论和方法在社会和行为科学中变得普遍。社会网络分析现在是当代社会学的主要范式之一，也被用于许多其他社会科学及形式科学。它与其他复杂网络一起构成了网络科学新兴领域的一部分。<br />
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==Overview 概览==<br />
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[[File:Barabasi Albert model.gif|thumb|left|图1：Evolution graph of a social network: [[Barabási–Albert model|Barabási model]].一个社会网络的演化图：巴拉巴西模型。]]<br />
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Barabási model.]]<br />
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The social network is a [[Scientific theory|theoretical]] [[Construct (philosophy of science)|construct]] useful in the [[social sciences]] to study relationships between individuals, [[social groups|groups]], [[formal organizations|organizations]], or even entire [[society|societies]] ([[social unit]]s, see [[Differentiation (sociology)|differentiation]]). The term is used to describe a [[social structure]] determined by such [[social interactions|interactions]]. The ties through which any given social unit connects represent the convergence of the various social contacts of that unit. This theoretical approach is, necessarily, relational. An [[axiom]] of the social network approach to understanding [[social interaction]] is that social phenomena should be primarily conceived and investigated through the properties of relations between and within units, instead of the properties of these units themselves. Thus, one common criticism of social network theory is that [[Agency (sociology)|individual agency]] is often ignored<ref name="jscott">Scott, John P. (2000). ''Social Network Analysis: A Handbook'' (2nd edition). Thousand Oaks, CA: Sage Publications.</ref> although this may not be the case in practice (see [[agent-based model]]ing). Precisely because many different types of relations, singular or in combination, form these network configurations, [[Network science|network analytics]] are useful to a broad range of research enterprises. In social science, these fields of study include, but are not limited to [[anthropology]], [[biology]], [[communication studies]], [[economics]], [[geography]], [[information science]], [[organizational studies]], [[social psychology]], [[sociology]], and [[sociolinguistics]].<br />
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The social network is a theoretical construct useful in the social sciences to study relationships between individuals, groups, organizations, or even entire societies (social units, see differentiation). The term is used to describe a social structure determined by such interactions. The ties through which any given social unit connects represent the convergence of the various social contacts of that unit. This theoretical approach is, necessarily, relational. An axiom of the social network approach to understanding social interaction is that social phenomena should be primarily conceived and investigated through the properties of relations between and within units, instead of the properties of these units themselves. Thus, one common criticism of social network theory is that individual agency is often ignored although this may not be the case in practice (see agent-based modeling). Precisely because many different types of relations, singular or in combination, form these network configurations, network analytics are useful to a broad range of research enterprises. In social science, these fields of study include, but are not limited to anthropology, biology, communication studies, economics, geography, information science, organizational studies, social psychology, sociology, and sociolinguistics.<br />
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社会网络是社会科学中研究个人、团体、组织甚至整个社会（社会单位，见分化）之间关系的理论构造。这个术语用来描述由这种相互作用决定的社会结构。任何一个特定的社会单元之间的联系都代表着这个单元各种社会联系的聚合。这种理论方法必然是相关的。理解社会互动的社会网络方法的一个公理是，社会现象应该主要通过单元之间和单元内部关系的性质来构思和研究，而不是这些单元本身的性质。因此，社会网络理论总被诟病的一点是其常忽视个体代理，而实践中可能并非如此（见基于主体的建模）。正是因为许多不同类型的关系（单独或组合形式）形成这些网络配置，网络分析在广泛的研究中有用。在社会科学中，这些研究领域包括但不限于'''<font color="#ff8000">人类学 Anthropology</font>'''、生物学、'''<font color="#ff8000">传播学 Communication Studies</font>'''、经济学、地理学、信息科学、组织学、社会心理学、社会学和'''<font color="#ff8000">社会语言学 Sociolinguistics</font>'''。<br />
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==History 历史==<br />
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In the late 1890s, both [[Émile Durkheim]] and [[Ferdinand Tönnies]] foreshadowed the idea of social networks in their theories and research of [[social group]]s. Tönnies argued that social groups can exist as personal and direct social ties that either link individuals who share values and belief (''[[Gemeinschaft]]'', German, commonly translated as "[[community]]") or impersonal, formal, and instrumental social links (''[[Gesellschaft]]'', German, commonly translated as "[[society]]").<ref>Tönnies, Ferdinand (1887). ''Gemeinschaft und Gesellschaft'', Leipzig: Fues's Verlag. (Translated, 1957 by Charles Price Loomis as ''Community and Society'', East Lansing: Michigan State University Press.)</ref> Durkheim gave a non-individualistic explanation of social facts, arguing that social phenomena arise when interacting individuals constitute a reality that can no longer be accounted for in terms of the properties of individual actors.<ref>Durkheim, Emile (1893). ''De la division du travail social: étude sur l'organisation des sociétés supérieures'', Paris: F. Alcan. (Translated, 1964, by Lewis A. Coser as ''The Division of Labor in Society'', New York: Free Press.)</ref> [[Georg Simmel]], writing at the turn of the twentieth century, pointed to the nature of networks and the effect of network size on interaction and examined the likelihood of interaction in loosely knit networks rather than groups.<ref>Simmel, Georg (1908). ''Soziologie'', Leipzig: Duncker & Humblot.</ref><br />
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In the late 1890s, both Émile Durkheim and Ferdinand Tönnies foreshadowed the idea of social networks in their theories and research of social groups. Tönnies argued that social groups can exist as personal and direct social ties that either link individuals who share values and belief (Gemeinschaft, German, commonly translated as "community") or impersonal, formal, and instrumental social links (Gesellschaft, German, commonly translated as "society"). Durkheim gave a non-individualistic explanation of social facts, arguing that social phenomena arise when interacting individuals constitute a reality that can no longer be accounted for in terms of the properties of individual actors. Georg Simmel, writing at the turn of the twentieth century, pointed to the nature of networks and the effect of network size on interaction and examined the likelihood of interaction in loosely knit networks rather than groups.<br />
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在19世纪90年代晚期，'''埃米尔·涂尔干 Émile Durkheim'''和'''斐迪南·滕尼斯 Ferdinand Tönnies'''在他们关于社会群体的理论和研究中都预示了社会网络的概念。滕尼斯认为，社会群体可以作为个人和直接的社会关系存在，这种关系或者将具有共同价值观和信仰的个人（德语 Gemeinschaft，通常翻译为“社区”）联系在一起，或者将非个人的、正式的和工具性的社会关系（德语 Gesellschaft，通常翻译为“社会”）联系在一起。涂尔干对社会事实给出了非个人主义的解释，认为当相互作用的个体构成一种再也不能用个体行为者的特性来解释的现实时，社会现象就产生了。格奥尔格·齐美尔在20世纪之交的著作中指出了网络的本质和网络规模对互动的影响，并研究了在松散的网络而非群体中互动的可能性。<br />
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[[File:Moreno Sociogram 2nd Grade.png|thumb|图2：Moreno's sociogram of a 2nd grade class 莫雷诺对一个二年级班级的社会关系图]]<br />
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Major developments in the field can be seen in the 1930s by several groups in psychology, anthropology, and mathematics working independently.<ref name=jscott /><ref>For a historical overview of the development of social network analysis, see: {{cite book|last1=Carrington|first1=Peter J.|last2=Scott|first2=John|chapter=Introduction|title=The Sage Handbook of Social Network Analysis| publisher=Sage|year=2011|isbn=978-1-84787-395-8|page=1|chapter-url=https://books.google.com/books?id=2chSmLzClXgC&pg=PA1}}</ref><ref>See also the diagram in {{cite book|author=Scott, John|title=Social Network Analysis: A Handbook|publisher=Sage|year=2000|isbn=978-0-7619-6339-4|page=8|url=https://books.google.com/books?id=Ww3_bKcz6kgC&pg=PA8}}</ref> In [[psychology]], in the 1930s, [[Jacob L. Moreno]] began systematic recording and analysis of social interaction in small groups, especially classrooms and work groups (see [[sociometry]]). In [[anthropology]], the foundation for social network theory is the theoretical and [[ethnography|ethnographic]] work of [[Bronislaw Malinowski]],<ref>Malinowski, Bronislaw (1913). ''The Family Among the Australian Aborigines: A Sociological Study''. London: University of London Press.</ref> [[Radcliffe-Brown|Alfred Radcliffe-Brown]],<ref>Radcliffe-Brown, Alfred Reginald (1930) ''The social organization of Australian tribes''. Sydney, Australia: University of Sydney ''Oceania'' monographs, No.1.</ref><ref>{{cite journal | last1 = Radcliffe-Brown | first1 = A. R. | year = 1940 | title = On social structure | url = | journal = Journal of the Royal Anthropological Institute | volume = 70 | issue = 1| pages = 1–12 | doi=10.2307/2844197| jstor = 2844197 }}</ref> and [[Claude Lévi-Strauss]].<ref>Lévi-Strauss, Claude ([1947]1967). ''Les structures élémentaires de la parenté''. Paris: La Haye, Mouton et Co. (Translated, 1969 by J. H. Bell, J. R. von Sturmer, and R. Needham, 1969, as ''The Elementary Structures of Kinship'', Boston: Beacon Press.)</ref> A group of social anthropologists associated with [[Max Gluckman]] and the [[Manchester school (anthropology)|Manchester School]], including [[John Arundel Barnes|John A. Barnes]],<ref>Barnes, John (1954). "Class and Committees in a Norwegian Island Parish". ''Human Relations'', (7): 39–58.</ref> [[J. Clyde Mitchell]] and [[Elizabeth Bott Spillius]],<ref>{{cite journal | last1 = Freeman | first1 = Linton C. | last2 = Wellman | first2 = Barry | year = 1995 | title = A note on the ancestoral Toronto home of social network analysis | url = | journal = Connections | volume = 18 | issue = 2| pages = 15–19 }}</ref><ref>{{cite journal | last1 = Savage | first1 = Mike | year = 2008 | title = Elizabeth Bott and the formation of modern British sociology | url = | journal = The Sociological Review | volume = 56 | issue = 4| pages = 579–605 | doi=10.1111/j.1467-954x.2008.00806.x}}</ref> often are credited with performing some of the first fieldwork from which network analyses were performed, investigating community networks in southern Africa, India and the United Kingdom.<ref name=jscott /> Concomitantly, British anthropologist [[Siegfried Frederick Nadel|S. F. Nadel]] codified a theory of social structure that was influential in later network analysis.<ref>Nadel, S. F. 1957. ''The Theory of Social Structure''. London: Cohen and West.</ref> In [[sociology]], the early (1930s) work of [[Talcott Parsons]] set the stage for taking a relational approach to understanding social structure.<ref>Parsons, Talcott ([1937] 1949). ''The Structure of Social Action: A Study in Social Theory with Special Reference to a Group of European Writers''. New York: The Free Press.</ref><ref>Parsons, Talcott (1951). ''The Social System''. New York: The Free Press.</ref> Later, drawing upon Parsons' theory, the work of sociologist [[Peter Blau]] provides a strong impetus for analyzing the relational ties of social units with his work on [[social exchange theory]].<ref>Blau, Peter (1956). ''Bureaucracy in Modern Society''. New York: Random House, Inc.</ref><ref>Blau, Peter (1960). "A Theory of Social Integration". ''The American Journal of Sociology'', (65)6: 545–556, (May).</ref><ref>Blau, Peter (1964). ''Exchange and Power in Social Life''.</ref><br />
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Major developments in the field can be seen in the 1930s by several groups in psychology, anthropology, and mathematics working independently. In psychology, in the 1930s, Jacob L. Moreno began systematic recording and analysis of social interaction in small groups, especially classrooms and work groups (see sociometry). In anthropology, the foundation for social network theory is the theoretical and ethnographic work of Bronislaw Malinowski, Alfred Radcliffe-Brown, and Claude Lévi-Strauss. A group of social anthropologists associated with Max Gluckman and the Manchester School, including John A. Barnes, J. Clyde Mitchell and Elizabeth Bott Spillius, often are credited with performing some of the first fieldwork from which network analyses were performed, investigating community networks in southern Africa, India and the United Kingdom. In sociology, the early (1930s) work of Talcott Parsons set the stage for taking a relational approach to understanding social structure. Later, drawing upon Parsons' theory, the work of sociologist Peter Blau provides a strong impetus for analyzing the relational ties of social units with his work on social exchange theory.<br />
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20世纪30年代，心理学、人类学和数学领域的几个独立研究小组已经看到了这一领域的重大发展。在心理学方面，在20世纪30年代，雅各布·L·莫雷诺开始系统地记录和分析小团体中的社会互动，尤其是课堂和工作团体中的社会互动（见'''<font color="#ff8000">社会测量 Sociometry）</font>'''。在人类学中，社会网络理论的基础是'''布罗尼斯拉夫·马林诺夫斯基 Bronislaw Malinowski'''，'''阿尔弗雷德·拉德克利夫-布朗 Alfred Radcliffe-Brown'''和'''克洛德·列维-斯特劳斯 Claude Lévi-Strauss'''的理论和人种学著作。包括'''约翰·A·巴恩斯 John A. Barnes'''、'''J·克莱德·米切尔 J. Clyde Mitchell'''和'''伊丽莎白·博特·斯皮利厄斯 Elizabeth Bott Spillius'''在内的一群与'''马克斯·格拉克曼 Max Gluckman'''和'''曼彻斯特学派 Manchester School'''有关的社会人类学家，经常被认为是执行了一些最初的实地工作，从而进行了网络分析，调查了南非、印度和英国的社区网络。在社会学方面，'''塔尔科特·帕森斯 Talcott Parsons'''的早期工作（1930年代）为采用关系方法理解社会结构奠定了基础。后来，社会学家'''彼得·布劳 Peter Blau'''的'''<font color="#ff8000">社会交换论 Social Exchange Theory</font>'''为分析社会单位之间的关系提供了强大的动力。<br />
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By the 1970s, a growing number of scholars worked to combine the different tracks and traditions. One group consisted of sociologist [[Harrison White]] and his students at the [[Harvard Department of Social Relations|Harvard University Department of Social Relations]]. Also independently active in the Harvard Social Relations department at the time were [[Charles Tilly]], who focused on networks in political and community sociology and social movements, and [[Stanley Milgram]], who developed the "six degrees of separation" thesis.<ref>{{cite web |url=http://www.semioticon.com/semiotix/semiotix14/sem-14-05.html |title=The Networked Individual: A Profile of Barry Wellman |author=Bernie Hogan}}</ref> [[Mark Granovetter]]<ref name="Introduction for the French Reader">{{cite journal | last1 = Granovetter | first1 = Mark | year = 2007 | title = Introduction for the French Reader | url = | journal = Sociologica | volume = 2 | issue = | pages = 1–8 }}</ref> and [[Barry Wellman]]<ref>Wellman, Barry (1988). "Structural analysis: From method and metaphor to theory and substance". pp. 19–61 in B. Wellman and S. D. Berkowitz (eds.) ''Social Structures: A Network Approach'', Cambridge, UK: Cambridge University Press.</ref> are among the former students of White who elaborated and championed the analysis of social networks.<ref name="Introduction for the French Reader"/><ref>Mullins, Nicholas. ''Theories and Theory Groups in Contemporary American Sociology''. New York: Harper and Row, 1973.</ref><ref>Tilly, Charles, ed. ''An Urban World''. Boston: Little Brown, 1974.</ref><ref>Wellman, Barry. 1988. "Structural Analysis: From Method and Metaphor to Theory and Substance". pp. 19–61 in ''Social Structures: A Network Approach'', edited by Barry Wellman and S. D. Berkowitz. Cambridge: Cambridge University Press.</ref><br />
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By the 1970s, a growing number of scholars worked to combine the different tracks and traditions. One group consisted of sociologist Harrison White and his students at the Harvard University Department of Social Relations. Also independently active in the Harvard Social Relations department at the time were Charles Tilly, who focused on networks in political and community sociology and social movements, and Stanley Milgram, who developed the "six degrees of separation" thesis. Mark Granovetter and Barry Wellman are among the former students of White who elaborated and championed the analysis of social networks.<br />
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到了20世纪70年代，越来越多的学者致力于将不同的轨迹和传统结合起来。其中一组由社会学家'''哈里森·怀特 Harrison White'''和他在哈佛大学社会关系系的学生组成。当时在哈佛大学社会关系系独立活动的还有专注于政治和社区社会学和社会运动的网络的'''查尔斯·堤利 Charles Tilly'''，还有发表了六度分隔理论论文的'''斯坦利·米尔格拉姆 Stanley Milgram'''。'''马克·格兰诺维特 Mark Granovetter'''和'''巴里·威尔曼 Barry Wellman'''是怀特以前的学生，他们阐述并支持对社交网络的分析。<br />
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Beginning in the late 1990s, social network analysis experienced work by sociologists, political scientists, and physicists such as [[Duncan J. Watts]], [[Albert-László Barabási]], [[Peter Bearman]], [[Nicholas A. Christakis]], [[James H. Fowler]], and others, developing and applying new models and methods to emerging data available about online social networks, as well as "digital traces" regarding face-to-face networks.<br />
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Beginning in the late 1990s, social network analysis experienced work by sociologists, political scientists, and physicists such as Duncan J. Watts, Albert-László Barabási, Peter Bearman, Nicholas A. Christakis, James H. Fowler, and others, developing and applying new models and methods to emerging data available about online social networks, as well as "digital traces" regarding face-to-face networks.<br />
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从20世纪90年代末开始，社会网络分析经历了社会学家、政治学家和物理学家的工作，如'''[[邓肯·瓦茨 Duncan J. Watts]]'''、'''[[艾伯特-拉斯洛·巴拉巴西 Albert-László Barabási]]'''、Peter Bearman、Nicholas A. Christakis、James H. Fowler等人，开发和应用新的模型和方法来获得有关在线社会网络的新兴数据，以及有关面对面网络的“数字痕迹”。<br />
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==Levels of analysis 分析层次==<br />
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[[File:Network self-organization stages.png|thumb|right|图3：Self-organization of a network, based on Nagler, Levina, & Timme, (2011) 基于Nagler, Levina和Timme的自组织网络，(2011)<ref>{{cite journal|author1=Nagler, Jan|author2=Anna Levina|author3=Marc Timme|year=2011|title=Impact of single links in competitive percolation|journal=Nature Physics|volume=7|issue=3|pages=265–270|doi=10.1038/nphys1860|arxiv=1103.0922|bibcode=2011NatPh...7..265N}}</ref>]]<br />
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[[File:Social Network Diagram (large).svg|right|thumb|Centrality]]<br />
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Centrality<br />
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中心性<br />
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In general, social networks are [[self-organization|self-organizing]], [[emergence|emergent]], and [[social complexity|complex]], such that a globally coherent pattern appears from the local interaction of the elements that make up the system.<ref>Newman, Mark, Albert-László Barabási and Duncan J. Watts (2006). ''The Structure and Dynamics of Networks'' (Princeton Studies in Complexity). Oxford: Princeton University Press.</ref><ref>{{cite journal | last1 = Wellman | first1 = Barry | year = 2008 | title = Review: The development of social network analysis: A study in the sociology of science | url = | journal = Contemporary Sociology | volume = 37 | issue = 3| pages = 221–222 | doi=10.1177/009430610803700308}}</ref> These patterns become more apparent as network size increases. However, a global network analysis<ref>{{cite book|last=Faust|first=Stanley Wasserman; Katherine|title=Social network analysis : methods and applications|year=1998|publisher=Cambridge Univ. Press|location=Cambridge [u.a.]|isbn=978-0521382694|edition=Reprint.}}</ref> of, for example, all [[interpersonal relationships]] in the world is not feasible and is likely to contain so much [[Information theory|information]] as to be uninformative. Practical limitations of computing power, ethics and participant recruitment and payment also limit the scope of a social network analysis.<ref name="Kadu12">Kadushin, C. (2012). ''Understanding social networks: Theories, concepts, and findings''. Oxford: Oxford University Press.</ref><ref>{{Cite journal| author=Granovetter, M.|title= Network sampling: Some first steps| year=1976 |pages=1287–1303 |volume=81| issue=6 |journal= American Journal of Sociology | doi=10.1086/226224}}</ref> The nuances of a local system may be lost in a large network analysis, hence the quality of information may be more important than its scale for understanding network properties. Thus, social networks are analyzed at the scale relevant to the researcher's theoretical question. Although [[level of analysis|levels of analysis]] are not necessarily [[Mutually exclusive events|mutually exclusive]], there are three general levels into which networks may fall: [[Microsociology|micro-level]], [[wikt:meso-|meso-level]], and [[Macrosociology|macro-level]].<br />
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In general, social networks are self-organizing, emergent, and complex, such that a globally coherent pattern appears from the local interaction of the elements that make up the system. These patterns become more apparent as network size increases. However, a global network analysis of, for example, all interpersonal relationships in the world is not feasible and is likely to contain so much information as to be uninformative. Practical limitations of computing power, ethics and participant recruitment and payment also limit the scope of a social network analysis. The nuances of a local system may be lost in a large network analysis, hence the quality of information may be more important than its scale for understanding network properties. Thus, social networks are analyzed at the scale relevant to the researcher's theoretical question. Although levels of analysis are not necessarily mutually exclusive, there are three general levels into which networks may fall: micro-level, meso-level, and macro-level.<br />
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一般来说，社会网络是自组织的、涌现的和复杂的，这样，一个全局一致的模式就会从组成系统的元素的局部交互中显现出来。随着网络规模的增大，这些模式变得更加明显。然而，一个全球网络分析（如世界上所有的人际关系）是不可行的，它可能包含太多的信息，以至于相当于没有提供信息。计算能力的实际限制、道德规范以及参与者的招聘和报酬也限制了社会网络分析的规模。局部系统的细微差别在大规模网络分析中可能会消失，因此对于理解网络属性来说，信息的质量可能比其规模更重要。因此，社会网络是在与研究者的理论问题相关的尺度上加以分析的。虽然分析层次不一定相互排斥，但网络可以分为三个一般层次: 微观、中观和宏观。<br />
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===Micro level 微观层面===<br />
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At the micro-level, social network research typically begins with an individual, [[Snowball sampling|snowballing]] as social relationships are traced, or may begin with a small group of individuals in a particular social context.<br />
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At the micro-level, social network research typically begins with an individual, snowballing as social relationships are traced, or may begin with a small group of individuals in a particular social context.<br />
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在微观层面上，社会网络研究通常从个人开始，随着社会关系的追踪而像滚雪球一样扩大，或者可能从特定社会背景下的一小群个体开始。<br />
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'''Dyadic level''': A [[Dyad (sociology)|dyad]] is a social relationship between two individuals. Network research on dyads may concentrate on [[Structural functionalism|structure]] of the relationship (e.g. multiplexity, strength), [[social equality]], and tendencies toward [[Reciprocity (social and political philosophy)|reciprocity/mutuality]].<br />
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Dyadic level: A dyad is a social relationship between two individuals. Network research on dyads may concentrate on structure of the relationship (e.g. multiplexity, strength), social equality, and tendencies toward reciprocity/mutuality.<br />
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二元层面：二元是两个个体之间的社会关系。网络对二元关系的研究可以集中在关系的结构上（如多样性、力量)、社会平等以及互惠互利的倾向。<br />
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'''Triadic level''': Add one individual to a dyad, and you have a [[Triadic relation|triad]]. Research at this level may concentrate on factors such as [[Independence number|balance]] and [[Vertex-transitive graph|transitivity]], as well as [[social equality]] and tendencies toward [[Reciprocity (social and political philosophy)|reciprocity/mutuality]].<ref name="Kadu12"/> In the [[balance theory]] of [[Fritz Heider]] the triad is the key to social dynamics. The discord in a rivalrous [[love triangle]] is an example of an unbalanced triad, likely to change to a balanced triad by a change in one of the relations. The dynamics of social friendships in society has been modeled by balancing triads. The study is carried forward with the theory of [[signed graph]]s.<br />
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Triadic level: Add one individual to a dyad, and you have a triad. Research at this level may concentrate on factors such as balance and transitivity, as well as social equality and tendencies toward reciprocity/mutuality. In the balance theory of Fritz Heider the triad is the key to social dynamics. The discord in a rivalrous love triangle is an example of an unbalanced triad, likely to change to a balanced triad by a change in one of the relations. The dynamics of social friendships in society has been modeled by balancing triads. The study is carried forward with the theory of signed graphs.<br />
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三元层面：再加一个个体到二元组中，就得到一个三元组。这一层次的研究可能集中在诸如平衡和传递性等因素，以及社会平等和互惠互利的倾向。在'''弗里茨 · 海德 Fritz Heider'''的'''平衡理论 Balance Theory'''中，三元组是社会动力学的关键。在敌对的三角恋中的不和谐是不平衡的三角关系的一个例子，很可能通过其中一种关系的改变而变成平衡的三角关系。社会中社会友谊的动态模型是通过平衡三角关系建立起来的。该研究利用了符号图理论。<br />
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'''Actor level''': The smallest unit of analysis in a social network is an individual in their social setting, i.e., an "actor" or "ego". Egonetwork analysis focuses on network characteristics such as size, relationship strength, density, [[centrality]], [[wikt:prestige|prestige]] and roles such as [[isolates|isolates, liaisons]], and [[Bridge (interpersonal)|bridges]].<ref name="Jone11"/> Such analyses, are most commonly used in the fields of [[psychology]] or [[Social psychology (sociology)|social psychology]], [[ethnographic]] [[kinship]] analysis or other [[genealogy|genealogical]] studies of relationships between individuals.<br />
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Actor level: The smallest unit of analysis in a social network is an individual in their social setting, i.e., an "actor" or "ego". Egonetwork analysis focuses on network characteristics such as size, relationship strength, density, centrality, prestige and roles such as isolates, liaisons, and bridges. Such analyses, are most commonly used in the fields of psychology or social psychology, ethnographic kinship analysis or other genealogical studies of relationships between individuals.<br />
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行为者层面：社会网络中最小的分析单位是社会环境中的个体，即“行为者”或“自我”。自我网络分析主要关注网络特征，例如大小、关系强度、密度、中心性、声望和隔离、联络和桥梁等角色。这种分析最常用于心理学或社会心理学、人种学亲属关系分析或其他个体关系的系谱研究领域。<br />
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'''Subset level''': [[Subset]] levels of network research problems begin at the micro-level, but may cross over into the meso-level of analysis. Subset level research may focus on [[Distance (graph theory)|distance]] and reachability, [[cliques]], [[Cohesion (social policy)|cohesive]] subgroups, or other [[Group action (sociology)|group actions]] or [[Group behaviour|behavior]].<ref>{{cite book | title="Graph Theoretical Approaches to Social Network Analysis". in Computational Complexity: Theory, Techniques, and Applications (Robert A. Meyers, ed.) | publisher=Springer | author=de Nooy, Wouter |year=2012 | pages=2864–2877 | isbn=978-1-4614-1800-9|doi=10.1007/978-1-4614-1800-9_176| chapter=Social Network Analysis, Graph Theoretical Approaches to }}</ref><br />
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Subset level: Subset levels of network research problems begin at the micro-level, but may cross over into the meso-level of analysis. Subset level research may focus on distance and reachability, cliques, cohesive subgroups, or other group actions or behavior.<br />
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子集层面：网络研究问题的子集级别开始于微观级别，但可能跨越到中观级别的分析。子集级别的研究可能集中在距离和可达性、派系、凝聚子群或其他群体行为或行为。<br />
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===Meso level 中观层面===<br />
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In general, meso-level theories begin with a [[Sample population|population]] size that falls between the micro- and macro-levels. However, meso-level may also refer to analyses that are specifically designed to reveal connections between micro- and macro-levels. Meso-level networks are low density and may exhibit causal processes distinct from interpersonal micro-level networks.<ref>{{cite journal | last1 = Hedström | first1 = Peter | last2 = Sandell | first2 = Rickard | last3 = Stern | first3 = Charlotta | year = 2000 | title = Mesolevel Networks and the Diffusion of Social Movements: The Case of the Swedish Social Democratic Party | url = http://www.nuffield.ox.ac.uk/users/hedstrom/ajs3.pdf | journal = American Journal of Sociology | volume = 106 | issue = 1| pages = 145–172 | doi = 10.1086/303109 }}</ref><br />
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In general, meso-level theories begin with a population size that falls between the micro- and macro-levels. However, meso-level may also refer to analyses that are specifically designed to reveal connections between micro- and macro-levels. Meso-level networks are low density and may exhibit causal processes distinct from interpersonal micro-level networks.<br />
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一般来说，中观层面的理论始于介于微观和宏观层面之间的规模。然而，中观层面也可以指专门为揭示微观和宏观层面之间的联系而设计的分析。中观层次的网络是密度低，可能表现出不同于人际微观层面网络的因果过程。<br />
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[[File:Social Red.jpg|thumb|right|图4：Social network diagram, meso-level 中观层面社会网络图]]<br />
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'''Organizations''': Formal [[organizations]] are [[social group]]s that distribute tasks for a collective [[goal]].<ref name="Riketta, M. 2007">{{cite journal | last1 = Riketta | first1 = M. | last2 = Nienber | first2 = S. | year = 2007 | title = Multiple identities and work motivation: The role of perceived compatibility between nested organizational units | url = | journal = British Journal of Management | volume = 18 | issue = | pages = S61–77 | doi=10.1111/j.1467-8551.2007.00526.x}}</ref> Network research on organizations may focus on either intra-organizational or inter-organizational ties in terms of [[Formal organization|formal]] or [[Informal organization|informal]] relationships. Intra-organizational networks themselves often contain multiple levels of analysis, especially in larger organizations with multiple branches, franchises or semi-autonomous departments. In these cases, research is often conducted at a work group level and organization level, focusing on the interplay between the two structures.<ref name="Riketta, M. 2007"/> Experiments with networked groups online have documented ways to optimize group-level coordination through diverse interventions, including the addition of autonomous agents to the groups.<ref>{{Cite journal|last=Shirado|first=Hirokazu|last2=Christakis|first2=Nicholas A|title=Locally noisy autonomous agents improve global human coordination in network experiments|journal=Nature|volume=545|issue=7654|pages=370–374|doi=10.1038/nature22332|pmid=28516927|pmc=5912653|bibcode=2017Natur.545..370S|year=2017}}</ref><br />
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Organizations: Formal organizations are social groups that distribute tasks for a collective goal. Network research on organizations may focus on either intra-organizational or inter-organizational ties in terms of formal or informal relationships. Intra-organizational networks themselves often contain multiple levels of analysis, especially in larger organizations with multiple branches, franchises or semi-autonomous departments. In these cases, research is often conducted at a work group level and organization level, focusing on the interplay between the two structures.<br />
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组织: 正规组织是为共同目标分配任务的社会团体。关于组织的网络研究可以侧重于正式或非正式关系方面的组织内或组织间联系。组织内网络本身往往包含多层次的分析，特别是在具有多个分支机构、特许权或半自治部门的较大组织中。在这些情况下，研究通常在工作组和组织层面进行，重点放在两个结构之间的相互作用。<br />
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'''Randomly distributed networks''': [[Exponential random graph models]] of social networks became state-of-the-art methods of social network analysis in the 1980s. This framework has the capacity to represent social-structural effects commonly observed in many human social networks, including general [[Degree (graph theory)|degree]]-based structural effects commonly observed in many human social networks as well as [[Reciprocity (social and political philosophy)|reciprocity]] and [[Transitive set|transitivity]], and at the node-level, [[homophily]] and [[Attribute-value system|attribute]]-based activity and popularity effects, as derived from explicit hypotheses about [[Dependency graph|dependencies]] among network ties. [[Parameter]]s are given in terms of the prevalence of small [[Induced subgraph|subgraph]] configurations in the network and can be interpreted as describing the combinations of local social processes from which a given network emerges. These probability models for networks on a given set of actors allow generalization beyond the restrictive dyadic independence assumption of micro-networks, allowing models to be built from theoretical structural foundations of social behavior.<ref>{{cite journal | last1 = Cranmer | first1 = Skyler J. | last2 = Desmarais | first2 = Bruce A. | year = 2011 | title = Inferential Network Analysis with Exponential Random Graph Models | url = | journal = Political Analysis | volume = 19 | issue = 1| pages = 66–86 | doi=10.1093/pan/mpq037| citeseerx = 10.1.1.623.751 }}</ref><br />
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Randomly distributed networks: Exponential random graph models of social networks became state-of-the-art methods of social network analysis in the 1980s. This framework has the capacity to represent social-structural effects commonly observed in many human social networks, including general degree-based structural effects commonly observed in many human social networks as well as reciprocity and transitivity, and at the node-level, homophily and attribute-based activity and popularity effects, as derived from explicit hypotheses about dependencies among network ties. Parameters are given in terms of the prevalence of small subgraph configurations in the network and can be interpreted as describing the combinations of local social processes from which a given network emerges. These probability models for networks on a given set of actors allow generalization beyond the restrictive dyadic independence assumption of micro-networks, allowing models to be built from theoretical structural foundations of social behavior.<br />
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随机分布的网络: 社会网络的'''<font color="#ff8000">指数随机图模型 Exponential random graph models</font>'''在20世纪80年代成为最先进的社会网络分析方法。这个框架有能力表示在许多人类社会网络中普遍观察到的社会结构效应，包括在许多人类社会网络中普遍观察到的基于程度的一般性结构效应以及互惠性和传递性，以及在节点一级、同相性和基于属性的活动和流行性效应，这些效应源于关于网络关系之间依赖性的明确假设。参数是根据网络中小型子图配置的流行程度给出的，可以解释为描述一个给定网络出现的局部社会过程的组合。这些网络的概率模型在给定的参与者集合上允许超越微型网络的限制性并元独立性假设的泛化，允许模型从社会行为的理论结构基础上建立。<br />
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[[File:Scale-free network sample.png|thumb|right|图5：Examples of a random network and a scale-free network. Each graph has 32 nodes and 32 links. Note the "hubs" (shaded) in the scale-free diagram (on the right).一个随机网络和一个无尺度网络的例子。每个图有32个顶点和32条边。注意无标度图中的“集线器”（阴影部分）（右侧）。]]<br />
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'''Scale-free networks''': A [[scale-free network]] is a [[complex network|network]] whose [[degree distribution]] follows a [[power law]], at least [[asymptotic]]ally. In [[network theory]] a scale-free ideal network is a [[random network]] with a [[degree distribution]] that unravels the size distribution of social groups.<ref>{{cite journal |author1=Moreira, André A.|author2=Demétrius R. Paula|author3=Raimundo N. Costa Filho|author4=José S. Andrade, Jr. |title=Competitive cluster growth in complex networks |year=2006 |doi=10.1103/PhysRevE.73.065101 |pmid=16906890 |journal=Physical Review E |volume=73 |issue=6 |pages=065101 |arxiv=cond-mat/0603272 |bibcode=2006PhRvE..73f5101M }}</ref> Specific characteristics of scale-free networks vary with the theories and analytical tools used to create them, however, in general, scale-free networks have some common characteristics. One notable characteristic in a scale-free network is the relative commonness of [[Vertex (graph theory)|vertices]] with a [[Maximum degree|degree]] that greatly exceeds the average. The highest-degree nodes are often called "hubs", and may serve specific purposes in their networks, although this depends greatly on the social context. Another general characteristic of scale-free networks is the [[clustering coefficient]] distribution, which decreases as the node degree increases. This distribution also follows a [[power law]].<ref>Barabási, Albert-László (2003). ''Linked: how everything is connected to everything else and what it means for business, science, and everyday life''. New York: Plum.</ref> The [[Barabási–Albert model|Barabási]] model of network evolution shown above is an example of a scale-free network.<br />
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Scale-free networks: A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. In network theory a scale-free ideal network is a random network with a degree distribution that unravels the size distribution of social groups. Specific characteristics of scale-free networks vary with the theories and analytical tools used to create them, however, in general, scale-free networks have some common characteristics. One notable characteristic in a scale-free network is the relative commonness of vertices with a degree that greatly exceeds the average. The highest-degree nodes are often called "hubs", and may serve specific purposes in their networks, although this depends greatly on the social context. Another general characteristic of scale-free networks is the clustering coefficient distribution, which decreases as the node degree increases. This distribution also follows a power law. The Barabási model of network evolution shown above is an example of a scale-free network.<br />
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'''<font color="#ff8000">[[无标度网络]] A Scale-free Network</font>''': 无标度网络网络是一个'''<font color="#ff8000">度分布 Degree Distribution</font>'''遵循'''<font color="#ff8000">幂律 Power Law</font>'''(至少是渐近的）的网络。在网络理论中，无标度理想网络是具有揭示了社会群体的规模分布的度分布的随机网络。无标度网络的具体特征随创建其的理论和分析工具的变化而变化，然而，一般来说，无标度网络具有一些共同的特征。无标度网络的一个显著特征是度远超均值的顶点的相对共性。最高度的节点通常被称为“'''<font color="#ff8000">枢纽 Hubs</font>'''” ，并且可能在其网络中服务于特定的目的，尽管这在很大程度上取决于社会环境。无标度网络的另一个一般特性是'''<font color="#ff8000">集聚系数 General Characteristic</font>'''分布，它随着节点度的增加而减少。该分布也遵循幂律。上面网络演化的巴拉巴西模型就是无标度网络的一个例子。<br />
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===Macro level 宏观层面===<br />
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Rather than tracing interpersonal interactions, macro-level analyses generally trace the outcomes of interactions, such as [[economic]] or other [[resource]] [[Transfer function|transfer]] interactions over a large [[Sample population|population]].<br />
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Rather than tracing interpersonal interactions, macro-level analyses generally trace the outcomes of interactions, such as economic or other resource transfer interactions over a large population.<br />
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宏观层面的分析不是追踪人际互动，而通常是追踪相互作用的结果，例如经济或其他资源转移在一大群体中的相互作用。<br />
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[[File:Diagram of a social network.jpg|thumb|right|图6: Diagram: section of a large-scale social network 一大型社会网络局部]]<br />
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'''Large-scale networks''': [[Large-scale macroeconometric model|Large-scale network]] is a term somewhat synonymous with "macro-level" as used, primarily, in [[social science|social]] and [[Behavioural sciences|behavioral]] sciences, in [[economics]]. Originally, the term was used extensively in the [[computer sciences]] (see [[Network mapping#Large-scale mapping project|large-scale network mapping]]).<br />
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Large-scale networks: Large-scale network is a term somewhat synonymous with "macro-level" as used, primarily, in social and behavioral sciences, in economics. Originally, the term was used extensively in the computer sciences (see large-scale network mapping).<br />
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大规模网络: 大规模网络是一个与“宏观层面”同义的术语，主要用于社会和行为科学，经济学。最初，这个术语在计算机科学中广泛使用（见大规模网络映射）。<br />
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'''Complex networks''': Most larger social networks display features of [[social complexity]], which involves substantial non-trivial features of [[network topology]], with patterns of complex connections between elements that are neither purely regular nor purely random (see, [[complexity science]], [[dynamical system]] and [[chaos theory]]), as do [[biological]], and [[Computer network|technological networks]]. Such [[complex network]] features include a heavy tail in the [[degree distribution]], a high [[clustering coefficient]], [[assortativity]] or disassortativity among vertices, [[community structure]] (see [[stochastic block model]]), and [[hierarchy|hierarchical structure]]. In the case of [[Agency (philosophy)|agency-directed]] networks these features also include [[Reciprocity in network|reciprocity]], triad significance profile (TSP, see [[network motif]]), and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as [[lattice graph|lattices]] and [[random graph]]s, do not show these features.<ref>{{cite journal|author=Strogatz, Steven H.|year=2001|title=Exploring complex networks|journal=Nature|volume=410|pages=268–276|doi=10.1038/35065725|pmid=11258382|issue=6825|bibcode=2001Natur.410..268S|doi-access=free}}</ref><br />
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Complex networks: Most larger social networks display features of social complexity, which involves substantial non-trivial features of network topology, with patterns of complex connections between elements that are neither purely regular nor purely random (see, complexity science, dynamical system and chaos theory), as do biological, and technological networks. Such complex network features include a heavy tail in the degree distribution, a high clustering coefficient, assortativity or disassortativity among vertices, community structure (see stochastic block model), and hierarchical structure. In the case of agency-directed networks these features also include reciprocity, triad significance profile (TSP, see network motif), and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as lattices and random graphs, do not show these features.<br />
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复杂网络: 大多数较大的社会网络呈现出社会复杂性的特征，包括'''<font color="#ff8000">网络拓扑 Network Topology</font>'''的大量非平凡特征，以及既不完全规则也不完全随机的元素之间的复杂连接模式（见[[复杂性科学]]、[[动力系统]]和[[混沌理论]]），生物和技术网络也是如此。这些复杂的网络特征包括度分布的重尾、高集聚系数、顶点之间的'''<font color="#ff8000">同配性 Assortativity</font>'''或非同配性、社区结构（见'''<font color="#ff8000">随机分块模型 Stochastic Block Model</font>'''）和层次结构。在主体导向网络的情况下，这些特征还包括互惠性、'''<font color="#32CD32">三重显著性特征</font>'''（TSP，见网络基序）及其他。相比之下，许多过去研究过的网络数学模型，如格和'''<font color="#ff8000">随机图 Random Graph</font>'''，并没有表现出这些特征。<br />
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==Theoretical links 理论联系==<br />
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===Imported theories 输入的理论===<br />
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Various theoretical frameworks have been imported for the use of social network analysis. The most prominent of these are [[Graph theory]], [[Balance theory]], Social comparison theory, and more recently, the [[Social identity approach]].<ref>{{cite book |last1=Kilduff|first1=M.|last2=Tsai|first2=W. |year=2003 |title= Social networks and organisations |publisher=Sage Publications}}</ref><br />
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Various theoretical frameworks have been imported for the use of social network analysis. The most prominent of these are Graph theory, Balance theory, Social comparison theory, and more recently, the Social identity approach.<br />
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为了使用社会网络分析，已经引入了各种理论框架。其中最突出的是[[图论]]、'''<font color="#ff8000">平衡理论 Balance theory</font>'''、社会比较论，以及最近的社会认同方法。<br />
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===Indigenous theories 本土理论===<br />
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Few complete theories have been produced from social network analysis. Two that have are [[role theory|structural role theory]] and [[heterophily|heterophily theory]].<br />
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Few complete theories have been produced from social network analysis. Two that have are structural role theory and heterophily theory.<br />
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很少有完整的理论产生于社会网络分析。现有两个为结构角色理论和异质性理论。<br />
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The basis of Heterophily Theory was the finding in one study that more numerous weak ties can be important in seeking information and innovation, as cliques have a tendency to have more homogeneous opinions as well as share many common traits. This homophilic tendency was the reason for the members of the cliques to be attracted together in the first place. However, being similar, each member of the clique would also know more or less what the other members knew. To find new information or insights, members of the clique will have to look beyond the clique to its other friends and acquaintances. This is what Granovetter called "the strength of weak ties".<ref>{{Cite journal | author=Granovetter, M. |year=1973 |title= The strength of weak ties |journal= American Journal of Sociology |volume=78 |issue=6 |pages= 1360–1380 |doi=10.1086/225469}}</ref><br />
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The basis of Heterophily Theory was the finding in one study that more numerous weak ties can be important in seeking information and innovation, as cliques have a tendency to have more homogeneous opinions as well as share many common traits. This homophilic tendency was the reason for the members of the cliques to be attracted together in the first place. However, being similar, each member of the clique would also know more or less what the other members knew. To find new information or insights, members of the clique will have to look beyond the clique to its other friends and acquaintances. This is what Granovetter called "the strength of weak ties".<br />
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异质性理论的基础是在一项研究中发现，更多的'''<font color="#ff8000">弱关系 Weak Tie</font>'''可以在寻求信息和创新方面发挥重要作用，因为小圈子倾向于有更同质化的观点，也有许多共同特征。这种亲同性倾向是小圈子成员被吸引到一起的首要原因。然而，由于相似，小圈子里的每一个成员或多或少都知道其他成员所知道的事情。为了获得新的信息或见解，小圈子里的成员不得不超越该圈子，关注其他的朋友及熟人。这就是格兰诺维特所说的“弱关系的力量”。<br />
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==Structural holes 结构洞==<br />
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In the context of networks, [[social capital]] exists where people have an advantage because of their location in a network. Contacts in a network provide information, opportunities and perspectives that can be beneficial to the central player in the network. Most social structures tend to be characterized by dense clusters of strong connections.<ref name="Burt 2004">{{cite journal|last=Burt|first=Ronald|title=Structural Holes and Good Ideas|journal=American Journal of Sociology|year=2004|doi=10.1086/421787|volume=110|issue=2|pages=349–399|citeseerx=10.1.1.388.2251}}</ref> Information within these clusters tends to be rather homogeneous and redundant. Non-redundant information is most often obtained through contacts in different clusters.<ref name="Burt 1992">{{cite book|last=Burt|first=Ronald|title=Structural Holes: The Social Structure of Competition|year=1992|publisher=Harvard University Press|location=Cambridge, MA}}</ref> When two separate clusters possess non-redundant information, there is said to be a structural hole between them.<ref name="Burt 1992"/> Thus, a network that bridges [[structural holes]] will provide network benefits that are in some degree additive, rather than overlapping. An ideal network structure has a vine and cluster structure, providing access to many different clusters and structural holes.<ref name="Burt 1992"/><br />
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In the context of networks, social capital exists where people have an advantage because of their location in a network. Contacts in a network provide information, opportunities and perspectives that can be beneficial to the central player in the network. Most social structures tend to be characterized by dense clusters of strong connections. Information within these clusters tends to be rather homogeneous and redundant. Non-redundant information is most often obtained through contacts in different clusters. When two separate clusters possess non-redundant information, there is said to be a structural hole between them. Thus, a network that bridges structural holes will provide network benefits that are in some degree additive, rather than overlapping. An ideal network structure has a vine and cluster structure, providing access to many different clusters and structural holes.<br />
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在网络的背景下，'''<font color="#ff8000">社会资本 Social Capital</font>'''存在于人们因其在网络中的位置而具有优势的地方。网络中的联系提供的信息、机会和观点可能有利于网络中的核心参与者。大多数社会结构往往以有强连接的密集集群为特征。这些集群中的信息往往是相当同质和冗余的。非冗余信息通常是通过不同集群中的联系获得的。当两个独立的集群拥有非冗余信息时，我们称它们之间存在一个'''<font color="#ff8000">结构洞 Structural Hole</font>'''。因此，一个'''<font color="#32CD32">连接</font>'''结构孔的网络在某种程度上提供附加而非重叠的的网络效益。理想的网络结构具有蔓生结构和集群结构，可访问许多不同集群和结构洞。<br />
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Networks rich in structural holes are a form of social capital in that they offer [[Information theory|information]] benefits. The main player in a network that bridges structural holes is able to access information from diverse sources and clusters.<ref name="Burt 1992"/> For example, in [[Business networking|business networks]], this is beneficial to an individual's career because he is more likely to hear of job openings and opportunities if his network spans a wide range of contacts in different industries/sectors. This concept is similar to Mark Granovetter's theory of [[Interpersonal ties|weak ties]], which rests on the basis that having a broad range of contacts is most effective for job attainment.<br />
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Networks rich in structural holes are a form of social capital in that they offer information benefits. The main player in a network that bridges structural holes is able to access information from diverse sources and clusters. For example, in business networks, this is beneficial to an individual's career because he is more likely to hear of job openings and opportunities if his network spans a wide range of contacts in different industries/sectors. This concept is similar to Mark Granovetter's theory of weak ties, which rests on the basis that having a broad range of contacts is most effective for job attainment.<br />
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富含结构洞的网络是社会资本的一种形式，因为它们提供信息利益。连接结构洞的网络中的主要参与者能够访问来自不同来源和集群的信息。例如，在'''<font color="#ff8000">商业社交 Business networking</font>'''中，这对个人的职业生涯是有益的，因为若其关系网涵盖不同行业 / 部门，则更有可能得知职位空缺和机会。这个概念类似于马克·格兰诺维特的弱关系理论，该理论的基础是拥有广泛的联系对于获得工作是最有效的。<br />
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==Research clusters 研究集群==<br />
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===Communication 传播学===<br />
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[[Communication Studies]] are often considered a part of both the social sciences and the humanities, drawing heavily on fields such as [[sociology]], [[psychology]], [[anthropology]], [[information science]], [[biology]], [[political science]], and [[economics]] as well as [[rhetoric]], [[literary studies]], and [[semiotics]]. Many communication concepts describe the transfer of information from one source to another, and can thus be conceived of in terms of a network.<br />
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Communication Studies are often considered a part of both the social sciences and the humanities, drawing heavily on fields such as sociology, psychology, anthropology, information science, biology, political science, and economics as well as rhetoric, literary studies, and semiotics. Many communication concepts describe the transfer of information from one source to another, and can thus be conceived of in terms of a network.<br />
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传播学通常被认为是社会科学和人文科学的一部分，主要集中在社会学、心理学、人类学、信息科学、生物学、政治学、经济学以及修辞学、文学研究和符号学等领域。许多通信概念描述了从一个源到另一个源的信息传输，因此可以从网络的角度来考虑。<br />
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===Community 社区===<br />
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In J.A. Barnes' day, a "[[community]]" referred to a specific geographic location and studies of community ties had to do with who talked, associated, traded, and attended church with whom. Today, however, there are extended "online" communities developed through [[telecommunications]] devices and [[social network services]]. Such devices and services require extensive and ongoing maintenance and analysis, often using [[network science]] methods. [[Community development]] studies, today, also make extensive use of such methods.<br />
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In J.A. Barnes' day, a "community" referred to a specific geographic location and studies of community ties had to do with who talked, associated, traded, and attended church with whom. Today, however, there are extended "online" communities developed through telecommunications devices and social network services. Such devices and services require extensive and ongoing maintenance and analysis, often using network science methods. Community development studies, today, also make extensive use of such methods.<br />
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在J.A. Barnes' day，“社区”指的是一个特定的地理位置和与谁交谈、联系、贸易和一同上教堂做礼拜有关的社区关系的研究。然而如今，通过电信设备和社交网络服务，有了扩展的“在线”社区。这样的设备和服务需要广泛和持续的维护和分析，通常使用网络科学方法。社区发展研究在如今也广泛使用这些方法。<br />
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===Complex networks 复杂网络===<br />
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[[Complex networks]] require methods specific to modelling and interpreting [[social complexity]] and [[complex adaptive system]]s, including techniques of [[dynamic network analysis]].<br />
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Complex networks require methods specific to modelling and interpreting social complexity and complex adaptive systems, including techniques of dynamic network analysis.<br />
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复杂网络需要建模和解释社会复杂性和复杂适应系统的特定方法，包括动态网络分析技术。<br />
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Mechanisms such as [[Dual-phase evolution#social networks|Dual-phase evolution]] explain how temporal changes in connectivity contribute to the formation of structure in social networks.<br />
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Mechanisms such as Dual-phase evolution explain how temporal changes in connectivity contribute to the formation of structure in social networks.<br />
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'''<font color="#ff8000">双相演化理论 Dual-phase Evolution</font>'''等机制解释了连接性的时间变化如何促进社会网络结构的形成。<br />
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===Criminal networks 犯罪网络===<br />
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In [[criminology]] and [[urban sociology]], much attention has been paid to the social networks among criminal actors. For example, Andrew Papachristos<ref>{{cite journal |last=Papachristos |first=Andrew |year=2009 |title=Murder by Structure: Dominance Relations and the Social Structure of Gang Homicide |journal=American Journal of Sociology |volume=115 |issue=1 |pages=74–128 |doi=10.2139/ssrn.855304 |url=http://www.papachristos.org/Publications_2_files/ajs_final_version.pdf |accessdate=29 March 2013 |url-status=dead |archiveurl=https://web.archive.org/web/20140407094725/http://www.papachristos.org/Publications_2_files/ajs_final_version.pdf |archivedate=7 April 2014 }}</ref> has studied gang murders as a series of exchanges between gangs. Murders can be seen to diffuse outwards from a single source, because weaker gangs cannot afford to kill members of stronger gangs in retaliation, but must commit other violent acts to maintain their reputation for strength.<br />
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In criminology and urban sociology, much attention has been paid to the social networks among criminal actors. For example, Andrew Papachristos has studied gang murders as a series of exchanges between gangs. Murders can be seen to diffuse outwards from a single source, because weaker gangs cannot afford to kill members of stronger gangs in retaliation, but must commit other violent acts to maintain their reputation for strength.<br />
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在'''<font color="#ff8000">犯罪学 Criminology</font>和'''<font color="#ff8000">城市社会学 Urban sociology</font>中，犯罪行为人之间的社会网络问题受到了广泛的关注。例如，'''安德鲁·帕帕克里斯托斯 Andrew Papachristos'''将帮派谋杀研究为帮派之间的一系列交流。谋杀可以视为从单一来源向外扩散，因为较弱的帮派无法承担为报复杀死较强帮派成员的代价，而必须采取其他暴力行动来维护其势力强大之名声。<br />
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===Diffusion of innovations 创新扩散理论===<br />
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[[Diffusion of innovations|Diffusion of ideas and innovations]] studies focus on the spread and use of ideas from one actor to another or one [[culture]] and another. This line of research seeks to explain why some become "early adopters" of ideas and innovations, and links social network structure with facilitating or impeding the spread of an innovation.<br />
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Diffusion of ideas and innovations studies focus on the spread and use of ideas from one actor to another or one culture and another. This line of research seeks to explain why some become "early adopters" of ideas and innovations, and links social network structure with facilitating or impeding the spread of an innovation.<br />
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'''<font color="#ff8000">创新扩散理论 Diffusion of Innovations</font>'''研究的重点是思想从一个行动者到另一个或一种文化到另一种的传播。这一系列的研究试图解释为什么有些人成为创意和创新的“早期接受者” ，并将社交网络结构与促进或阻碍创新的传播联系起来。<br />
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===Demography 人口学===<br />
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In [[demography]], the study of social networks has led to new sampling methods for estimating and reaching populations that are hard to enumerate (for example, homeless people or intravenous drug users.) For example, respondent driven sampling is a network-based sampling technique that relies on respondents to a survey recommending further respondents.<br />
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In demography, the study of social networks has led to new sampling methods for estimating and reaching populations that are hard to enumerate (for example, homeless people or intravenous drug users.) For example, respondent driven sampling is a network-based sampling technique that relies on respondents to a survey recommending further respondents.<br />
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在'''<font color="#ff8000">人口学 Demography</font>'''方面，对社会网络的研究导致了新的抽样方法，用于估计和覆盖难以统计的人群（如无家可归者或静脉注射毒品者）。例如，受访者驱动的抽样依赖于调查的受访者推荐更多的受访者，是一种基于网络的抽样技术。<br />
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===Economic sociology 经济社会学===<br />
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The field of [[sociology]] focuses almost entirely on networks of outcomes of social interactions. More narrowly, [[economic sociology]] considers behavioral interactions of individuals and groups through [[social capital]] and social "markets". Sociologists, such as Mark Granovetter, have developed core principles about the interactions of social structure, information, ability to punish or reward, and trust that frequently recur in their analyses of political, economic and other institutions. Granovetter examines how social structures and social networks can affect economic outcomes like hiring, price, productivity and innovation and describes sociologists' contributions to analyzing the impact of social structure and networks on the economy.<ref>{{cite journal |last=Granovetter |first=Mark |year=2005 |title=The Impact of Social Structure on Economic Outcomes |journal=The Journal of Economic Perspectives |jstor=4134991 |volume=19 |issue=1 |pages=33–50|doi=10.1257/0895330053147958 }}</ref><br />
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The field of sociology focuses almost entirely on networks of outcomes of social interactions. More narrowly, economic sociology considers behavioral interactions of individuals and groups through social capital and social "markets". Sociologists, such as Mark Granovetter, have developed core principles about the interactions of social structure, information, ability to punish or reward, and trust that frequently recur in their analyses of political, economic and other institutions. Granovetter examines how social structures and social networks can affect economic outcomes like hiring, price, productivity and innovation and describes sociologists' contributions to analyzing the impact of social structure and networks on the economy.<br />
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社会学领域几乎完全关注社会互动的结果网络。更狭义地说，'''<font color="#ff8000">经济社会学 Economic Sociology</font>'''通过社会资本和社会“市场”考虑个人和群体的行为互动。社会学家，如马克·格兰诺维特，已经研究出关于社会结构、信息、奖惩能力和信任相互作用的核心原则，这些原则在他们对政治、经济和其他制度的分析中经常出现。格兰诺维特研究了社会结构和社会网络如何影响经济结果，如雇佣、价格、生产力和创新，并描述了社会学家对分析社会结构和网络对经济的影响的贡献。<br />
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===Health care 卫生保健===<br />
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Analysis of social networks is increasingly incorporated into [[health care analytics]], not only in [[epidemiology|epidemiological]] studies but also in models of [[Health Communication|patient communication]] and education, disease prevention, mental health diagnosis and treatment, and in the study of health care organizations and [[health care systems|systems]].<ref>Levy, Judith and Bernice Pescosolido (2002). ''Social Networks and Health''. Boston, MA: JAI Press.</ref><br />
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Analysis of social networks is increasingly incorporated into health care analytics, not only in epidemiological studies but also in models of patient communication and education, disease prevention, mental health diagnosis and treatment, and in the study of health care organizations and systems.<br />
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社会网络分析越来越多地被纳入卫生保健分析——不仅在'''<font color="#ff8000">流行病学 Epidemiology</font>'''研究中，而且在病人沟通和教育、疾病预防、心理健康诊断和治疗模型中，以及在卫生保健组织和系统的研究中。<br />
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===Human ecology 人类生态学===<br />
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[[Human ecology]] is an [[interdisciplinary]] and [[transdisciplinary]] study of the relationship between [[human]]s and their [[natural environment|natural]], [[Social environment|social]], and [[built environment]]s. The scientific philosophy of human ecology has a diffuse history with connections to [[geography]], [[sociology]], [[psychology]], [[anthropology]], [[zoology]], and natural [[ecology]].<ref>Crona, Beatrice and Klaus Hubacek (eds.) (2010). [http://www.ecologyandsociety.org/issues/view.php?sf=48 "Special Issue: Social network analysis in natural resource governance"]. ''Ecology and Society'', 48.</ref><ref>Ernstson, Henrich (2010). "Reading list: Using social network analysis (SNA) in social-ecological studies". [http://rs.resalliance.org/2010/11/03/reading-list-using-social-network-analysis-sna-in-social-ecological-studies/ ''Resilience Science'']</ref><br />
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Human ecology is an interdisciplinary and transdisciplinary study of the relationship between humans and their natural, social, and built environments. The scientific philosophy of human ecology has a diffuse history with connections to geography, sociology, psychology, anthropology, zoology, and natural ecology.<br />
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'''<font color="#ff8000">人类生态学 Human ecology</font>'''是研究人类与其自然环境、社会环境和'''<font color="#ff8000">建成环境 Built environment</font>'''之间关系的一门跨学科科学。人类生态学的科学哲学与地理学、社会学、心理学、人类学、动物学和自然生态学有着密切的联系。<br />
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===Language and linguistics 语言与语言学===<br />
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Studies of [[language]] and [[linguistics]], particularly [[evolutionary linguistics]], focus on the development of [[Morphology (linguistics)|linguistic forms]] and transfer of changes, [[Phonology|sounds]] or words, from one language system to another through networks of social interaction. Social networks are also important in [[language shift]], as groups of people add and/or abandon languages to their repertoire.<br />
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Studies of language and linguistics, particularly evolutionary linguistics, focus on the development of linguistic forms and transfer of changes, sounds or words, from one language system to another through networks of social interaction. Social networks are also important in language shift, as groups of people add and/or abandon languages to their repertoire.<br />
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语言学和语言学的研究，特别是'''<font color="#ff8000">演化语言学 Evolutionary Linguistics</font>'''，关注通过社会互动网络从一个语言系统转移到另一个语言系统时，语言形式的发展以及声音或词语的变化。社交网络在语言转换中也很重要，因为一些人群增加或者放弃了他们的语言。<br />
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===Literary networks 文学网络===<br />
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In the study of literary systems, network analysis has been applied by Anheier, Gerhards and Romo,<ref>{{cite journal | last1 = Anheier | first1 = H. K. | last2 = Romo | first2 = F. P. | year = 1995 | title = Forms of capital and social structure of fields: examining Bourdieu's social topography | url = | journal = American Journal of Sociology | volume = 100 | issue = 4| pages = 859–903 | doi=10.1086/230603}}</ref> De Nooy,<ref>{{cite journal | last1 = De Nooy | first1 = W | year = 2003| title = Fields and networks: Correspondence analysis and social network analysis in the framework of Field Theory | url = | journal = Poetics | volume = 31 | issue = 5–6| pages = 305–327 | doi = 10.1016/S0304-422X(03)00035-4 }}</ref> and Senekal,<ref>{{cite journal | last1 = Senekal | first1 = B. A. | year = 2012 | title = Die Afrikaanse literêre sisteem: ʼn Eksperimentele benadering met behulp van Sosiale-netwerk-analise (SNA) | url = | journal = LitNet Akademies | volume = 9 | issue = | page = 3 }}</ref> to study various aspects of how literature functions. The basic premise is that polysystem theory, which has been around since the writings of [[Even-Zohar]], can be integrated with network theory and the relationships between different actors in the literary network, e.g. writers, critics, publishers, literary histories, etc., can be mapped using [[Computer graphics (computer science)|visualization]] from SNA.<br />
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In the study of literary systems, network analysis has been applied by Anheier, Gerhards and Romo, De Nooy, and Senekal, to study various aspects of how literature functions. The basic premise is that polysystem theory, which has been around since the writings of Even-Zohar, can be integrated with network theory and the relationships between different actors in the literary network, e.g. writers, critics, publishers, literary histories, etc., can be mapped using visualization from SNA.<br />
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在文学体系的研究中，网络分析被'''阿海尔 Anheier'''、'''格尔哈兹 Gerhards'''和'''罗姆 Romo'''、'''努伊 De Nooy'''和Senekal应用于研究文学如何运作的各个方面。其基本前提是将'''文-佐哈尔 Even-Zohar'''著述以来就存在的多元系统理论可以与网络理论相结合，以及文学网络中不同行为者（如作家、评论家、出版商、文学史等）之间的关系可以通过SNA可视化映射。<br />
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===Organizational studies 组织研究===<br />
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Research studies of [[Formal organization|formal]] or [[informal organization]] [[Social relation|relationships]], [[organizational communication]], [[economics]], [[economic sociology]], and other [[resource]] [[Transfer function|transfers]]. Social networks have also been used to examine how organizations interact with each other, characterizing the many [[Interlocking directorate|informal connections]] that link executives together, as well as associations and connections between individual employees at different organizations.<ref>{{cite journal | last1 = Podolny | first1 = J. M. | last2 = Baron | first2 = J. N. | year = 1997 | title = Resources and relationships: Social networks and mobility in the workplace | url = | journal = American Sociological Review | volume = 62 | issue = 5| pages = 673–693 | doi=10.2307/2657354| jstor = 2657354 | citeseerx = 10.1.1.114.6822 }}</ref> Intra-organizational networks have been found to affect [[organizational commitment]],<ref>{{cite journal | last1 = Lee | first1 = J. | last2 = Kim | first2 = S. | year = 2011 | title = Exploring the role of social networks in affective organizational commitment: Network centrality, strength of ties, and structural holes | url = | journal = The American Review of Public Administration | volume = 41 | issue = 2| pages = 205–223 | doi=10.1177/0275074010373803}}</ref> [[organizational identification]],<ref name="Jone11">{{cite journal | last1 = Jones | first1 = C. | last2 = Volpe | first2 = E.H. | year = 2011 | title = Organizational identification: Extending our understanding of social identities through social networks | url = | journal = Journal of Organizational Behavior | volume = 32 | issue = 3| pages = 413–434 | doi=10.1002/job.694}}</ref> [[Organizational citizenship behavior|interpersonal citizenship behaviour]].<ref>{{cite journal | last1 = Bowler | first1 = W. M. | last2 = Brass | first2 = D. J. | year = 2011 | title = Relational correlates of interpersonal citizenship behaviour: A social network perspective | doi = 10.1037/0021-9010.91.1.70 | pmid = 16435939 | journal = Journal of Applied Psychology | volume = 91 | issue = 1| pages = 70–82 | citeseerx = 10.1.1.516.8746 }}</ref><br />
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Research studies of formal or informal organization relationships, organizational communication, economics, economic sociology, and other resource transfers. Social networks have also been used to examine how organizations interact with each other, characterizing the many informal connections that link executives together, as well as associations and connections between individual employees at different organizations. Intra-organizational networks have been found to affect organizational commitment, organizational identification, interpersonal citizenship behaviour.<br />
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研究正式或非正式组织关系、'''<font color="#ff8000">组织沟通 Organizational Communication</font>'''、经济、经济社会学和其他资源转移。社交网络也被用来研究组织之间如何相互作用，描述了许多将高管联系在一起的非正式联系，以及不同组织的个体雇员之间的联系。研究发现，组织内网络对'''<font color="#ff8000">组织承诺 Organizational Commitment</font>'''、组织认同、人际公民行为有影响。<br />
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===Social capital 社会资本===<br />
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[[Social capital]] is a form of [[Capital (economics)|economic]] and [[cultural capital]] in which social networks are central, [[Stock and flow|transactions]] are marked by [[Reciprocity (social psychology)|reciprocity]], [[Trust (social sciences)|trust]], and [[cooperation]], and [[Market (economics)|market]] [[Agent (economics)|agents]] produce [[goods and services]] not mainly for themselves, but for a [[common good]]. [[Social capital]] is split into three dimensions: the structural, the relational and the cognitive dimension. The structural dimension describes how partners interact with each other and which specific partners meet in a social network. Also The structural dimension of social capital indicates the level of ties among organizations.<ref>(Claridge, 2018).</ref>. This dimension is highly connected to the relational dimension which refers to trustworthiness, norms, expectations and idenfications of the bonds between partners.The relational dimension explains the nature of these ties which is mainly illustrated by the level of trust accorded to the network of organizations. <ref>(Claridge, 2018).</ref> The cognitive dimension analyses the extent to which organizations share common goals and objectives as a result of their ties and interactions. <ref>(Claridge, 2018).</ref><br />
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Social capital is a form of economic and cultural capital in which social networks are central, transactions are marked by reciprocity, trust, and cooperation, and market agents produce goods and services not mainly for themselves, but for a common good. Social capital is split into three dimensions: the structural, the relational and the cognitive dimension. The structural dimension describes how partners interact with each other and which specific partners meet in a social network. Also The structural dimension of social capital indicates the level of ties among organizations. This dimension is highly connected to the relational dimension which refers to trustworthiness, norms, expectations and idenfications of the bonds between partners. The relational dimension explains the nature of these ties which is mainly illustrated by the level of trust accorded to the network of organizations. The cognitive dimension analyses the extent to which organizations share common goals and objectives as a result of their ties and interactions. <br />
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社会资本是一种以社会网络为中心的经济和文化资本，交易以互惠、信任和合作为特征，市场主体生产的商品和服务主要不是为了自已，而是为了共同的利益。社会资本分为三个维度: 结构维度、关系维度和认知维度。结构维度描述了合作伙伴之间如何相互作用，以及哪些特定的合作伙伴在社交网络中相遇。社会资本的结构维度反映了组织之间的关系水平。这个维度与关系维度高度相关，关系维度指的是伙伴之间关系的可信度、规范、期望和认同度。 关系维度解释了这些联系的本质，主要表现在对组织网络的信任程度上。认知维度分析组织在多大程度上因其联系和相互作用而共享共同的目标和目的。<br />
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[[Social capital]] is a sociological concept about the value of [[social relation]]s and the role of cooperation and confidence to achieve positive outcomes. The term refers to the value one can get from their social ties. For example, newly arrived immigrants can make use of their social ties to established migrants to acquire jobs they may otherwise have trouble getting (e.g., because of unfamiliarity with the local language). A positive relationship exists between social capital and the intensity of social network use.<ref>{{cite journal|last=Sebastián|first=Valenzuela|author2=Namsu Park |author3=Kerk F. Kee |title=Is There Social Capital in a Social Network Site? Facebook Use and College Students' Life Satisfaction, Trust, and Participation|journal=Journal of Computer-Mediated Communication|year=2009|volume=14|issue=4|pages=875–901|doi=10.1111/j.1083-6101.2009.01474.x|doi-access=free}}</ref><ref>{{cite journal | title=Social Connectivity in America: Changes in Adult Friendship Network Size from 2002 to 2007 |author1=Wang, Hua |author2=Barry Wellman |lastauthoramp=yes | journal=American Behavioral Scientist | year=2010 | volume=53 | issue=8 | pages=1148–1169 | doi=10.1177/0002764209356247|url=https://semanticscholar.org/paper/402068f74efb35c093e7fd8c0f8268521219e242 }}</ref> In a dynamic framework, higher activity in a network feeds into higher social capital which itself encourages more activity.<ref name="GaudeulGiannetti2013">{{cite journal|last1=Gaudeul|first1=Alexia|last2=Giannetti|first2=Caterina|title=The role of reciprocation in social network formation, with an application to LiveJournal|journal=Social Networks|volume=35|issue=3|year=2013|pages=317–330|issn=03788733|doi=10.1016/j.socnet.2013.03.003}}</ref><br />
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Social capital is a sociological concept about the value of social relations and the role of cooperation and confidence to achieve positive outcomes. The term refers to the value one can get from their social ties. For example, newly arrived immigrants can make use of their social ties to established migrants to acquire jobs they may otherwise have trouble getting (e.g., because of unfamiliarity with the local language). A positive relationship exists between social capital and the intensity of social network use. In a dynamic framework, higher activity in a network feeds into higher social capital which itself encourages more activity.<br />
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社会资本是一个关于社会关系的价值、以及合作与信心对取得积极成果的作用的社会学概念。该术语指的是一个人可以从其社会关系中获得的价值。例如，新来的移民可以利用他们与先到的移民的社会关系，获得可能本难以获得的工作（如因为不熟悉当地语言）。社会资本与社会网络使用强度存在正相关关系。在一个动态的框架中，网络中更高的活动会产生更高的社会资本，而社会资本本身又会激励更多的活动。<br />
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=== Advertising 广告===<br />
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This particular cluster focuses on brand-image and promotional strategy effectiveness, taking into account the impact of customer participation on sales and brand-image. This is gauged through techniques such as sentiment analysis which rely on mathematical areas of study such as data mining and analytics. This area of research produces vast numbers of commercial applications as the main goal of any study is to understand consumer behaviour and drive sales.<br />
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This particular cluster focuses on brand-image and promotional strategy effectiveness, taking into account the impact of customer participation on sales and brand-image. This is gauged through techniques such as sentiment analysis which rely on mathematical areas of study such as data mining and analytics. This area of research produces vast numbers of commercial applications as the main goal of any study is to understand consumer behaviour and drive sales.<br />
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这个特别的集群侧重于品牌形象和促销策略的有效性，同时考虑到顾客参与对销售和品牌形象的影响。这是通过诸如基于数学领域的研究（如数据挖掘和分析）的情感分析等技术来衡量的。这一研究领域产生了大量的商业应用，因为任何研究的主要目标都是了解消费者行为并推动销售。<br />
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===Network position and benefits 网络地位和利益===<br />
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In many [[Formal organizations|organizations]], members tend to focus their activities inside their own groups, which stifles creativity and restricts opportunities. A player whose network bridges structural holes has an advantage in detecting and developing rewarding opportunities.<ref name="Burt 2004"/> Such a player can mobilize social capital by acting as a "broker" of information between two clusters that otherwise would not have been in contact, thus providing access to new ideas, opinions and opportunities. British philosopher and political economist [[John Stuart Mill]], writes, "it is hardly possible to overrate the value ... of placing human beings in contact with persons dissimilar to themselves.... Such communication [is] one of the primary sources of progress."<ref>{{cite book|last=Mill|first=John|title=Principles of Political Economy|year=1909|publisher=William J Ashley|location=Library of Economics and Liberty}}</ref> Thus, a player with a network rich in structural holes can add value to an organization through new ideas and opportunities. This in turn, helps an individual's career development and advancement.<br />
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In many organizations, members tend to focus their activities inside their own groups, which stifles creativity and restricts opportunities. A player whose network bridges structural holes has an advantage in detecting and developing rewarding opportunities. Thus, a player with a network rich in structural holes can add value to an organization through new ideas and opportunities. This in turn, helps an individual's career development and advancement.<br />
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在许多组织中，成员倾向于把他们的活动集中在他们自己的团队中，这抑制了创造力并限制了机会。一个网络连接结构洞的玩家在发现和开发机会方面有优势。因此，一个拥有富含结构洞的网络的玩家可以通过新的想法和机会为组织增加价值。这反过来又有助于个人的职业发展和晋升。<br />
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A social capital broker also reaps control benefits of being the facilitator of information flow between contacts. In the case of consulting firm Eden McCallum, the founders were able to advance their careers by bridging their connections with former big three consulting firm consultants and mid-size industry firms.<ref>{{cite journal|last=Gardner|first=Heidi|author2=Eccles, Robert |title=Eden McCallum: A Network Based Consulting Firm|journal=Harvard Business School Review|year=2011}}</ref> By bridging structural holes and mobilizing social capital, players can advance their careers by executing new opportunities between contacts.<br />
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A social capital broker also reaps control benefits of being the facilitator of information flow between contacts. In the case of consulting firm Eden McCallum, the founders were able to advance their careers by bridging their connections with former big three consulting firm consultants and mid-size industry firms. By bridging structural holes and mobilizing social capital, players can advance their careers by executing new opportunities between contacts.<br />
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社会资本中介人还可以从促进联系人之间的信息流动中获得控制权利。在咨询公司 Eden McCallum 的案例中，创始人能够通过与前三大咨询公司顾问和中等规模的行业公司建立联系来发展其事业。通过连接结构洞和调动社会资本，玩家可以通过在联系人之间执行新的机会来发展其事业。<br />
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There has been research that both substantiates and refutes the benefits of information brokerage. A study of high tech Chinese firms by Zhixing Xiao found that the control benefits of structural holes are "dissonant to the dominant firm-wide spirit of cooperation and the information benefits cannot materialize due to the communal sharing values" of such organizations.<ref>Xiao, Zhixing; Tsui, Anne (2007). "When Brokers May Not Work: The Cultural Contingency of [http://iniciarsesionentrar.com/facebook/ Social] Capital in Chinese High-tech Firms". ''Administrative Science Quarterly''.</ref> However, this study only analyzed Chinese firms, which tend to have strong communal sharing values. Information and control benefits of structural holes are still valuable in firms that are not quite as inclusive and cooperative on the firm-wide level. In 2004, Ronald Burt studied 673 managers who ran the supply chain for one of America's largest electronics companies. He found that managers who often discussed issues with other groups were better paid, received more positive job evaluations and were more likely to be promoted.<ref name="Burt 2004"/> Thus, bridging structural holes can be beneficial to an organization, and in turn, to an individual's career.<br />
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There has been research that both substantiates and refutes the benefits of information brokerage. A study of high tech Chinese firms by Zhixing Xiao found that the control benefits of structural holes are "dissonant to the dominant firm-wide spirit of cooperation and the information benefits cannot materialize due to the communal sharing values" of such organizations. However, this study only analyzed Chinese firms, which tend to have strong communal sharing values. Information and control benefits of structural holes are still valuable in firms that are not quite as inclusive and cooperative on the firm-wide level. In 2004, Ronald Burt studied 673 managers who ran the supply chain for one of America's largest electronics companies. He found that managers who often discussed issues with other groups were better paid, received more positive job evaluations and were more likely to be promoted. Thus, bridging structural holes can be beneficial to an organization, and in turn, to an individual's career.<br />
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有研究证实和反驳了信息中介的好处。Zhixing Xiao 对中国高科技企业的一项研究发现，结构洞的控制效益“与占主导地位的企业合作精神不协调，且因这类组织的共享价值观，信息效益不能实现”。然而，这项研究只分析了往往有很强的公共共享价值观的中国企业。结构洞的信息和控制利益对于那些在整个公司层面上不那么具有包容性和合作性的公司来说仍然是有价值的。2004年，'''罗纳德·伯特 Ronald Burt'''调查了为美国最大的电子公司之一管理供应链的673名管理者。他发现，那些经常与其他团体讨论问题的管理人员收入更高，得到的工作评价更积极，也更有可能获得晋升。因此，'''<font color="#32CD32">连接</font>'''结构洞对组织有益，反过来对个人的职业生涯也有益。<br />
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===Social media 社交媒体===<br />
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{{main article|Social media}}<br />
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[[Computer network]]s combined with social networking software produces a new medium for social interaction.<ref>{{Cite book|title=The International Encyclopedia of Media Effects|last=Amichai-Hamburger|first=Yair|last2=Hayat|first2=Tsahi|date=2017|publisher=John Wiley & Sons, Inc.|isbn=9781118783764|language=en|doi=10.1002/9781118783764.wbieme0170}}</ref> A relationship over a computerized [[social networking service]] can be characterized by context, direction, and strength. The content of a relation refers to the resource that is exchanged. In a [[computer mediated communication]] context, social pairs exchange different kinds of information, including sending a data file or a computer program as well as providing emotional support or arranging a meeting. With the rise of [[electronic commerce]], information exchanged may also correspond to exchanges of money, goods or services in the "real" world.<ref>{{cite journal | title=Studying Online Social Networks | last1=Garton|first1=Laura|first2=Caroline |last2=Haythornthwaite|author2-link=Caroline Haythornthwaite|first3=Barry|last3=Wellman|author3-link=Barry Wellman | journal=Journal of Computer-Mediated Communication |date=23 June 2006 | volume=3 | issue=1 |pages = 0| doi=10.1111/j.1083-6101.1997.tb00062.x}}</ref> [[Social network analysis]] methods have become essential to examining these types of computer mediated communication.<br />
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Computer networks combined with social networking software produces a new medium for social interaction. A relationship over a computerized social networking service can be characterized by context, direction, and strength. The content of a relation refers to the resource that is exchanged. In a computer mediated communication context, social pairs exchange different kinds of information, including sending a data file or a computer program as well as providing emotional support or arranging a meeting. With the rise of electronic commerce, information exchanged may also correspond to exchanges of money, goods or services in the "real" world. Social network analysis methods have become essential to examining these types of computer mediated communication.<br />
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计算机网络与社交网络软件的结合为社交互动提供了一种新的媒介。计算机化的社交网络服务上的关系可以通过上下文、方向和强度来描述。关系的内容指的是交换的资源。在'''<font color="#ff8000">电脑中介传播 Computer-mediated Communication</font>'''环境中，社会对交换不同种类的信息，包括发送数据文件或计算机程序，以及提供情感支持或安排会议。随着'''<font color="#ff8000">电子商务 Electronic Commerce</font>'''的兴起，交换的信息也可能对应于“现实”世界中的货币、商品或服务的交换。社会网络分析方法已经成为检验此类计算机为媒介的交流的必要手段。<br />
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In addition, the sheer size and the volatile nature of [[social media]] has given rise to new network metrics. A key concern with networks extracted from social media is the lack of robustness of network metrics given missing data.<ref>{{cite journal | last1 = Wei | first1 = Wei | last2 = Joseph | first2 = Kenneth | last3 = Liu | first3 = Huan | last4 = Carley | first4 = Kathleen M. | year = 2016 | title = Exploring Characteristics of Suspended Users and Network Stability on Twitter | url = | journal = Social Network Analysis and Mining | volume = 6 | issue = | page = 51 | doi = 10.1007/s13278-016-0358-5 }}</ref><br />
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In addition, the sheer size and the volatile nature of social media has given rise to new network metrics. A key concern with networks extracted from social media is the lack of robustness of network metrics given missing data.<br />
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此外，社交媒体的庞大规模和不稳定性也催生了新的网络指标。从社交媒体中提取网络的一个关键问题是，在缺失数据的情况下，网络指标缺乏鲁棒性。<br />
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=='''<font color="#32CD32">See also 参阅</font>'''==<br />
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* [[Bibliography of sociology]] 社会学参考书目<br />
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* [[Business networking]] 商业社交<br />
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* [[Collective network]] 集体网络<br />
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* [[International Network for Social Network Analysis]] 国际社会网络学会<br />
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* [[Network society]] 网络社会<br />
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* [[Network theory]] 网络理论<br />
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*[[Network science]] 网络科学<br />
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* [[Semiotics of social networking]] 社交网络符号学<br />
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* [[Scientific collaboration network]] 科学合作网络<br />
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* [[Social network analysis]] 社会网络分析<br />
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* [[Social network (sociolinguistics)]] 社会网络<br />
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* [[Social networking service]] 社会网络服务<br />
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* [[Social web]] 社交网络<br />
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* [[Structural fold]] 结构折叠<br />
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==References 参考资料==<br />
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==Further reading 进一步阅读==<br />
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* {{cite journal |last1=Aneja |first1=Nagender |last2=Gambhir |first2=Sapna |title=Ad-hoc Social Network: A Comprehensive Survey |url=http://www.ijser.org/researchpaper%5CAd-hoc-Social-Network-A-Comprehensive-Survey.pdf |journal=International Journal of Scientific & Engineering Research |volume=4 |issue=8 |date=August 2013 |pages=156–160|issn=2229-5518}}<br />
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* {{cite book |last=Barabási |first=Albert-László |year=2003 |title=Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life |publisher=Plum |isbn=978-0-452-28439-5 |url-access=registration |url=https://archive.org/details/linkedhoweveryth00bara }}<br />
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* {{cite book |last=Barnett |first=George A. |year=2011 |title=Encyclopedia of Social Networks |publisher=Sage |isbn=978-1-4129-7911-5}}<br />
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* {{cite book |last=Estrada |first=E |year=2011 |title=The Structure of Complex Networks: Theory and Applications |publisher=Oxford University Press |isbn=978-0-199-59175-6}}<br />
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* {{cite book |last=Ferguson |first=Niall |year=2018 |title=The Square and the Tower: Networks and Power, from the Freemasons to Facebook |isbn=978-0735222915 |publisher=Penguin Press}}<br />
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* {{cite book |last=Freeman |first=Linton C. |year=2004 |title=The Development of Social Network Analysis: A Study in the Sociology of Science |publisher=Empirical Press |isbn=978-1-59457-714-7}}<br />
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* {{cite book |last=Kadushin |first=Charles |year=2012 |title=Understanding Social Networks: Theories, Concepts, and Findings |location= |publisher=Oxford University Press |isbn=978-0-19-537946-4}}<br />
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* {{cite book |last1=Mauro |first1=Rios |last2=Petrella |first2=Carlos |year=2014 |title=La Quimera de las Redes Sociales |trans-title=The Chimera of Social Networks |language=Spanish |publisher=Bubok España |isbn=978-9974-99-637-3}}<br />
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* {{cite book |last1=Rainie |first1=Lee |last2=Wellman |first2=Barry |year=2012 |title=Networked: The New Social Operating System |url=https://archive.org/details/networkednewsoci0000rain |url-access=registration |location=Cambridge, Mass. |publisher=MIT Press |isbn=978-0262017190}}<br />
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* {{cite book |last=Scott |first=John |year=1991 |title=Social Network Analysis: A Handbook |publisher=Sage |isbn=978-0-7619-6338-7}}<br />
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* {{cite book |last1=Wasserman |first1=Stanley |last2=Faust |first2=Katherine |year=1994 |title=Social Network Analysis: Methods and Applications |publisher=Cambridge University Press |isbn=978-0-521-38269-4 |series=Structural Analysis in the Social Sciences }}<br />
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* {{cite book |last1=Wellman |first1=Barry |last2=Berkowitz |first2=S. D. |year=1988 |title=Social Structures: A Network Approach |series=Structural Analysis in the Social Sciences |publisher=Cambridge University Press |isbn=978-0-521-24441-1}}<br />
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==External links 外部链接==<br />
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{{Commons category|Social networks}}<br />
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===Organizations 组织===<br />
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* [http://www.insna.org/ International Network for Social Network Analysis]<br />
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===Peer-reviewed journals 同行评审期刊===<br />
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* ''[http://www.sciencedirect.com/science/journal/03788733 Social Networks]''<br />
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* ''[http://journals.cambridge.org/action/displaySpecialPage?pageId=3656 Network Science]''<br />
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* ''[http://www.cmu.edu/joss/content/articles/volindex.html Journal of Social Structure]''<br />
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* ''[http://www.tandfonline.com/toc/gmas20/current Journal of Mathematical Sociology]''<br />
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* ''[https://www.springer.com/computer/database+management+%26+information+retrieval/journal/13278 Social Network Analysis and Mining (SNAM)]''<br />
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* {{cite journal |title=INSNA - ''Connections'' Journal |journal=Connections Bulletin of the International Network for Social Network Analysis |location=Toronto |publisher=International Network for Social Network Analysis |url=http://www.insna.org/pubs/connections/ |issn=0226-1766 |url-status=dead |archiveurl=https://web.archive.org/web/20130718063641/http://www.insna.org/pubs/connections/ |archivedate=2013-07-18 }}<br />
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===Textbooks and educational resources 课本及教育资源===<br />
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* ''[http://www.cs.cornell.edu/home/kleinber/networks-book/ Networks, Crowds, and Markets]'' (2010) by D. Easley & J. Kleinberg<br />
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* ''[http://faculty.ucr.edu/~hanneman/nettext/ Introduction to Social Networks Methods]'' (2005) by R. Hanneman & M. Riddle<br />
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* [http://www.analytictech.com/networks/ Social Network Analysis Instructional Web Site] by S. Borgatti<br />
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* ''[https://web.archive.org/web/20160304063126/http://www.novagob.org/file/view/153259/ebook-gratuito-sobre-redes-sociales-virtuales Guide for virtual social networks for public administrations]'' (2015) by Mauro D. Ríos (in Spanish)<br />
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===Data sets 数据集===<br />
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* [http://pajek.imfm.si/doku.php?id=data:urls:index Pajek's list of lists of datasets]<br />
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* [http://networkdata.ics.uci.edu/index.html UC Irvine Network Data Repository]<br />
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* [http://snap.stanford.edu/data/ Stanford Large Network Dataset Collection]<br />
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* [http://www-personal.umich.edu/~mejn/netdata/ M.E.J. Newman datasets]<br />
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* [http://vlado.fmf.uni-lj.si/pub/networks/data/ Pajek datasets]<br />
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* [http://wiki.gephi.org/index.php?title=Datasets#Social_networks Gephi datasets]<br />
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* [http://konect.uni-koblenz.de/networks KONECT – Koblenz network collection]<br />
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* [http://www.stats.ox.ac.uk/~snijders/siena/ RSiena datasets]<br />
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==编者推荐==<br />
集智俱乐部：[https://swarma.org/?p=18646 社会网络分析的四个维度：研究方法、应用和软件工具综述 | 网络科学论文速递25篇]<br />
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[http://arxiv.org/abs/2002.09485 The Four Dimensions of Social Network Analysis: An Overview of Research Methods, Applications, and Software Tools]：近年来，基于社交网络的应用呈指数级增长，其中一个方面的原因是，这个应用领域提供了一个特别肥沃的地方，可以测试和开发最先进的计算技术，从 Web 中提取有价值的信息。这项工作的主要贡献有三个方面：(1) 我们对社会网络分析 (SNA) 的最新发展水平进行了文献综述；(2) 我们提出了一套基于 SNA 四个基本特征（或维度）的新度量标准；(3) 最后，我们对一套流行的 SNA 工具和框架进行了定量分析。我们还进行了一项科学计量学研究，以检测该领域中最活跃的研究领域和应用领域。本工作提出了四个不同维度的定义，即(a)模式、(b)知识发现、(c)信息融合、(d)集成、可伸缩性和可视化，用于定义一组新的度量（称为度），以评估 SNA 的不同软件工具和框架（根据之前的度量对一组 20 个 SNA-软件工具进行分析和排序）。这些维度连同定义的程度，可以评价和衡量社会网络技术的成熟度，寻求对它们的定量评估，从而揭示这一活跃领域的挑战和未来发展趋势。<br />
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范畴: 系统论<br />
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<small>This page was moved from [[wikipedia:en:Social network]]. Its edit history can be viewed at [[社会网络/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E7%A4%BE%E4%BC%9A%E7%BD%91%E7%BB%9C&diff=12521社会网络2020-08-14T04:06:37Z<p>Dorr：</p>
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<div>已由Dorr初步翻译<br />
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此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Social structure made up of a set of social actors}}<br />
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{{about|the theoretical concept as used in the social and behavioral sciences|social networking sites|Social networking service|the 2010 movie|The Social Network|other uses|Social network (disambiguation)}}<br />
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{{Network Science}}<br />
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A '''social network ''' is a [[social structure]] made up of a set of [[social]] actors (such as individuals or organizations), sets of [[Dyad (sociology)|dyadic]] ties, and other [[Social relation|social interactions]] between actors. The social network perspective provides a set of methods for analyzing the structure of whole social entities as well as a variety of theories explaining the patterns observed in these structures.<ref name="WF94CH1">{{cite book|last1=Wasserman |first1=Stanley | author-link1= Stanley Wasserman|last2=Faust |first2=Katherine |year=1994 |title=Social Network Analysis: Methods and Applications |isbn=9780521387071 |chapter=Social Network Analysis in the Social and Behavioral Sciences |pages=1–27 |publisher=Cambridge University Press}}</ref> The study of these structures uses [[social network analysis]] to identify local and global patterns, locate influential entities, and examine network dynamics.<br />
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A social network is a social structure made up of a set of social actors (such as individuals or organizations), sets of dyadic ties, and other social interactions between actors. The social network perspective provides a set of methods for analyzing the structure of whole social entities as well as a variety of theories explaining the patterns observed in these structures. The study of these structures uses social network analysis to identify local and global patterns, locate influential entities, and examine network dynamics.<br />
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'''<font color="#ff8000">社会网络 Social Network</font>'''是一种由一组社会行为者（个人或组织）、一组二元关系以及行为者之间的其他社会互动组成的社会结构。社会网络视角为分析整个社会实体的结构提供了一套方法，也为解释在这些结构中观察到的模式提供了各种理论。对这些结构的研究使用'''<font color="#ff8000">社会网络分析 Social Network Analysis</font>'''以确定本地及全球模式，定位有影响力的实体，并审查网络动态。<br />
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Social networks and the analysis of them is an inherently [[Interdisciplinarity|interdisciplinary]] academic field which emerged from [[social psychology]], [[sociology]], [[statistics]], and [[graph theory]]. [[Georg Simmel]] authored early structural theories in sociology emphasizing the dynamics of triads and "web of group affiliations".<ref>{{cite book|last1=Scott |first1=W. Richard |last2=Davis |first2=Gerald F. |author-link1= William Richard Scott|author-link2= Gerald F. Davis|title=Organizations and Organizing |chapter=Networks In and Around Organizations |year=2003 |isbn=978-0-13-195893-7 |publisher=Pearson Prentice Hall}}</ref> [[Jacob Moreno]] is credited with developing the first [[sociogram]]s in the 1930s to study interpersonal relationships. These approaches were mathematically formalized in the 1950s and theories and methods of social networks became pervasive in the [[Social science|social and behavioral sciences]] by the 1980s.<ref name="WF94CH1"/><ref name="Freeman History">{{cite book|last=Freeman |first=Linton |year=2004 |publisher=Empirical Press |isbn=978-1-59457-714-7 |title=The Development of Social Network Analysis: A Study in the Sociology of Science}}</ref> [[Social network analysis]] is now one of the major paradigms in contemporary sociology, and is also employed in a number of other social and formal sciences. Together with other [[complex network]]s, it forms part of the nascent field of [[network science]].<ref>{{cite journal|journal=Science |year=2009 |volume=323 |number=5916 |pages=892–895 |doi=10.1126/science.1165821 |pmid=19213908 |title=Network Analysis in the Social Sciences |first1=Stephen P. |last1=Borgatti |first2=Ajay |last2=Mehra |first3=Daniel J. |last3=Brass |first4=Giuseppe |last4=Labianca|bibcode=2009Sci...323..892B |citeseerx=10.1.1.536.5568 }}</ref><ref name="EK">{{cite book|title=Networks, Crowds, and Markets: Reasoning about a Highly Connected World |first1=David |last1=Easley |first2=Jon |last2=Kleinberg |publisher=Cambridge University Press |year=2010 |chapter=Overview |pages=1–20 |isbn=978-0-521-19533-1}}</ref><br />
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Social networks and the analysis of them is an inherently interdisciplinary academic field which emerged from social psychology, sociology, statistics, and graph theory. Georg Simmel authored early structural theories in sociology emphasizing the dynamics of triads and "web of group affiliations". Jacob Moreno is credited with developing the first sociograms in the 1930s to study interpersonal relationships. These approaches were mathematically formalized in the 1950s and theories and methods of social networks became pervasive in the social and behavioral sciences by the 1980s. Social network analysis is now one of the major paradigms in contemporary sociology, and is also employed in a number of other social and formal sciences. Together with other complex networks, it forms part of the nascent field of network science.<br />
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社会网络及其分析是社会心理学、社会学、统计学和图论中内在的跨学科学术领域。'''格奥尔格·齐美尔 Georg Simmel'''是早期社会学结构主义理论的作者，他强调三合会和“群体关系网”的动态性。'''雅各布·莫雷诺 Jacob Moreno'''被认为是在20世纪30年代发展了第一份'''<font color="#ff8000">社交关系图 Sociogram</font>'''以研究人际关系的人。这些方法在20世纪50年代得到数学化，到20世纪80年代，社会网络的理论和方法在社会和行为科学中变得普遍。社会网络分析现在是当代社会学的主要范式之一，也被用于许多其他社会科学及形式科学。它与其他复杂网络一起构成了网络科学新兴领域的一部分。<br />
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==Overview 概览==<br />
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[[File:Barabasi Albert model.gif|thumb|left|图1：Evolution graph of a social network: [[Barabási–Albert model|Barabási model]].一个社会网络的演化图：巴拉巴西模型。]]<br />
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Barabási model.]]<br />
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The social network is a [[Scientific theory|theoretical]] [[Construct (philosophy of science)|construct]] useful in the [[social sciences]] to study relationships between individuals, [[social groups|groups]], [[formal organizations|organizations]], or even entire [[society|societies]] ([[social unit]]s, see [[Differentiation (sociology)|differentiation]]). The term is used to describe a [[social structure]] determined by such [[social interactions|interactions]]. The ties through which any given social unit connects represent the convergence of the various social contacts of that unit. This theoretical approach is, necessarily, relational. An [[axiom]] of the social network approach to understanding [[social interaction]] is that social phenomena should be primarily conceived and investigated through the properties of relations between and within units, instead of the properties of these units themselves. Thus, one common criticism of social network theory is that [[Agency (sociology)|individual agency]] is often ignored<ref name="jscott">Scott, John P. (2000). ''Social Network Analysis: A Handbook'' (2nd edition). Thousand Oaks, CA: Sage Publications.</ref> although this may not be the case in practice (see [[agent-based model]]ing). Precisely because many different types of relations, singular or in combination, form these network configurations, [[Network science|network analytics]] are useful to a broad range of research enterprises. In social science, these fields of study include, but are not limited to [[anthropology]], [[biology]], [[communication studies]], [[economics]], [[geography]], [[information science]], [[organizational studies]], [[social psychology]], [[sociology]], and [[sociolinguistics]].<br />
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The social network is a theoretical construct useful in the social sciences to study relationships between individuals, groups, organizations, or even entire societies (social units, see differentiation). The term is used to describe a social structure determined by such interactions. The ties through which any given social unit connects represent the convergence of the various social contacts of that unit. This theoretical approach is, necessarily, relational. An axiom of the social network approach to understanding social interaction is that social phenomena should be primarily conceived and investigated through the properties of relations between and within units, instead of the properties of these units themselves. Thus, one common criticism of social network theory is that individual agency is often ignored although this may not be the case in practice (see agent-based modeling). Precisely because many different types of relations, singular or in combination, form these network configurations, network analytics are useful to a broad range of research enterprises. In social science, these fields of study include, but are not limited to anthropology, biology, communication studies, economics, geography, information science, organizational studies, social psychology, sociology, and sociolinguistics.<br />
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社会网络是社会科学中研究个人、团体、组织甚至整个社会（社会单位，见分化）之间关系的理论构造。这个术语用来描述由这种相互作用决定的社会结构。任何一个特定的社会单元之间的联系都代表着这个单元各种社会联系的聚合。这种理论方法必然是相关的。理解社会互动的社会网络方法的一个公理是，社会现象应该主要通过单元之间和单元内部关系的性质来构思和研究，而不是这些单元本身的性质。因此，社会网络理论总被诟病的一点是其常忽视个体代理，而实践中可能并非如此（见基于主体的建模）。正是因为许多不同类型的关系（单独或组合形式）形成这些网络配置，网络分析在广泛的研究中有用。在社会科学中，这些研究领域包括但不限于'''<font color="#ff8000">人类学 Anthropology</font>'''、生物学、'''<font color="#ff8000">传播学 Communication Studies</font>'''、经济学、地理学、信息科学、组织学、社会心理学、社会学和'''<font color="#ff8000">社会语言学 Sociolinguistics</font>'''。<br />
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==History 历史==<br />
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In the late 1890s, both [[Émile Durkheim]] and [[Ferdinand Tönnies]] foreshadowed the idea of social networks in their theories and research of [[social group]]s. Tönnies argued that social groups can exist as personal and direct social ties that either link individuals who share values and belief (''[[Gemeinschaft]]'', German, commonly translated as "[[community]]") or impersonal, formal, and instrumental social links (''[[Gesellschaft]]'', German, commonly translated as "[[society]]").<ref>Tönnies, Ferdinand (1887). ''Gemeinschaft und Gesellschaft'', Leipzig: Fues's Verlag. (Translated, 1957 by Charles Price Loomis as ''Community and Society'', East Lansing: Michigan State University Press.)</ref> Durkheim gave a non-individualistic explanation of social facts, arguing that social phenomena arise when interacting individuals constitute a reality that can no longer be accounted for in terms of the properties of individual actors.<ref>Durkheim, Emile (1893). ''De la division du travail social: étude sur l'organisation des sociétés supérieures'', Paris: F. Alcan. (Translated, 1964, by Lewis A. Coser as ''The Division of Labor in Society'', New York: Free Press.)</ref> [[Georg Simmel]], writing at the turn of the twentieth century, pointed to the nature of networks and the effect of network size on interaction and examined the likelihood of interaction in loosely knit networks rather than groups.<ref>Simmel, Georg (1908). ''Soziologie'', Leipzig: Duncker & Humblot.</ref><br />
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In the late 1890s, both Émile Durkheim and Ferdinand Tönnies foreshadowed the idea of social networks in their theories and research of social groups. Tönnies argued that social groups can exist as personal and direct social ties that either link individuals who share values and belief (Gemeinschaft, German, commonly translated as "community") or impersonal, formal, and instrumental social links (Gesellschaft, German, commonly translated as "society"). Durkheim gave a non-individualistic explanation of social facts, arguing that social phenomena arise when interacting individuals constitute a reality that can no longer be accounted for in terms of the properties of individual actors. Georg Simmel, writing at the turn of the twentieth century, pointed to the nature of networks and the effect of network size on interaction and examined the likelihood of interaction in loosely knit networks rather than groups.<br />
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在19世纪90年代晚期，'''埃米尔·涂尔干 Émile Durkheim'''和'''斐迪南·滕尼斯 Ferdinand Tönnies'''在他们关于社会群体的理论和研究中都预示了社会网络的概念。滕尼斯认为，社会群体可以作为个人和直接的社会关系存在，这种关系或者将具有共同价值观和信仰的个人（德语 Gemeinschaft，通常翻译为“社区”）联系在一起，或者将非个人的、正式的和工具性的社会关系（德语 Gesellschaft，通常翻译为“社会”）联系在一起。涂尔干对社会事实给出了非个人主义的解释，认为当相互作用的个体构成一种再也不能用个体行为者的特性来解释的现实时，社会现象就产生了。格奥尔格·齐美尔在20世纪之交的著作中指出了网络的本质和网络规模对互动的影响，并研究了在松散的网络而非群体中互动的可能性。<br />
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[[File:Moreno Sociogram 2nd Grade.png|thumb|图2：Moreno's sociogram of a 2nd grade class 莫雷诺对一个二年级班级的社会关系图]]<br />
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Major developments in the field can be seen in the 1930s by several groups in psychology, anthropology, and mathematics working independently.<ref name=jscott /><ref>For a historical overview of the development of social network analysis, see: {{cite book|last1=Carrington|first1=Peter J.|last2=Scott|first2=John|chapter=Introduction|title=The Sage Handbook of Social Network Analysis| publisher=Sage|year=2011|isbn=978-1-84787-395-8|page=1|chapter-url=https://books.google.com/books?id=2chSmLzClXgC&pg=PA1}}</ref><ref>See also the diagram in {{cite book|author=Scott, John|title=Social Network Analysis: A Handbook|publisher=Sage|year=2000|isbn=978-0-7619-6339-4|page=8|url=https://books.google.com/books?id=Ww3_bKcz6kgC&pg=PA8}}</ref> In [[psychology]], in the 1930s, [[Jacob L. Moreno]] began systematic recording and analysis of social interaction in small groups, especially classrooms and work groups (see [[sociometry]]). In [[anthropology]], the foundation for social network theory is the theoretical and [[ethnography|ethnographic]] work of [[Bronislaw Malinowski]],<ref>Malinowski, Bronislaw (1913). ''The Family Among the Australian Aborigines: A Sociological Study''. London: University of London Press.</ref> [[Radcliffe-Brown|Alfred Radcliffe-Brown]],<ref>Radcliffe-Brown, Alfred Reginald (1930) ''The social organization of Australian tribes''. Sydney, Australia: University of Sydney ''Oceania'' monographs, No.1.</ref><ref>{{cite journal | last1 = Radcliffe-Brown | first1 = A. R. | year = 1940 | title = On social structure | url = | journal = Journal of the Royal Anthropological Institute | volume = 70 | issue = 1| pages = 1–12 | doi=10.2307/2844197| jstor = 2844197 }}</ref> and [[Claude Lévi-Strauss]].<ref>Lévi-Strauss, Claude ([1947]1967). ''Les structures élémentaires de la parenté''. Paris: La Haye, Mouton et Co. (Translated, 1969 by J. H. Bell, J. R. von Sturmer, and R. Needham, 1969, as ''The Elementary Structures of Kinship'', Boston: Beacon Press.)</ref> A group of social anthropologists associated with [[Max Gluckman]] and the [[Manchester school (anthropology)|Manchester School]], including [[John Arundel Barnes|John A. Barnes]],<ref>Barnes, John (1954). "Class and Committees in a Norwegian Island Parish". ''Human Relations'', (7): 39–58.</ref> [[J. Clyde Mitchell]] and [[Elizabeth Bott Spillius]],<ref>{{cite journal | last1 = Freeman | first1 = Linton C. | last2 = Wellman | first2 = Barry | year = 1995 | title = A note on the ancestoral Toronto home of social network analysis | url = | journal = Connections | volume = 18 | issue = 2| pages = 15–19 }}</ref><ref>{{cite journal | last1 = Savage | first1 = Mike | year = 2008 | title = Elizabeth Bott and the formation of modern British sociology | url = | journal = The Sociological Review | volume = 56 | issue = 4| pages = 579–605 | doi=10.1111/j.1467-954x.2008.00806.x}}</ref> often are credited with performing some of the first fieldwork from which network analyses were performed, investigating community networks in southern Africa, India and the United Kingdom.<ref name=jscott /> Concomitantly, British anthropologist [[Siegfried Frederick Nadel|S. F. Nadel]] codified a theory of social structure that was influential in later network analysis.<ref>Nadel, S. F. 1957. ''The Theory of Social Structure''. London: Cohen and West.</ref> In [[sociology]], the early (1930s) work of [[Talcott Parsons]] set the stage for taking a relational approach to understanding social structure.<ref>Parsons, Talcott ([1937] 1949). ''The Structure of Social Action: A Study in Social Theory with Special Reference to a Group of European Writers''. New York: The Free Press.</ref><ref>Parsons, Talcott (1951). ''The Social System''. New York: The Free Press.</ref> Later, drawing upon Parsons' theory, the work of sociologist [[Peter Blau]] provides a strong impetus for analyzing the relational ties of social units with his work on [[social exchange theory]].<ref>Blau, Peter (1956). ''Bureaucracy in Modern Society''. New York: Random House, Inc.</ref><ref>Blau, Peter (1960). "A Theory of Social Integration". ''The American Journal of Sociology'', (65)6: 545–556, (May).</ref><ref>Blau, Peter (1964). ''Exchange and Power in Social Life''.</ref><br />
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Major developments in the field can be seen in the 1930s by several groups in psychology, anthropology, and mathematics working independently. In psychology, in the 1930s, Jacob L. Moreno began systematic recording and analysis of social interaction in small groups, especially classrooms and work groups (see sociometry). In anthropology, the foundation for social network theory is the theoretical and ethnographic work of Bronislaw Malinowski, Alfred Radcliffe-Brown, and Claude Lévi-Strauss. A group of social anthropologists associated with Max Gluckman and the Manchester School, including John A. Barnes, J. Clyde Mitchell and Elizabeth Bott Spillius, often are credited with performing some of the first fieldwork from which network analyses were performed, investigating community networks in southern Africa, India and the United Kingdom. In sociology, the early (1930s) work of Talcott Parsons set the stage for taking a relational approach to understanding social structure. Later, drawing upon Parsons' theory, the work of sociologist Peter Blau provides a strong impetus for analyzing the relational ties of social units with his work on social exchange theory.<br />
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20世纪30年代，心理学、人类学和数学领域的几个独立研究小组已经看到了这一领域的重大发展。在心理学方面，在20世纪30年代，雅各布·L·莫雷诺开始系统地记录和分析小团体中的社会互动，尤其是课堂和工作团体中的社会互动（见'''<font color="#ff8000">社会测量 Sociometry）</font>'''。在人类学中，社会网络理论的基础是'''布罗尼斯拉夫·马林诺夫斯基 Bronislaw Malinowski'''，'''阿尔弗雷德·拉德克利夫-布朗 Alfred Radcliffe-Brown'''和'''克洛德·列维-斯特劳斯 Claude Lévi-Strauss'''的理论和人种学著作。包括'''约翰·A·巴恩斯 John A. Barnes'''、'''J·克莱德·米切尔 J. Clyde Mitchell'''和'''伊丽莎白·博特·斯皮利厄斯 Elizabeth Bott Spillius'''在内的一群与'''马克斯·格拉克曼 Max Gluckman'''和'''曼彻斯特学派 Manchester School'''有关的社会人类学家，经常被认为是执行了一些最初的实地工作，从而进行了网络分析，调查了南非、印度和英国的社区网络。在社会学方面，'''塔尔科特·帕森斯 Talcott Parsons'''的早期工作（1930年代）为采用关系方法理解社会结构奠定了基础。后来，社会学家'''彼得·布劳 Peter Blau'''的'''<font color="#ff8000">社会交换论 Social Exchange Theory</font>'''为分析社会单位之间的关系提供了强大的动力。<br />
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By the 1970s, a growing number of scholars worked to combine the different tracks and traditions. One group consisted of sociologist [[Harrison White]] and his students at the [[Harvard Department of Social Relations|Harvard University Department of Social Relations]]. Also independently active in the Harvard Social Relations department at the time were [[Charles Tilly]], who focused on networks in political and community sociology and social movements, and [[Stanley Milgram]], who developed the "six degrees of separation" thesis.<ref>{{cite web |url=http://www.semioticon.com/semiotix/semiotix14/sem-14-05.html |title=The Networked Individual: A Profile of Barry Wellman |author=Bernie Hogan}}</ref> [[Mark Granovetter]]<ref name="Introduction for the French Reader">{{cite journal | last1 = Granovetter | first1 = Mark | year = 2007 | title = Introduction for the French Reader | url = | journal = Sociologica | volume = 2 | issue = | pages = 1–8 }}</ref> and [[Barry Wellman]]<ref>Wellman, Barry (1988). "Structural analysis: From method and metaphor to theory and substance". pp. 19–61 in B. Wellman and S. D. Berkowitz (eds.) ''Social Structures: A Network Approach'', Cambridge, UK: Cambridge University Press.</ref> are among the former students of White who elaborated and championed the analysis of social networks.<ref name="Introduction for the French Reader"/><ref>Mullins, Nicholas. ''Theories and Theory Groups in Contemporary American Sociology''. New York: Harper and Row, 1973.</ref><ref>Tilly, Charles, ed. ''An Urban World''. Boston: Little Brown, 1974.</ref><ref>Wellman, Barry. 1988. "Structural Analysis: From Method and Metaphor to Theory and Substance". pp. 19–61 in ''Social Structures: A Network Approach'', edited by Barry Wellman and S. D. Berkowitz. Cambridge: Cambridge University Press.</ref><br />
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By the 1970s, a growing number of scholars worked to combine the different tracks and traditions. One group consisted of sociologist Harrison White and his students at the Harvard University Department of Social Relations. Also independently active in the Harvard Social Relations department at the time were Charles Tilly, who focused on networks in political and community sociology and social movements, and Stanley Milgram, who developed the "six degrees of separation" thesis. Mark Granovetter and Barry Wellman are among the former students of White who elaborated and championed the analysis of social networks.<br />
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到了20世纪70年代，越来越多的学者致力于将不同的轨迹和传统结合起来。其中一组由社会学家'''哈里森·怀特 Harrison White'''和他在哈佛大学社会关系系的学生组成。当时在哈佛大学社会关系系独立活动的还有专注于政治和社区社会学和社会运动的网络的'''查尔斯·堤利 Charles Tilly'''，还有发表了六度分隔理论论文的'''斯坦利·米尔格拉姆 Stanley Milgram'''。'''马克·格兰诺维特 Mark Granovetter'''和'''巴里·威尔曼 Barry Wellman'''是怀特以前的学生，他们阐述并支持对社交网络的分析。<br />
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Beginning in the late 1990s, social network analysis experienced work by sociologists, political scientists, and physicists such as [[Duncan J. Watts]], [[Albert-László Barabási]], [[Peter Bearman]], [[Nicholas A. Christakis]], [[James H. Fowler]], and others, developing and applying new models and methods to emerging data available about online social networks, as well as "digital traces" regarding face-to-face networks.<br />
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Beginning in the late 1990s, social network analysis experienced work by sociologists, political scientists, and physicists such as Duncan J. Watts, Albert-László Barabási, Peter Bearman, Nicholas A. Christakis, James H. Fowler, and others, developing and applying new models and methods to emerging data available about online social networks, as well as "digital traces" regarding face-to-face networks.<br />
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从20世纪90年代末开始，社会网络分析经历了社会学家、政治学家和物理学家的工作，如'''[[邓肯·瓦茨 Duncan J. Watts]]'''、'''[[艾伯特-拉斯洛·巴拉巴西 Albert-László Barabási]]'''、Peter Bearman、Nicholas A. Christakis、James H. Fowler等人，开发和应用新的模型和方法来获得有关在线社会网络的新兴数据，以及有关面对面网络的“数字痕迹”。<br />
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==Levels of analysis 分析层次==<br />
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[[File:Network self-organization stages.png|thumb|right|图3：Self-organization of a network, based on Nagler, Levina, & Timme, (2011) 基于Nagler, Levina和Timme的自组织网络，(2011)<ref>{{cite journal|author1=Nagler, Jan|author2=Anna Levina|author3=Marc Timme|year=2011|title=Impact of single links in competitive percolation|journal=Nature Physics|volume=7|issue=3|pages=265–270|doi=10.1038/nphys1860|arxiv=1103.0922|bibcode=2011NatPh...7..265N}}</ref>]]<br />
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[[File:Social Network Diagram (large).svg|right|thumb|Centrality]]<br />
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Centrality<br />
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中心性<br />
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In general, social networks are [[self-organization|self-organizing]], [[emergence|emergent]], and [[social complexity|complex]], such that a globally coherent pattern appears from the local interaction of the elements that make up the system.<ref>Newman, Mark, Albert-László Barabási and Duncan J. Watts (2006). ''The Structure and Dynamics of Networks'' (Princeton Studies in Complexity). Oxford: Princeton University Press.</ref><ref>{{cite journal | last1 = Wellman | first1 = Barry | year = 2008 | title = Review: The development of social network analysis: A study in the sociology of science | url = | journal = Contemporary Sociology | volume = 37 | issue = 3| pages = 221–222 | doi=10.1177/009430610803700308}}</ref> These patterns become more apparent as network size increases. However, a global network analysis<ref>{{cite book|last=Faust|first=Stanley Wasserman; Katherine|title=Social network analysis : methods and applications|year=1998|publisher=Cambridge Univ. Press|location=Cambridge [u.a.]|isbn=978-0521382694|edition=Reprint.}}</ref> of, for example, all [[interpersonal relationships]] in the world is not feasible and is likely to contain so much [[Information theory|information]] as to be uninformative. Practical limitations of computing power, ethics and participant recruitment and payment also limit the scope of a social network analysis.<ref name="Kadu12">Kadushin, C. (2012). ''Understanding social networks: Theories, concepts, and findings''. Oxford: Oxford University Press.</ref><ref>{{Cite journal| author=Granovetter, M.|title= Network sampling: Some first steps| year=1976 |pages=1287–1303 |volume=81| issue=6 |journal= American Journal of Sociology | doi=10.1086/226224}}</ref> The nuances of a local system may be lost in a large network analysis, hence the quality of information may be more important than its scale for understanding network properties. Thus, social networks are analyzed at the scale relevant to the researcher's theoretical question. Although [[level of analysis|levels of analysis]] are not necessarily [[Mutually exclusive events|mutually exclusive]], there are three general levels into which networks may fall: [[Microsociology|micro-level]], [[wikt:meso-|meso-level]], and [[Macrosociology|macro-level]].<br />
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In general, social networks are self-organizing, emergent, and complex, such that a globally coherent pattern appears from the local interaction of the elements that make up the system. These patterns become more apparent as network size increases. However, a global network analysis of, for example, all interpersonal relationships in the world is not feasible and is likely to contain so much information as to be uninformative. Practical limitations of computing power, ethics and participant recruitment and payment also limit the scope of a social network analysis. The nuances of a local system may be lost in a large network analysis, hence the quality of information may be more important than its scale for understanding network properties. Thus, social networks are analyzed at the scale relevant to the researcher's theoretical question. Although levels of analysis are not necessarily mutually exclusive, there are three general levels into which networks may fall: micro-level, meso-level, and macro-level.<br />
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一般来说，社会网络是自组织的、涌现的和复杂的，这样，一个全局一致的模式就会从组成系统的元素的局部交互中显现出来。随着网络规模的增大，这些模式变得更加明显。然而，一个全球网络分析（如世界上所有的人际关系）是不可行的，它可能包含太多的信息，以至于相当于没有提供信息。计算能力的实际限制、道德规范以及参与者的招聘和报酬也限制了社会网络分析的规模。局部系统的细微差别在大规模网络分析中可能会消失，因此对于理解网络属性来说，信息的质量可能比其规模更重要。因此，社会网络是在与研究者的理论问题相关的尺度上加以分析的。虽然分析层次不一定相互排斥，但网络可以分为三个一般层次: 微观、中观和宏观。<br />
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===Micro level 微观层面===<br />
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At the micro-level, social network research typically begins with an individual, [[Snowball sampling|snowballing]] as social relationships are traced, or may begin with a small group of individuals in a particular social context.<br />
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At the micro-level, social network research typically begins with an individual, snowballing as social relationships are traced, or may begin with a small group of individuals in a particular social context.<br />
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在微观层面上，社会网络研究通常从个人开始，随着社会关系的追踪而像滚雪球一样扩大，或者可能从特定社会背景下的一小群个体开始。<br />
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'''Dyadic level''': A [[Dyad (sociology)|dyad]] is a social relationship between two individuals. Network research on dyads may concentrate on [[Structural functionalism|structure]] of the relationship (e.g. multiplexity, strength), [[social equality]], and tendencies toward [[Reciprocity (social and political philosophy)|reciprocity/mutuality]].<br />
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Dyadic level: A dyad is a social relationship between two individuals. Network research on dyads may concentrate on structure of the relationship (e.g. multiplexity, strength), social equality, and tendencies toward reciprocity/mutuality.<br />
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二元层面：二元是两个个体之间的社会关系。网络对二元关系的研究可以集中在关系的结构上（如多样性、力量)、社会平等以及互惠互利的倾向。<br />
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'''Triadic level''': Add one individual to a dyad, and you have a [[Triadic relation|triad]]. Research at this level may concentrate on factors such as [[Independence number|balance]] and [[Vertex-transitive graph|transitivity]], as well as [[social equality]] and tendencies toward [[Reciprocity (social and political philosophy)|reciprocity/mutuality]].<ref name="Kadu12"/> In the [[balance theory]] of [[Fritz Heider]] the triad is the key to social dynamics. The discord in a rivalrous [[love triangle]] is an example of an unbalanced triad, likely to change to a balanced triad by a change in one of the relations. The dynamics of social friendships in society has been modeled by balancing triads. The study is carried forward with the theory of [[signed graph]]s.<br />
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Triadic level: Add one individual to a dyad, and you have a triad. Research at this level may concentrate on factors such as balance and transitivity, as well as social equality and tendencies toward reciprocity/mutuality. In the balance theory of Fritz Heider the triad is the key to social dynamics. The discord in a rivalrous love triangle is an example of an unbalanced triad, likely to change to a balanced triad by a change in one of the relations. The dynamics of social friendships in society has been modeled by balancing triads. The study is carried forward with the theory of signed graphs.<br />
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三元层面：再加一个个体到二元组中，就得到一个三元组。这一层次的研究可能集中在诸如平衡和传递性等因素，以及社会平等和互惠互利的倾向。在'''弗里茨 · 海德 Fritz Heider'''的'''平衡理论 Balance Theory'''中，三元组是社会动力学的关键。在敌对的三角恋中的不和谐是不平衡的三角关系的一个例子，很可能通过其中一种关系的改变而变成平衡的三角关系。社会中社会友谊的动态模型是通过平衡三角关系建立起来的。该研究利用了符号图理论。<br />
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'''Actor level''': The smallest unit of analysis in a social network is an individual in their social setting, i.e., an "actor" or "ego". Egonetwork analysis focuses on network characteristics such as size, relationship strength, density, [[centrality]], [[wikt:prestige|prestige]] and roles such as [[isolates|isolates, liaisons]], and [[Bridge (interpersonal)|bridges]].<ref name="Jone11"/> Such analyses, are most commonly used in the fields of [[psychology]] or [[Social psychology (sociology)|social psychology]], [[ethnographic]] [[kinship]] analysis or other [[genealogy|genealogical]] studies of relationships between individuals.<br />
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Actor level: The smallest unit of analysis in a social network is an individual in their social setting, i.e., an "actor" or "ego". Egonetwork analysis focuses on network characteristics such as size, relationship strength, density, centrality, prestige and roles such as isolates, liaisons, and bridges. Such analyses, are most commonly used in the fields of psychology or social psychology, ethnographic kinship analysis or other genealogical studies of relationships between individuals.<br />
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行为者层面：社会网络中最小的分析单位是社会环境中的个体，即“行为者”或“自我”。自我网络分析主要关注网络特征，例如大小、关系强度、密度、中心性、声望和隔离、联络和桥梁等角色。这种分析最常用于心理学或社会心理学、人种学亲属关系分析或其他个体关系的系谱研究领域。<br />
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'''Subset level''': [[Subset]] levels of network research problems begin at the micro-level, but may cross over into the meso-level of analysis. Subset level research may focus on [[Distance (graph theory)|distance]] and reachability, [[cliques]], [[Cohesion (social policy)|cohesive]] subgroups, or other [[Group action (sociology)|group actions]] or [[Group behaviour|behavior]].<ref>{{cite book | title="Graph Theoretical Approaches to Social Network Analysis". in Computational Complexity: Theory, Techniques, and Applications (Robert A. Meyers, ed.) | publisher=Springer | author=de Nooy, Wouter |year=2012 | pages=2864–2877 | isbn=978-1-4614-1800-9|doi=10.1007/978-1-4614-1800-9_176| chapter=Social Network Analysis, Graph Theoretical Approaches to }}</ref><br />
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Subset level: Subset levels of network research problems begin at the micro-level, but may cross over into the meso-level of analysis. Subset level research may focus on distance and reachability, cliques, cohesive subgroups, or other group actions or behavior.<br />
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子集层面：网络研究问题的子集级别开始于微观级别，但可能跨越到中观级别的分析。子集级别的研究可能集中在距离和可达性、派系、凝聚子群或其他群体行为或行为。<br />
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===Meso level 中观层面===<br />
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In general, meso-level theories begin with a [[Sample population|population]] size that falls between the micro- and macro-levels. However, meso-level may also refer to analyses that are specifically designed to reveal connections between micro- and macro-levels. Meso-level networks are low density and may exhibit causal processes distinct from interpersonal micro-level networks.<ref>{{cite journal | last1 = Hedström | first1 = Peter | last2 = Sandell | first2 = Rickard | last3 = Stern | first3 = Charlotta | year = 2000 | title = Mesolevel Networks and the Diffusion of Social Movements: The Case of the Swedish Social Democratic Party | url = http://www.nuffield.ox.ac.uk/users/hedstrom/ajs3.pdf | journal = American Journal of Sociology | volume = 106 | issue = 1| pages = 145–172 | doi = 10.1086/303109 }}</ref><br />
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In general, meso-level theories begin with a population size that falls between the micro- and macro-levels. However, meso-level may also refer to analyses that are specifically designed to reveal connections between micro- and macro-levels. Meso-level networks are low density and may exhibit causal processes distinct from interpersonal micro-level networks.<br />
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一般来说，中观层面的理论始于介于微观和宏观层面之间的规模。然而，中观层面也可以指专门为揭示微观和宏观层面之间的联系而设计的分析。中观层次的网络是密度低，可能表现出不同于人际微观层面网络的因果过程。<br />
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[[File:Social Red.jpg|thumb|right|图4：Social network diagram, meso-level 中观层面社会网络图]]<br />
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'''Organizations''': Formal [[organizations]] are [[social group]]s that distribute tasks for a collective [[goal]].<ref name="Riketta, M. 2007">{{cite journal | last1 = Riketta | first1 = M. | last2 = Nienber | first2 = S. | year = 2007 | title = Multiple identities and work motivation: The role of perceived compatibility between nested organizational units | url = | journal = British Journal of Management | volume = 18 | issue = | pages = S61–77 | doi=10.1111/j.1467-8551.2007.00526.x}}</ref> Network research on organizations may focus on either intra-organizational or inter-organizational ties in terms of [[Formal organization|formal]] or [[Informal organization|informal]] relationships. Intra-organizational networks themselves often contain multiple levels of analysis, especially in larger organizations with multiple branches, franchises or semi-autonomous departments. In these cases, research is often conducted at a work group level and organization level, focusing on the interplay between the two structures.<ref name="Riketta, M. 2007"/> Experiments with networked groups online have documented ways to optimize group-level coordination through diverse interventions, including the addition of autonomous agents to the groups.<ref>{{Cite journal|last=Shirado|first=Hirokazu|last2=Christakis|first2=Nicholas A|title=Locally noisy autonomous agents improve global human coordination in network experiments|journal=Nature|volume=545|issue=7654|pages=370–374|doi=10.1038/nature22332|pmid=28516927|pmc=5912653|bibcode=2017Natur.545..370S|year=2017}}</ref><br />
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Organizations: Formal organizations are social groups that distribute tasks for a collective goal. Network research on organizations may focus on either intra-organizational or inter-organizational ties in terms of formal or informal relationships. Intra-organizational networks themselves often contain multiple levels of analysis, especially in larger organizations with multiple branches, franchises or semi-autonomous departments. In these cases, research is often conducted at a work group level and organization level, focusing on the interplay between the two structures.<br />
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组织: 正规组织是为共同目标分配任务的社会团体。关于组织的网络研究可以侧重于正式或非正式关系方面的组织内或组织间联系。组织内网络本身往往包含多层次的分析，特别是在具有多个分支机构、特许权或半自治部门的较大组织中。在这些情况下，研究通常在工作组和组织层面进行，重点放在两个结构之间的相互作用。<br />
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'''Randomly distributed networks''': [[Exponential random graph models]] of social networks became state-of-the-art methods of social network analysis in the 1980s. This framework has the capacity to represent social-structural effects commonly observed in many human social networks, including general [[Degree (graph theory)|degree]]-based structural effects commonly observed in many human social networks as well as [[Reciprocity (social and political philosophy)|reciprocity]] and [[Transitive set|transitivity]], and at the node-level, [[homophily]] and [[Attribute-value system|attribute]]-based activity and popularity effects, as derived from explicit hypotheses about [[Dependency graph|dependencies]] among network ties. [[Parameter]]s are given in terms of the prevalence of small [[Induced subgraph|subgraph]] configurations in the network and can be interpreted as describing the combinations of local social processes from which a given network emerges. These probability models for networks on a given set of actors allow generalization beyond the restrictive dyadic independence assumption of micro-networks, allowing models to be built from theoretical structural foundations of social behavior.<ref>{{cite journal | last1 = Cranmer | first1 = Skyler J. | last2 = Desmarais | first2 = Bruce A. | year = 2011 | title = Inferential Network Analysis with Exponential Random Graph Models | url = | journal = Political Analysis | volume = 19 | issue = 1| pages = 66–86 | doi=10.1093/pan/mpq037| citeseerx = 10.1.1.623.751 }}</ref><br />
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Randomly distributed networks: Exponential random graph models of social networks became state-of-the-art methods of social network analysis in the 1980s. This framework has the capacity to represent social-structural effects commonly observed in many human social networks, including general degree-based structural effects commonly observed in many human social networks as well as reciprocity and transitivity, and at the node-level, homophily and attribute-based activity and popularity effects, as derived from explicit hypotheses about dependencies among network ties. Parameters are given in terms of the prevalence of small subgraph configurations in the network and can be interpreted as describing the combinations of local social processes from which a given network emerges. These probability models for networks on a given set of actors allow generalization beyond the restrictive dyadic independence assumption of micro-networks, allowing models to be built from theoretical structural foundations of social behavior.<br />
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随机分布的网络: 社会网络的'''<font color="#ff8000">指数随机图模型 Exponential random graph models</font>'''在20世纪80年代成为最先进的社会网络分析方法。这个框架有能力表示在许多人类社会网络中普遍观察到的社会结构效应，包括在许多人类社会网络中普遍观察到的基于程度的一般性结构效应以及互惠性和传递性，以及在节点一级、同相性和基于属性的活动和流行性效应，这些效应源于关于网络关系之间依赖性的明确假设。参数是根据网络中小型子图配置的流行程度给出的，可以解释为描述一个给定网络出现的局部社会过程的组合。这些网络的概率模型在给定的参与者集合上允许超越微型网络的限制性并元独立性假设的泛化，允许模型从社会行为的理论结构基础上建立。<br />
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[[File:Scale-free network sample.png|thumb|right|图5：Examples of a random network and a scale-free network. Each graph has 32 nodes and 32 links. Note the "hubs" (shaded) in the scale-free diagram (on the right).一个随机网络和一个无尺度网络的例子。每个图有32个顶点和32条边。注意无标度图中的“集线器”（阴影部分）（右侧）。]]<br />
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'''Scale-free networks''': A [[scale-free network]] is a [[complex network|network]] whose [[degree distribution]] follows a [[power law]], at least [[asymptotic]]ally. In [[network theory]] a scale-free ideal network is a [[random network]] with a [[degree distribution]] that unravels the size distribution of social groups.<ref>{{cite journal |author1=Moreira, André A.|author2=Demétrius R. Paula|author3=Raimundo N. Costa Filho|author4=José S. Andrade, Jr. |title=Competitive cluster growth in complex networks |year=2006 |doi=10.1103/PhysRevE.73.065101 |pmid=16906890 |journal=Physical Review E |volume=73 |issue=6 |pages=065101 |arxiv=cond-mat/0603272 |bibcode=2006PhRvE..73f5101M }}</ref> Specific characteristics of scale-free networks vary with the theories and analytical tools used to create them, however, in general, scale-free networks have some common characteristics. One notable characteristic in a scale-free network is the relative commonness of [[Vertex (graph theory)|vertices]] with a [[Maximum degree|degree]] that greatly exceeds the average. The highest-degree nodes are often called "hubs", and may serve specific purposes in their networks, although this depends greatly on the social context. Another general characteristic of scale-free networks is the [[clustering coefficient]] distribution, which decreases as the node degree increases. This distribution also follows a [[power law]].<ref>Barabási, Albert-László (2003). ''Linked: how everything is connected to everything else and what it means for business, science, and everyday life''. New York: Plum.</ref> The [[Barabási–Albert model|Barabási]] model of network evolution shown above is an example of a scale-free network.<br />
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Scale-free networks: A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. In network theory a scale-free ideal network is a random network with a degree distribution that unravels the size distribution of social groups. Specific characteristics of scale-free networks vary with the theories and analytical tools used to create them, however, in general, scale-free networks have some common characteristics. One notable characteristic in a scale-free network is the relative commonness of vertices with a degree that greatly exceeds the average. The highest-degree nodes are often called "hubs", and may serve specific purposes in their networks, although this depends greatly on the social context. Another general characteristic of scale-free networks is the clustering coefficient distribution, which decreases as the node degree increases. This distribution also follows a power law. The Barabási model of network evolution shown above is an example of a scale-free network.<br />
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'''<font color="#ff8000">[[无标度网络]] A Scale-free Network</font>''': 无标度网络网络是一个'''<font color="#ff8000">度分布 Degree Distribution</font>'''遵循'''<font color="#ff8000">幂律 Power Law</font>'''(至少是渐近的）的网络。在网络理论中，无标度理想网络是具有揭示了社会群体的规模分布的度分布的随机网络。无标度网络的具体特征随创建其的理论和分析工具的变化而变化，然而，一般来说，无标度网络具有一些共同的特征。无标度网络的一个显著特征是度远超均值的顶点的相对共性。最高度的节点通常被称为“'''<font color="#ff8000">枢纽 Hubs</font>'''” ，并且可能在其网络中服务于特定的目的，尽管这在很大程度上取决于社会环境。无标度网络的另一个一般特性是'''<font color="#ff8000">集聚系数 General Characteristic</font>'''分布，它随着节点度的增加而减少。该分布也遵循幂律。上面网络演化的巴拉巴西模型就是无标度网络的一个例子。<br />
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===Macro level 宏观层面===<br />
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Rather than tracing interpersonal interactions, macro-level analyses generally trace the outcomes of interactions, such as [[economic]] or other [[resource]] [[Transfer function|transfer]] interactions over a large [[Sample population|population]].<br />
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Rather than tracing interpersonal interactions, macro-level analyses generally trace the outcomes of interactions, such as economic or other resource transfer interactions over a large population.<br />
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宏观层面的分析不是追踪人际互动，而通常是追踪相互作用的结果，例如经济或其他资源转移在一大群体中的相互作用。<br />
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[[File:Diagram of a social network.jpg|thumb|right|图6: Diagram: section of a large-scale social network 一大型社会网络局部]]<br />
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'''Large-scale networks''': [[Large-scale macroeconometric model|Large-scale network]] is a term somewhat synonymous with "macro-level" as used, primarily, in [[social science|social]] and [[Behavioural sciences|behavioral]] sciences, in [[economics]]. Originally, the term was used extensively in the [[computer sciences]] (see [[Network mapping#Large-scale mapping project|large-scale network mapping]]).<br />
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Large-scale networks: Large-scale network is a term somewhat synonymous with "macro-level" as used, primarily, in social and behavioral sciences, in economics. Originally, the term was used extensively in the computer sciences (see large-scale network mapping).<br />
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大规模网络: 大规模网络是一个与“宏观层面”同义的术语，主要用于社会和行为科学，经济学。最初，这个术语在计算机科学中广泛使用（见大规模网络映射）。<br />
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'''Complex networks''': Most larger social networks display features of [[social complexity]], which involves substantial non-trivial features of [[network topology]], with patterns of complex connections between elements that are neither purely regular nor purely random (see, [[complexity science]], [[dynamical system]] and [[chaos theory]]), as do [[biological]], and [[Computer network|technological networks]]. Such [[complex network]] features include a heavy tail in the [[degree distribution]], a high [[clustering coefficient]], [[assortativity]] or disassortativity among vertices, [[community structure]] (see [[stochastic block model]]), and [[hierarchy|hierarchical structure]]. In the case of [[Agency (philosophy)|agency-directed]] networks these features also include [[Reciprocity in network|reciprocity]], triad significance profile (TSP, see [[network motif]]), and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as [[lattice graph|lattices]] and [[random graph]]s, do not show these features.<ref>{{cite journal|author=Strogatz, Steven H.|year=2001|title=Exploring complex networks|journal=Nature|volume=410|pages=268–276|doi=10.1038/35065725|pmid=11258382|issue=6825|bibcode=2001Natur.410..268S|doi-access=free}}</ref><br />
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Complex networks: Most larger social networks display features of social complexity, which involves substantial non-trivial features of network topology, with patterns of complex connections between elements that are neither purely regular nor purely random (see, complexity science, dynamical system and chaos theory), as do biological, and technological networks. Such complex network features include a heavy tail in the degree distribution, a high clustering coefficient, assortativity or disassortativity among vertices, community structure (see stochastic block model), and hierarchical structure. In the case of agency-directed networks these features also include reciprocity, triad significance profile (TSP, see network motif), and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as lattices and random graphs, do not show these features.<br />
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复杂网络: 大多数较大的社会网络呈现出社会复杂性的特征，包括'''<font color="#ff8000">网络拓扑 Network Topology</font>'''的大量非平凡特征，以及既不完全规则也不完全随机的元素之间的复杂连接模式（见[[复杂性科学]]、[[动力系统]]和[[混沌理论]]），生物和技术网络也是如此。这些复杂的网络特征包括度分布的重尾、高集聚系数、顶点之间的'''<font color="#ff8000">同配性 Assortativity</font>'''或非同配性、社区结构（见'''<font color="#ff8000">随机分块模型 Stochastic Block Model</font>'''）和层次结构。在主体导向网络的情况下，这些特征还包括互惠性、<font color="#32CD32">三重显著性特征</font>（TSP，见网络基序）及其他。相比之下，许多过去研究过的网络数学模型，如格和'''<font color="#ff8000">随机图 Random Graph</font>'''，并没有表现出这些特征。<br />
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==Theoretical links 理论联系==<br />
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===Imported theories 输入的理论===<br />
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Various theoretical frameworks have been imported for the use of social network analysis. The most prominent of these are [[Graph theory]], [[Balance theory]], Social comparison theory, and more recently, the [[Social identity approach]].<ref>{{cite book |last1=Kilduff|first1=M.|last2=Tsai|first2=W. |year=2003 |title= Social networks and organisations |publisher=Sage Publications}}</ref><br />
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Various theoretical frameworks have been imported for the use of social network analysis. The most prominent of these are Graph theory, Balance theory, Social comparison theory, and more recently, the Social identity approach.<br />
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为了使用社会网络分析，已经引入了各种理论框架。其中最突出的是[[图论]]、'''<font color="#ff8000">平衡理论 Balance theory</font>'''、社会比较论，以及最近的社会认同方法。<br />
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===Indigenous theories 本土理论===<br />
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Few complete theories have been produced from social network analysis. Two that have are [[role theory|structural role theory]] and [[heterophily|heterophily theory]].<br />
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Few complete theories have been produced from social network analysis. Two that have are structural role theory and heterophily theory.<br />
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很少有完整的理论产生于社会网络分析。现有两个为结构角色理论和异质性理论。<br />
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The basis of Heterophily Theory was the finding in one study that more numerous weak ties can be important in seeking information and innovation, as cliques have a tendency to have more homogeneous opinions as well as share many common traits. This homophilic tendency was the reason for the members of the cliques to be attracted together in the first place. However, being similar, each member of the clique would also know more or less what the other members knew. To find new information or insights, members of the clique will have to look beyond the clique to its other friends and acquaintances. This is what Granovetter called "the strength of weak ties".<ref>{{Cite journal | author=Granovetter, M. |year=1973 |title= The strength of weak ties |journal= American Journal of Sociology |volume=78 |issue=6 |pages= 1360–1380 |doi=10.1086/225469}}</ref><br />
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The basis of Heterophily Theory was the finding in one study that more numerous weak ties can be important in seeking information and innovation, as cliques have a tendency to have more homogeneous opinions as well as share many common traits. This homophilic tendency was the reason for the members of the cliques to be attracted together in the first place. However, being similar, each member of the clique would also know more or less what the other members knew. To find new information or insights, members of the clique will have to look beyond the clique to its other friends and acquaintances. This is what Granovetter called "the strength of weak ties".<br />
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异质性理论的基础是在一项研究中发现，更多的'''<font color="#ff8000">弱关系 Weak Tie</font>'''可以在寻求信息和创新方面发挥重要作用，因为小圈子倾向于有更同质化的观点，也有许多共同特征。这种亲同性倾向是小圈子成员被吸引到一起的首要原因。然而，由于相似，小圈子里的每一个成员或多或少都知道其他成员所知道的事情。为了获得新的信息或见解，小圈子里的成员不得不超越该圈子，关注其他的朋友及熟人。这就是格兰诺维特所说的“弱关系的力量”。<br />
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==Structural holes 结构洞==<br />
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In the context of networks, [[social capital]] exists where people have an advantage because of their location in a network. Contacts in a network provide information, opportunities and perspectives that can be beneficial to the central player in the network. Most social structures tend to be characterized by dense clusters of strong connections.<ref name="Burt 2004">{{cite journal|last=Burt|first=Ronald|title=Structural Holes and Good Ideas|journal=American Journal of Sociology|year=2004|doi=10.1086/421787|volume=110|issue=2|pages=349–399|citeseerx=10.1.1.388.2251}}</ref> Information within these clusters tends to be rather homogeneous and redundant. Non-redundant information is most often obtained through contacts in different clusters.<ref name="Burt 1992">{{cite book|last=Burt|first=Ronald|title=Structural Holes: The Social Structure of Competition|year=1992|publisher=Harvard University Press|location=Cambridge, MA}}</ref> When two separate clusters possess non-redundant information, there is said to be a structural hole between them.<ref name="Burt 1992"/> Thus, a network that bridges [[structural holes]] will provide network benefits that are in some degree additive, rather than overlapping. An ideal network structure has a vine and cluster structure, providing access to many different clusters and structural holes.<ref name="Burt 1992"/><br />
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In the context of networks, social capital exists where people have an advantage because of their location in a network. Contacts in a network provide information, opportunities and perspectives that can be beneficial to the central player in the network. Most social structures tend to be characterized by dense clusters of strong connections. Information within these clusters tends to be rather homogeneous and redundant. Non-redundant information is most often obtained through contacts in different clusters. When two separate clusters possess non-redundant information, there is said to be a structural hole between them. Thus, a network that bridges structural holes will provide network benefits that are in some degree additive, rather than overlapping. An ideal network structure has a vine and cluster structure, providing access to many different clusters and structural holes.<br />
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在网络的背景下，'''<font color="#ff8000">社会资本 Social Capital</font>'''存在于人们因其在网络中的位置而具有优势的地方。网络中的联系提供的信息、机会和观点可能有利于网络中的核心参与者。大多数社会结构往往以有强连接的密集集群为特征。这些集群中的信息往往是相当同质和冗余的。非冗余信息通常是通过不同集群中的联系获得的。当两个独立的集群拥有非冗余信息时，我们称它们之间存在一个'''<font color="#ff8000">结构洞 Structural Hole</font>'''。因此，一个'''<font color="#32CD32">连接</font>'''结构孔的网络在某种程度上提供附加而非重叠的的网络效益。理想的网络结构具有蔓生结构和集群结构，可访问许多不同集群和结构洞。<br />
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Networks rich in structural holes are a form of social capital in that they offer [[Information theory|information]] benefits. The main player in a network that bridges structural holes is able to access information from diverse sources and clusters.<ref name="Burt 1992"/> For example, in [[Business networking|business networks]], this is beneficial to an individual's career because he is more likely to hear of job openings and opportunities if his network spans a wide range of contacts in different industries/sectors. This concept is similar to Mark Granovetter's theory of [[Interpersonal ties|weak ties]], which rests on the basis that having a broad range of contacts is most effective for job attainment.<br />
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Networks rich in structural holes are a form of social capital in that they offer information benefits. The main player in a network that bridges structural holes is able to access information from diverse sources and clusters. For example, in business networks, this is beneficial to an individual's career because he is more likely to hear of job openings and opportunities if his network spans a wide range of contacts in different industries/sectors. This concept is similar to Mark Granovetter's theory of weak ties, which rests on the basis that having a broad range of contacts is most effective for job attainment.<br />
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富含结构洞的网络是社会资本的一种形式，因为它们提供信息利益。连接结构洞的网络中的主要参与者能够访问来自不同来源和集群的信息。例如，在'''<font color="#ff8000">商业社交 Business networking</font>'''中，这对个人的职业生涯是有益的，因为若其关系网涵盖不同行业 / 部门，则更有可能得知职位空缺和机会。这个概念类似于马克·格兰诺维特的弱关系理论，该理论的基础是拥有广泛的联系对于获得工作是最有效的。<br />
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==Research clusters 研究集群==<br />
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===Communication 传播学===<br />
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[[Communication Studies]] are often considered a part of both the social sciences and the humanities, drawing heavily on fields such as [[sociology]], [[psychology]], [[anthropology]], [[information science]], [[biology]], [[political science]], and [[economics]] as well as [[rhetoric]], [[literary studies]], and [[semiotics]]. Many communication concepts describe the transfer of information from one source to another, and can thus be conceived of in terms of a network.<br />
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Communication Studies are often considered a part of both the social sciences and the humanities, drawing heavily on fields such as sociology, psychology, anthropology, information science, biology, political science, and economics as well as rhetoric, literary studies, and semiotics. Many communication concepts describe the transfer of information from one source to another, and can thus be conceived of in terms of a network.<br />
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传播学通常被认为是社会科学和人文科学的一部分，主要集中在社会学、心理学、人类学、信息科学、生物学、政治学、经济学以及修辞学、文学研究和符号学等领域。许多通信概念描述了从一个源到另一个源的信息传输，因此可以从网络的角度来考虑。<br />
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===Community 社区===<br />
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In J.A. Barnes' day, a "[[community]]" referred to a specific geographic location and studies of community ties had to do with who talked, associated, traded, and attended church with whom. Today, however, there are extended "online" communities developed through [[telecommunications]] devices and [[social network services]]. Such devices and services require extensive and ongoing maintenance and analysis, often using [[network science]] methods. [[Community development]] studies, today, also make extensive use of such methods.<br />
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In J.A. Barnes' day, a "community" referred to a specific geographic location and studies of community ties had to do with who talked, associated, traded, and attended church with whom. Today, however, there are extended "online" communities developed through telecommunications devices and social network services. Such devices and services require extensive and ongoing maintenance and analysis, often using network science methods. Community development studies, today, also make extensive use of such methods.<br />
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在J.A. Barnes' day，“社区”指的是一个特定的地理位置和与谁交谈、联系、贸易和一同上教堂做礼拜有关的社区关系的研究。然而如今，通过电信设备和社交网络服务，有了扩展的“在线”社区。这样的设备和服务需要广泛和持续的维护和分析，通常使用网络科学方法。社区发展研究在如今也广泛使用这些方法。<br />
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===Complex networks 复杂网络===<br />
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[[Complex networks]] require methods specific to modelling and interpreting [[social complexity]] and [[complex adaptive system]]s, including techniques of [[dynamic network analysis]].<br />
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Complex networks require methods specific to modelling and interpreting social complexity and complex adaptive systems, including techniques of dynamic network analysis.<br />
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复杂网络需要建模和解释社会复杂性和复杂适应系统的特定方法，包括动态网络分析技术。<br />
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Mechanisms such as [[Dual-phase evolution#social networks|Dual-phase evolution]] explain how temporal changes in connectivity contribute to the formation of structure in social networks.<br />
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Mechanisms such as Dual-phase evolution explain how temporal changes in connectivity contribute to the formation of structure in social networks.<br />
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'''<font color="#ff8000">双相演化理论 Dual-phase Evolution</font>'''等机制解释了连接性的时间变化如何促进社会网络结构的形成。<br />
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===Criminal networks 犯罪网络===<br />
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In [[criminology]] and [[urban sociology]], much attention has been paid to the social networks among criminal actors. For example, Andrew Papachristos<ref>{{cite journal |last=Papachristos |first=Andrew |year=2009 |title=Murder by Structure: Dominance Relations and the Social Structure of Gang Homicide |journal=American Journal of Sociology |volume=115 |issue=1 |pages=74–128 |doi=10.2139/ssrn.855304 |url=http://www.papachristos.org/Publications_2_files/ajs_final_version.pdf |accessdate=29 March 2013 |url-status=dead |archiveurl=https://web.archive.org/web/20140407094725/http://www.papachristos.org/Publications_2_files/ajs_final_version.pdf |archivedate=7 April 2014 }}</ref> has studied gang murders as a series of exchanges between gangs. Murders can be seen to diffuse outwards from a single source, because weaker gangs cannot afford to kill members of stronger gangs in retaliation, but must commit other violent acts to maintain their reputation for strength.<br />
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In criminology and urban sociology, much attention has been paid to the social networks among criminal actors. For example, Andrew Papachristos has studied gang murders as a series of exchanges between gangs. Murders can be seen to diffuse outwards from a single source, because weaker gangs cannot afford to kill members of stronger gangs in retaliation, but must commit other violent acts to maintain their reputation for strength.<br />
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在'''<font color="#ff8000">犯罪学 Criminology</font>和'''<font color="#ff8000">城市社会学 Urban sociology</font>中，犯罪行为人之间的社会网络问题受到了广泛的关注。例如，'''安德鲁·帕帕克里斯托斯 Andrew Papachristos'''将帮派谋杀研究为帮派之间的一系列交流。谋杀可以视为从单一来源向外扩散，因为较弱的帮派无法承担为报复杀死较强帮派成员的代价，而必须采取其他暴力行动来维护其势力强大之名声。<br />
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===Diffusion of innovations 创新扩散理论===<br />
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[[Diffusion of innovations|Diffusion of ideas and innovations]] studies focus on the spread and use of ideas from one actor to another or one [[culture]] and another. This line of research seeks to explain why some become "early adopters" of ideas and innovations, and links social network structure with facilitating or impeding the spread of an innovation.<br />
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Diffusion of ideas and innovations studies focus on the spread and use of ideas from one actor to another or one culture and another. This line of research seeks to explain why some become "early adopters" of ideas and innovations, and links social network structure with facilitating or impeding the spread of an innovation.<br />
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'''<font color="#ff8000">创新扩散理论 Diffusion of Innovations</font>'''研究的重点是思想从一个行动者到另一个或一种文化到另一种的传播。这一系列的研究试图解释为什么有些人成为创意和创新的“早期接受者” ，并将社交网络结构与促进或阻碍创新的传播联系起来。<br />
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===Demography 人口学===<br />
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In [[demography]], the study of social networks has led to new sampling methods for estimating and reaching populations that are hard to enumerate (for example, homeless people or intravenous drug users.) For example, respondent driven sampling is a network-based sampling technique that relies on respondents to a survey recommending further respondents.<br />
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In demography, the study of social networks has led to new sampling methods for estimating and reaching populations that are hard to enumerate (for example, homeless people or intravenous drug users.) For example, respondent driven sampling is a network-based sampling technique that relies on respondents to a survey recommending further respondents.<br />
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在'''<font color="#ff8000">人口学 Demography</font>'''方面，对社会网络的研究导致了新的抽样方法，用于估计和覆盖难以统计的人群（如无家可归者或静脉注射毒品者）。例如，受访者驱动的抽样依赖于调查的受访者推荐更多的受访者，是一种基于网络的抽样技术。<br />
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===Economic sociology 经济社会学===<br />
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The field of [[sociology]] focuses almost entirely on networks of outcomes of social interactions. More narrowly, [[economic sociology]] considers behavioral interactions of individuals and groups through [[social capital]] and social "markets". Sociologists, such as Mark Granovetter, have developed core principles about the interactions of social structure, information, ability to punish or reward, and trust that frequently recur in their analyses of political, economic and other institutions. Granovetter examines how social structures and social networks can affect economic outcomes like hiring, price, productivity and innovation and describes sociologists' contributions to analyzing the impact of social structure and networks on the economy.<ref>{{cite journal |last=Granovetter |first=Mark |year=2005 |title=The Impact of Social Structure on Economic Outcomes |journal=The Journal of Economic Perspectives |jstor=4134991 |volume=19 |issue=1 |pages=33–50|doi=10.1257/0895330053147958 }}</ref><br />
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The field of sociology focuses almost entirely on networks of outcomes of social interactions. More narrowly, economic sociology considers behavioral interactions of individuals and groups through social capital and social "markets". Sociologists, such as Mark Granovetter, have developed core principles about the interactions of social structure, information, ability to punish or reward, and trust that frequently recur in their analyses of political, economic and other institutions. Granovetter examines how social structures and social networks can affect economic outcomes like hiring, price, productivity and innovation and describes sociologists' contributions to analyzing the impact of social structure and networks on the economy.<br />
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社会学领域几乎完全关注社会互动的结果网络。更狭义地说，'''<font color="#ff8000">经济社会学 Economic Sociology</font>'''通过社会资本和社会“市场”考虑个人和群体的行为互动。社会学家，如马克·格兰诺维特，已经研究出关于社会结构、信息、奖惩能力和信任相互作用的核心原则，这些原则在他们对政治、经济和其他制度的分析中经常出现。格兰诺维特研究了社会结构和社会网络如何影响经济结果，如雇佣、价格、生产力和创新，并描述了社会学家对分析社会结构和网络对经济的影响的贡献。<br />
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===Health care 卫生保健===<br />
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Analysis of social networks is increasingly incorporated into [[health care analytics]], not only in [[epidemiology|epidemiological]] studies but also in models of [[Health Communication|patient communication]] and education, disease prevention, mental health diagnosis and treatment, and in the study of health care organizations and [[health care systems|systems]].<ref>Levy, Judith and Bernice Pescosolido (2002). ''Social Networks and Health''. Boston, MA: JAI Press.</ref><br />
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Analysis of social networks is increasingly incorporated into health care analytics, not only in epidemiological studies but also in models of patient communication and education, disease prevention, mental health diagnosis and treatment, and in the study of health care organizations and systems.<br />
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社会网络分析越来越多地被纳入卫生保健分析——不仅在'''<font color="#ff8000">流行病学 Epidemiology</font>'''研究中，而且在病人沟通和教育、疾病预防、心理健康诊断和治疗模型中，以及在卫生保健组织和系统的研究中。<br />
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===Human ecology 人类生态学===<br />
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[[Human ecology]] is an [[interdisciplinary]] and [[transdisciplinary]] study of the relationship between [[human]]s and their [[natural environment|natural]], [[Social environment|social]], and [[built environment]]s. The scientific philosophy of human ecology has a diffuse history with connections to [[geography]], [[sociology]], [[psychology]], [[anthropology]], [[zoology]], and natural [[ecology]].<ref>Crona, Beatrice and Klaus Hubacek (eds.) (2010). [http://www.ecologyandsociety.org/issues/view.php?sf=48 "Special Issue: Social network analysis in natural resource governance"]. ''Ecology and Society'', 48.</ref><ref>Ernstson, Henrich (2010). "Reading list: Using social network analysis (SNA) in social-ecological studies". [http://rs.resalliance.org/2010/11/03/reading-list-using-social-network-analysis-sna-in-social-ecological-studies/ ''Resilience Science'']</ref><br />
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Human ecology is an interdisciplinary and transdisciplinary study of the relationship between humans and their natural, social, and built environments. The scientific philosophy of human ecology has a diffuse history with connections to geography, sociology, psychology, anthropology, zoology, and natural ecology.<br />
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'''<font color="#ff8000">人类生态学 Human ecology</font>'''是研究人类与其自然环境、社会环境和'''<font color="#ff8000">建成环境 Built environment</font>'''之间关系的一门跨学科科学。人类生态学的科学哲学与地理学、社会学、心理学、人类学、动物学和自然生态学有着密切的联系。<br />
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===Language and linguistics 语言与语言学===<br />
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Studies of [[language]] and [[linguistics]], particularly [[evolutionary linguistics]], focus on the development of [[Morphology (linguistics)|linguistic forms]] and transfer of changes, [[Phonology|sounds]] or words, from one language system to another through networks of social interaction. Social networks are also important in [[language shift]], as groups of people add and/or abandon languages to their repertoire.<br />
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Studies of language and linguistics, particularly evolutionary linguistics, focus on the development of linguistic forms and transfer of changes, sounds or words, from one language system to another through networks of social interaction. Social networks are also important in language shift, as groups of people add and/or abandon languages to their repertoire.<br />
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语言学和语言学的研究，特别是'''<font color="#ff8000">演化语言学 Evolutionary Linguistics</font>'''，关注通过社会互动网络从一个语言系统转移到另一个语言系统时，语言形式的发展以及声音或词语的变化。社交网络在语言转换中也很重要，因为一些人群增加或者放弃了他们的语言。<br />
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===Literary networks 文学网络===<br />
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In the study of literary systems, network analysis has been applied by Anheier, Gerhards and Romo,<ref>{{cite journal | last1 = Anheier | first1 = H. K. | last2 = Romo | first2 = F. P. | year = 1995 | title = Forms of capital and social structure of fields: examining Bourdieu's social topography | url = | journal = American Journal of Sociology | volume = 100 | issue = 4| pages = 859–903 | doi=10.1086/230603}}</ref> De Nooy,<ref>{{cite journal | last1 = De Nooy | first1 = W | year = 2003| title = Fields and networks: Correspondence analysis and social network analysis in the framework of Field Theory | url = | journal = Poetics | volume = 31 | issue = 5–6| pages = 305–327 | doi = 10.1016/S0304-422X(03)00035-4 }}</ref> and Senekal,<ref>{{cite journal | last1 = Senekal | first1 = B. A. | year = 2012 | title = Die Afrikaanse literêre sisteem: ʼn Eksperimentele benadering met behulp van Sosiale-netwerk-analise (SNA) | url = | journal = LitNet Akademies | volume = 9 | issue = | page = 3 }}</ref> to study various aspects of how literature functions. The basic premise is that polysystem theory, which has been around since the writings of [[Even-Zohar]], can be integrated with network theory and the relationships between different actors in the literary network, e.g. writers, critics, publishers, literary histories, etc., can be mapped using [[Computer graphics (computer science)|visualization]] from SNA.<br />
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In the study of literary systems, network analysis has been applied by Anheier, Gerhards and Romo, De Nooy, and Senekal, to study various aspects of how literature functions. The basic premise is that polysystem theory, which has been around since the writings of Even-Zohar, can be integrated with network theory and the relationships between different actors in the literary network, e.g. writers, critics, publishers, literary histories, etc., can be mapped using visualization from SNA.<br />
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在文学体系的研究中，网络分析被'''阿海尔 Anheier'''、'''格尔哈兹 Gerhards'''和'''罗姆 Romo'''、'''努伊 De Nooy'''和Senekal应用于研究文学如何运作的各个方面。其基本前提是将'''文-佐哈尔 Even-Zohar'''著述以来就存在的多元系统理论可以与网络理论相结合，以及文学网络中不同行为者（如作家、评论家、出版商、文学史等）之间的关系可以通过SNA可视化映射。<br />
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===Organizational studies 组织研究===<br />
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Research studies of [[Formal organization|formal]] or [[informal organization]] [[Social relation|relationships]], [[organizational communication]], [[economics]], [[economic sociology]], and other [[resource]] [[Transfer function|transfers]]. Social networks have also been used to examine how organizations interact with each other, characterizing the many [[Interlocking directorate|informal connections]] that link executives together, as well as associations and connections between individual employees at different organizations.<ref>{{cite journal | last1 = Podolny | first1 = J. M. | last2 = Baron | first2 = J. N. | year = 1997 | title = Resources and relationships: Social networks and mobility in the workplace | url = | journal = American Sociological Review | volume = 62 | issue = 5| pages = 673–693 | doi=10.2307/2657354| jstor = 2657354 | citeseerx = 10.1.1.114.6822 }}</ref> Intra-organizational networks have been found to affect [[organizational commitment]],<ref>{{cite journal | last1 = Lee | first1 = J. | last2 = Kim | first2 = S. | year = 2011 | title = Exploring the role of social networks in affective organizational commitment: Network centrality, strength of ties, and structural holes | url = | journal = The American Review of Public Administration | volume = 41 | issue = 2| pages = 205–223 | doi=10.1177/0275074010373803}}</ref> [[organizational identification]],<ref name="Jone11">{{cite journal | last1 = Jones | first1 = C. | last2 = Volpe | first2 = E.H. | year = 2011 | title = Organizational identification: Extending our understanding of social identities through social networks | url = | journal = Journal of Organizational Behavior | volume = 32 | issue = 3| pages = 413–434 | doi=10.1002/job.694}}</ref> [[Organizational citizenship behavior|interpersonal citizenship behaviour]].<ref>{{cite journal | last1 = Bowler | first1 = W. M. | last2 = Brass | first2 = D. J. | year = 2011 | title = Relational correlates of interpersonal citizenship behaviour: A social network perspective | doi = 10.1037/0021-9010.91.1.70 | pmid = 16435939 | journal = Journal of Applied Psychology | volume = 91 | issue = 1| pages = 70–82 | citeseerx = 10.1.1.516.8746 }}</ref><br />
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Research studies of formal or informal organization relationships, organizational communication, economics, economic sociology, and other resource transfers. Social networks have also been used to examine how organizations interact with each other, characterizing the many informal connections that link executives together, as well as associations and connections between individual employees at different organizations. Intra-organizational networks have been found to affect organizational commitment, organizational identification, interpersonal citizenship behaviour.<br />
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研究正式或非正式组织关系、'''<font color="#ff8000">组织沟通 Organizational Communication</font>'''、经济、经济社会学和其他资源转移。社交网络也被用来研究组织之间如何相互作用，描述了许多将高管联系在一起的非正式联系，以及不同组织的个体雇员之间的联系。研究发现，组织内网络对'''<font color="#ff8000">组织承诺 Organizational Commitment</font>'''、组织认同、人际公民行为有影响。<br />
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===Social capital 社会资本===<br />
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[[Social capital]] is a form of [[Capital (economics)|economic]] and [[cultural capital]] in which social networks are central, [[Stock and flow|transactions]] are marked by [[Reciprocity (social psychology)|reciprocity]], [[Trust (social sciences)|trust]], and [[cooperation]], and [[Market (economics)|market]] [[Agent (economics)|agents]] produce [[goods and services]] not mainly for themselves, but for a [[common good]]. [[Social capital]] is split into three dimensions: the structural, the relational and the cognitive dimension. The structural dimension describes how partners interact with each other and which specific partners meet in a social network. Also The structural dimension of social capital indicates the level of ties among organizations.<ref>(Claridge, 2018).</ref>. This dimension is highly connected to the relational dimension which refers to trustworthiness, norms, expectations and idenfications of the bonds between partners.The relational dimension explains the nature of these ties which is mainly illustrated by the level of trust accorded to the network of organizations. <ref>(Claridge, 2018).</ref> The cognitive dimension analyses the extent to which organizations share common goals and objectives as a result of their ties and interactions. <ref>(Claridge, 2018).</ref><br />
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Social capital is a form of economic and cultural capital in which social networks are central, transactions are marked by reciprocity, trust, and cooperation, and market agents produce goods and services not mainly for themselves, but for a common good. Social capital is split into three dimensions: the structural, the relational and the cognitive dimension. The structural dimension describes how partners interact with each other and which specific partners meet in a social network. Also The structural dimension of social capital indicates the level of ties among organizations. This dimension is highly connected to the relational dimension which refers to trustworthiness, norms, expectations and idenfications of the bonds between partners. The relational dimension explains the nature of these ties which is mainly illustrated by the level of trust accorded to the network of organizations. The cognitive dimension analyses the extent to which organizations share common goals and objectives as a result of their ties and interactions. <br />
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社会资本是一种以社会网络为中心的经济和文化资本，交易以互惠、信任和合作为特征，市场主体生产的商品和服务主要不是为了自已，而是为了共同的利益。社会资本分为三个维度: 结构维度、关系维度和认知维度。结构维度描述了合作伙伴之间如何相互作用，以及哪些特定的合作伙伴在社交网络中相遇。社会资本的结构维度反映了组织之间的关系水平。这个维度与关系维度高度相关，关系维度指的是伙伴之间关系的可信度、规范、期望和认同度。 关系维度解释了这些联系的本质，主要表现在对组织网络的信任程度上。认知维度分析组织在多大程度上因其联系和相互作用而共享共同的目标和目的。<br />
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[[Social capital]] is a sociological concept about the value of [[social relation]]s and the role of cooperation and confidence to achieve positive outcomes. The term refers to the value one can get from their social ties. For example, newly arrived immigrants can make use of their social ties to established migrants to acquire jobs they may otherwise have trouble getting (e.g., because of unfamiliarity with the local language). A positive relationship exists between social capital and the intensity of social network use.<ref>{{cite journal|last=Sebastián|first=Valenzuela|author2=Namsu Park |author3=Kerk F. Kee |title=Is There Social Capital in a Social Network Site? Facebook Use and College Students' Life Satisfaction, Trust, and Participation|journal=Journal of Computer-Mediated Communication|year=2009|volume=14|issue=4|pages=875–901|doi=10.1111/j.1083-6101.2009.01474.x|doi-access=free}}</ref><ref>{{cite journal | title=Social Connectivity in America: Changes in Adult Friendship Network Size from 2002 to 2007 |author1=Wang, Hua |author2=Barry Wellman |lastauthoramp=yes | journal=American Behavioral Scientist | year=2010 | volume=53 | issue=8 | pages=1148–1169 | doi=10.1177/0002764209356247|url=https://semanticscholar.org/paper/402068f74efb35c093e7fd8c0f8268521219e242 }}</ref> In a dynamic framework, higher activity in a network feeds into higher social capital which itself encourages more activity.<ref name="GaudeulGiannetti2013">{{cite journal|last1=Gaudeul|first1=Alexia|last2=Giannetti|first2=Caterina|title=The role of reciprocation in social network formation, with an application to LiveJournal|journal=Social Networks|volume=35|issue=3|year=2013|pages=317–330|issn=03788733|doi=10.1016/j.socnet.2013.03.003}}</ref><br />
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Social capital is a sociological concept about the value of social relations and the role of cooperation and confidence to achieve positive outcomes. The term refers to the value one can get from their social ties. For example, newly arrived immigrants can make use of their social ties to established migrants to acquire jobs they may otherwise have trouble getting (e.g., because of unfamiliarity with the local language). A positive relationship exists between social capital and the intensity of social network use. In a dynamic framework, higher activity in a network feeds into higher social capital which itself encourages more activity.<br />
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社会资本是一个关于社会关系的价值、以及合作与信心对取得积极成果的作用的社会学概念。该术语指的是一个人可以从其社会关系中获得的价值。例如，新来的移民可以利用他们与先到的移民的社会关系，获得可能本难以获得的工作（如因为不熟悉当地语言）。社会资本与社会网络使用强度存在正相关关系。在一个动态的框架中，网络中更高的活动会产生更高的社会资本，而社会资本本身又会激励更多的活动。<br />
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=== Advertising 广告===<br />
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This particular cluster focuses on brand-image and promotional strategy effectiveness, taking into account the impact of customer participation on sales and brand-image. This is gauged through techniques such as sentiment analysis which rely on mathematical areas of study such as data mining and analytics. This area of research produces vast numbers of commercial applications as the main goal of any study is to understand consumer behaviour and drive sales.<br />
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This particular cluster focuses on brand-image and promotional strategy effectiveness, taking into account the impact of customer participation on sales and brand-image. This is gauged through techniques such as sentiment analysis which rely on mathematical areas of study such as data mining and analytics. This area of research produces vast numbers of commercial applications as the main goal of any study is to understand consumer behaviour and drive sales.<br />
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这个特别的集群侧重于品牌形象和促销策略的有效性，同时考虑到顾客参与对销售和品牌形象的影响。这是通过诸如基于数学领域的研究（如数据挖掘和分析）的情感分析等技术来衡量的。这一研究领域产生了大量的商业应用，因为任何研究的主要目标都是了解消费者行为并推动销售。<br />
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===Network position and benefits 网络地位和利益===<br />
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In many [[Formal organizations|organizations]], members tend to focus their activities inside their own groups, which stifles creativity and restricts opportunities. A player whose network bridges structural holes has an advantage in detecting and developing rewarding opportunities.<ref name="Burt 2004"/> Such a player can mobilize social capital by acting as a "broker" of information between two clusters that otherwise would not have been in contact, thus providing access to new ideas, opinions and opportunities. British philosopher and political economist [[John Stuart Mill]], writes, "it is hardly possible to overrate the value ... of placing human beings in contact with persons dissimilar to themselves.... Such communication [is] one of the primary sources of progress."<ref>{{cite book|last=Mill|first=John|title=Principles of Political Economy|year=1909|publisher=William J Ashley|location=Library of Economics and Liberty}}</ref> Thus, a player with a network rich in structural holes can add value to an organization through new ideas and opportunities. This in turn, helps an individual's career development and advancement.<br />
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In many organizations, members tend to focus their activities inside their own groups, which stifles creativity and restricts opportunities. A player whose network bridges structural holes has an advantage in detecting and developing rewarding opportunities. Thus, a player with a network rich in structural holes can add value to an organization through new ideas and opportunities. This in turn, helps an individual's career development and advancement.<br />
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在许多组织中，成员倾向于把他们的活动集中在他们自己的团队中，这抑制了创造力并限制了机会。一个网络连接结构洞的玩家在发现和开发机会方面有优势。因此，一个拥有富含结构洞的网络的玩家可以通过新的想法和机会为组织增加价值。这反过来又有助于个人的职业发展和晋升。<br />
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A social capital broker also reaps control benefits of being the facilitator of information flow between contacts. In the case of consulting firm Eden McCallum, the founders were able to advance their careers by bridging their connections with former big three consulting firm consultants and mid-size industry firms.<ref>{{cite journal|last=Gardner|first=Heidi|author2=Eccles, Robert |title=Eden McCallum: A Network Based Consulting Firm|journal=Harvard Business School Review|year=2011}}</ref> By bridging structural holes and mobilizing social capital, players can advance their careers by executing new opportunities between contacts.<br />
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A social capital broker also reaps control benefits of being the facilitator of information flow between contacts. In the case of consulting firm Eden McCallum, the founders were able to advance their careers by bridging their connections with former big three consulting firm consultants and mid-size industry firms. By bridging structural holes and mobilizing social capital, players can advance their careers by executing new opportunities between contacts.<br />
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社会资本中介人还可以从促进联系人之间的信息流动中获得控制权利。在咨询公司 Eden McCallum 的案例中，创始人能够通过与前三大咨询公司顾问和中等规模的行业公司建立联系来发展其事业。通过连接结构洞和调动社会资本，玩家可以通过在联系人之间执行新的机会来发展其事业。<br />
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There has been research that both substantiates and refutes the benefits of information brokerage. A study of high tech Chinese firms by Zhixing Xiao found that the control benefits of structural holes are "dissonant to the dominant firm-wide spirit of cooperation and the information benefits cannot materialize due to the communal sharing values" of such organizations.<ref>Xiao, Zhixing; Tsui, Anne (2007). "When Brokers May Not Work: The Cultural Contingency of [http://iniciarsesionentrar.com/facebook/ Social] Capital in Chinese High-tech Firms". ''Administrative Science Quarterly''.</ref> However, this study only analyzed Chinese firms, which tend to have strong communal sharing values. Information and control benefits of structural holes are still valuable in firms that are not quite as inclusive and cooperative on the firm-wide level. In 2004, Ronald Burt studied 673 managers who ran the supply chain for one of America's largest electronics companies. He found that managers who often discussed issues with other groups were better paid, received more positive job evaluations and were more likely to be promoted.<ref name="Burt 2004"/> Thus, bridging structural holes can be beneficial to an organization, and in turn, to an individual's career.<br />
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There has been research that both substantiates and refutes the benefits of information brokerage. A study of high tech Chinese firms by Zhixing Xiao found that the control benefits of structural holes are "dissonant to the dominant firm-wide spirit of cooperation and the information benefits cannot materialize due to the communal sharing values" of such organizations. However, this study only analyzed Chinese firms, which tend to have strong communal sharing values. Information and control benefits of structural holes are still valuable in firms that are not quite as inclusive and cooperative on the firm-wide level. In 2004, Ronald Burt studied 673 managers who ran the supply chain for one of America's largest electronics companies. He found that managers who often discussed issues with other groups were better paid, received more positive job evaluations and were more likely to be promoted. Thus, bridging structural holes can be beneficial to an organization, and in turn, to an individual's career.<br />
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有研究证实和反驳了信息中介的好处。Zhixing Xiao 对中国高科技企业的一项研究发现，结构洞的控制效益“与占主导地位的企业合作精神不协调，且因这类组织的共享价值观，信息效益不能实现”。然而，这项研究只分析了往往有很强的公共共享价值观的中国企业。结构洞的信息和控制利益对于那些在整个公司层面上不那么具有包容性和合作性的公司来说仍然是有价值的。2004年，'''罗纳德·伯特 Ronald Burt'''调查了为美国最大的电子公司之一管理供应链的673名管理者。他发现，那些经常与其他团体讨论问题的管理人员收入更高，得到的工作评价更积极，也更有可能获得晋升。因此，'''<font color="#32CD32">连接</font>'''结构洞对组织有益，反过来对个人的职业生涯也有益。<br />
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===Social media 社交媒体===<br />
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[[Computer network]]s combined with social networking software produces a new medium for social interaction.<ref>{{Cite book|title=The International Encyclopedia of Media Effects|last=Amichai-Hamburger|first=Yair|last2=Hayat|first2=Tsahi|date=2017|publisher=John Wiley & Sons, Inc.|isbn=9781118783764|language=en|doi=10.1002/9781118783764.wbieme0170}}</ref> A relationship over a computerized [[social networking service]] can be characterized by context, direction, and strength. The content of a relation refers to the resource that is exchanged. In a [[computer mediated communication]] context, social pairs exchange different kinds of information, including sending a data file or a computer program as well as providing emotional support or arranging a meeting. With the rise of [[electronic commerce]], information exchanged may also correspond to exchanges of money, goods or services in the "real" world.<ref>{{cite journal | title=Studying Online Social Networks | last1=Garton|first1=Laura|first2=Caroline |last2=Haythornthwaite|author2-link=Caroline Haythornthwaite|first3=Barry|last3=Wellman|author3-link=Barry Wellman | journal=Journal of Computer-Mediated Communication |date=23 June 2006 | volume=3 | issue=1 |pages = 0| doi=10.1111/j.1083-6101.1997.tb00062.x}}</ref> [[Social network analysis]] methods have become essential to examining these types of computer mediated communication.<br />
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Computer networks combined with social networking software produces a new medium for social interaction. A relationship over a computerized social networking service can be characterized by context, direction, and strength. The content of a relation refers to the resource that is exchanged. In a computer mediated communication context, social pairs exchange different kinds of information, including sending a data file or a computer program as well as providing emotional support or arranging a meeting. With the rise of electronic commerce, information exchanged may also correspond to exchanges of money, goods or services in the "real" world. Social network analysis methods have become essential to examining these types of computer mediated communication.<br />
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计算机网络与社交网络软件的结合为社交互动提供了一种新的媒介。计算机化的社交网络服务上的关系可以通过上下文、方向和强度来描述。关系的内容指的是交换的资源。在'''<font color="#ff8000">电脑中介传播 Computer-mediated Communication</font>'''环境中，社会对交换不同种类的信息，包括发送数据文件或计算机程序，以及提供情感支持或安排会议。随着'''<font color="#ff8000">电子商务 Electronic Commerce</font>'''的兴起，交换的信息也可能对应于“现实”世界中的货币、商品或服务的交换。社会网络分析方法已经成为检验此类计算机为媒介的交流的必要手段。<br />
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In addition, the sheer size and the volatile nature of [[social media]] has given rise to new network metrics. A key concern with networks extracted from social media is the lack of robustness of network metrics given missing data.<ref>{{cite journal | last1 = Wei | first1 = Wei | last2 = Joseph | first2 = Kenneth | last3 = Liu | first3 = Huan | last4 = Carley | first4 = Kathleen M. | year = 2016 | title = Exploring Characteristics of Suspended Users and Network Stability on Twitter | url = | journal = Social Network Analysis and Mining | volume = 6 | issue = | page = 51 | doi = 10.1007/s13278-016-0358-5 }}</ref><br />
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In addition, the sheer size and the volatile nature of social media has given rise to new network metrics. A key concern with networks extracted from social media is the lack of robustness of network metrics given missing data.<br />
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此外，社交媒体的庞大规模和不稳定性也催生了新的网络指标。从社交媒体中提取网络的一个关键问题是，在缺失数据的情况下，网络指标缺乏鲁棒性。<br />
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=='''<font color="#32CD32">See also 参阅</font>'''==<br />
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* [[Bibliography of sociology]] 社会学参考书目<br />
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* [[Business networking]] 商业社交<br />
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* [[Collective network]] 集体网络<br />
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* [[International Network for Social Network Analysis]] 国际社会网络学会<br />
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* [[Network society]] 网络社会<br />
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* [[Network theory]] 网络理论<br />
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*[[Network science]] 网络科学<br />
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* [[Semiotics of social networking]] 社交网络符号学<br />
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* [[Scientific collaboration network]] 科学合作网络<br />
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* [[Social network analysis]] 社会网络分析<br />
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* [[Social network (sociolinguistics)]] 社会网络<br />
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* [[Social networking service]] 社会网络服务<br />
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* [[Social web]] 社交网络<br />
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* [[Structural fold]] 结构折叠<br />
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==References 参考资料==<br />
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{{reflist}}<br />
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==Further reading 进一步阅读==<br />
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* {{cite journal |last1=Aneja |first1=Nagender |last2=Gambhir |first2=Sapna |title=Ad-hoc Social Network: A Comprehensive Survey |url=http://www.ijser.org/researchpaper%5CAd-hoc-Social-Network-A-Comprehensive-Survey.pdf |journal=International Journal of Scientific & Engineering Research |volume=4 |issue=8 |date=August 2013 |pages=156–160|issn=2229-5518}}<br />
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* {{cite book |last=Barabási |first=Albert-László |year=2003 |title=Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life |publisher=Plum |isbn=978-0-452-28439-5 |url-access=registration |url=https://archive.org/details/linkedhoweveryth00bara }}<br />
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* {{cite book |last=Barnett |first=George A. |year=2011 |title=Encyclopedia of Social Networks |publisher=Sage |isbn=978-1-4129-7911-5}}<br />
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* {{cite book |last=Estrada |first=E |year=2011 |title=The Structure of Complex Networks: Theory and Applications |publisher=Oxford University Press |isbn=978-0-199-59175-6}}<br />
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* {{cite book |last=Ferguson |first=Niall |year=2018 |title=The Square and the Tower: Networks and Power, from the Freemasons to Facebook |isbn=978-0735222915 |publisher=Penguin Press}}<br />
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* {{cite book |last=Freeman |first=Linton C. |year=2004 |title=The Development of Social Network Analysis: A Study in the Sociology of Science |publisher=Empirical Press |isbn=978-1-59457-714-7}}<br />
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* {{cite book |last=Kadushin |first=Charles |year=2012 |title=Understanding Social Networks: Theories, Concepts, and Findings |location= |publisher=Oxford University Press |isbn=978-0-19-537946-4}}<br />
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* {{cite book |last1=Mauro |first1=Rios |last2=Petrella |first2=Carlos |year=2014 |title=La Quimera de las Redes Sociales |trans-title=The Chimera of Social Networks |language=Spanish |publisher=Bubok España |isbn=978-9974-99-637-3}}<br />
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* {{cite book |last1=Rainie |first1=Lee |last2=Wellman |first2=Barry |year=2012 |title=Networked: The New Social Operating System |url=https://archive.org/details/networkednewsoci0000rain |url-access=registration |location=Cambridge, Mass. |publisher=MIT Press |isbn=978-0262017190}}<br />
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* {{cite book |last=Scott |first=John |year=1991 |title=Social Network Analysis: A Handbook |publisher=Sage |isbn=978-0-7619-6338-7}}<br />
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* {{cite book |last1=Wasserman |first1=Stanley |last2=Faust |first2=Katherine |year=1994 |title=Social Network Analysis: Methods and Applications |publisher=Cambridge University Press |isbn=978-0-521-38269-4 |series=Structural Analysis in the Social Sciences }}<br />
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* {{cite book |last1=Wellman |first1=Barry |last2=Berkowitz |first2=S. D. |year=1988 |title=Social Structures: A Network Approach |series=Structural Analysis in the Social Sciences |publisher=Cambridge University Press |isbn=978-0-521-24441-1}}<br />
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==External links 外部链接==<br />
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{{Commons category|Social networks}}<br />
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===Organizations 组织===<br />
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* [http://www.insna.org/ International Network for Social Network Analysis]<br />
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===Peer-reviewed journals 同行评审期刊===<br />
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* ''[http://www.sciencedirect.com/science/journal/03788733 Social Networks]''<br />
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* ''[http://journals.cambridge.org/action/displaySpecialPage?pageId=3656 Network Science]''<br />
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* ''[http://www.cmu.edu/joss/content/articles/volindex.html Journal of Social Structure]''<br />
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* ''[http://www.tandfonline.com/toc/gmas20/current Journal of Mathematical Sociology]''<br />
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* ''[https://www.springer.com/computer/database+management+%26+information+retrieval/journal/13278 Social Network Analysis and Mining (SNAM)]''<br />
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* {{cite journal |title=INSNA - ''Connections'' Journal |journal=Connections Bulletin of the International Network for Social Network Analysis |location=Toronto |publisher=International Network for Social Network Analysis |url=http://www.insna.org/pubs/connections/ |issn=0226-1766 |url-status=dead |archiveurl=https://web.archive.org/web/20130718063641/http://www.insna.org/pubs/connections/ |archivedate=2013-07-18 }}<br />
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===Textbooks and educational resources 课本及教育资源===<br />
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* ''[http://www.cs.cornell.edu/home/kleinber/networks-book/ Networks, Crowds, and Markets]'' (2010) by D. Easley & J. Kleinberg<br />
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* ''[http://faculty.ucr.edu/~hanneman/nettext/ Introduction to Social Networks Methods]'' (2005) by R. Hanneman & M. Riddle<br />
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* [http://www.analytictech.com/networks/ Social Network Analysis Instructional Web Site] by S. Borgatti<br />
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* ''[https://web.archive.org/web/20160304063126/http://www.novagob.org/file/view/153259/ebook-gratuito-sobre-redes-sociales-virtuales Guide for virtual social networks for public administrations]'' (2015) by Mauro D. Ríos (in Spanish)<br />
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===Data sets 数据集===<br />
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* [http://pajek.imfm.si/doku.php?id=data:urls:index Pajek's list of lists of datasets]<br />
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* [http://networkdata.ics.uci.edu/index.html UC Irvine Network Data Repository]<br />
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* [http://snap.stanford.edu/data/ Stanford Large Network Dataset Collection]<br />
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* [http://www-personal.umich.edu/~mejn/netdata/ M.E.J. Newman datasets]<br />
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* [http://vlado.fmf.uni-lj.si/pub/networks/data/ Pajek datasets]<br />
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* [http://wiki.gephi.org/index.php?title=Datasets#Social_networks Gephi datasets]<br />
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* [http://konect.uni-koblenz.de/networks KONECT – Koblenz network collection]<br />
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* [http://www.stats.ox.ac.uk/~snijders/siena/ RSiena datasets]<br />
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==编者推荐==<br />
集智俱乐部：[https://swarma.org/?p=18646 社会网络分析的四个维度：研究方法、应用和软件工具综述 | 网络科学论文速递25篇]<br />
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[http://arxiv.org/abs/2002.09485 The Four Dimensions of Social Network Analysis: An Overview of Research Methods, Applications, and Software Tools]：近年来，基于社交网络的应用呈指数级增长，其中一个方面的原因是，这个应用领域提供了一个特别肥沃的地方，可以测试和开发最先进的计算技术，从 Web 中提取有价值的信息。这项工作的主要贡献有三个方面：(1) 我们对社会网络分析 (SNA) 的最新发展水平进行了文献综述；(2) 我们提出了一套基于 SNA 四个基本特征（或维度）的新度量标准；(3) 最后，我们对一套流行的 SNA 工具和框架进行了定量分析。我们还进行了一项科学计量学研究，以检测该领域中最活跃的研究领域和应用领域。本工作提出了四个不同维度的定义，即(a)模式、(b)知识发现、(c)信息融合、(d)集成、可伸缩性和可视化，用于定义一组新的度量（称为度），以评估 SNA 的不同软件工具和框架（根据之前的度量对一组 20 个 SNA-软件工具进行分析和排序）。这些维度连同定义的程度，可以评价和衡量社会网络技术的成熟度，寻求对它们的定量评估，从而揭示这一活跃领域的挑战和未来发展趋势。<br />
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<small>This page was moved from [[wikipedia:en:Social network]]. Its edit history can be viewed at [[社会网络/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E7%A4%BE%E4%BC%9A%E7%BD%91%E7%BB%9C&diff=12505社会网络2020-08-13T16:16:06Z<p>Dorr：</p>
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<div>已由Dorr初步翻译<br />
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此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Social structure made up of a set of social actors}}<br />
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{{about|the theoretical concept as used in the social and behavioral sciences|social networking sites|Social networking service|the 2010 movie|The Social Network|other uses|Social network (disambiguation)}}<br />
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A '''social network ''' is a [[social structure]] made up of a set of [[social]] actors (such as individuals or organizations), sets of [[Dyad (sociology)|dyadic]] ties, and other [[Social relation|social interactions]] between actors. The social network perspective provides a set of methods for analyzing the structure of whole social entities as well as a variety of theories explaining the patterns observed in these structures.<ref name="WF94CH1">{{cite book|last1=Wasserman |first1=Stanley | author-link1= Stanley Wasserman|last2=Faust |first2=Katherine |year=1994 |title=Social Network Analysis: Methods and Applications |isbn=9780521387071 |chapter=Social Network Analysis in the Social and Behavioral Sciences |pages=1–27 |publisher=Cambridge University Press}}</ref> The study of these structures uses [[social network analysis]] to identify local and global patterns, locate influential entities, and examine network dynamics.<br />
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A social network is a social structure made up of a set of social actors (such as individuals or organizations), sets of dyadic ties, and other social interactions between actors. The social network perspective provides a set of methods for analyzing the structure of whole social entities as well as a variety of theories explaining the patterns observed in these structures. The study of these structures uses social network analysis to identify local and global patterns, locate influential entities, and examine network dynamics.<br />
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'''<font color="#ff8000">社会网络 Social Network</font>'''是一种由一组社会行为者（个人或组织）、一组二元关系以及行为者之间的其他社会互动组成的社会结构。社会网络视角为分析整个社会实体的结构提供了一套方法，也为解释在这些结构中观察到的模式提供了各种理论。对这些结构的研究使用'''<font color="#ff8000">社会网络分析 Social Network Analysis</font>'''以确定本地及全球模式，定位有影响力的实体，并审查网络动态。<br />
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Social networks and the analysis of them is an inherently [[Interdisciplinarity|interdisciplinary]] academic field which emerged from [[social psychology]], [[sociology]], [[statistics]], and [[graph theory]]. [[Georg Simmel]] authored early structural theories in sociology emphasizing the dynamics of triads and "web of group affiliations".<ref>{{cite book|last1=Scott |first1=W. Richard |last2=Davis |first2=Gerald F. |author-link1= William Richard Scott|author-link2= Gerald F. Davis|title=Organizations and Organizing |chapter=Networks In and Around Organizations |year=2003 |isbn=978-0-13-195893-7 |publisher=Pearson Prentice Hall}}</ref> [[Jacob Moreno]] is credited with developing the first [[sociogram]]s in the 1930s to study interpersonal relationships. These approaches were mathematically formalized in the 1950s and theories and methods of social networks became pervasive in the [[Social science|social and behavioral sciences]] by the 1980s.<ref name="WF94CH1"/><ref name="Freeman History">{{cite book|last=Freeman |first=Linton |year=2004 |publisher=Empirical Press |isbn=978-1-59457-714-7 |title=The Development of Social Network Analysis: A Study in the Sociology of Science}}</ref> [[Social network analysis]] is now one of the major paradigms in contemporary sociology, and is also employed in a number of other social and formal sciences. Together with other [[complex network]]s, it forms part of the nascent field of [[network science]].<ref>{{cite journal|journal=Science |year=2009 |volume=323 |number=5916 |pages=892–895 |doi=10.1126/science.1165821 |pmid=19213908 |title=Network Analysis in the Social Sciences |first1=Stephen P. |last1=Borgatti |first2=Ajay |last2=Mehra |first3=Daniel J. |last3=Brass |first4=Giuseppe |last4=Labianca|bibcode=2009Sci...323..892B |citeseerx=10.1.1.536.5568 }}</ref><ref name="EK">{{cite book|title=Networks, Crowds, and Markets: Reasoning about a Highly Connected World |first1=David |last1=Easley |first2=Jon |last2=Kleinberg |publisher=Cambridge University Press |year=2010 |chapter=Overview |pages=1–20 |isbn=978-0-521-19533-1}}</ref><br />
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Social networks and the analysis of them is an inherently interdisciplinary academic field which emerged from social psychology, sociology, statistics, and graph theory. Georg Simmel authored early structural theories in sociology emphasizing the dynamics of triads and "web of group affiliations". Jacob Moreno is credited with developing the first sociograms in the 1930s to study interpersonal relationships. These approaches were mathematically formalized in the 1950s and theories and methods of social networks became pervasive in the social and behavioral sciences by the 1980s. Social network analysis is now one of the major paradigms in contemporary sociology, and is also employed in a number of other social and formal sciences. Together with other complex networks, it forms part of the nascent field of network science.<br />
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社会网络及其分析是社会心理学、社会学、统计学和图论中内在的跨学科学术领域。'''格奥尔格·齐美尔 Georg Simmel'''是早期社会学结构主义理论的作者，他强调三合会和“群体关系网”的动态性。'''雅各布·莫雷诺 Jacob Moreno'''被认为是在20世纪30年代发展了第一份'''<font color="#ff8000">社交关系图 Sociogram</font>'''以研究人际关系的人。这些方法在20世纪50年代得到数学化，到20世纪80年代，社会网络的理论和方法在社会和行为科学中变得普遍。社会网络分析现在是当代社会学的主要范式之一，也被用于许多其他社会科学及形式科学。它与其他复杂网络一起构成了网络科学新兴领域的一部分。<br />
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==Overview 概览==<br />
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[[File:Barabasi Albert model.gif|thumb|left|图1：Evolution graph of a social network: [[Barabási–Albert model|Barabási model]].一个社会网络的演化图：巴拉巴西模型。]]<br />
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Barabási model.]]<br />
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Barabási model.]]<br />
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The social network is a [[Scientific theory|theoretical]] [[Construct (philosophy of science)|construct]] useful in the [[social sciences]] to study relationships between individuals, [[social groups|groups]], [[formal organizations|organizations]], or even entire [[society|societies]] ([[social unit]]s, see [[Differentiation (sociology)|differentiation]]). The term is used to describe a [[social structure]] determined by such [[social interactions|interactions]]. The ties through which any given social unit connects represent the convergence of the various social contacts of that unit. This theoretical approach is, necessarily, relational. An [[axiom]] of the social network approach to understanding [[social interaction]] is that social phenomena should be primarily conceived and investigated through the properties of relations between and within units, instead of the properties of these units themselves. Thus, one common criticism of social network theory is that [[Agency (sociology)|individual agency]] is often ignored<ref name="jscott">Scott, John P. (2000). ''Social Network Analysis: A Handbook'' (2nd edition). Thousand Oaks, CA: Sage Publications.</ref> although this may not be the case in practice (see [[agent-based model]]ing). Precisely because many different types of relations, singular or in combination, form these network configurations, [[Network science|network analytics]] are useful to a broad range of research enterprises. In social science, these fields of study include, but are not limited to [[anthropology]], [[biology]], [[communication studies]], [[economics]], [[geography]], [[information science]], [[organizational studies]], [[social psychology]], [[sociology]], and [[sociolinguistics]].<br />
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The social network is a theoretical construct useful in the social sciences to study relationships between individuals, groups, organizations, or even entire societies (social units, see differentiation). The term is used to describe a social structure determined by such interactions. The ties through which any given social unit connects represent the convergence of the various social contacts of that unit. This theoretical approach is, necessarily, relational. An axiom of the social network approach to understanding social interaction is that social phenomena should be primarily conceived and investigated through the properties of relations between and within units, instead of the properties of these units themselves. Thus, one common criticism of social network theory is that individual agency is often ignored although this may not be the case in practice (see agent-based modeling). Precisely because many different types of relations, singular or in combination, form these network configurations, network analytics are useful to a broad range of research enterprises. In social science, these fields of study include, but are not limited to anthropology, biology, communication studies, economics, geography, information science, organizational studies, social psychology, sociology, and sociolinguistics.<br />
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社会网络是社会科学中研究个人、团体、组织甚至整个社会（社会单位，见分化）之间关系的理论构造。这个术语用来描述由这种相互作用决定的社会结构。任何一个特定的社会单元之间的联系都代表着这个单元各种社会联系的聚合。这种理论方法必然是相关的。理解社会互动的社会网络方法的一个公理是，社会现象应该主要通过单元之间和单元内部关系的性质来构思和研究，而不是这些单元本身的性质。因此，社会网络理论总被诟病的一点是其常忽视个体代理，而实践中可能并非如此（见基于主体的建模）。正是因为许多不同类型的关系（单独或组合形式）形成这些网络配置，网络分析在广泛的研究中有用。在社会科学中，这些研究领域包括但不限于'''<font color="#ff8000">人类学 Anthropology</font>'''、生物学、'''<font color="#ff8000">传播学 Communication Studies</font>'''、经济学、地理学、信息科学、组织学、社会心理学、社会学和'''<font color="#ff8000">社会语言学 Sociolinguistics</font>'''。<br />
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==History 历史==<br />
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In the late 1890s, both [[Émile Durkheim]] and [[Ferdinand Tönnies]] foreshadowed the idea of social networks in their theories and research of [[social group]]s. Tönnies argued that social groups can exist as personal and direct social ties that either link individuals who share values and belief (''[[Gemeinschaft]]'', German, commonly translated as "[[community]]") or impersonal, formal, and instrumental social links (''[[Gesellschaft]]'', German, commonly translated as "[[society]]").<ref>Tönnies, Ferdinand (1887). ''Gemeinschaft und Gesellschaft'', Leipzig: Fues's Verlag. (Translated, 1957 by Charles Price Loomis as ''Community and Society'', East Lansing: Michigan State University Press.)</ref> Durkheim gave a non-individualistic explanation of social facts, arguing that social phenomena arise when interacting individuals constitute a reality that can no longer be accounted for in terms of the properties of individual actors.<ref>Durkheim, Emile (1893). ''De la division du travail social: étude sur l'organisation des sociétés supérieures'', Paris: F. Alcan. (Translated, 1964, by Lewis A. Coser as ''The Division of Labor in Society'', New York: Free Press.)</ref> [[Georg Simmel]], writing at the turn of the twentieth century, pointed to the nature of networks and the effect of network size on interaction and examined the likelihood of interaction in loosely knit networks rather than groups.<ref>Simmel, Georg (1908). ''Soziologie'', Leipzig: Duncker & Humblot.</ref><br />
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In the late 1890s, both Émile Durkheim and Ferdinand Tönnies foreshadowed the idea of social networks in their theories and research of social groups. Tönnies argued that social groups can exist as personal and direct social ties that either link individuals who share values and belief (Gemeinschaft, German, commonly translated as "community") or impersonal, formal, and instrumental social links (Gesellschaft, German, commonly translated as "society"). Durkheim gave a non-individualistic explanation of social facts, arguing that social phenomena arise when interacting individuals constitute a reality that can no longer be accounted for in terms of the properties of individual actors. Georg Simmel, writing at the turn of the twentieth century, pointed to the nature of networks and the effect of network size on interaction and examined the likelihood of interaction in loosely knit networks rather than groups.<br />
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在19世纪90年代晚期，'''埃米尔·涂尔干 Émile Durkheim'''和'''斐迪南·滕尼斯 Ferdinand Tönnies'''在他们关于社会群体的理论和研究中都预示了社会网络的概念。滕尼斯认为，社会群体可以作为个人和直接的社会关系存在，这种关系或者将具有共同价值观和信仰的个人（德语 Gemeinschaft，通常翻译为“社区”）联系在一起，或者将非个人的、正式的和工具性的社会关系（德语 Gesellschaft，通常翻译为“社会”）联系在一起。涂尔干对社会事实给出了非个人主义的解释，认为当相互作用的个体构成一种再也不能用个体行为者的特性来解释的现实时，社会现象就产生了。格奥尔格·齐美尔在20世纪之交的著作中指出了网络的本质和网络规模对互动的影响，并研究了在松散的网络而非群体中互动的可能性。<br />
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[[File:Moreno Sociogram 2nd Grade.png|thumb|图2：Moreno's sociogram of a 2nd grade class 莫雷诺对一个二年级班级的社会关系图]]<br />
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Major developments in the field can be seen in the 1930s by several groups in psychology, anthropology, and mathematics working independently.<ref name=jscott /><ref>For a historical overview of the development of social network analysis, see: {{cite book|last1=Carrington|first1=Peter J.|last2=Scott|first2=John|chapter=Introduction|title=The Sage Handbook of Social Network Analysis| publisher=Sage|year=2011|isbn=978-1-84787-395-8|page=1|chapter-url=https://books.google.com/books?id=2chSmLzClXgC&pg=PA1}}</ref><ref>See also the diagram in {{cite book|author=Scott, John|title=Social Network Analysis: A Handbook|publisher=Sage|year=2000|isbn=978-0-7619-6339-4|page=8|url=https://books.google.com/books?id=Ww3_bKcz6kgC&pg=PA8}}</ref> In [[psychology]], in the 1930s, [[Jacob L. Moreno]] began systematic recording and analysis of social interaction in small groups, especially classrooms and work groups (see [[sociometry]]). In [[anthropology]], the foundation for social network theory is the theoretical and [[ethnography|ethnographic]] work of [[Bronislaw Malinowski]],<ref>Malinowski, Bronislaw (1913). ''The Family Among the Australian Aborigines: A Sociological Study''. London: University of London Press.</ref> [[Radcliffe-Brown|Alfred Radcliffe-Brown]],<ref>Radcliffe-Brown, Alfred Reginald (1930) ''The social organization of Australian tribes''. Sydney, Australia: University of Sydney ''Oceania'' monographs, No.1.</ref><ref>{{cite journal | last1 = Radcliffe-Brown | first1 = A. R. | year = 1940 | title = On social structure | url = | journal = Journal of the Royal Anthropological Institute | volume = 70 | issue = 1| pages = 1–12 | doi=10.2307/2844197| jstor = 2844197 }}</ref> and [[Claude Lévi-Strauss]].<ref>Lévi-Strauss, Claude ([1947]1967). ''Les structures élémentaires de la parenté''. Paris: La Haye, Mouton et Co. (Translated, 1969 by J. H. Bell, J. R. von Sturmer, and R. Needham, 1969, as ''The Elementary Structures of Kinship'', Boston: Beacon Press.)</ref> A group of social anthropologists associated with [[Max Gluckman]] and the [[Manchester school (anthropology)|Manchester School]], including [[John Arundel Barnes|John A. Barnes]],<ref>Barnes, John (1954). "Class and Committees in a Norwegian Island Parish". ''Human Relations'', (7): 39–58.</ref> [[J. Clyde Mitchell]] and [[Elizabeth Bott Spillius]],<ref>{{cite journal | last1 = Freeman | first1 = Linton C. | last2 = Wellman | first2 = Barry | year = 1995 | title = A note on the ancestoral Toronto home of social network analysis | url = | journal = Connections | volume = 18 | issue = 2| pages = 15–19 }}</ref><ref>{{cite journal | last1 = Savage | first1 = Mike | year = 2008 | title = Elizabeth Bott and the formation of modern British sociology | url = | journal = The Sociological Review | volume = 56 | issue = 4| pages = 579–605 | doi=10.1111/j.1467-954x.2008.00806.x}}</ref> often are credited with performing some of the first fieldwork from which network analyses were performed, investigating community networks in southern Africa, India and the United Kingdom.<ref name=jscott /> Concomitantly, British anthropologist [[Siegfried Frederick Nadel|S. F. Nadel]] codified a theory of social structure that was influential in later network analysis.<ref>Nadel, S. F. 1957. ''The Theory of Social Structure''. London: Cohen and West.</ref> In [[sociology]], the early (1930s) work of [[Talcott Parsons]] set the stage for taking a relational approach to understanding social structure.<ref>Parsons, Talcott ([1937] 1949). ''The Structure of Social Action: A Study in Social Theory with Special Reference to a Group of European Writers''. New York: The Free Press.</ref><ref>Parsons, Talcott (1951). ''The Social System''. New York: The Free Press.</ref> Later, drawing upon Parsons' theory, the work of sociologist [[Peter Blau]] provides a strong impetus for analyzing the relational ties of social units with his work on [[social exchange theory]].<ref>Blau, Peter (1956). ''Bureaucracy in Modern Society''. New York: Random House, Inc.</ref><ref>Blau, Peter (1960). "A Theory of Social Integration". ''The American Journal of Sociology'', (65)6: 545–556, (May).</ref><ref>Blau, Peter (1964). ''Exchange and Power in Social Life''.</ref><br />
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Major developments in the field can be seen in the 1930s by several groups in psychology, anthropology, and mathematics working independently. In psychology, in the 1930s, Jacob L. Moreno began systematic recording and analysis of social interaction in small groups, especially classrooms and work groups (see sociometry). In anthropology, the foundation for social network theory is the theoretical and ethnographic work of Bronislaw Malinowski, Alfred Radcliffe-Brown, and Claude Lévi-Strauss. A group of social anthropologists associated with Max Gluckman and the Manchester School, including John A. Barnes, J. Clyde Mitchell and Elizabeth Bott Spillius, often are credited with performing some of the first fieldwork from which network analyses were performed, investigating community networks in southern Africa, India and the United Kingdom. In sociology, the early (1930s) work of Talcott Parsons set the stage for taking a relational approach to understanding social structure. Later, drawing upon Parsons' theory, the work of sociologist Peter Blau provides a strong impetus for analyzing the relational ties of social units with his work on social exchange theory.<br />
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20世纪30年代，心理学、人类学和数学领域的几个独立研究小组已经看到了这一领域的重大发展。在心理学方面，在20世纪30年代，雅各布·L·莫雷诺开始系统地记录和分析小团体中的社会互动，尤其是课堂和工作团体中的社会互动（见'''<font color="#ff8000">社会测量 Sociometry）</font>'''。在人类学中，社会网络理论的基础是'''布罗尼斯拉夫·马林诺夫斯基 Bronislaw Malinowski'''，'''阿尔弗雷德·拉德克利夫-布朗 Alfred Radcliffe-Brown'''和'''克洛德·列维-斯特劳斯 Claude Lévi-Strauss'''的理论和人种学著作。包括'''约翰·A·巴恩斯 John A. Barnes'''、'''J·克莱德·米切尔 J. Clyde Mitchell'''和'''伊丽莎白·博特·斯皮利厄斯 Elizabeth Bott Spillius'''在内的一群与'''马克斯·格拉克曼 Max Gluckman'''和'''曼彻斯特学派 Manchester School'''有关的社会人类学家，经常被认为是执行了一些最初的实地工作，从而进行了网络分析，调查了南非、印度和英国的社区网络。在社会学方面，'''塔尔科特·帕森斯 Talcott Parsons'''的早期工作（1930年代）为采用关系方法理解社会结构奠定了基础。后来，社会学家'''彼得·布劳 Peter Blau'''的'''<font color="#ff8000">社会交换论 Social Exchange Theory</font>'''为分析社会单位之间的关系提供了强大的动力。<br />
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By the 1970s, a growing number of scholars worked to combine the different tracks and traditions. One group consisted of sociologist [[Harrison White]] and his students at the [[Harvard Department of Social Relations|Harvard University Department of Social Relations]]. Also independently active in the Harvard Social Relations department at the time were [[Charles Tilly]], who focused on networks in political and community sociology and social movements, and [[Stanley Milgram]], who developed the "six degrees of separation" thesis.<ref>{{cite web |url=http://www.semioticon.com/semiotix/semiotix14/sem-14-05.html |title=The Networked Individual: A Profile of Barry Wellman |author=Bernie Hogan}}</ref> [[Mark Granovetter]]<ref name="Introduction for the French Reader">{{cite journal | last1 = Granovetter | first1 = Mark | year = 2007 | title = Introduction for the French Reader | url = | journal = Sociologica | volume = 2 | issue = | pages = 1–8 }}</ref> and [[Barry Wellman]]<ref>Wellman, Barry (1988). "Structural analysis: From method and metaphor to theory and substance". pp. 19–61 in B. Wellman and S. D. Berkowitz (eds.) ''Social Structures: A Network Approach'', Cambridge, UK: Cambridge University Press.</ref> are among the former students of White who elaborated and championed the analysis of social networks.<ref name="Introduction for the French Reader"/><ref>Mullins, Nicholas. ''Theories and Theory Groups in Contemporary American Sociology''. New York: Harper and Row, 1973.</ref><ref>Tilly, Charles, ed. ''An Urban World''. Boston: Little Brown, 1974.</ref><ref>Wellman, Barry. 1988. "Structural Analysis: From Method and Metaphor to Theory and Substance". pp. 19–61 in ''Social Structures: A Network Approach'', edited by Barry Wellman and S. D. Berkowitz. Cambridge: Cambridge University Press.</ref><br />
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By the 1970s, a growing number of scholars worked to combine the different tracks and traditions. One group consisted of sociologist Harrison White and his students at the Harvard University Department of Social Relations. Also independently active in the Harvard Social Relations department at the time were Charles Tilly, who focused on networks in political and community sociology and social movements, and Stanley Milgram, who developed the "six degrees of separation" thesis. Mark Granovetter and Barry Wellman are among the former students of White who elaborated and championed the analysis of social networks.<br />
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到了20世纪70年代，越来越多的学者致力于将不同的轨迹和传统结合起来。其中一组由社会学家'''哈里森·怀特 Harrison White'''和他在哈佛大学社会关系系的学生组成。当时在哈佛大学社会关系系独立活动的还有专注于政治和社区社会学和社会运动的网络的'''查尔斯·堤利 Charles Tilly'''，还有发表了六度分隔理论论文的'''斯坦利·米尔格拉姆 Stanley Milgram'''。'''马克·格兰诺维特 Mark Granovetter'''和'''巴里·威尔曼 Barry Wellman'''是怀特以前的学生，他们阐述并支持对社交网络的分析。<br />
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Beginning in the late 1990s, social network analysis experienced work by sociologists, political scientists, and physicists such as [[Duncan J. Watts]], [[Albert-László Barabási]], [[Peter Bearman]], [[Nicholas A. Christakis]], [[James H. Fowler]], and others, developing and applying new models and methods to emerging data available about online social networks, as well as "digital traces" regarding face-to-face networks.<br />
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Beginning in the late 1990s, social network analysis experienced work by sociologists, political scientists, and physicists such as Duncan J. Watts, Albert-László Barabási, Peter Bearman, Nicholas A. Christakis, James H. Fowler, and others, developing and applying new models and methods to emerging data available about online social networks, as well as "digital traces" regarding face-to-face networks.<br />
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从20世纪90年代末开始，社会网络分析经历了社会学家、政治学家和物理学家的工作，如'''[[邓肯·瓦茨 Duncan J. Watts]]'''、'''[[艾伯特-拉斯洛·巴拉巴西 Albert-László Barabási]]'''、Peter Bearman、Nicholas A. Christakis、James H. Fowler等人，开发和应用新的模型和方法来获得有关在线社会网络的新兴数据，以及有关面对面网络的“数字痕迹”。<br />
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==Levels of analysis 分析层次==<br />
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[[File:Network self-organization stages.png|thumb|right|图3：Self-organization of a network, based on Nagler, Levina, & Timme, (2011) 基于Nagler, Levina和Timme的自组织网络，(2011)<ref>{{cite journal|author1=Nagler, Jan|author2=Anna Levina|author3=Marc Timme|year=2011|title=Impact of single links in competitive percolation|journal=Nature Physics|volume=7|issue=3|pages=265–270|doi=10.1038/nphys1860|arxiv=1103.0922|bibcode=2011NatPh...7..265N}}</ref>]]<br />
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[[File:Social Network Diagram (large).svg|right|thumb|Centrality]]<br />
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Centrality<br />
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中心性<br />
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In general, social networks are [[self-organization|self-organizing]], [[emergence|emergent]], and [[social complexity|complex]], such that a globally coherent pattern appears from the local interaction of the elements that make up the system.<ref>Newman, Mark, Albert-László Barabási and Duncan J. Watts (2006). ''The Structure and Dynamics of Networks'' (Princeton Studies in Complexity). Oxford: Princeton University Press.</ref><ref>{{cite journal | last1 = Wellman | first1 = Barry | year = 2008 | title = Review: The development of social network analysis: A study in the sociology of science | url = | journal = Contemporary Sociology | volume = 37 | issue = 3| pages = 221–222 | doi=10.1177/009430610803700308}}</ref> These patterns become more apparent as network size increases. However, a global network analysis<ref>{{cite book|last=Faust|first=Stanley Wasserman; Katherine|title=Social network analysis : methods and applications|year=1998|publisher=Cambridge Univ. Press|location=Cambridge [u.a.]|isbn=978-0521382694|edition=Reprint.}}</ref> of, for example, all [[interpersonal relationships]] in the world is not feasible and is likely to contain so much [[Information theory|information]] as to be uninformative. Practical limitations of computing power, ethics and participant recruitment and payment also limit the scope of a social network analysis.<ref name="Kadu12">Kadushin, C. (2012). ''Understanding social networks: Theories, concepts, and findings''. Oxford: Oxford University Press.</ref><ref>{{Cite journal| author=Granovetter, M.|title= Network sampling: Some first steps| year=1976 |pages=1287–1303 |volume=81| issue=6 |journal= American Journal of Sociology | doi=10.1086/226224}}</ref> The nuances of a local system may be lost in a large network analysis, hence the quality of information may be more important than its scale for understanding network properties. Thus, social networks are analyzed at the scale relevant to the researcher's theoretical question. Although [[level of analysis|levels of analysis]] are not necessarily [[Mutually exclusive events|mutually exclusive]], there are three general levels into which networks may fall: [[Microsociology|micro-level]], [[wikt:meso-|meso-level]], and [[Macrosociology|macro-level]].<br />
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In general, social networks are self-organizing, emergent, and complex, such that a globally coherent pattern appears from the local interaction of the elements that make up the system. These patterns become more apparent as network size increases. However, a global network analysis of, for example, all interpersonal relationships in the world is not feasible and is likely to contain so much information as to be uninformative. Practical limitations of computing power, ethics and participant recruitment and payment also limit the scope of a social network analysis. The nuances of a local system may be lost in a large network analysis, hence the quality of information may be more important than its scale for understanding network properties. Thus, social networks are analyzed at the scale relevant to the researcher's theoretical question. Although levels of analysis are not necessarily mutually exclusive, there are three general levels into which networks may fall: micro-level, meso-level, and macro-level.<br />
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一般来说，社会网络是自组织的、涌现的和复杂的，这样，一个全局一致的模式就会从组成系统的元素的局部交互中显现出来。随着网络规模的增大，这些模式变得更加明显。然而，一个全球网络分析（如世界上所有的人际关系）是不可行的，它可能包含太多的信息，以至于相当于没有提供信息。计算能力的实际限制、道德规范以及参与者的招聘和报酬也限制了社会网络分析的规模。局部系统的细微差别在大规模网络分析中可能会消失，因此对于理解网络属性来说，信息的质量可能比其规模更重要。因此，社会网络是在与研究者的理论问题相关的尺度上加以分析的。虽然分析层次不一定相互排斥，但网络可以分为三个一般层次: 微观、中观和宏观。<br />
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===Micro level 微观层面===<br />
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At the micro-level, social network research typically begins with an individual, [[Snowball sampling|snowballing]] as social relationships are traced, or may begin with a small group of individuals in a particular social context.<br />
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At the micro-level, social network research typically begins with an individual, snowballing as social relationships are traced, or may begin with a small group of individuals in a particular social context.<br />
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在微观层面上，社会网络研究通常从个人开始，随着社会关系的追踪而像滚雪球一样扩大，或者可能从特定社会背景下的一小群个体开始。<br />
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'''Dyadic level''': A [[Dyad (sociology)|dyad]] is a social relationship between two individuals. Network research on dyads may concentrate on [[Structural functionalism|structure]] of the relationship (e.g. multiplexity, strength), [[social equality]], and tendencies toward [[Reciprocity (social and political philosophy)|reciprocity/mutuality]].<br />
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Dyadic level: A dyad is a social relationship between two individuals. Network research on dyads may concentrate on structure of the relationship (e.g. multiplexity, strength), social equality, and tendencies toward reciprocity/mutuality.<br />
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二元层面：二元是两个个体之间的社会关系。网络对二元关系的研究可以集中在关系的结构上（如多样性、力量)、社会平等以及互惠互利的倾向。<br />
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'''Triadic level''': Add one individual to a dyad, and you have a [[Triadic relation|triad]]. Research at this level may concentrate on factors such as [[Independence number|balance]] and [[Vertex-transitive graph|transitivity]], as well as [[social equality]] and tendencies toward [[Reciprocity (social and political philosophy)|reciprocity/mutuality]].<ref name="Kadu12"/> In the [[balance theory]] of [[Fritz Heider]] the triad is the key to social dynamics. The discord in a rivalrous [[love triangle]] is an example of an unbalanced triad, likely to change to a balanced triad by a change in one of the relations. The dynamics of social friendships in society has been modeled by balancing triads. The study is carried forward with the theory of [[signed graph]]s.<br />
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Triadic level: Add one individual to a dyad, and you have a triad. Research at this level may concentrate on factors such as balance and transitivity, as well as social equality and tendencies toward reciprocity/mutuality. In the balance theory of Fritz Heider the triad is the key to social dynamics. The discord in a rivalrous love triangle is an example of an unbalanced triad, likely to change to a balanced triad by a change in one of the relations. The dynamics of social friendships in society has been modeled by balancing triads. The study is carried forward with the theory of signed graphs.<br />
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三元层面：再加一个个体到二元组中，就得到一个三元组。这一层次的研究可能集中在诸如平衡和传递性等因素，以及社会平等和互惠互利的倾向。在'''弗里茨 · 海德 Fritz Heider'''的'''平衡理论 Balance Theory'''中，三元组是社会动力学的关键。在敌对的三角恋中的不和谐是不平衡的三角关系的一个例子，很可能通过其中一种关系的改变而变成平衡的三角关系。社会中社会友谊的动态模型是通过平衡三角关系建立起来的。该研究利用了符号图理论。<br />
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'''Actor level''': The smallest unit of analysis in a social network is an individual in their social setting, i.e., an "actor" or "ego". Egonetwork analysis focuses on network characteristics such as size, relationship strength, density, [[centrality]], [[wikt:prestige|prestige]] and roles such as [[isolates|isolates, liaisons]], and [[Bridge (interpersonal)|bridges]].<ref name="Jone11"/> Such analyses, are most commonly used in the fields of [[psychology]] or [[Social psychology (sociology)|social psychology]], [[ethnographic]] [[kinship]] analysis or other [[genealogy|genealogical]] studies of relationships between individuals.<br />
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Actor level: The smallest unit of analysis in a social network is an individual in their social setting, i.e., an "actor" or "ego". Egonetwork analysis focuses on network characteristics such as size, relationship strength, density, centrality, prestige and roles such as isolates, liaisons, and bridges. Such analyses, are most commonly used in the fields of psychology or social psychology, ethnographic kinship analysis or other genealogical studies of relationships between individuals.<br />
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行为者层面：社会网络中最小的分析单位是社会环境中的个体，即“行为者”或“自我”。自我网络分析主要关注网络特征，例如大小、关系强度、密度、中心性、声望和隔离、联络和桥梁等角色。这种分析最常用于心理学或社会心理学、人种学亲属关系分析或其他个体关系的系谱研究领域。<br />
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'''Subset level''': [[Subset]] levels of network research problems begin at the micro-level, but may cross over into the meso-level of analysis. Subset level research may focus on [[Distance (graph theory)|distance]] and reachability, [[cliques]], [[Cohesion (social policy)|cohesive]] subgroups, or other [[Group action (sociology)|group actions]] or [[Group behaviour|behavior]].<ref>{{cite book | title="Graph Theoretical Approaches to Social Network Analysis". in Computational Complexity: Theory, Techniques, and Applications (Robert A. Meyers, ed.) | publisher=Springer | author=de Nooy, Wouter |year=2012 | pages=2864–2877 | isbn=978-1-4614-1800-9|doi=10.1007/978-1-4614-1800-9_176| chapter=Social Network Analysis, Graph Theoretical Approaches to }}</ref><br />
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Subset level: Subset levels of network research problems begin at the micro-level, but may cross over into the meso-level of analysis. Subset level research may focus on distance and reachability, cliques, cohesive subgroups, or other group actions or behavior.<br />
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子集层面：网络研究问题的子集级别开始于微观级别，但可能跨越到中观级别的分析。子集级别的研究可能集中在距离和可达性、派系、凝聚子群或其他群体行为或行为。<br />
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===Meso level 中观层面===<br />
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In general, meso-level theories begin with a [[Sample population|population]] size that falls between the micro- and macro-levels. However, meso-level may also refer to analyses that are specifically designed to reveal connections between micro- and macro-levels. Meso-level networks are low density and may exhibit causal processes distinct from interpersonal micro-level networks.<ref>{{cite journal | last1 = Hedström | first1 = Peter | last2 = Sandell | first2 = Rickard | last3 = Stern | first3 = Charlotta | year = 2000 | title = Mesolevel Networks and the Diffusion of Social Movements: The Case of the Swedish Social Democratic Party | url = http://www.nuffield.ox.ac.uk/users/hedstrom/ajs3.pdf | journal = American Journal of Sociology | volume = 106 | issue = 1| pages = 145–172 | doi = 10.1086/303109 }}</ref><br />
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In general, meso-level theories begin with a population size that falls between the micro- and macro-levels. However, meso-level may also refer to analyses that are specifically designed to reveal connections between micro- and macro-levels. Meso-level networks are low density and may exhibit causal processes distinct from interpersonal micro-level networks.<br />
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一般来说，中观层面的理论始于介于微观和宏观层面之间的规模。然而，中观层面也可以指专门为揭示微观和宏观层面之间的联系而设计的分析。中观层次的网络是密度低，可能表现出不同于人际微观层面网络的因果过程。<br />
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[[File:Social Red.jpg|thumb|right|图4：Social network diagram, meso-level 中观层面社会网络图]]<br />
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'''Organizations''': Formal [[organizations]] are [[social group]]s that distribute tasks for a collective [[goal]].<ref name="Riketta, M. 2007">{{cite journal | last1 = Riketta | first1 = M. | last2 = Nienber | first2 = S. | year = 2007 | title = Multiple identities and work motivation: The role of perceived compatibility between nested organizational units | url = | journal = British Journal of Management | volume = 18 | issue = | pages = S61–77 | doi=10.1111/j.1467-8551.2007.00526.x}}</ref> Network research on organizations may focus on either intra-organizational or inter-organizational ties in terms of [[Formal organization|formal]] or [[Informal organization|informal]] relationships. Intra-organizational networks themselves often contain multiple levels of analysis, especially in larger organizations with multiple branches, franchises or semi-autonomous departments. In these cases, research is often conducted at a work group level and organization level, focusing on the interplay between the two structures.<ref name="Riketta, M. 2007"/> Experiments with networked groups online have documented ways to optimize group-level coordination through diverse interventions, including the addition of autonomous agents to the groups.<ref>{{Cite journal|last=Shirado|first=Hirokazu|last2=Christakis|first2=Nicholas A|title=Locally noisy autonomous agents improve global human coordination in network experiments|journal=Nature|volume=545|issue=7654|pages=370–374|doi=10.1038/nature22332|pmid=28516927|pmc=5912653|bibcode=2017Natur.545..370S|year=2017}}</ref><br />
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Organizations: Formal organizations are social groups that distribute tasks for a collective goal. Network research on organizations may focus on either intra-organizational or inter-organizational ties in terms of formal or informal relationships. Intra-organizational networks themselves often contain multiple levels of analysis, especially in larger organizations with multiple branches, franchises or semi-autonomous departments. In these cases, research is often conducted at a work group level and organization level, focusing on the interplay between the two structures.<br />
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组织: 正规组织是为共同目标分配任务的社会团体。关于组织的网络研究可以侧重于正式或非正式关系方面的组织内或组织间联系。组织内网络本身往往包含多层次的分析，特别是在具有多个分支机构、特许权或半自治部门的较大组织中。在这些情况下，研究通常在工作组和组织层面进行，重点放在两个结构之间的相互作用。<br />
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'''Randomly distributed networks''': [[Exponential random graph models]] of social networks became state-of-the-art methods of social network analysis in the 1980s. This framework has the capacity to represent social-structural effects commonly observed in many human social networks, including general [[Degree (graph theory)|degree]]-based structural effects commonly observed in many human social networks as well as [[Reciprocity (social and political philosophy)|reciprocity]] and [[Transitive set|transitivity]], and at the node-level, [[homophily]] and [[Attribute-value system|attribute]]-based activity and popularity effects, as derived from explicit hypotheses about [[Dependency graph|dependencies]] among network ties. [[Parameter]]s are given in terms of the prevalence of small [[Induced subgraph|subgraph]] configurations in the network and can be interpreted as describing the combinations of local social processes from which a given network emerges. These probability models for networks on a given set of actors allow generalization beyond the restrictive dyadic independence assumption of micro-networks, allowing models to be built from theoretical structural foundations of social behavior.<ref>{{cite journal | last1 = Cranmer | first1 = Skyler J. | last2 = Desmarais | first2 = Bruce A. | year = 2011 | title = Inferential Network Analysis with Exponential Random Graph Models | url = | journal = Political Analysis | volume = 19 | issue = 1| pages = 66–86 | doi=10.1093/pan/mpq037| citeseerx = 10.1.1.623.751 }}</ref><br />
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Randomly distributed networks: Exponential random graph models of social networks became state-of-the-art methods of social network analysis in the 1980s. This framework has the capacity to represent social-structural effects commonly observed in many human social networks, including general degree-based structural effects commonly observed in many human social networks as well as reciprocity and transitivity, and at the node-level, homophily and attribute-based activity and popularity effects, as derived from explicit hypotheses about dependencies among network ties. Parameters are given in terms of the prevalence of small subgraph configurations in the network and can be interpreted as describing the combinations of local social processes from which a given network emerges. These probability models for networks on a given set of actors allow generalization beyond the restrictive dyadic independence assumption of micro-networks, allowing models to be built from theoretical structural foundations of social behavior.<br />
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随机分布的网络: 社会网络的'''<font color="#ff8000">指数随机图模型 Exponential random graph models</font>'''在20世纪80年代成为最先进的社会网络分析方法。这个框架有能力表示在许多人类社会网络中普遍观察到的社会结构效应，包括在许多人类社会网络中普遍观察到的基于程度的一般性结构效应以及互惠性和传递性，以及在节点一级、同相性和基于属性的活动和流行性效应，这些效应源于关于网络关系之间依赖性的明确假设。参数是根据网络中小型子图配置的流行程度给出的，可以解释为描述一个给定网络出现的局部社会过程的组合。这些网络的概率模型在给定的参与者集合上允许超越微型网络的限制性并元独立性假设的泛化，允许模型从社会行为的理论结构基础上建立。<br />
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[[File:Scale-free network sample.png|thumb|right|图5：Examples of a random network and a scale-free network. Each graph has 32 nodes and 32 links. Note the "hubs" (shaded) in the scale-free diagram (on the right).一个随机网络和一个无尺度网络的例子。每个图有32个顶点和32条边。注意无标度图中的“集线器”（阴影部分）（右侧）。]]<br />
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'''Scale-free networks''': A [[scale-free network]] is a [[complex network|network]] whose [[degree distribution]] follows a [[power law]], at least [[asymptotic]]ally. In [[network theory]] a scale-free ideal network is a [[random network]] with a [[degree distribution]] that unravels the size distribution of social groups.<ref>{{cite journal |author1=Moreira, André A.|author2=Demétrius R. Paula|author3=Raimundo N. Costa Filho|author4=José S. Andrade, Jr. |title=Competitive cluster growth in complex networks |year=2006 |doi=10.1103/PhysRevE.73.065101 |pmid=16906890 |journal=Physical Review E |volume=73 |issue=6 |pages=065101 |arxiv=cond-mat/0603272 |bibcode=2006PhRvE..73f5101M }}</ref> Specific characteristics of scale-free networks vary with the theories and analytical tools used to create them, however, in general, scale-free networks have some common characteristics. One notable characteristic in a scale-free network is the relative commonness of [[Vertex (graph theory)|vertices]] with a [[Maximum degree|degree]] that greatly exceeds the average. The highest-degree nodes are often called "hubs", and may serve specific purposes in their networks, although this depends greatly on the social context. Another general characteristic of scale-free networks is the [[clustering coefficient]] distribution, which decreases as the node degree increases. This distribution also follows a [[power law]].<ref>Barabási, Albert-László (2003). ''Linked: how everything is connected to everything else and what it means for business, science, and everyday life''. New York: Plum.</ref> The [[Barabási–Albert model|Barabási]] model of network evolution shown above is an example of a scale-free network.<br />
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Scale-free networks: A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. In network theory a scale-free ideal network is a random network with a degree distribution that unravels the size distribution of social groups. Specific characteristics of scale-free networks vary with the theories and analytical tools used to create them, however, in general, scale-free networks have some common characteristics. One notable characteristic in a scale-free network is the relative commonness of vertices with a degree that greatly exceeds the average. The highest-degree nodes are often called "hubs", and may serve specific purposes in their networks, although this depends greatly on the social context. Another general characteristic of scale-free networks is the clustering coefficient distribution, which decreases as the node degree increases. This distribution also follows a power law. The Barabási model of network evolution shown above is an example of a scale-free network.<br />
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'''<font color="#ff8000">[[无标度网络]] A Scale-free Network</font>''': 无标度网络网络是一个'''<font color="#ff8000">度分布 Degree Distribution</font>'''遵循'''<font color="#ff8000">幂律 Power Law</font>'''(至少是渐近的）的网络。在网络理论中，无标度理想网络是具有揭示了社会群体的规模分布的度分布的随机网络。无标度网络的具体特征随创建其的理论和分析工具的变化而变化，然而，一般来说，无标度网络具有一些共同的特征。无标度网络的一个显著特征是度远超均值的顶点的相对共性。最高度的节点通常被称为“'''<font color="#ff8000">枢纽 Hubs</font>'''” ，并且可能在其网络中服务于特定的目的，尽管这在很大程度上取决于社会环境。无标度网络的另一个一般特性是'''<font color="#ff8000">集聚系数 General Characteristic</font>'''分布，它随着节点度的增加而减少。该分布也遵循幂律。上面网络演化的巴拉巴西模型就是无标度网络的一个例子。<br />
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===Macro level 宏观层面===<br />
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Rather than tracing interpersonal interactions, macro-level analyses generally trace the outcomes of interactions, such as [[economic]] or other [[resource]] [[Transfer function|transfer]] interactions over a large [[Sample population|population]].<br />
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Rather than tracing interpersonal interactions, macro-level analyses generally trace the outcomes of interactions, such as economic or other resource transfer interactions over a large population.<br />
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宏观层面的分析不是追踪人际互动，而通常是追踪相互作用的结果，例如经济或其他资源转移在一大群体中的相互作用。<br />
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[[File:Diagram of a social network.jpg|thumb|right|图6: Diagram: section of a large-scale social network 一大型社会网络局部]]<br />
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'''Large-scale networks''': [[Large-scale macroeconometric model|Large-scale network]] is a term somewhat synonymous with "macro-level" as used, primarily, in [[social science|social]] and [[Behavioural sciences|behavioral]] sciences, in [[economics]]. Originally, the term was used extensively in the [[computer sciences]] (see [[Network mapping#Large-scale mapping project|large-scale network mapping]]).<br />
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Large-scale networks: Large-scale network is a term somewhat synonymous with "macro-level" as used, primarily, in social and behavioral sciences, in economics. Originally, the term was used extensively in the computer sciences (see large-scale network mapping).<br />
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大规模网络: 大规模网络是一个与“宏观层面”同义的术语，主要用于社会和行为科学，经济学。最初，这个术语在计算机科学中广泛使用（见大规模网络映射）。<br />
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'''Complex networks''': Most larger social networks display features of [[social complexity]], which involves substantial non-trivial features of [[network topology]], with patterns of complex connections between elements that are neither purely regular nor purely random (see, [[complexity science]], [[dynamical system]] and [[chaos theory]]), as do [[biological]], and [[Computer network|technological networks]]. Such [[complex network]] features include a heavy tail in the [[degree distribution]], a high [[clustering coefficient]], [[assortativity]] or disassortativity among vertices, [[community structure]] (see [[stochastic block model]]), and [[hierarchy|hierarchical structure]]. In the case of [[Agency (philosophy)|agency-directed]] networks these features also include [[Reciprocity in network|reciprocity]], triad significance profile (TSP, see [[network motif]]), and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as [[lattice graph|lattices]] and [[random graph]]s, do not show these features.<ref>{{cite journal|author=Strogatz, Steven H.|year=2001|title=Exploring complex networks|journal=Nature|volume=410|pages=268–276|doi=10.1038/35065725|pmid=11258382|issue=6825|bibcode=2001Natur.410..268S|doi-access=free}}</ref><br />
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Complex networks: Most larger social networks display features of social complexity, which involves substantial non-trivial features of network topology, with patterns of complex connections between elements that are neither purely regular nor purely random (see, complexity science, dynamical system and chaos theory), as do biological, and technological networks. Such complex network features include a heavy tail in the degree distribution, a high clustering coefficient, assortativity or disassortativity among vertices, community structure (see stochastic block model), and hierarchical structure. In the case of agency-directed networks these features also include reciprocity, triad significance profile (TSP, see network motif), and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as lattices and random graphs, do not show these features.<br />
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复杂网络: 大多数较大的社会网络呈现出社会复杂性的特征，包括'''<font color="#ff8000">网络拓扑 Network Topology</font>'''的大量非平凡特征，以及既不完全规则也不完全随机的元素之间的复杂连接模式（见[[复杂性科学]]、[[动力系统]]和[[混沌理论]]），生物和技术网络也是如此。这些复杂的网络特征包括度分布的重尾、高集聚系数、顶点之间的'''<font color="#ff8000">同配性 Assortativity</font>'''或非同配性、社区结构（见'''<font color="#ff8000">随机分块模型 Stochastic Block Model</font>'''）和层次结构。在主体导向网络的情况下，这些特征还包括互惠性、<font color="#32CD32">三重显著性特征</font>（TSP，见网络基序）及其他。相比之下，许多过去研究过的网络数学模型，如格和'''<font color="#ff8000">随机图 Random Graph</font>'''，并没有表现出这些特征。<br />
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==Theoretical links 理论联系==<br />
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===Imported theories 输入的理论===<br />
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Various theoretical frameworks have been imported for the use of social network analysis. The most prominent of these are [[Graph theory]], [[Balance theory]], Social comparison theory, and more recently, the [[Social identity approach]].<ref>{{cite book |last1=Kilduff|first1=M.|last2=Tsai|first2=W. |year=2003 |title= Social networks and organisations |publisher=Sage Publications}}</ref><br />
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Various theoretical frameworks have been imported for the use of social network analysis. The most prominent of these are Graph theory, Balance theory, Social comparison theory, and more recently, the Social identity approach.<br />
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为了使用社会网络分析，已经引入了各种理论框架。其中最突出的是[[图论]]、'''<font color="#ff8000">平衡理论 Balance theory</font>'''、社会比较论，以及最近的社会认同方法。<br />
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===Indigenous theories 本土理论===<br />
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Few complete theories have been produced from social network analysis. Two that have are [[role theory|structural role theory]] and [[heterophily|heterophily theory]].<br />
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Few complete theories have been produced from social network analysis. Two that have are structural role theory and heterophily theory.<br />
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很少有完整的理论产生于社会网络分析。现有两个为结构角色理论和异质性理论。<br />
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The basis of Heterophily Theory was the finding in one study that more numerous weak ties can be important in seeking information and innovation, as cliques have a tendency to have more homogeneous opinions as well as share many common traits. This homophilic tendency was the reason for the members of the cliques to be attracted together in the first place. However, being similar, each member of the clique would also know more or less what the other members knew. To find new information or insights, members of the clique will have to look beyond the clique to its other friends and acquaintances. This is what Granovetter called "the strength of weak ties".<ref>{{Cite journal | author=Granovetter, M. |year=1973 |title= The strength of weak ties |journal= American Journal of Sociology |volume=78 |issue=6 |pages= 1360–1380 |doi=10.1086/225469}}</ref><br />
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The basis of Heterophily Theory was the finding in one study that more numerous weak ties can be important in seeking information and innovation, as cliques have a tendency to have more homogeneous opinions as well as share many common traits. This homophilic tendency was the reason for the members of the cliques to be attracted together in the first place. However, being similar, each member of the clique would also know more or less what the other members knew. To find new information or insights, members of the clique will have to look beyond the clique to its other friends and acquaintances. This is what Granovetter called "the strength of weak ties".<br />
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异质性理论的基础是在一项研究中发现，更多的'''<font color="#ff8000">弱关系 Weak Tie</font>'''可以在寻求信息和创新方面发挥重要作用，因为小圈子倾向于有更同质化的观点，也有许多共同特征。这种亲同性倾向是小圈子成员被吸引到一起的首要原因。然而，由于相似，小圈子里的每一个成员或多或少都知道其他成员所知道的事情。为了获得新的信息或见解，小圈子里的成员不得不超越该圈子，关注其他的朋友及熟人。这就是格兰诺维特所说的“弱关系的力量”。<br />
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==Structural holes 结构洞==<br />
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In the context of networks, [[social capital]] exists where people have an advantage because of their location in a network. Contacts in a network provide information, opportunities and perspectives that can be beneficial to the central player in the network. Most social structures tend to be characterized by dense clusters of strong connections.<ref name="Burt 2004">{{cite journal|last=Burt|first=Ronald|title=Structural Holes and Good Ideas|journal=American Journal of Sociology|year=2004|doi=10.1086/421787|volume=110|issue=2|pages=349–399|citeseerx=10.1.1.388.2251}}</ref> Information within these clusters tends to be rather homogeneous and redundant. Non-redundant information is most often obtained through contacts in different clusters.<ref name="Burt 1992">{{cite book|last=Burt|first=Ronald|title=Structural Holes: The Social Structure of Competition|year=1992|publisher=Harvard University Press|location=Cambridge, MA}}</ref> When two separate clusters possess non-redundant information, there is said to be a structural hole between them.<ref name="Burt 1992"/> Thus, a network that bridges [[structural holes]] will provide network benefits that are in some degree additive, rather than overlapping. An ideal network structure has a vine and cluster structure, providing access to many different clusters and structural holes.<ref name="Burt 1992"/><br />
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In the context of networks, social capital exists where people have an advantage because of their location in a network. Contacts in a network provide information, opportunities and perspectives that can be beneficial to the central player in the network. Most social structures tend to be characterized by dense clusters of strong connections. Information within these clusters tends to be rather homogeneous and redundant. Non-redundant information is most often obtained through contacts in different clusters. When two separate clusters possess non-redundant information, there is said to be a structural hole between them. Thus, a network that bridges structural holes will provide network benefits that are in some degree additive, rather than overlapping. An ideal network structure has a vine and cluster structure, providing access to many different clusters and structural holes.<br />
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在网络的背景下，'''<font color="#ff8000">社会资本 Social Capital</font>'''存在于人们因其在网络中的位置而具有优势的地方。网络中的联系提供的信息、机会和观点可能有利于网络中的核心参与者。大多数社会结构往往以有强连接的密集集群为特征。这些集群中的信息往往是相当同质和冗余的。非冗余信息通常是通过不同集群中的联系获得的。当两个独立的集群拥有非冗余信息时，我们称它们之间存在一个'''<font color="#ff8000">结构洞 Structural Hole</font>'''。因此，一个连接结构孔的网络在某种程度上提供附加而非重叠的的网络效益。理想的网络结构具有蔓生结构和集群结构，可访问许多不同集群和结构洞。<br />
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Networks rich in structural holes are a form of social capital in that they offer [[Information theory|information]] benefits. The main player in a network that bridges structural holes is able to access information from diverse sources and clusters.<ref name="Burt 1992"/> For example, in [[Business networking|business networks]], this is beneficial to an individual's career because he is more likely to hear of job openings and opportunities if his network spans a wide range of contacts in different industries/sectors. This concept is similar to Mark Granovetter's theory of [[Interpersonal ties|weak ties]], which rests on the basis that having a broad range of contacts is most effective for job attainment.<br />
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Networks rich in structural holes are a form of social capital in that they offer information benefits. The main player in a network that bridges structural holes is able to access information from diverse sources and clusters. For example, in business networks, this is beneficial to an individual's career because he is more likely to hear of job openings and opportunities if his network spans a wide range of contacts in different industries/sectors. This concept is similar to Mark Granovetter's theory of weak ties, which rests on the basis that having a broad range of contacts is most effective for job attainment.<br />
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富含结构洞的网络是社会资本的一种形式，因为它们提供信息利益。连接结构洞的网络中的主要参与者能够访问来自不同来源和集群的信息。例如，在'''<font color="#ff8000">商业社交 Business networking</font>'''中，这对个人的职业生涯是有益的，因为若其关系网涵盖不同行业 / 部门，则更有可能得知职位空缺和机会。这个概念类似于马克·格兰诺维特的弱关系理论，该理论的基础是拥有广泛的联系对于获得工作是最有效的。<br />
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==Research clusters 研究集群==<br />
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===Communication 传播学===<br />
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[[Communication Studies]] are often considered a part of both the social sciences and the humanities, drawing heavily on fields such as [[sociology]], [[psychology]], [[anthropology]], [[information science]], [[biology]], [[political science]], and [[economics]] as well as [[rhetoric]], [[literary studies]], and [[semiotics]]. Many communication concepts describe the transfer of information from one source to another, and can thus be conceived of in terms of a network.<br />
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Communication Studies are often considered a part of both the social sciences and the humanities, drawing heavily on fields such as sociology, psychology, anthropology, information science, biology, political science, and economics as well as rhetoric, literary studies, and semiotics. Many communication concepts describe the transfer of information from one source to another, and can thus be conceived of in terms of a network.<br />
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传播学通常被认为是社会科学和人文科学的一部分，主要集中在社会学、心理学、人类学、信息科学、生物学、政治学、经济学以及修辞学、文学研究和符号学等领域。许多通信概念描述了从一个源到另一个源的信息传输，因此可以从网络的角度来考虑。<br />
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===Community 社区===<br />
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In J.A. Barnes' day, a "[[community]]" referred to a specific geographic location and studies of community ties had to do with who talked, associated, traded, and attended church with whom. Today, however, there are extended "online" communities developed through [[telecommunications]] devices and [[social network services]]. Such devices and services require extensive and ongoing maintenance and analysis, often using [[network science]] methods. [[Community development]] studies, today, also make extensive use of such methods.<br />
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In J.A. Barnes' day, a "community" referred to a specific geographic location and studies of community ties had to do with who talked, associated, traded, and attended church with whom. Today, however, there are extended "online" communities developed through telecommunications devices and social network services. Such devices and services require extensive and ongoing maintenance and analysis, often using network science methods. Community development studies, today, also make extensive use of such methods.<br />
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在J.A. Barnes' day，“社区”指的是一个特定的地理位置和与谁交谈、联系、贸易和一同上教堂做礼拜有关的社区关系的研究。然而如今，通过电信设备和社交网络服务，有了扩展的“在线”社区。这样的设备和服务需要广泛和持续的维护和分析，通常使用网络科学方法。社区发展研究在如今也广泛使用这些方法。<br />
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===Complex networks 复杂网络===<br />
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[[Complex networks]] require methods specific to modelling and interpreting [[social complexity]] and [[complex adaptive system]]s, including techniques of [[dynamic network analysis]].<br />
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Complex networks require methods specific to modelling and interpreting social complexity and complex adaptive systems, including techniques of dynamic network analysis.<br />
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复杂网络需要建模和解释社会复杂性和复杂适应系统的特定方法，包括动态网络分析技术。<br />
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Mechanisms such as [[Dual-phase evolution#social networks|Dual-phase evolution]] explain how temporal changes in connectivity contribute to the formation of structure in social networks.<br />
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Mechanisms such as Dual-phase evolution explain how temporal changes in connectivity contribute to the formation of structure in social networks.<br />
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'''<font color="#ff8000">双相演化理论 Dual-phase Evolution</font>'''等机制解释了连接性的时间变化如何促进社会网络结构的形成。<br />
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===Criminal networks 犯罪网络===<br />
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In [[criminology]] and [[urban sociology]], much attention has been paid to the social networks among criminal actors. For example, Andrew Papachristos<ref>{{cite journal |last=Papachristos |first=Andrew |year=2009 |title=Murder by Structure: Dominance Relations and the Social Structure of Gang Homicide |journal=American Journal of Sociology |volume=115 |issue=1 |pages=74–128 |doi=10.2139/ssrn.855304 |url=http://www.papachristos.org/Publications_2_files/ajs_final_version.pdf |accessdate=29 March 2013 |url-status=dead |archiveurl=https://web.archive.org/web/20140407094725/http://www.papachristos.org/Publications_2_files/ajs_final_version.pdf |archivedate=7 April 2014 }}</ref> has studied gang murders as a series of exchanges between gangs. Murders can be seen to diffuse outwards from a single source, because weaker gangs cannot afford to kill members of stronger gangs in retaliation, but must commit other violent acts to maintain their reputation for strength.<br />
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In criminology and urban sociology, much attention has been paid to the social networks among criminal actors. For example, Andrew Papachristos has studied gang murders as a series of exchanges between gangs. Murders can be seen to diffuse outwards from a single source, because weaker gangs cannot afford to kill members of stronger gangs in retaliation, but must commit other violent acts to maintain their reputation for strength.<br />
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在'''<font color="#ff8000">犯罪学 Criminology</font>和'''<font color="#ff8000">城市社会学 Urban sociology</font>中，犯罪行为人之间的社会网络问题受到了广泛的关注。例如，'''安德鲁·帕帕克里斯托斯 Andrew Papachristos'''将帮派谋杀研究为帮派之间的一系列交流。谋杀可以视为从单一来源向外扩散，因为较弱的帮派无法承担为报复杀死较强帮派成员的代价，而必须采取其他暴力行动来维护其势力强大之名声。<br />
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===Diffusion of innovations 创新扩散理论===<br />
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[[Diffusion of innovations|Diffusion of ideas and innovations]] studies focus on the spread and use of ideas from one actor to another or one [[culture]] and another. This line of research seeks to explain why some become "early adopters" of ideas and innovations, and links social network structure with facilitating or impeding the spread of an innovation.<br />
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Diffusion of ideas and innovations studies focus on the spread and use of ideas from one actor to another or one culture and another. This line of research seeks to explain why some become "early adopters" of ideas and innovations, and links social network structure with facilitating or impeding the spread of an innovation.<br />
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'''<font color="#ff8000">创新扩散理论 Diffusion of Innovations</font>'''研究的重点是思想从一个行动者到另一个或一种文化到另一种的传播。这一系列的研究试图解释为什么有些人成为创意和创新的“早期接受者” ，并将社交网络结构与促进或阻碍创新的传播联系起来。<br />
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===Demography 人口学===<br />
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In [[demography]], the study of social networks has led to new sampling methods for estimating and reaching populations that are hard to enumerate (for example, homeless people or intravenous drug users.) For example, respondent driven sampling is a network-based sampling technique that relies on respondents to a survey recommending further respondents.<br />
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In demography, the study of social networks has led to new sampling methods for estimating and reaching populations that are hard to enumerate (for example, homeless people or intravenous drug users.) For example, respondent driven sampling is a network-based sampling technique that relies on respondents to a survey recommending further respondents.<br />
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在'''<font color="#ff8000">人口学 Demography</font>'''方面，对社会网络的研究导致了新的抽样方法，用于估计和覆盖难以统计的人群（如无家可归者或静脉注射毒品者）。例如，受访者驱动的抽样依赖于调查的受访者推荐更多的受访者，是一种基于网络的抽样技术。<br />
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===Economic sociology 经济社会学===<br />
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The field of [[sociology]] focuses almost entirely on networks of outcomes of social interactions. More narrowly, [[economic sociology]] considers behavioral interactions of individuals and groups through [[social capital]] and social "markets". Sociologists, such as Mark Granovetter, have developed core principles about the interactions of social structure, information, ability to punish or reward, and trust that frequently recur in their analyses of political, economic and other institutions. Granovetter examines how social structures and social networks can affect economic outcomes like hiring, price, productivity and innovation and describes sociologists' contributions to analyzing the impact of social structure and networks on the economy.<ref>{{cite journal |last=Granovetter |first=Mark |year=2005 |title=The Impact of Social Structure on Economic Outcomes |journal=The Journal of Economic Perspectives |jstor=4134991 |volume=19 |issue=1 |pages=33–50|doi=10.1257/0895330053147958 }}</ref><br />
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The field of sociology focuses almost entirely on networks of outcomes of social interactions. More narrowly, economic sociology considers behavioral interactions of individuals and groups through social capital and social "markets". Sociologists, such as Mark Granovetter, have developed core principles about the interactions of social structure, information, ability to punish or reward, and trust that frequently recur in their analyses of political, economic and other institutions. Granovetter examines how social structures and social networks can affect economic outcomes like hiring, price, productivity and innovation and describes sociologists' contributions to analyzing the impact of social structure and networks on the economy.<br />
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社会学领域几乎完全关注社会互动的结果网络。更狭义地说，'''<font color="#ff8000">经济社会学 Economic Sociology</font>'''通过社会资本和社会“市场”考虑个人和群体的行为互动。社会学家，如马克·格兰诺维特，已经研究出关于社会结构、信息、奖惩能力和信任相互作用的核心原则，这些原则在他们对政治、经济和其他制度的分析中经常出现。格兰诺维特研究了社会结构和社会网络如何影响经济结果，如雇佣、价格、生产力和创新，并描述了社会学家对分析社会结构和网络对经济的影响的贡献。<br />
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===Health care 卫生保健===<br />
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Analysis of social networks is increasingly incorporated into [[health care analytics]], not only in [[epidemiology|epidemiological]] studies but also in models of [[Health Communication|patient communication]] and education, disease prevention, mental health diagnosis and treatment, and in the study of health care organizations and [[health care systems|systems]].<ref>Levy, Judith and Bernice Pescosolido (2002). ''Social Networks and Health''. Boston, MA: JAI Press.</ref><br />
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Analysis of social networks is increasingly incorporated into health care analytics, not only in epidemiological studies but also in models of patient communication and education, disease prevention, mental health diagnosis and treatment, and in the study of health care organizations and systems.<br />
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社会网络分析越来越多地被纳入卫生保健分析——不仅在'''<font color="#ff8000">流行病学 Epidemiology</font>'''研究中，而且在病人沟通和教育、疾病预防、心理健康诊断和治疗模型中，以及在卫生保健组织和系统的研究中。<br />
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===Human ecology 人类生态学===<br />
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[[Human ecology]] is an [[interdisciplinary]] and [[transdisciplinary]] study of the relationship between [[human]]s and their [[natural environment|natural]], [[Social environment|social]], and [[built environment]]s. The scientific philosophy of human ecology has a diffuse history with connections to [[geography]], [[sociology]], [[psychology]], [[anthropology]], [[zoology]], and natural [[ecology]].<ref>Crona, Beatrice and Klaus Hubacek (eds.) (2010). [http://www.ecologyandsociety.org/issues/view.php?sf=48 "Special Issue: Social network analysis in natural resource governance"]. ''Ecology and Society'', 48.</ref><ref>Ernstson, Henrich (2010). "Reading list: Using social network analysis (SNA) in social-ecological studies". [http://rs.resalliance.org/2010/11/03/reading-list-using-social-network-analysis-sna-in-social-ecological-studies/ ''Resilience Science'']</ref><br />
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Human ecology is an interdisciplinary and transdisciplinary study of the relationship between humans and their natural, social, and built environments. The scientific philosophy of human ecology has a diffuse history with connections to geography, sociology, psychology, anthropology, zoology, and natural ecology.<br />
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'''<font color="#ff8000">人类生态学 Human ecology</font>'''是研究人类与其自然环境、社会环境和'''<font color="#ff8000">建成环境 Built environment</font>'''之间关系的一门跨学科科学。人类生态学的科学哲学与地理学、社会学、心理学、人类学、动物学和自然生态学有着密切的联系。<br />
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===Language and linguistics 语言与语言学===<br />
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Studies of [[language]] and [[linguistics]], particularly [[evolutionary linguistics]], focus on the development of [[Morphology (linguistics)|linguistic forms]] and transfer of changes, [[Phonology|sounds]] or words, from one language system to another through networks of social interaction. Social networks are also important in [[language shift]], as groups of people add and/or abandon languages to their repertoire.<br />
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Studies of language and linguistics, particularly evolutionary linguistics, focus on the development of linguistic forms and transfer of changes, sounds or words, from one language system to another through networks of social interaction. Social networks are also important in language shift, as groups of people add and/or abandon languages to their repertoire.<br />
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语言学和语言学的研究，特别是'''<font color="#ff8000">演化语言学 Evolutionary Linguistics</font>'''，关注通过社会互动网络从一个语言系统转移到另一个语言系统时，语言形式的发展以及声音或词语的变化。社交网络在语言转换中也很重要，因为一些人群增加或者放弃了他们的语言。<br />
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===Literary networks 文学网络===<br />
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In the study of literary systems, network analysis has been applied by Anheier, Gerhards and Romo,<ref>{{cite journal | last1 = Anheier | first1 = H. K. | last2 = Romo | first2 = F. P. | year = 1995 | title = Forms of capital and social structure of fields: examining Bourdieu's social topography | url = | journal = American Journal of Sociology | volume = 100 | issue = 4| pages = 859–903 | doi=10.1086/230603}}</ref> De Nooy,<ref>{{cite journal | last1 = De Nooy | first1 = W | year = 2003| title = Fields and networks: Correspondence analysis and social network analysis in the framework of Field Theory | url = | journal = Poetics | volume = 31 | issue = 5–6| pages = 305–327 | doi = 10.1016/S0304-422X(03)00035-4 }}</ref> and Senekal,<ref>{{cite journal | last1 = Senekal | first1 = B. A. | year = 2012 | title = Die Afrikaanse literêre sisteem: ʼn Eksperimentele benadering met behulp van Sosiale-netwerk-analise (SNA) | url = | journal = LitNet Akademies | volume = 9 | issue = | page = 3 }}</ref> to study various aspects of how literature functions. The basic premise is that polysystem theory, which has been around since the writings of [[Even-Zohar]], can be integrated with network theory and the relationships between different actors in the literary network, e.g. writers, critics, publishers, literary histories, etc., can be mapped using [[Computer graphics (computer science)|visualization]] from SNA.<br />
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In the study of literary systems, network analysis has been applied by Anheier, Gerhards and Romo, De Nooy, and Senekal, to study various aspects of how literature functions. The basic premise is that polysystem theory, which has been around since the writings of Even-Zohar, can be integrated with network theory and the relationships between different actors in the literary network, e.g. writers, critics, publishers, literary histories, etc., can be mapped using visualization from SNA.<br />
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在文学体系的研究中，网络分析被'''阿海尔 Anheier'''、'''格尔哈兹 Gerhards'''和'''罗姆 Romo'''、'''努伊 De Nooy'''和Senekal应用于研究文学如何运作的各个方面。其基本前提是将'''文-佐哈尔 Even-Zohar'''著述以来就存在的多元系统理论可以与网络理论相结合，以及文学网络中不同行为者（如作家、评论家、出版商、文学史等）之间的关系可以通过SNA可视化映射。<br />
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===Organizational studies 组织研究===<br />
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Research studies of [[Formal organization|formal]] or [[informal organization]] [[Social relation|relationships]], [[organizational communication]], [[economics]], [[economic sociology]], and other [[resource]] [[Transfer function|transfers]]. Social networks have also been used to examine how organizations interact with each other, characterizing the many [[Interlocking directorate|informal connections]] that link executives together, as well as associations and connections between individual employees at different organizations.<ref>{{cite journal | last1 = Podolny | first1 = J. M. | last2 = Baron | first2 = J. N. | year = 1997 | title = Resources and relationships: Social networks and mobility in the workplace | url = | journal = American Sociological Review | volume = 62 | issue = 5| pages = 673–693 | doi=10.2307/2657354| jstor = 2657354 | citeseerx = 10.1.1.114.6822 }}</ref> Intra-organizational networks have been found to affect [[organizational commitment]],<ref>{{cite journal | last1 = Lee | first1 = J. | last2 = Kim | first2 = S. | year = 2011 | title = Exploring the role of social networks in affective organizational commitment: Network centrality, strength of ties, and structural holes | url = | journal = The American Review of Public Administration | volume = 41 | issue = 2| pages = 205–223 | doi=10.1177/0275074010373803}}</ref> [[organizational identification]],<ref name="Jone11">{{cite journal | last1 = Jones | first1 = C. | last2 = Volpe | first2 = E.H. | year = 2011 | title = Organizational identification: Extending our understanding of social identities through social networks | url = | journal = Journal of Organizational Behavior | volume = 32 | issue = 3| pages = 413–434 | doi=10.1002/job.694}}</ref> [[Organizational citizenship behavior|interpersonal citizenship behaviour]].<ref>{{cite journal | last1 = Bowler | first1 = W. M. | last2 = Brass | first2 = D. J. | year = 2011 | title = Relational correlates of interpersonal citizenship behaviour: A social network perspective | doi = 10.1037/0021-9010.91.1.70 | pmid = 16435939 | journal = Journal of Applied Psychology | volume = 91 | issue = 1| pages = 70–82 | citeseerx = 10.1.1.516.8746 }}</ref><br />
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Research studies of formal or informal organization relationships, organizational communication, economics, economic sociology, and other resource transfers. Social networks have also been used to examine how organizations interact with each other, characterizing the many informal connections that link executives together, as well as associations and connections between individual employees at different organizations. Intra-organizational networks have been found to affect organizational commitment, organizational identification, interpersonal citizenship behaviour.<br />
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研究正式或非正式组织关系、'''<font color="#ff8000">组织沟通 Organizational Communication</font>'''、经济、经济社会学和其他资源转移。社交网络也被用来研究组织之间如何相互作用，描述了许多将高管联系在一起的非正式联系，以及不同组织的个体雇员之间的联系。研究发现，组织内网络对'''<font color="#ff8000">组织承诺 Organizational Commitment</font>'''、组织认同、人际公民行为有影响。<br />
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===Social capital 社会资本===<br />
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[[Social capital]] is a form of [[Capital (economics)|economic]] and [[cultural capital]] in which social networks are central, [[Stock and flow|transactions]] are marked by [[Reciprocity (social psychology)|reciprocity]], [[Trust (social sciences)|trust]], and [[cooperation]], and [[Market (economics)|market]] [[Agent (economics)|agents]] produce [[goods and services]] not mainly for themselves, but for a [[common good]]. [[Social capital]] is split into three dimensions: the structural, the relational and the cognitive dimension. The structural dimension describes how partners interact with each other and which specific partners meet in a social network. Also The structural dimension of social capital indicates the level of ties among organizations.<ref>(Claridge, 2018).</ref>. This dimension is highly connected to the relational dimension which refers to trustworthiness, norms, expectations and idenfications of the bonds between partners.The relational dimension explains the nature of these ties which is mainly illustrated by the level of trust accorded to the network of organizations. <ref>(Claridge, 2018).</ref> The cognitive dimension analyses the extent to which organizations share common goals and objectives as a result of their ties and interactions. <ref>(Claridge, 2018).</ref><br />
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Social capital is a form of economic and cultural capital in which social networks are central, transactions are marked by reciprocity, trust, and cooperation, and market agents produce goods and services not mainly for themselves, but for a common good. Social capital is split into three dimensions: the structural, the relational and the cognitive dimension. The structural dimension describes how partners interact with each other and which specific partners meet in a social network. Also The structural dimension of social capital indicates the level of ties among organizations. This dimension is highly connected to the relational dimension which refers to trustworthiness, norms, expectations and idenfications of the bonds between partners. The relational dimension explains the nature of these ties which is mainly illustrated by the level of trust accorded to the network of organizations. The cognitive dimension analyses the extent to which organizations share common goals and objectives as a result of their ties and interactions. <br />
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社会资本是一种以社会网络为中心的经济和文化资本，交易以互惠、信任和合作为特征，市场主体生产的商品和服务主要不是为了自已，而是为了共同的利益。社会资本分为三个维度: 结构维度、关系维度和认知维度。结构维度描述了合作伙伴之间如何相互作用，以及哪些特定的合作伙伴在社交网络中相遇。社会资本的结构维度反映了组织之间的关系水平。这个维度与关系维度高度相关，关系维度指的是伙伴之间关系的可信度、规范、期望和认同度。 关系维度解释了这些联系的本质，主要表现在对组织网络的信任程度上。认知维度分析组织在多大程度上因其联系和相互作用而共享共同的目标和目的。<br />
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[[Social capital]] is a sociological concept about the value of [[social relation]]s and the role of cooperation and confidence to achieve positive outcomes. The term refers to the value one can get from their social ties. For example, newly arrived immigrants can make use of their social ties to established migrants to acquire jobs they may otherwise have trouble getting (e.g., because of unfamiliarity with the local language). A positive relationship exists between social capital and the intensity of social network use.<ref>{{cite journal|last=Sebastián|first=Valenzuela|author2=Namsu Park |author3=Kerk F. Kee |title=Is There Social Capital in a Social Network Site? Facebook Use and College Students' Life Satisfaction, Trust, and Participation|journal=Journal of Computer-Mediated Communication|year=2009|volume=14|issue=4|pages=875–901|doi=10.1111/j.1083-6101.2009.01474.x|doi-access=free}}</ref><ref>{{cite journal | title=Social Connectivity in America: Changes in Adult Friendship Network Size from 2002 to 2007 |author1=Wang, Hua |author2=Barry Wellman |lastauthoramp=yes | journal=American Behavioral Scientist | year=2010 | volume=53 | issue=8 | pages=1148–1169 | doi=10.1177/0002764209356247|url=https://semanticscholar.org/paper/402068f74efb35c093e7fd8c0f8268521219e242 }}</ref> In a dynamic framework, higher activity in a network feeds into higher social capital which itself encourages more activity.<ref name="GaudeulGiannetti2013">{{cite journal|last1=Gaudeul|first1=Alexia|last2=Giannetti|first2=Caterina|title=The role of reciprocation in social network formation, with an application to LiveJournal|journal=Social Networks|volume=35|issue=3|year=2013|pages=317–330|issn=03788733|doi=10.1016/j.socnet.2013.03.003}}</ref><br />
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Social capital is a sociological concept about the value of social relations and the role of cooperation and confidence to achieve positive outcomes. The term refers to the value one can get from their social ties. For example, newly arrived immigrants can make use of their social ties to established migrants to acquire jobs they may otherwise have trouble getting (e.g., because of unfamiliarity with the local language). A positive relationship exists between social capital and the intensity of social network use. In a dynamic framework, higher activity in a network feeds into higher social capital which itself encourages more activity.<br />
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社会资本是一个关于社会关系的价值、以及合作与信心对取得积极成果的作用的社会学概念。该术语指的是一个人可以从其社会关系中获得的价值。例如，新来的移民可以利用他们与先到的移民的社会关系，获得可能本难以获得的工作（如因为不熟悉当地语言）。社会资本与社会网络使用强度存在正相关关系。在一个动态的框架中，网络中更高的活动会产生更高的社会资本，而社会资本本身又会激励更多的活动。<br />
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=== Advertising 广告===<br />
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This particular cluster focuses on brand-image and promotional strategy effectiveness, taking into account the impact of customer participation on sales and brand-image. This is gauged through techniques such as sentiment analysis which rely on mathematical areas of study such as data mining and analytics. This area of research produces vast numbers of commercial applications as the main goal of any study is to understand consumer behaviour and drive sales.<br />
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This particular cluster focuses on brand-image and promotional strategy effectiveness, taking into account the impact of customer participation on sales and brand-image. This is gauged through techniques such as sentiment analysis which rely on mathematical areas of study such as data mining and analytics. This area of research produces vast numbers of commercial applications as the main goal of any study is to understand consumer behaviour and drive sales.<br />
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这个集群侧重于品牌形象和促销策略的有效性，同时考虑到顾客参与对销售和品牌形象的影响。这是通过诸如情感分析等技术来衡量的，这些技术依赖于数学领域的研究，如数据挖掘和分析。这一研究领域产生了大量的商业应用，因为任何研究的主要目标都是了解消费者行为并推动销售。<br />
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===Network position and benefits 网络地位和利益===<br />
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In many [[Formal organizations|organizations]], members tend to focus their activities inside their own groups, which stifles creativity and restricts opportunities. A player whose network bridges structural holes has an advantage in detecting and developing rewarding opportunities.<ref name="Burt 2004"/> Such a player can mobilize social capital by acting as a "broker" of information between two clusters that otherwise would not have been in contact, thus providing access to new ideas, opinions and opportunities. British philosopher and political economist [[John Stuart Mill]], writes, "it is hardly possible to overrate the value ... of placing human beings in contact with persons dissimilar to themselves.... Such communication [is] one of the primary sources of progress."<ref>{{cite book|last=Mill|first=John|title=Principles of Political Economy|year=1909|publisher=William J Ashley|location=Library of Economics and Liberty}}</ref> Thus, a player with a network rich in structural holes can add value to an organization through new ideas and opportunities. This in turn, helps an individual's career development and advancement.<br />
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In many organizations, members tend to focus their activities inside their own groups, which stifles creativity and restricts opportunities. A player whose network bridges structural holes has an advantage in detecting and developing rewarding opportunities. Thus, a player with a network rich in structural holes can add value to an organization through new ideas and opportunities. This in turn, helps an individual's career development and advancement.<br />
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在许多组织中，成员倾向于把他们的活动集中在他们自己的团队中，这抑制了创造力和限制了机会。一个玩家的网络桥梁结构洞有优势，在检测和发展奖励机会。因此，一个拥有丰富结构性漏洞的网络的玩家可以通过新的想法和机会为组织增加价值。这反过来又有助于个人的职业发展和晋升。<br />
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A social capital broker also reaps control benefits of being the facilitator of information flow between contacts. In the case of consulting firm Eden McCallum, the founders were able to advance their careers by bridging their connections with former big three consulting firm consultants and mid-size industry firms.<ref>{{cite journal|last=Gardner|first=Heidi|author2=Eccles, Robert |title=Eden McCallum: A Network Based Consulting Firm|journal=Harvard Business School Review|year=2011}}</ref> By bridging structural holes and mobilizing social capital, players can advance their careers by executing new opportunities between contacts.<br />
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A social capital broker also reaps control benefits of being the facilitator of information flow between contacts. In the case of consulting firm Eden McCallum, the founders were able to advance their careers by bridging their connections with former big three consulting firm consultants and mid-size industry firms. By bridging structural holes and mobilizing social capital, players can advance their careers by executing new opportunities between contacts.<br />
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社会资本经纪人还可以从促进联系人之间的信息流动中获得控制权利。在咨询公司 Eden McCallum 的案例中，创始人能够通过与前三大咨询公司顾问和中等规模的行业公司建立联系来提升自己的职业生涯。通过弥合结构漏洞和调动社会资本，玩家可以通过在联系人之间执行新的机会来提升自己的职业生涯。<br />
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There has been research that both substantiates and refutes the benefits of information brokerage. A study of high tech Chinese firms by Zhixing Xiao found that the control benefits of structural holes are "dissonant to the dominant firm-wide spirit of cooperation and the information benefits cannot materialize due to the communal sharing values" of such organizations.<ref>Xiao, Zhixing; Tsui, Anne (2007). "When Brokers May Not Work: The Cultural Contingency of [http://iniciarsesionentrar.com/facebook/ Social] Capital in Chinese High-tech Firms". ''Administrative Science Quarterly''.</ref> However, this study only analyzed Chinese firms, which tend to have strong communal sharing values. Information and control benefits of structural holes are still valuable in firms that are not quite as inclusive and cooperative on the firm-wide level. In 2004, Ronald Burt studied 673 managers who ran the supply chain for one of America's largest electronics companies. He found that managers who often discussed issues with other groups were better paid, received more positive job evaluations and were more likely to be promoted.<ref name="Burt 2004"/> Thus, bridging structural holes can be beneficial to an organization, and in turn, to an individual's career.<br />
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There has been research that both substantiates and refutes the benefits of information brokerage. A study of high tech Chinese firms by Zhixing Xiao found that the control benefits of structural holes are "dissonant to the dominant firm-wide spirit of cooperation and the information benefits cannot materialize due to the communal sharing values" of such organizations. However, this study only analyzed Chinese firms, which tend to have strong communal sharing values. Information and control benefits of structural holes are still valuable in firms that are not quite as inclusive and cooperative on the firm-wide level. In 2004, Ronald Burt studied 673 managers who ran the supply chain for one of America's largest electronics companies. He found that managers who often discussed issues with other groups were better paid, received more positive job evaluations and were more likely to be promoted. Thus, bridging structural holes can be beneficial to an organization, and in turn, to an individual's career.<br />
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有研究证实和反驳了信息经纪的好处。知行晓对中国高科技企业的一项研究发现，结构性漏洞的控制效益“与占主导地位的企业合作精神不协调，由于这类组织的共享价值观，信息效益不能实现”。然而，这项研究只分析了中国企业，这些企业往往有很强的公共共享价值观。结构性漏洞的信息和控制利益对于那些在整个公司层面上不那么具有包容性和合作性的公司来说仍然是有价值的。2004年，罗纳德 · 伯特研究了673名管理者，他们为美国最大的电子公司之一管理供应链。他发现，那些经常与其他团体讨论问题的管理人员收入更高，得到的工作评价更积极，也更有可能获得晋升。因此，架起结构性漏洞对组织有益，反过来对个人的职业生涯也有益。<br />
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===Social media 社交媒体===<br />
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[[Computer network]]s combined with social networking software produces a new medium for social interaction.<ref>{{Cite book|title=The International Encyclopedia of Media Effects|last=Amichai-Hamburger|first=Yair|last2=Hayat|first2=Tsahi|date=2017|publisher=John Wiley & Sons, Inc.|isbn=9781118783764|language=en|doi=10.1002/9781118783764.wbieme0170}}</ref> A relationship over a computerized [[social networking service]] can be characterized by context, direction, and strength. The content of a relation refers to the resource that is exchanged. In a [[computer mediated communication]] context, social pairs exchange different kinds of information, including sending a data file or a computer program as well as providing emotional support or arranging a meeting. With the rise of [[electronic commerce]], information exchanged may also correspond to exchanges of money, goods or services in the "real" world.<ref>{{cite journal | title=Studying Online Social Networks | last1=Garton|first1=Laura|first2=Caroline |last2=Haythornthwaite|author2-link=Caroline Haythornthwaite|first3=Barry|last3=Wellman|author3-link=Barry Wellman | journal=Journal of Computer-Mediated Communication |date=23 June 2006 | volume=3 | issue=1 |pages = 0| doi=10.1111/j.1083-6101.1997.tb00062.x}}</ref> [[Social network analysis]] methods have become essential to examining these types of computer mediated communication.<br />
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Computer networks combined with social networking software produces a new medium for social interaction. A relationship over a computerized social networking service can be characterized by context, direction, and strength. The content of a relation refers to the resource that is exchanged. In a computer mediated communication context, social pairs exchange different kinds of information, including sending a data file or a computer program as well as providing emotional support or arranging a meeting. With the rise of electronic commerce, information exchanged may also correspond to exchanges of money, goods or services in the "real" world. Social network analysis methods have become essential to examining these types of computer mediated communication.<br />
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计算机网络与社交网络软件的结合为社交互动提供了一种新的媒介。通过计算机化的社交网络服务建立起来的关系可以是拥有属性、方向和力量。关系的内容指的是交换的资源。在计算机中介的交流环境中，社会对交换不同种类的信息，包括发送数据文件或计算机程序，以及提供情感支持或安排会议。随着电子商务的兴起，交换的信息也可能对应于“现实”世界中的货币、商品或服务的交换。社会网络分析方法已经成为检验这些类型的计算机中介交流的必要手段。<br />
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In addition, the sheer size and the volatile nature of [[social media]] has given rise to new network metrics. A key concern with networks extracted from social media is the lack of robustness of network metrics given missing data.<ref>{{cite journal | last1 = Wei | first1 = Wei | last2 = Joseph | first2 = Kenneth | last3 = Liu | first3 = Huan | last4 = Carley | first4 = Kathleen M. | year = 2016 | title = Exploring Characteristics of Suspended Users and Network Stability on Twitter | url = | journal = Social Network Analysis and Mining | volume = 6 | issue = | page = 51 | doi = 10.1007/s13278-016-0358-5 }}</ref><br />
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In addition, the sheer size and the volatile nature of social media has given rise to new network metrics. A key concern with networks extracted from social media is the lack of robustness of network metrics given missing data.<br />
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此外，社交媒体的庞大规模和不稳定性也催生了新的网络指标。从社交媒体中提取网络的一个关键问题是，在缺失数据的情况下，网络指标缺乏健壮性。<br />
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==See also 参阅==<br />
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* [[Bibliography of sociology]] 社会学参考书目<br />
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* [[Business networking]] 商业社交<br />
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* [[Collective network]] 集体网络<br />
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* [[International Network for Social Network Analysis]] 国际社会网络学会<br />
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* [[Network society]] 网络社会<br />
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* [[Network theory]] 网络理论<br />
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*[[Network science]] 网络科学<br />
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* [[Semiotics of social networking]]<br />
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* [[Social network analysis]] 社会网络分析<br />
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* [[Social network (sociolinguistics)]] 社会网络<br />
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* [[Social networking service]] 社会网络服务<br />
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* [[Social web]] 社交网络<br />
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* [[Structural fold]] 结构折叠<br />
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==References==<br />
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==References==<br />
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参考资料<br />
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==Further reading==<br />
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==Further reading==<br />
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* {{cite journal |last1=Aneja |first1=Nagender |last2=Gambhir |first2=Sapna |title=Ad-hoc Social Network: A Comprehensive Survey |url=http://www.ijser.org/researchpaper%5CAd-hoc-Social-Network-A-Comprehensive-Survey.pdf |journal=International Journal of Scientific & Engineering Research |volume=4 |issue=8 |date=August 2013 |pages=156–160|issn=2229-5518}}<br />
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* {{cite book |last=Barabási |first=Albert-László |year=2003 |title=Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life |publisher=Plum |isbn=978-0-452-28439-5 |url-access=registration |url=https://archive.org/details/linkedhoweveryth00bara }}<br />
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* {{cite book |last=Barnett |first=George A. |year=2011 |title=Encyclopedia of Social Networks |publisher=Sage |isbn=978-1-4129-7911-5}}<br />
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* {{cite book |last=Estrada |first=E |year=2011 |title=The Structure of Complex Networks: Theory and Applications |publisher=Oxford University Press |isbn=978-0-199-59175-6}}<br />
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* {{cite book |last=Ferguson |first=Niall |year=2018 |title=The Square and the Tower: Networks and Power, from the Freemasons to Facebook |isbn=978-0735222915 |publisher=Penguin Press}}<br />
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* {{cite book |last=Freeman |first=Linton C. |year=2004 |title=The Development of Social Network Analysis: A Study in the Sociology of Science |publisher=Empirical Press |isbn=978-1-59457-714-7}}<br />
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* {{cite book |last=Kadushin |first=Charles |year=2012 |title=Understanding Social Networks: Theories, Concepts, and Findings |location= |publisher=Oxford University Press |isbn=978-0-19-537946-4}}<br />
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* {{cite book |last1=Mauro |first1=Rios |last2=Petrella |first2=Carlos |year=2014 |title=La Quimera de las Redes Sociales |trans-title=The Chimera of Social Networks |language=Spanish |publisher=Bubok España |isbn=978-9974-99-637-3}}<br />
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* {{cite book |last1=Rainie |first1=Lee |last2=Wellman |first2=Barry |year=2012 |title=Networked: The New Social Operating System |url=https://archive.org/details/networkednewsoci0000rain |url-access=registration |location=Cambridge, Mass. |publisher=MIT Press |isbn=978-0262017190}}<br />
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* {{cite book |last=Scott |first=John |year=1991 |title=Social Network Analysis: A Handbook |publisher=Sage |isbn=978-0-7619-6338-7}}<br />
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* {{cite book |last1=Wasserman |first1=Stanley |last2=Faust |first2=Katherine |year=1994 |title=Social Network Analysis: Methods and Applications |publisher=Cambridge University Press |isbn=978-0-521-38269-4 |series=Structural Analysis in the Social Sciences }}<br />
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* {{cite book |last1=Wellman |first1=Barry |last2=Berkowitz |first2=S. D. |year=1988 |title=Social Structures: A Network Approach |series=Structural Analysis in the Social Sciences |publisher=Cambridge University Press |isbn=978-0-521-24441-1}}<br />
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==External links 外部链接==<br />
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* [http://www.insna.org/ International Network for Social Network Analysis]<br />
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同行评审期刊<br />
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* ''[http://www.sciencedirect.com/science/journal/03788733 Social Networks]''<br />
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* ''[http://journals.cambridge.org/action/displaySpecialPage?pageId=3656 Network Science]''<br />
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* ''[http://www.cmu.edu/joss/content/articles/volindex.html Journal of Social Structure]''<br />
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* ''[http://www.tandfonline.com/toc/gmas20/current Journal of Mathematical Sociology]''<br />
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* ''[https://www.springer.com/computer/database+management+%26+information+retrieval/journal/13278 Social Network Analysis and Mining (SNAM)]''<br />
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* {{cite journal |title=INSNA - ''Connections'' Journal |journal=Connections Bulletin of the International Network for Social Network Analysis |location=Toronto |publisher=International Network for Social Network Analysis |url=http://www.insna.org/pubs/connections/ |issn=0226-1766 |url-status=dead |archiveurl=https://web.archive.org/web/20130718063641/http://www.insna.org/pubs/connections/ |archivedate=2013-07-18 }}<br />
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===Textbooks and educational resources===<br />
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课本及教育资源<br />
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* ''[http://www.cs.cornell.edu/home/kleinber/networks-book/ Networks, Crowds, and Markets]'' (2010) by D. Easley & J. Kleinberg<br />
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* ''[http://faculty.ucr.edu/~hanneman/nettext/ Introduction to Social Networks Methods]'' (2005) by R. Hanneman & M. Riddle<br />
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* [http://www.analytictech.com/networks/ Social Network Analysis Instructional Web Site] by S. Borgatti<br />
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* ''[https://web.archive.org/web/20160304063126/http://www.novagob.org/file/view/153259/ebook-gratuito-sobre-redes-sociales-virtuales Guide for virtual social networks for public administrations]'' (2015) by Mauro D. Ríos (in Spanish)<br />
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===Data sets===<br />
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数据集<br />
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* [http://pajek.imfm.si/doku.php?id=data:urls:index Pajek's list of lists of datasets]<br />
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* [http://networkdata.ics.uci.edu/index.html UC Irvine Network Data Repository]<br />
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* [http://snap.stanford.edu/data/ Stanford Large Network Dataset Collection]<br />
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* [http://www-personal.umich.edu/~mejn/netdata/ M.E.J. Newman datasets]<br />
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* [http://vlado.fmf.uni-lj.si/pub/networks/data/ Pajek datasets]<br />
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* [http://wiki.gephi.org/index.php?title=Datasets#Social_networks Gephi datasets]<br />
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* [http://konect.uni-koblenz.de/networks KONECT – Koblenz network collection]<br />
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* [http://www.stats.ox.ac.uk/~snijders/siena/ RSiena datasets]<br />
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==编者推荐==<br />
集智俱乐部：[https://swarma.org/?p=18646 社会网络分析的四个维度：研究方法、应用和软件工具综述 | 网络科学论文速递25篇]<br />
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[http://arxiv.org/abs/2002.09485 The Four Dimensions of Social Network Analysis: An Overview of Research Methods, Applications, and Software Tools]：近年来，基于社交网络的应用呈指数级增长，其中一个方面的原因是，这个应用领域提供了一个特别肥沃的地方，可以测试和开发最先进的计算技术，从 Web 中提取有价值的信息。这项工作的主要贡献有三个方面：(1) 我们对社会网络分析 (SNA) 的最新发展水平进行了文献综述；(2) 我们提出了一套基于 SNA 四个基本特征（或维度）的新度量标准；(3) 最后，我们对一套流行的 SNA 工具和框架进行了定量分析。我们还进行了一项科学计量学研究，以检测该领域中最活跃的研究领域和应用领域。本工作提出了四个不同维度的定义，即(a)模式、(b)知识发现、(c)信息融合、(d)集成、可伸缩性和可视化，用于定义一组新的度量（称为度），以评估 SNA 的不同软件工具和框架（根据之前的度量对一组 20 个 SNA-软件工具进行分析和排序）。这些维度连同定义的程度，可以评价和衡量社会网络技术的成熟度，寻求对它们的定量评估，从而揭示这一活跃领域的挑战和未来发展趋势。<br />
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范畴: 系统论<br />
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<small>This page was moved from [[wikipedia:en:Social network]]. Its edit history can be viewed at [[社会网络/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E7%A4%BE%E4%BC%9A%E7%BD%91%E7%BB%9C&diff=12073社会网络2020-08-11T14:30:48Z<p>Dorr：</p>
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<div>已由Dorr初步翻译<br />
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此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Social structure made up of a set of social actors}}<br />
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{{about|the theoretical concept as used in the social and behavioral sciences|social networking sites|Social networking service|the 2010 movie|The Social Network|other uses|Social network (disambiguation)}}<br />
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A '''social network ''' is a [[social structure]] made up of a set of [[social]] actors (such as individuals or organizations), sets of [[Dyad (sociology)|dyadic]] ties, and other [[Social relation|social interactions]] between actors. The social network perspective provides a set of methods for analyzing the structure of whole social entities as well as a variety of theories explaining the patterns observed in these structures.<ref name="WF94CH1">{{cite book|last1=Wasserman |first1=Stanley | author-link1= Stanley Wasserman|last2=Faust |first2=Katherine |year=1994 |title=Social Network Analysis: Methods and Applications |isbn=9780521387071 |chapter=Social Network Analysis in the Social and Behavioral Sciences |pages=1–27 |publisher=Cambridge University Press}}</ref> The study of these structures uses [[social network analysis]] to identify local and global patterns, locate influential entities, and examine network dynamics.<br />
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A social network is a social structure made up of a set of social actors (such as individuals or organizations), sets of dyadic ties, and other social interactions between actors. The social network perspective provides a set of methods for analyzing the structure of whole social entities as well as a variety of theories explaining the patterns observed in these structures. The study of these structures uses social network analysis to identify local and global patterns, locate influential entities, and examine network dynamics.<br />
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'''<font color="#ff8000">社会网络 Social Network</font>'''是一种由一组社会行为者（个人或组织）、一组二元关系以及行为者之间的其他社会互动组成的社会结构。社会网络视角为分析整个社会实体的结构提供了一套方法，也为解释在这些结构中观察到的模式提供了各种理论。对这些结构的研究使用'''<font color="#ff8000">社会网络分析 Social Network Analysis</font>'''以确定本地及全球模式，定位有影响力的实体，并审查网络动态。<br />
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Social networks and the analysis of them is an inherently [[Interdisciplinarity|interdisciplinary]] academic field which emerged from [[social psychology]], [[sociology]], [[statistics]], and [[graph theory]]. [[Georg Simmel]] authored early structural theories in sociology emphasizing the dynamics of triads and "web of group affiliations".<ref>{{cite book|last1=Scott |first1=W. Richard |last2=Davis |first2=Gerald F. |author-link1= William Richard Scott|author-link2= Gerald F. Davis|title=Organizations and Organizing |chapter=Networks In and Around Organizations |year=2003 |isbn=978-0-13-195893-7 |publisher=Pearson Prentice Hall}}</ref> [[Jacob Moreno]] is credited with developing the first [[sociogram]]s in the 1930s to study interpersonal relationships. These approaches were mathematically formalized in the 1950s and theories and methods of social networks became pervasive in the [[Social science|social and behavioral sciences]] by the 1980s.<ref name="WF94CH1"/><ref name="Freeman History">{{cite book|last=Freeman |first=Linton |year=2004 |publisher=Empirical Press |isbn=978-1-59457-714-7 |title=The Development of Social Network Analysis: A Study in the Sociology of Science}}</ref> [[Social network analysis]] is now one of the major paradigms in contemporary sociology, and is also employed in a number of other social and formal sciences. Together with other [[complex network]]s, it forms part of the nascent field of [[network science]].<ref>{{cite journal|journal=Science |year=2009 |volume=323 |number=5916 |pages=892–895 |doi=10.1126/science.1165821 |pmid=19213908 |title=Network Analysis in the Social Sciences |first1=Stephen P. |last1=Borgatti |first2=Ajay |last2=Mehra |first3=Daniel J. |last3=Brass |first4=Giuseppe |last4=Labianca|bibcode=2009Sci...323..892B |citeseerx=10.1.1.536.5568 }}</ref><ref name="EK">{{cite book|title=Networks, Crowds, and Markets: Reasoning about a Highly Connected World |first1=David |last1=Easley |first2=Jon |last2=Kleinberg |publisher=Cambridge University Press |year=2010 |chapter=Overview |pages=1–20 |isbn=978-0-521-19533-1}}</ref><br />
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Social networks and the analysis of them is an inherently interdisciplinary academic field which emerged from social psychology, sociology, statistics, and graph theory. Georg Simmel authored early structural theories in sociology emphasizing the dynamics of triads and "web of group affiliations". Jacob Moreno is credited with developing the first sociograms in the 1930s to study interpersonal relationships. These approaches were mathematically formalized in the 1950s and theories and methods of social networks became pervasive in the social and behavioral sciences by the 1980s. Social network analysis is now one of the major paradigms in contemporary sociology, and is also employed in a number of other social and formal sciences. Together with other complex networks, it forms part of the nascent field of network science.<br />
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社会网络及其分析是社会心理学、社会学、统计学和图论中内在的跨学科学术领域。'''格奥尔格·齐美尔 Georg Simmel'''是早期社会学结构主义理论的作者，他强调三合会和“群体关系网”的动态性。'''雅各布·莫雷诺 Jacob Moreno'''被认为是在20世纪30年代发展了第一份'''<font color="#ff8000">社交关系图 Sociogram</font>'''以研究人际关系的人。这些方法在20世纪50年代得到数学化，到20世纪80年代，社会网络的理论和方法在社会和行为科学中变得普遍。社会网络分析现在是当代社会学的主要范式之一，也被用于许多其他社会科学及形式科学。它与其他复杂网络一起构成了网络科学新兴领域的一部分。<br />
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==Overview 概览==<br />
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[[File:Barabasi Albert model.gif|thumb|left|Evolution graph of a social network: [[Barabási–Albert model|Barabási model]].图一：一个社会网络的演化图：巴拉巴西模型。]]<br />
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Barabási model.]]<br />
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Barabási model.]]<br />
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The social network is a [[Scientific theory|theoretical]] [[Construct (philosophy of science)|construct]] useful in the [[social sciences]] to study relationships between individuals, [[social groups|groups]], [[formal organizations|organizations]], or even entire [[society|societies]] ([[social unit]]s, see [[Differentiation (sociology)|differentiation]]). The term is used to describe a [[social structure]] determined by such [[social interactions|interactions]]. The ties through which any given social unit connects represent the convergence of the various social contacts of that unit. This theoretical approach is, necessarily, relational. An [[axiom]] of the social network approach to understanding [[social interaction]] is that social phenomena should be primarily conceived and investigated through the properties of relations between and within units, instead of the properties of these units themselves. Thus, one common criticism of social network theory is that [[Agency (sociology)|individual agency]] is often ignored<ref name="jscott">Scott, John P. (2000). ''Social Network Analysis: A Handbook'' (2nd edition). Thousand Oaks, CA: Sage Publications.</ref> although this may not be the case in practice (see [[agent-based model]]ing). Precisely because many different types of relations, singular or in combination, form these network configurations, [[Network science|network analytics]] are useful to a broad range of research enterprises. In social science, these fields of study include, but are not limited to [[anthropology]], [[biology]], [[communication studies]], [[economics]], [[geography]], [[information science]], [[organizational studies]], [[social psychology]], [[sociology]], and [[sociolinguistics]].<br />
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The social network is a theoretical construct useful in the social sciences to study relationships between individuals, groups, organizations, or even entire societies (social units, see differentiation). The term is used to describe a social structure determined by such interactions. The ties through which any given social unit connects represent the convergence of the various social contacts of that unit. This theoretical approach is, necessarily, relational. An axiom of the social network approach to understanding social interaction is that social phenomena should be primarily conceived and investigated through the properties of relations between and within units, instead of the properties of these units themselves. Thus, one common criticism of social network theory is that individual agency is often ignored although this may not be the case in practice (see agent-based modeling). Precisely because many different types of relations, singular or in combination, form these network configurations, network analytics are useful to a broad range of research enterprises. In social science, these fields of study include, but are not limited to anthropology, biology, communication studies, economics, geography, information science, organizational studies, social psychology, sociology, and sociolinguistics.<br />
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社会网络是社会科学中研究个人、团体、组织甚至整个社会（社会单位，见分化）之间关系的理论构造。这个术语用来描述由这种相互作用决定的社会结构。任何一个特定的社会单元之间的联系都代表着这个单元各种社会联系的聚合。这种理论方法必然是相关的。理解社会互动的社会网络方法的一个公理是，社会现象应该主要通过单元之间和单元内部关系的性质来构思和研究，而不是这些单元本身的性质。因此，社会网络理论总被诟病的一点是其常忽视个体代理，而实践中可能并非如此（见基于主体的建模）。正是因为许多不同类型的关系（单独或组合形式）形成这些网络配置，网络分析在广泛的研究中有用。在社会科学中，这些研究领域包括但不限于'''<font color="#ff8000">人类学 Anthropology</font>'''、生物学、'''<font color="#ff8000">传播学 Communication Studies</font>'''、经济学、地理学、信息科学、组织学、社会心理学、社会学和'''<font color="#ff8000">社会语言学 Sociolinguistics</font>'''。<br />
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==History 历史==<br />
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In the late 1890s, both [[Émile Durkheim]] and [[Ferdinand Tönnies]] foreshadowed the idea of social networks in their theories and research of [[social group]]s. Tönnies argued that social groups can exist as personal and direct social ties that either link individuals who share values and belief (''[[Gemeinschaft]]'', German, commonly translated as "[[community]]") or impersonal, formal, and instrumental social links (''[[Gesellschaft]]'', German, commonly translated as "[[society]]").<ref>Tönnies, Ferdinand (1887). ''Gemeinschaft und Gesellschaft'', Leipzig: Fues's Verlag. (Translated, 1957 by Charles Price Loomis as ''Community and Society'', East Lansing: Michigan State University Press.)</ref> Durkheim gave a non-individualistic explanation of social facts, arguing that social phenomena arise when interacting individuals constitute a reality that can no longer be accounted for in terms of the properties of individual actors.<ref>Durkheim, Emile (1893). ''De la division du travail social: étude sur l'organisation des sociétés supérieures'', Paris: F. Alcan. (Translated, 1964, by Lewis A. Coser as ''The Division of Labor in Society'', New York: Free Press.)</ref> [[Georg Simmel]], writing at the turn of the twentieth century, pointed to the nature of networks and the effect of network size on interaction and examined the likelihood of interaction in loosely knit networks rather than groups.<ref>Simmel, Georg (1908). ''Soziologie'', Leipzig: Duncker & Humblot.</ref><br />
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In the late 1890s, both Émile Durkheim and Ferdinand Tönnies foreshadowed the idea of social networks in their theories and research of social groups. Tönnies argued that social groups can exist as personal and direct social ties that either link individuals who share values and belief (Gemeinschaft, German, commonly translated as "community") or impersonal, formal, and instrumental social links (Gesellschaft, German, commonly translated as "society"). Durkheim gave a non-individualistic explanation of social facts, arguing that social phenomena arise when interacting individuals constitute a reality that can no longer be accounted for in terms of the properties of individual actors. Georg Simmel, writing at the turn of the twentieth century, pointed to the nature of networks and the effect of network size on interaction and examined the likelihood of interaction in loosely knit networks rather than groups.<br />
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在19世纪90年代晚期，'''埃米尔·涂尔干 Émile Durkheim'''和'''斐迪南·滕尼斯 Ferdinand Tönnies'''在他们关于社会群体的理论和研究中都预示了社会网络的概念。滕尼斯认为，社会群体可以作为个人和直接的社会关系存在，这种关系或者将具有共同价值观和信仰的个人（德语 Gemeinschaft，通常翻译为“社区”）联系在一起，或者将非个人的、正式的和工具性的社会关系（德语 Gesellschaft，通常翻译为“社会”）联系在一起。涂尔干对社会事实给出了非个人主义的解释，认为当相互作用的个体构成一种再也不能用个体行为者的特性来解释的现实时，社会现象就产生了。格奥尔格·齐美尔在20世纪之交的著作中指出了网络的本质和网络规模对互动的影响，并研究了在松散的网络而非群体中互动的可能性。<br />
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[[File:Moreno Sociogram 2nd Grade.png|thumb|Moreno's sociogram of a 2nd grade class 图二：莫雷诺对一个二年级班级的社会关系图]]<br />
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Major developments in the field can be seen in the 1930s by several groups in psychology, anthropology, and mathematics working independently.<ref name=jscott /><ref>For a historical overview of the development of social network analysis, see: {{cite book|last1=Carrington|first1=Peter J.|last2=Scott|first2=John|chapter=Introduction|title=The Sage Handbook of Social Network Analysis| publisher=Sage|year=2011|isbn=978-1-84787-395-8|page=1|chapter-url=https://books.google.com/books?id=2chSmLzClXgC&pg=PA1}}</ref><ref>See also the diagram in {{cite book|author=Scott, John|title=Social Network Analysis: A Handbook|publisher=Sage|year=2000|isbn=978-0-7619-6339-4|page=8|url=https://books.google.com/books?id=Ww3_bKcz6kgC&pg=PA8}}</ref> In [[psychology]], in the 1930s, [[Jacob L. Moreno]] began systematic recording and analysis of social interaction in small groups, especially classrooms and work groups (see [[sociometry]]). In [[anthropology]], the foundation for social network theory is the theoretical and [[ethnography|ethnographic]] work of [[Bronislaw Malinowski]],<ref>Malinowski, Bronislaw (1913). ''The Family Among the Australian Aborigines: A Sociological Study''. London: University of London Press.</ref> [[Radcliffe-Brown|Alfred Radcliffe-Brown]],<ref>Radcliffe-Brown, Alfred Reginald (1930) ''The social organization of Australian tribes''. Sydney, Australia: University of Sydney ''Oceania'' monographs, No.1.</ref><ref>{{cite journal | last1 = Radcliffe-Brown | first1 = A. R. | year = 1940 | title = On social structure | url = | journal = Journal of the Royal Anthropological Institute | volume = 70 | issue = 1| pages = 1–12 | doi=10.2307/2844197| jstor = 2844197 }}</ref> and [[Claude Lévi-Strauss]].<ref>Lévi-Strauss, Claude ([1947]1967). ''Les structures élémentaires de la parenté''. Paris: La Haye, Mouton et Co. (Translated, 1969 by J. H. Bell, J. R. von Sturmer, and R. Needham, 1969, as ''The Elementary Structures of Kinship'', Boston: Beacon Press.)</ref> A group of social anthropologists associated with [[Max Gluckman]] and the [[Manchester school (anthropology)|Manchester School]], including [[John Arundel Barnes|John A. Barnes]],<ref>Barnes, John (1954). "Class and Committees in a Norwegian Island Parish". ''Human Relations'', (7): 39–58.</ref> [[J. Clyde Mitchell]] and [[Elizabeth Bott Spillius]],<ref>{{cite journal | last1 = Freeman | first1 = Linton C. | last2 = Wellman | first2 = Barry | year = 1995 | title = A note on the ancestoral Toronto home of social network analysis | url = | journal = Connections | volume = 18 | issue = 2| pages = 15–19 }}</ref><ref>{{cite journal | last1 = Savage | first1 = Mike | year = 2008 | title = Elizabeth Bott and the formation of modern British sociology | url = | journal = The Sociological Review | volume = 56 | issue = 4| pages = 579–605 | doi=10.1111/j.1467-954x.2008.00806.x}}</ref> often are credited with performing some of the first fieldwork from which network analyses were performed, investigating community networks in southern Africa, India and the United Kingdom.<ref name=jscott /> Concomitantly, British anthropologist [[Siegfried Frederick Nadel|S. F. Nadel]] codified a theory of social structure that was influential in later network analysis.<ref>Nadel, S. F. 1957. ''The Theory of Social Structure''. London: Cohen and West.</ref> In [[sociology]], the early (1930s) work of [[Talcott Parsons]] set the stage for taking a relational approach to understanding social structure.<ref>Parsons, Talcott ([1937] 1949). ''The Structure of Social Action: A Study in Social Theory with Special Reference to a Group of European Writers''. New York: The Free Press.</ref><ref>Parsons, Talcott (1951). ''The Social System''. New York: The Free Press.</ref> Later, drawing upon Parsons' theory, the work of sociologist [[Peter Blau]] provides a strong impetus for analyzing the relational ties of social units with his work on [[social exchange theory]].<ref>Blau, Peter (1956). ''Bureaucracy in Modern Society''. New York: Random House, Inc.</ref><ref>Blau, Peter (1960). "A Theory of Social Integration". ''The American Journal of Sociology'', (65)6: 545–556, (May).</ref><ref>Blau, Peter (1964). ''Exchange and Power in Social Life''.</ref><br />
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Major developments in the field can be seen in the 1930s by several groups in psychology, anthropology, and mathematics working independently. In psychology, in the 1930s, Jacob L. Moreno began systematic recording and analysis of social interaction in small groups, especially classrooms and work groups (see sociometry). In anthropology, the foundation for social network theory is the theoretical and ethnographic work of Bronislaw Malinowski, Alfred Radcliffe-Brown, and Claude Lévi-Strauss. A group of social anthropologists associated with Max Gluckman and the Manchester School, including John A. Barnes, J. Clyde Mitchell and Elizabeth Bott Spillius, often are credited with performing some of the first fieldwork from which network analyses were performed, investigating community networks in southern Africa, India and the United Kingdom. In sociology, the early (1930s) work of Talcott Parsons set the stage for taking a relational approach to understanding social structure. Later, drawing upon Parsons' theory, the work of sociologist Peter Blau provides a strong impetus for analyzing the relational ties of social units with his work on social exchange theory.<br />
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20世纪30年代，心理学、人类学和数学领域的几个独立研究小组已经看到了这一领域的重大发展。在心理学方面，在20世纪30年代，雅各布·L·莫雷诺开始系统地记录和分析小团体中的社会互动，尤其是课堂和工作团体中的社会互动（见'''<font color="#ff8000">社会测量 Sociometry）</font>'''。在人类学中，社会网络理论的基础是'''布罗尼斯拉夫·马林诺夫斯基 Bronislaw Malinowski'''，'''阿尔弗雷德·拉德克利夫-布朗 Alfred Radcliffe-Brown'''和'''克洛德·列维-斯特劳斯 Claude Lévi-Strauss'''的理论和人种学著作。包括'''约翰·A·巴恩斯 John A. Barnes'''、'''J·克莱德·米切尔 J. Clyde Mitchell'''和'''伊丽莎白·博特·斯皮利厄斯 Elizabeth Bott Spillius'''在内的一群与'''马克斯·格拉克曼 Max Gluckman'''和'''曼彻斯特学派 Manchester School'''有关的社会人类学家，经常被认为是执行了一些最初的实地工作，从而进行了网络分析，调查了南非、印度和英国的社区网络。在社会学方面，'''塔尔科特·帕森斯 Talcott Parsons'''的早期工作（1930年代）为采用关系方法理解社会结构奠定了基础。后来，社会学家'''彼得·布劳 Peter Blau'''的'''<font color="#ff8000">社会交换论 Social Exchange Theory</font>'''为分析社会单位之间的关系提供了强大的动力。<br />
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By the 1970s, a growing number of scholars worked to combine the different tracks and traditions. One group consisted of sociologist [[Harrison White]] and his students at the [[Harvard Department of Social Relations|Harvard University Department of Social Relations]]. Also independently active in the Harvard Social Relations department at the time were [[Charles Tilly]], who focused on networks in political and community sociology and social movements, and [[Stanley Milgram]], who developed the "six degrees of separation" thesis.<ref>{{cite web |url=http://www.semioticon.com/semiotix/semiotix14/sem-14-05.html |title=The Networked Individual: A Profile of Barry Wellman |author=Bernie Hogan}}</ref> [[Mark Granovetter]]<ref name="Introduction for the French Reader">{{cite journal | last1 = Granovetter | first1 = Mark | year = 2007 | title = Introduction for the French Reader | url = | journal = Sociologica | volume = 2 | issue = | pages = 1–8 }}</ref> and [[Barry Wellman]]<ref>Wellman, Barry (1988). "Structural analysis: From method and metaphor to theory and substance". pp. 19–61 in B. Wellman and S. D. Berkowitz (eds.) ''Social Structures: A Network Approach'', Cambridge, UK: Cambridge University Press.</ref> are among the former students of White who elaborated and championed the analysis of social networks.<ref name="Introduction for the French Reader"/><ref>Mullins, Nicholas. ''Theories and Theory Groups in Contemporary American Sociology''. New York: Harper and Row, 1973.</ref><ref>Tilly, Charles, ed. ''An Urban World''. Boston: Little Brown, 1974.</ref><ref>Wellman, Barry. 1988. "Structural Analysis: From Method and Metaphor to Theory and Substance". pp. 19–61 in ''Social Structures: A Network Approach'', edited by Barry Wellman and S. D. Berkowitz. Cambridge: Cambridge University Press.</ref><br />
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By the 1970s, a growing number of scholars worked to combine the different tracks and traditions. One group consisted of sociologist Harrison White and his students at the Harvard University Department of Social Relations. Also independently active in the Harvard Social Relations department at the time were Charles Tilly, who focused on networks in political and community sociology and social movements, and Stanley Milgram, who developed the "six degrees of separation" thesis. Mark Granovetter and Barry Wellman are among the former students of White who elaborated and championed the analysis of social networks.<br />
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到了20世纪70年代，越来越多的学者致力于将不同的轨迹和传统结合起来。其中一组由社会学家'''哈里森·怀特 Harrison White'''和他在哈佛大学社会关系系的学生组成。当时在哈佛大学社会关系系独立活动的还有专注于政治和社区社会学和社会运动的网络的'''查尔斯·堤利 Charles Tilly'''，还有发表了六度分隔理论论文的'''斯坦利·米尔格拉姆 Stanley Milgram'''。'''马克·格兰诺维特 Mark Granovetter'''和'''巴里·威尔曼 Barry Wellman'''是怀特以前的学生，他们阐述并支持对社交网络的分析。<br />
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Beginning in the late 1990s, social network analysis experienced work by sociologists, political scientists, and physicists such as [[Duncan J. Watts]], [[Albert-László Barabási]], [[Peter Bearman]], [[Nicholas A. Christakis]], [[James H. Fowler]], and others, developing and applying new models and methods to emerging data available about online social networks, as well as "digital traces" regarding face-to-face networks.<br />
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Beginning in the late 1990s, social network analysis experienced work by sociologists, political scientists, and physicists such as Duncan J. Watts, Albert-László Barabási, Peter Bearman, Nicholas A. Christakis, James H. Fowler, and others, developing and applying new models and methods to emerging data available about online social networks, as well as "digital traces" regarding face-to-face networks.<br />
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从20世纪90年代末开始，社会网络分析经历了社会学家、政治学家和物理学家的工作，如'''[[邓肯·瓦茨 Duncan J. Watts]]'''、'''[[艾伯特-拉斯洛·巴拉巴西 Albert-László Barabási]]'''、Peter Bearman、Nicholas A. Christakis、James H. Fowler等人，开发和应用新的模型和方法来获得有关在线社会网络的新兴数据，以及有关面对面网络的“数字痕迹”。<br />
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==Levels of analysis 分析层次==<br />
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[[File:Network self-organization stages.png|thumb|right|Self-organization of a network, based on Nagler, Levina, & Timme, (2011) 图三：基于Nagler, Levina和Timme的自组织网络，(2011)<ref>{{cite journal|author1=Nagler, Jan|author2=Anna Levina|author3=Marc Timme|year=2011|title=Impact of single links in competitive percolation|journal=Nature Physics|volume=7|issue=3|pages=265–270|doi=10.1038/nphys1860|arxiv=1103.0922|bibcode=2011NatPh...7..265N}}</ref>]]<br />
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[[File:Social Network Diagram (large).svg|right|thumb|Centrality]]<br />
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Centrality<br />
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中心性<br />
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In general, social networks are [[self-organization|self-organizing]], [[emergence|emergent]], and [[social complexity|complex]], such that a globally coherent pattern appears from the local interaction of the elements that make up the system.<ref>Newman, Mark, Albert-László Barabási and Duncan J. Watts (2006). ''The Structure and Dynamics of Networks'' (Princeton Studies in Complexity). Oxford: Princeton University Press.</ref><ref>{{cite journal | last1 = Wellman | first1 = Barry | year = 2008 | title = Review: The development of social network analysis: A study in the sociology of science | url = | journal = Contemporary Sociology | volume = 37 | issue = 3| pages = 221–222 | doi=10.1177/009430610803700308}}</ref> These patterns become more apparent as network size increases. However, a global network analysis<ref>{{cite book|last=Faust|first=Stanley Wasserman; Katherine|title=Social network analysis : methods and applications|year=1998|publisher=Cambridge Univ. Press|location=Cambridge [u.a.]|isbn=978-0521382694|edition=Reprint.}}</ref> of, for example, all [[interpersonal relationships]] in the world is not feasible and is likely to contain so much [[Information theory|information]] as to be uninformative. Practical limitations of computing power, ethics and participant recruitment and payment also limit the scope of a social network analysis.<ref name="Kadu12">Kadushin, C. (2012). ''Understanding social networks: Theories, concepts, and findings''. Oxford: Oxford University Press.</ref><ref>{{Cite journal| author=Granovetter, M.|title= Network sampling: Some first steps| year=1976 |pages=1287–1303 |volume=81| issue=6 |journal= American Journal of Sociology | doi=10.1086/226224}}</ref> The nuances of a local system may be lost in a large network analysis, hence the quality of information may be more important than its scale for understanding network properties. Thus, social networks are analyzed at the scale relevant to the researcher's theoretical question. Although [[level of analysis|levels of analysis]] are not necessarily [[Mutually exclusive events|mutually exclusive]], there are three general levels into which networks may fall: [[Microsociology|micro-level]], [[wikt:meso-|meso-level]], and [[Macrosociology|macro-level]].<br />
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In general, social networks are self-organizing, emergent, and complex, such that a globally coherent pattern appears from the local interaction of the elements that make up the system. These patterns become more apparent as network size increases. However, a global network analysis of, for example, all interpersonal relationships in the world is not feasible and is likely to contain so much information as to be uninformative. Practical limitations of computing power, ethics and participant recruitment and payment also limit the scope of a social network analysis. The nuances of a local system may be lost in a large network analysis, hence the quality of information may be more important than its scale for understanding network properties. Thus, social networks are analyzed at the scale relevant to the researcher's theoretical question. Although levels of analysis are not necessarily mutually exclusive, there are three general levels into which networks may fall: micro-level, meso-level, and macro-level.<br />
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一般来说，社会网络是自组织的、涌现的和复杂的，这样，一个全局一致的模式就会从组成系统的元素的局部交互中显现出来。随着网络规模的增大，这些模式变得更加明显。然而，一个全球网络分析（如世界上所有的人际关系）是不可行的，它可能包含太多的信息，以至于相当于没有提供信息。计算能力的实际限制、道德规范以及参与者的招聘和报酬也限制了社会网络分析的规模。局部系统的细微差别在大规模网络分析中可能会消失，因此对于理解网络属性来说，信息的质量可能比其规模更重要。因此，社会网络是在与研究者的理论问题相关的尺度上加以分析的。虽然分析层次不一定相互排斥，但网络可以分为三个一般层次: 微观、中观和宏观。<br />
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===Micro level 微观层面===<br />
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At the micro-level, social network research typically begins with an individual, [[Snowball sampling|snowballing]] as social relationships are traced, or may begin with a small group of individuals in a particular social context.<br />
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At the micro-level, social network research typically begins with an individual, snowballing as social relationships are traced, or may begin with a small group of individuals in a particular social context.<br />
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在微观层面上，社会网络研究通常从个人开始，随着社会关系的追踪而像滚雪球一样扩大，或者可能从特定社会背景下的一小群个体开始。<br />
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'''Dyadic level''': A [[Dyad (sociology)|dyad]] is a social relationship between two individuals. Network research on dyads may concentrate on [[Structural functionalism|structure]] of the relationship (e.g. multiplexity, strength), [[social equality]], and tendencies toward [[Reciprocity (social and political philosophy)|reciprocity/mutuality]].<br />
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Dyadic level: A dyad is a social relationship between two individuals. Network research on dyads may concentrate on structure of the relationship (e.g. multiplexity, strength), social equality, and tendencies toward reciprocity/mutuality.<br />
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二元层面：二元是两个个体之间的社会关系。网络对二元关系的研究可以集中在关系的结构上（如多样性、力量)、社会平等以及互惠互利的倾向。<br />
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'''Triadic level''': Add one individual to a dyad, and you have a [[Triadic relation|triad]]. Research at this level may concentrate on factors such as [[Independence number|balance]] and [[Vertex-transitive graph|transitivity]], as well as [[social equality]] and tendencies toward [[Reciprocity (social and political philosophy)|reciprocity/mutuality]].<ref name="Kadu12"/> In the [[balance theory]] of [[Fritz Heider]] the triad is the key to social dynamics. The discord in a rivalrous [[love triangle]] is an example of an unbalanced triad, likely to change to a balanced triad by a change in one of the relations. The dynamics of social friendships in society has been modeled by balancing triads. The study is carried forward with the theory of [[signed graph]]s.<br />
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Triadic level: Add one individual to a dyad, and you have a triad. Research at this level may concentrate on factors such as balance and transitivity, as well as social equality and tendencies toward reciprocity/mutuality. In the balance theory of Fritz Heider the triad is the key to social dynamics. The discord in a rivalrous love triangle is an example of an unbalanced triad, likely to change to a balanced triad by a change in one of the relations. The dynamics of social friendships in society has been modeled by balancing triads. The study is carried forward with the theory of signed graphs.<br />
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三元层面：再加一个个体到二元组中，就得到一个三元组。这一层次的研究可能集中在诸如平衡和传递性等因素，以及社会平等和互惠互利的倾向。在'''弗里茨 · 海德 Fritz Heider'''的'''平衡理论 Balance Theory'''中，三元组是社会动力学的关键。在敌对的三角恋中的不和谐是不平衡的三角关系的一个例子，很可能通过其中一种关系的改变而变成平衡的三角关系。社会中社会友谊的动态模型是通过平衡三角关系建立起来的。该研究利用了符号图理论。<br />
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'''Actor level''': The smallest unit of analysis in a social network is an individual in their social setting, i.e., an "actor" or "ego". Egonetwork analysis focuses on network characteristics such as size, relationship strength, density, [[centrality]], [[wikt:prestige|prestige]] and roles such as [[isolates|isolates, liaisons]], and [[Bridge (interpersonal)|bridges]].<ref name="Jone11"/> Such analyses, are most commonly used in the fields of [[psychology]] or [[Social psychology (sociology)|social psychology]], [[ethnographic]] [[kinship]] analysis or other [[genealogy|genealogical]] studies of relationships between individuals.<br />
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Actor level: The smallest unit of analysis in a social network is an individual in their social setting, i.e., an "actor" or "ego". Egonetwork analysis focuses on network characteristics such as size, relationship strength, density, centrality, prestige and roles such as isolates, liaisons, and bridges. Such analyses, are most commonly used in the fields of psychology or social psychology, ethnographic kinship analysis or other genealogical studies of relationships between individuals.<br />
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行为者层面：社会网络中最小的分析单位是社会环境中的个体，即“行为者”或“自我”。自我网络分析主要关注网络特征，例如大小、关系强度、密度、中心性、声望和隔离、联络和桥梁等角色。这种分析最常用于心理学或社会心理学、人种学亲属关系分析或其他个体关系的系谱研究领域。<br />
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'''Subset level''': [[Subset]] levels of network research problems begin at the micro-level, but may cross over into the meso-level of analysis. Subset level research may focus on [[Distance (graph theory)|distance]] and reachability, [[cliques]], [[Cohesion (social policy)|cohesive]] subgroups, or other [[Group action (sociology)|group actions]] or [[Group behaviour|behavior]].<ref>{{cite book | title="Graph Theoretical Approaches to Social Network Analysis". in Computational Complexity: Theory, Techniques, and Applications (Robert A. Meyers, ed.) | publisher=Springer | author=de Nooy, Wouter |year=2012 | pages=2864–2877 | isbn=978-1-4614-1800-9|doi=10.1007/978-1-4614-1800-9_176| chapter=Social Network Analysis, Graph Theoretical Approaches to }}</ref><br />
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Subset level: Subset levels of network research problems begin at the micro-level, but may cross over into the meso-level of analysis. Subset level research may focus on distance and reachability, cliques, cohesive subgroups, or other group actions or behavior.<br />
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子集层面：网络研究问题的子集级别开始于微观级别，但可能跨越到中观级别的分析。子集级别的研究可能集中在距离和可达性、派系、凝聚子群或其他群体行为或行为。<br />
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===Meso level 中观层面===<br />
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In general, meso-level theories begin with a [[Sample population|population]] size that falls between the micro- and macro-levels. However, meso-level may also refer to analyses that are specifically designed to reveal connections between micro- and macro-levels. Meso-level networks are low density and may exhibit causal processes distinct from interpersonal micro-level networks.<ref>{{cite journal | last1 = Hedström | first1 = Peter | last2 = Sandell | first2 = Rickard | last3 = Stern | first3 = Charlotta | year = 2000 | title = Mesolevel Networks and the Diffusion of Social Movements: The Case of the Swedish Social Democratic Party | url = http://www.nuffield.ox.ac.uk/users/hedstrom/ajs3.pdf | journal = American Journal of Sociology | volume = 106 | issue = 1| pages = 145–172 | doi = 10.1086/303109 }}</ref><br />
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In general, meso-level theories begin with a population size that falls between the micro- and macro-levels. However, meso-level may also refer to analyses that are specifically designed to reveal connections between micro- and macro-levels. Meso-level networks are low density and may exhibit causal processes distinct from interpersonal micro-level networks.<br />
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一般来说，中观层面的理论始于介于微观和宏观层面之间的规模。然而，中观层面也可以指专门为揭示微观和宏观层面之间的联系而设计的分析。中观层次的网络是密度低，可能表现出不同于人际微观层面网络的因果过程。<br />
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[[File:Social Red.jpg|thumb|right|Social network diagram, meso-level 图四：中观层而社会网络图]]<br />
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'''Organizations''': Formal [[organizations]] are [[social group]]s that distribute tasks for a collective [[goal]].<ref name="Riketta, M. 2007">{{cite journal | last1 = Riketta | first1 = M. | last2 = Nienber | first2 = S. | year = 2007 | title = Multiple identities and work motivation: The role of perceived compatibility between nested organizational units | url = | journal = British Journal of Management | volume = 18 | issue = | pages = S61–77 | doi=10.1111/j.1467-8551.2007.00526.x}}</ref> Network research on organizations may focus on either intra-organizational or inter-organizational ties in terms of [[Formal organization|formal]] or [[Informal organization|informal]] relationships. Intra-organizational networks themselves often contain multiple levels of analysis, especially in larger organizations with multiple branches, franchises or semi-autonomous departments. In these cases, research is often conducted at a work group level and organization level, focusing on the interplay between the two structures.<ref name="Riketta, M. 2007"/> Experiments with networked groups online have documented ways to optimize group-level coordination through diverse interventions, including the addition of autonomous agents to the groups.<ref>{{Cite journal|last=Shirado|first=Hirokazu|last2=Christakis|first2=Nicholas A|title=Locally noisy autonomous agents improve global human coordination in network experiments|journal=Nature|volume=545|issue=7654|pages=370–374|doi=10.1038/nature22332|pmid=28516927|pmc=5912653|bibcode=2017Natur.545..370S|year=2017}}</ref><br />
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Organizations: Formal organizations are social groups that distribute tasks for a collective goal. Network research on organizations may focus on either intra-organizational or inter-organizational ties in terms of formal or informal relationships. Intra-organizational networks themselves often contain multiple levels of analysis, especially in larger organizations with multiple branches, franchises or semi-autonomous departments. In these cases, research is often conducted at a work group level and organization level, focusing on the interplay between the two structures.<br />
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组织: 正规组织是为共同目标分配任务的社会团体。关于组织的网络研究可以侧重于正式或非正式关系方面的组织内或组织间联系。组织内网络本身往往包含多层次的分析，特别是在具有多个分支机构、特许权或半自治部门的较大组织中。在这些情况下，研究通常在工作组和组织层面进行，重点放在两个结构之间的相互作用。<br />
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'''Randomly distributed networks''': [[Exponential random graph models]] of social networks became state-of-the-art methods of social network analysis in the 1980s. This framework has the capacity to represent social-structural effects commonly observed in many human social networks, including general [[Degree (graph theory)|degree]]-based structural effects commonly observed in many human social networks as well as [[Reciprocity (social and political philosophy)|reciprocity]] and [[Transitive set|transitivity]], and at the node-level, [[homophily]] and [[Attribute-value system|attribute]]-based activity and popularity effects, as derived from explicit hypotheses about [[Dependency graph|dependencies]] among network ties. [[Parameter]]s are given in terms of the prevalence of small [[Induced subgraph|subgraph]] configurations in the network and can be interpreted as describing the combinations of local social processes from which a given network emerges. These probability models for networks on a given set of actors allow generalization beyond the restrictive dyadic independence assumption of micro-networks, allowing models to be built from theoretical structural foundations of social behavior.<ref>{{cite journal | last1 = Cranmer | first1 = Skyler J. | last2 = Desmarais | first2 = Bruce A. | year = 2011 | title = Inferential Network Analysis with Exponential Random Graph Models | url = | journal = Political Analysis | volume = 19 | issue = 1| pages = 66–86 | doi=10.1093/pan/mpq037| citeseerx = 10.1.1.623.751 }}</ref><br />
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Randomly distributed networks: Exponential random graph models of social networks became state-of-the-art methods of social network analysis in the 1980s. This framework has the capacity to represent social-structural effects commonly observed in many human social networks, including general degree-based structural effects commonly observed in many human social networks as well as reciprocity and transitivity, and at the node-level, homophily and attribute-based activity and popularity effects, as derived from explicit hypotheses about dependencies among network ties. Parameters are given in terms of the prevalence of small subgraph configurations in the network and can be interpreted as describing the combinations of local social processes from which a given network emerges. These probability models for networks on a given set of actors allow generalization beyond the restrictive dyadic independence assumption of micro-networks, allowing models to be built from theoretical structural foundations of social behavior.<br />
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随机分布的网络: 社会网络的'''<font color="#ff8000">指数随机图模型 Exponential random graph models</font>'''在20世纪80年代成为最先进的社会网络分析方法。这个框架有能力表示在许多人类社会网络中普遍观察到的社会结构效应，包括在许多人类社会网络中普遍观察到的基于程度的一般性结构效应以及互惠性和传递性，以及在节点一级、同相性和基于属性的活动和流行性效应，这些效应源于关于网络关系之间依赖性的明确假设。参数是根据网络中小型子图配置的流行程度给出的，可以解释为描述一个给定网络出现的局部社会过程的组合。这些网络的概率模型在给定的参与者集合上允许超越微型网络的限制性并元独立性假设的泛化，允许模型从社会行为的理论结构基础上建立。<br />
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[[File:Scale-free network sample.png|thumb|right|Examples of a random network and a scale-free network. Each graph has 32 nodes and 32 links. Note the "hubs" (shaded) in the scale-free diagram (on the right).图五：一个随机网络和一个无尺度网络的例子。每个图有32个顶点和32条边。注意无标度图中的“集线器”（阴影部分）（右侧）。]]<br />
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'''Scale-free networks''': A [[scale-free network]] is a [[complex network|network]] whose [[degree distribution]] follows a [[power law]], at least [[asymptotic]]ally. In [[network theory]] a scale-free ideal network is a [[random network]] with a [[degree distribution]] that unravels the size distribution of social groups.<ref>{{cite journal |author1=Moreira, André A.|author2=Demétrius R. Paula|author3=Raimundo N. Costa Filho|author4=José S. Andrade, Jr. |title=Competitive cluster growth in complex networks |year=2006 |doi=10.1103/PhysRevE.73.065101 |pmid=16906890 |journal=Physical Review E |volume=73 |issue=6 |pages=065101 |arxiv=cond-mat/0603272 |bibcode=2006PhRvE..73f5101M }}</ref> Specific characteristics of scale-free networks vary with the theories and analytical tools used to create them, however, in general, scale-free networks have some common characteristics. One notable characteristic in a scale-free network is the relative commonness of [[Vertex (graph theory)|vertices]] with a [[Maximum degree|degree]] that greatly exceeds the average. The highest-degree nodes are often called "hubs", and may serve specific purposes in their networks, although this depends greatly on the social context. Another general characteristic of scale-free networks is the [[clustering coefficient]] distribution, which decreases as the node degree increases. This distribution also follows a [[power law]].<ref>Barabási, Albert-László (2003). ''Linked: how everything is connected to everything else and what it means for business, science, and everyday life''. New York: Plum.</ref> The [[Barabási–Albert model|Barabási]] model of network evolution shown above is an example of a scale-free network.<br />
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Scale-free networks: A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. In network theory a scale-free ideal network is a random network with a degree distribution that unravels the size distribution of social groups. Specific characteristics of scale-free networks vary with the theories and analytical tools used to create them, however, in general, scale-free networks have some common characteristics. One notable characteristic in a scale-free network is the relative commonness of vertices with a degree that greatly exceeds the average. The highest-degree nodes are often called "hubs", and may serve specific purposes in their networks, although this depends greatly on the social context. Another general characteristic of scale-free networks is the clustering coefficient distribution, which decreases as the node degree increases. This distribution also follows a power law. The Barabási model of network evolution shown above is an example of a scale-free network.<br />
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'''<font color="#ff8000">[[无标度网络]] A Scale-free Network</font>''': 无标度网络网络是一个'''<font color="#ff8000">度分布 Degree Distribution</font>'''遵循'''<font color="#ff8000">幂律 Power Law</font>'''(至少是渐近的）的网络。在网络理论中，无标度理想网络是具有揭示了社会群体的规模分布的度分布的随机网络。无标度网络的具体特征随创建其的理论和分析工具的变化而变化，然而，一般来说，无标度网络具有一些共同的特征。无标度网络的一个显著特征是度远超均值的顶点的相对共性。最高度的节点通常被称为“'''<font color="#ff8000">枢纽 Hubs</font>'''” ，并且可能在其网络中服务于特定的目的，尽管这在很大程度上取决于社会环境。无标度网络的另一个一般特性是'''<font color="#ff8000">集聚系数 General Characteristic</font>'''分布，它随着节点度的增加而减少。该分布也遵循幂律。上面网络演化的巴拉巴西模型就是无标度网络的一个例子。<br />
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===Macro level 宏观层面===<br />
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Rather than tracing interpersonal interactions, macro-level analyses generally trace the outcomes of interactions, such as [[economic]] or other [[resource]] [[Transfer function|transfer]] interactions over a large [[Sample population|population]].<br />
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Rather than tracing interpersonal interactions, macro-level analyses generally trace the outcomes of interactions, such as economic or other resource transfer interactions over a large population.<br />
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宏观层面的分析不是追踪人际互动，而通常是追踪相互作用的结果，例如经济或其他资源转移在一大群体中的相互作用。<br />
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[[File:Diagram of a social network.jpg|thumb|right|Diagram: section of a large-scale social network 图六: 一大型社会网络局部]]<br />
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'''Large-scale networks''': [[Large-scale macroeconometric model|Large-scale network]] is a term somewhat synonymous with "macro-level" as used, primarily, in [[social science|social]] and [[Behavioural sciences|behavioral]] sciences, in [[economics]]. Originally, the term was used extensively in the [[computer sciences]] (see [[Network mapping#Large-scale mapping project|large-scale network mapping]]).<br />
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Large-scale networks: Large-scale network is a term somewhat synonymous with "macro-level" as used, primarily, in social and behavioral sciences, in economics. Originally, the term was used extensively in the computer sciences (see large-scale network mapping).<br />
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大规模网络: 大规模网络是一个与“宏观层面”同义的术语，主要用于社会和行为科学，经济学。最初，这个术语在计算机科学中广泛使用（见大规模网络映射）。<br />
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'''Complex networks''': Most larger social networks display features of [[social complexity]], which involves substantial non-trivial features of [[network topology]], with patterns of complex connections between elements that are neither purely regular nor purely random (see, [[complexity science]], [[dynamical system]] and [[chaos theory]]), as do [[biological]], and [[Computer network|technological networks]]. Such [[complex network]] features include a heavy tail in the [[degree distribution]], a high [[clustering coefficient]], [[assortativity]] or disassortativity among vertices, [[community structure]] (see [[stochastic block model]]), and [[hierarchy|hierarchical structure]]. In the case of [[Agency (philosophy)|agency-directed]] networks these features also include [[Reciprocity in network|reciprocity]], triad significance profile (TSP, see [[network motif]]), and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as [[lattice graph|lattices]] and [[random graph]]s, do not show these features.<ref>{{cite journal|author=Strogatz, Steven H.|year=2001|title=Exploring complex networks|journal=Nature|volume=410|pages=268–276|doi=10.1038/35065725|pmid=11258382|issue=6825|bibcode=2001Natur.410..268S|doi-access=free}}</ref><br />
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Complex networks: Most larger social networks display features of social complexity, which involves substantial non-trivial features of network topology, with patterns of complex connections between elements that are neither purely regular nor purely random (see, complexity science, dynamical system and chaos theory), as do biological, and technological networks. Such complex network features include a heavy tail in the degree distribution, a high clustering coefficient, assortativity or disassortativity among vertices, community structure (see stochastic block model), and hierarchical structure. In the case of agency-directed networks these features also include reciprocity, triad significance profile (TSP, see network motif), and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as lattices and random graphs, do not show these features.<br />
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复杂网络: 大多数较大的社会网络呈现出社会复杂性的特征，包括'''<font color="#ff8000">网络拓扑 Network Topology</font>'''的大量非平凡特征，以及既不完全规则也不完全随机的元素之间的复杂连接模式（见[[复杂性科学]]、[[动力系统]]和[[混沌理论]]），生物和技术网络也是如此。这些复杂的网络特征包括度分布的重尾、高集聚系数、顶点之间的'''<font color="#ff8000">同配性 Assortativity</font>'''或非同配性、社区结构（见'''<font color="#ff8000">随机分块模型 Stochastic Block Model</font>'''）和层次结构。在主体导向网络的情况下，这些特征还包括互惠性、三重显著性特征（TSP，见网络基序）及其他。相比之下，许多过去研究过的网络数学模型，如格和'''<font color="#ff8000">随机图 Random Graph</font>'''，并没有表现出这些特征。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）三重显著性特征不知翻译<br />
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==Theoretical links 理论联系==<br />
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===Imported theories 输入的理论===<br />
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Various theoretical frameworks have been imported for the use of social network analysis. The most prominent of these are [[Graph theory]], [[Balance theory]], Social comparison theory, and more recently, the [[Social identity approach]].<ref>{{cite book |last1=Kilduff|first1=M.|last2=Tsai|first2=W. |year=2003 |title= Social networks and organisations |publisher=Sage Publications}}</ref><br />
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Various theoretical frameworks have been imported for the use of social network analysis. The most prominent of these are Graph theory, Balance theory, Social comparison theory, and more recently, the Social identity approach.<br />
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为了使用社会网络分析，已经引入了各种理论框架。其中最突出的是[[图论]]、'''<font color="#ff8000">平衡理论 Balance theory</font>'''、社会比较论，以及最近的社会认同方法。<br />
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===Indigenous theories 本土理论===<br />
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Few complete theories have been produced from social network analysis. Two that have are [[role theory|structural role theory]] and [[heterophily|heterophily theory]].<br />
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Few complete theories have been produced from social network analysis. Two that have are structural role theory and heterophily theory.<br />
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很少有完整的理论产生于社会网络分析。现有两个为结构角色理论和异质性理论。<br />
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The basis of Heterophily Theory was the finding in one study that more numerous weak ties can be important in seeking information and innovation, as cliques have a tendency to have more homogeneous opinions as well as share many common traits. This homophilic tendency was the reason for the members of the cliques to be attracted together in the first place. However, being similar, each member of the clique would also know more or less what the other members knew. To find new information or insights, members of the clique will have to look beyond the clique to its other friends and acquaintances. This is what Granovetter called "the strength of weak ties".<ref>{{Cite journal | author=Granovetter, M. |year=1973 |title= The strength of weak ties |journal= American Journal of Sociology |volume=78 |issue=6 |pages= 1360–1380 |doi=10.1086/225469}}</ref><br />
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The basis of Heterophily Theory was the finding in one study that more numerous weak ties can be important in seeking information and innovation, as cliques have a tendency to have more homogeneous opinions as well as share many common traits. This homophilic tendency was the reason for the members of the cliques to be attracted together in the first place. However, being similar, each member of the clique would also know more or less what the other members knew. To find new information or insights, members of the clique will have to look beyond the clique to its other friends and acquaintances. This is what Granovetter called "the strength of weak ties".<br />
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异质性理论的基础是在一项研究中发现，更多的'''<font color="#ff8000">弱关系 Weak Tie</font>'''可以在寻求信息和创新方面发挥重要作用，因为小圈子倾向于有更同质化的观点，也有许多共同特征。这种亲同性倾向是小圈子成员被吸引到一起的首要原因。然而，由于相似，小圈子里的每一个成员或多或少都知道其他成员所知道的事情。为了获得新的信息或见解，小圈子里的成员不得不超越该圈子，关注其他的朋友及熟人。这就是格兰诺维特所说的“弱关系的力量”。<br />
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==Structural holes 结构洞==<br />
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In the context of networks, [[social capital]] exists where people have an advantage because of their location in a network. Contacts in a network provide information, opportunities and perspectives that can be beneficial to the central player in the network. Most social structures tend to be characterized by dense clusters of strong connections.<ref name="Burt 2004">{{cite journal|last=Burt|first=Ronald|title=Structural Holes and Good Ideas|journal=American Journal of Sociology|year=2004|doi=10.1086/421787|volume=110|issue=2|pages=349–399|citeseerx=10.1.1.388.2251}}</ref> Information within these clusters tends to be rather homogeneous and redundant. Non-redundant information is most often obtained through contacts in different clusters.<ref name="Burt 1992">{{cite book|last=Burt|first=Ronald|title=Structural Holes: The Social Structure of Competition|year=1992|publisher=Harvard University Press|location=Cambridge, MA}}</ref> When two separate clusters possess non-redundant information, there is said to be a structural hole between them.<ref name="Burt 1992"/> Thus, a network that bridges [[structural holes]] will provide network benefits that are in some degree additive, rather than overlapping. An ideal network structure has a vine and cluster structure, providing access to many different clusters and structural holes.<ref name="Burt 1992"/><br />
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In the context of networks, social capital exists where people have an advantage because of their location in a network. Contacts in a network provide information, opportunities and perspectives that can be beneficial to the central player in the network. Most social structures tend to be characterized by dense clusters of strong connections. Information within these clusters tends to be rather homogeneous and redundant. Non-redundant information is most often obtained through contacts in different clusters. When two separate clusters possess non-redundant information, there is said to be a structural hole between them. Thus, a network that bridges structural holes will provide network benefits that are in some degree additive, rather than overlapping. An ideal network structure has a vine and cluster structure, providing access to many different clusters and structural holes.<br />
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在网络的背景下，'''<font color="#ff8000">社会资本 Social Capital</font>'''存在于人们因其在网络中的位置而具有优势的地方。网络中的联系提供的信息、机会和观点可能有利于网络中的核心参与者。大多数社会结构往往以有强连接的密集集群为特征。这些集群中的信息往往是相当同质和冗余的。非冗余信息通常是通过不同集群中的联系获得的。当两个独立的集群拥有非冗余信息时，我们称它们之间存在一个'''<font color="#ff8000">结构洞 Structural Hole</font>'''。因此，一个连接结构孔的网络在某种程度上提供附加而非重叠的的网络效益。理想的网络结构具有蔓生结构和集群结构，可访问许多不同集群和结构洞。<br />
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Networks rich in structural holes are a form of social capital in that they offer [[Information theory|information]] benefits. The main player in a network that bridges structural holes is able to access information from diverse sources and clusters.<ref name="Burt 1992"/> For example, in [[Business networking|business networks]], this is beneficial to an individual's career because he is more likely to hear of job openings and opportunities if his network spans a wide range of contacts in different industries/sectors. This concept is similar to Mark Granovetter's theory of [[Interpersonal ties|weak ties]], which rests on the basis that having a broad range of contacts is most effective for job attainment.<br />
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Networks rich in structural holes are a form of social capital in that they offer information benefits. The main player in a network that bridges structural holes is able to access information from diverse sources and clusters. For example, in business networks, this is beneficial to an individual's career because he is more likely to hear of job openings and opportunities if his network spans a wide range of contacts in different industries/sectors. This concept is similar to Mark Granovetter's theory of weak ties, which rests on the basis that having a broad range of contacts is most effective for job attainment.<br />
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富含结构洞的网络是社会资本的一种形式，因为它们提供信息利益。连接结构洞的网络中的主要参与者能够访问来自不同来源和集群的信息。例如，在'''<font color="#ff8000">商业社交 Business networking</font>'''中，这对个人的职业生涯是有益的，因为若其关系网涵盖不同行业 / 部门，则更有可能得知职位空缺和机会。这个概念类似于马克·格兰诺维特的弱关系理论，该理论的基础是拥有广泛的联系对于获得工作是最有效的。<br />
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==Research clusters 研究集群==<br />
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===Communication 传播学===<br />
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[[Communication Studies]] are often considered a part of both the social sciences and the humanities, drawing heavily on fields such as [[sociology]], [[psychology]], [[anthropology]], [[information science]], [[biology]], [[political science]], and [[economics]] as well as [[rhetoric]], [[literary studies]], and [[semiotics]]. Many communication concepts describe the transfer of information from one source to another, and can thus be conceived of in terms of a network.<br />
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Communication Studies are often considered a part of both the social sciences and the humanities, drawing heavily on fields such as sociology, psychology, anthropology, information science, biology, political science, and economics as well as rhetoric, literary studies, and semiotics. Many communication concepts describe the transfer of information from one source to another, and can thus be conceived of in terms of a network.<br />
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传播学通常被认为是社会科学和人文科学的一部分，主要集中在社会学、心理学、人类学、信息科学、生物学、政治学、经济学以及修辞学、文学研究和符号学等领域。许多通信概念描述了从一个源到另一个源的信息传输，因此可以从网络的角度来考虑。<br />
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===Community 社区===<br />
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In J.A. Barnes' day, a "[[community]]" referred to a specific geographic location and studies of community ties had to do with who talked, associated, traded, and attended church with whom. Today, however, there are extended "online" communities developed through [[telecommunications]] devices and [[social network services]]. Such devices and services require extensive and ongoing maintenance and analysis, often using [[network science]] methods. [[Community development]] studies, today, also make extensive use of such methods.<br />
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In J.A. Barnes' day, a "community" referred to a specific geographic location and studies of community ties had to do with who talked, associated, traded, and attended church with whom. Today, however, there are extended "online" communities developed through telecommunications devices and social network services. Such devices and services require extensive and ongoing maintenance and analysis, often using network science methods. Community development studies, today, also make extensive use of such methods.<br />
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在J.A. Barnes' day，“社区”指的是一个特定的地理位置和社区关系的研究与谁交谈，联系，贸易和参加教会与谁。然而，今天，通过电信设备和社交网络服务，有了扩展的“在线”社区。这样的设备和服务需要广泛和持续的维护和分析，通常使用网络科学的方法。社区发展研究，今天，也广泛使用这些方法。<br />
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===Complex networks 复杂网络===<br />
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[[Complex networks]] require methods specific to modelling and interpreting [[social complexity]] and [[complex adaptive system]]s, including techniques of [[dynamic network analysis]].<br />
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Complex networks require methods specific to modelling and interpreting social complexity and complex adaptive systems, including techniques of dynamic network analysis.<br />
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复杂网络需要建模和解释社会复杂性和复杂适应系统的特定方法，包括动态网络分析技术。<br />
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Mechanisms such as [[Dual-phase evolution#social networks|Dual-phase evolution]] explain how temporal changes in connectivity contribute to the formation of structure in social networks.<br />
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Mechanisms such as Dual-phase evolution explain how temporal changes in connectivity contribute to the formation of structure in social networks.<br />
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双阶段进化等机制解释了连接性的时间变化如何促进社会网络结构的形成。<br />
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===Criminal networks 犯罪网络===<br />
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In [[criminology]] and [[urban sociology]], much attention has been paid to the social networks among criminal actors. For example, Andrew Papachristos<ref>{{cite journal |last=Papachristos |first=Andrew |year=2009 |title=Murder by Structure: Dominance Relations and the Social Structure of Gang Homicide |journal=American Journal of Sociology |volume=115 |issue=1 |pages=74–128 |doi=10.2139/ssrn.855304 |url=http://www.papachristos.org/Publications_2_files/ajs_final_version.pdf |accessdate=29 March 2013 |url-status=dead |archiveurl=https://web.archive.org/web/20140407094725/http://www.papachristos.org/Publications_2_files/ajs_final_version.pdf |archivedate=7 April 2014 }}</ref> has studied gang murders as a series of exchanges between gangs. Murders can be seen to diffuse outwards from a single source, because weaker gangs cannot afford to kill members of stronger gangs in retaliation, but must commit other violent acts to maintain their reputation for strength.<br />
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In criminology and urban sociology, much attention has been paid to the social networks among criminal actors. For example, Andrew Papachristos has studied gang murders as a series of exchanges between gangs. Murders can be seen to diffuse outwards from a single source, because weaker gangs cannot afford to kill members of stronger gangs in retaliation, but must commit other violent acts to maintain their reputation for strength.<br />
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在犯罪学和城市社会学中，犯罪行为人之间的社会网络问题受到了广泛的关注。例如，安德鲁 · 帕帕克里斯托斯将帮派谋杀研究为帮派之间的一系列交流。谋杀可以被视为从单一来源向外扩散，因为较弱的帮派无法承担杀死较强帮派成员进行报复的代价，而必须采取其他暴力行动来维护他们强大的声誉。<br />
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===Diffusion of innovations 创新产品渗透理论===<br />
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[[Diffusion of innovations|Diffusion of ideas and innovations]] studies focus on the spread and use of ideas from one actor to another or one [[culture]] and another. This line of research seeks to explain why some become "early adopters" of ideas and innovations, and links social network structure with facilitating or impeding the spread of an innovation.<br />
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Diffusion of ideas and innovations studies focus on the spread and use of ideas from one actor to another or one culture and another. This line of research seeks to explain why some become "early adopters" of ideas and innovations, and links social network structure with facilitating or impeding the spread of an innovation.<br />
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思想和创新的传播研究的重点是思想从一个行动者传播到另一个行动者或一种文化和另一种文化。这一系列的研究试图解释为什么有些人成为创意和创新的“早期接受者” ，并将社交网络结构与促进或阻碍创新的传播联系起来。<br />
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===Demography 人口统计===<br />
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In [[demography]], the study of social networks has led to new sampling methods for estimating and reaching populations that are hard to enumerate (for example, homeless people or intravenous drug users.) For example, respondent driven sampling is a network-based sampling technique that relies on respondents to a survey recommending further respondents.<br />
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In demography, the study of social networks has led to new sampling methods for estimating and reaching populations that are hard to enumerate (for example, homeless people or intravenous drug users.) For example, respondent driven sampling is a network-based sampling technique that relies on respondents to a survey recommending further respondents.<br />
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在人口统计学方面，对社会网络的研究导致了新的抽样方法，用于估计和覆盖难以统计的人群(例如，无家可归者或静脉注射毒品者)例如，受访者驱动的抽样是一种基于网络的抽样技术，依赖于调查的受访者推荐更多的受访者。<br />
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===Economic sociology 经济社会学===<br />
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The field of [[sociology]] focuses almost entirely on networks of outcomes of social interactions. More narrowly, [[economic sociology]] considers behavioral interactions of individuals and groups through [[social capital]] and social "markets". Sociologists, such as Mark Granovetter, have developed core principles about the interactions of social structure, information, ability to punish or reward, and trust that frequently recur in their analyses of political, economic and other institutions. Granovetter examines how social structures and social networks can affect economic outcomes like hiring, price, productivity and innovation and describes sociologists' contributions to analyzing the impact of social structure and networks on the economy.<ref>{{cite journal |last=Granovetter |first=Mark |year=2005 |title=The Impact of Social Structure on Economic Outcomes |journal=The Journal of Economic Perspectives |jstor=4134991 |volume=19 |issue=1 |pages=33–50|doi=10.1257/0895330053147958 }}</ref><br />
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The field of sociology focuses almost entirely on networks of outcomes of social interactions. More narrowly, economic sociology considers behavioral interactions of individuals and groups through social capital and social "markets". Sociologists, such as Mark Granovetter, have developed core principles about the interactions of social structure, information, ability to punish or reward, and trust that frequently recur in their analyses of political, economic and other institutions. Granovetter examines how social structures and social networks can affect economic outcomes like hiring, price, productivity and innovation and describes sociologists' contributions to analyzing the impact of social structure and networks on the economy.<br />
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社会学领域几乎完全关注社会互动的结果网络。更狭义地说，经济社会学通过社会资本和社会“市场”考虑个人和群体的行为互动。社会学家，如马克·格兰诺维特，已经发展出关于社会结构、信息、惩罚或奖励能力和信任相互作用的核心原则，这些原则在他们对政治、经济和其他制度的分析中经常出现。格兰诺维特研究了社会结构和社会网络如何影响经济结果，如雇佣、价格、生产力和创新，并描述了社会学家对分析社会结构和网络对经济的影响的贡献。<br />
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===Health care 医疗保健===<br />
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Analysis of social networks is increasingly incorporated into [[health care analytics]], not only in [[epidemiology|epidemiological]] studies but also in models of [[Health Communication|patient communication]] and education, disease prevention, mental health diagnosis and treatment, and in the study of health care organizations and [[health care systems|systems]].<ref>Levy, Judith and Bernice Pescosolido (2002). ''Social Networks and Health''. Boston, MA: JAI Press.</ref><br />
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Analysis of social networks is increasingly incorporated into health care analytics, not only in epidemiological studies but also in models of patient communication and education, disease prevention, mental health diagnosis and treatment, and in the study of health care organizations and systems.<br />
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社会网络分析不仅在流行病学研究中，而且在病人沟通和教育模型、疾病预防、心理健康诊断和治疗以及卫生保健组织和系统的研究中，越来越多地被纳入卫生保健分析中。<br />
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===Human ecology 人类生态学===<br />
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[[Human ecology]] is an [[interdisciplinary]] and [[transdisciplinary]] study of the relationship between [[human]]s and their [[natural environment|natural]], [[Social environment|social]], and [[built environment]]s. The scientific philosophy of human ecology has a diffuse history with connections to [[geography]], [[sociology]], [[psychology]], [[anthropology]], [[zoology]], and natural [[ecology]].<ref>Crona, Beatrice and Klaus Hubacek (eds.) (2010). [http://www.ecologyandsociety.org/issues/view.php?sf=48 "Special Issue: Social network analysis in natural resource governance"]. ''Ecology and Society'', 48.</ref><ref>Ernstson, Henrich (2010). "Reading list: Using social network analysis (SNA) in social-ecological studies". [http://rs.resalliance.org/2010/11/03/reading-list-using-social-network-analysis-sna-in-social-ecological-studies/ ''Resilience Science'']</ref><br />
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Human ecology is an interdisciplinary and transdisciplinary study of the relationship between humans and their natural, social, and built environments. The scientific philosophy of human ecology has a diffuse history with connections to geography, sociology, psychology, anthropology, zoology, and natural ecology.<br />
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人类生态学是一门跨学科、跨学科的学科，研究人类与自然、社会和人造环境之间的关系。人类生态学的科学哲学与地理学、社会学、心理学、人类学、动物学和自然生态学有着密切的联系。<br />
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===Language and linguistics 语言与语言学===<br />
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Studies of [[language]] and [[linguistics]], particularly [[evolutionary linguistics]], focus on the development of [[Morphology (linguistics)|linguistic forms]] and transfer of changes, [[Phonology|sounds]] or words, from one language system to another through networks of social interaction. Social networks are also important in [[language shift]], as groups of people add and/or abandon languages to their repertoire.<br />
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Studies of language and linguistics, particularly evolutionary linguistics, focus on the development of linguistic forms and transfer of changes, sounds or words, from one language system to another through networks of social interaction. Social networks are also important in language shift, as groups of people add and/or abandon languages to their repertoire.<br />
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语言学和语言学的研究，特别是演化语言学，集中在语言形式的发展和变化，声音或词汇的转移，从一个语言系统到另一个通过社会互动的网络。社交网络在语言转换中也很重要，因为一群人增加或者放弃了他们的语言。<br />
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===Literary networks 文学网络===<br />
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In the study of literary systems, network analysis has been applied by Anheier, Gerhards and Romo,<ref>{{cite journal | last1 = Anheier | first1 = H. K. | last2 = Romo | first2 = F. P. | year = 1995 | title = Forms of capital and social structure of fields: examining Bourdieu's social topography | url = | journal = American Journal of Sociology | volume = 100 | issue = 4| pages = 859–903 | doi=10.1086/230603}}</ref> De Nooy,<ref>{{cite journal | last1 = De Nooy | first1 = W | year = 2003| title = Fields and networks: Correspondence analysis and social network analysis in the framework of Field Theory | url = | journal = Poetics | volume = 31 | issue = 5–6| pages = 305–327 | doi = 10.1016/S0304-422X(03)00035-4 }}</ref> and Senekal,<ref>{{cite journal | last1 = Senekal | first1 = B. A. | year = 2012 | title = Die Afrikaanse literêre sisteem: ʼn Eksperimentele benadering met behulp van Sosiale-netwerk-analise (SNA) | url = | journal = LitNet Akademies | volume = 9 | issue = | page = 3 }}</ref> to study various aspects of how literature functions. The basic premise is that polysystem theory, which has been around since the writings of [[Even-Zohar]], can be integrated with network theory and the relationships between different actors in the literary network, e.g. writers, critics, publishers, literary histories, etc., can be mapped using [[Computer graphics (computer science)|visualization]] from SNA.<br />
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In the study of literary systems, network analysis has been applied by Anheier, Gerhards and Romo, De Nooy, and Senekal, to study various aspects of how literature functions. The basic premise is that polysystem theory, which has been around since the writings of Even-Zohar, can be integrated with network theory and the relationships between different actors in the literary network, e.g. writers, critics, publishers, literary histories, etc., can be mapped using visualization from SNA.<br />
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在文学体系的研究中，网络分析被阿海尔、格尔哈兹和罗姆、努伊和 Senekal 应用于研究文学功能的各个方面。其基本前提是多元系统理论可以与网络理论以及文学网络中不同行为者之间的关系相结合。作家、评论家、出版商、文学史等等，都可以通过使用 SNA 的可视化来绘制地图。<br />
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===Organizational studies 组织研究===<br />
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Research studies of [[Formal organization|formal]] or [[informal organization]] [[Social relation|relationships]], [[organizational communication]], [[economics]], [[economic sociology]], and other [[resource]] [[Transfer function|transfers]]. Social networks have also been used to examine how organizations interact with each other, characterizing the many [[Interlocking directorate|informal connections]] that link executives together, as well as associations and connections between individual employees at different organizations.<ref>{{cite journal | last1 = Podolny | first1 = J. M. | last2 = Baron | first2 = J. N. | year = 1997 | title = Resources and relationships: Social networks and mobility in the workplace | url = | journal = American Sociological Review | volume = 62 | issue = 5| pages = 673–693 | doi=10.2307/2657354| jstor = 2657354 | citeseerx = 10.1.1.114.6822 }}</ref> Intra-organizational networks have been found to affect [[organizational commitment]],<ref>{{cite journal | last1 = Lee | first1 = J. | last2 = Kim | first2 = S. | year = 2011 | title = Exploring the role of social networks in affective organizational commitment: Network centrality, strength of ties, and structural holes | url = | journal = The American Review of Public Administration | volume = 41 | issue = 2| pages = 205–223 | doi=10.1177/0275074010373803}}</ref> [[organizational identification]],<ref name="Jone11">{{cite journal | last1 = Jones | first1 = C. | last2 = Volpe | first2 = E.H. | year = 2011 | title = Organizational identification: Extending our understanding of social identities through social networks | url = | journal = Journal of Organizational Behavior | volume = 32 | issue = 3| pages = 413–434 | doi=10.1002/job.694}}</ref> [[Organizational citizenship behavior|interpersonal citizenship behaviour]].<ref>{{cite journal | last1 = Bowler | first1 = W. M. | last2 = Brass | first2 = D. J. | year = 2011 | title = Relational correlates of interpersonal citizenship behaviour: A social network perspective | doi = 10.1037/0021-9010.91.1.70 | pmid = 16435939 | journal = Journal of Applied Psychology | volume = 91 | issue = 1| pages = 70–82 | citeseerx = 10.1.1.516.8746 }}</ref><br />
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Research studies of formal or informal organization relationships, organizational communication, economics, economic sociology, and other resource transfers. Social networks have also been used to examine how organizations interact with each other, characterizing the many informal connections that link executives together, as well as associations and connections between individual employees at different organizations. Intra-organizational networks have been found to affect organizational commitment, organizational identification, interpersonal citizenship behaviour.<br />
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研究正式或非正式的组织关系，组织沟通，经济，经济社会学和其他资源转移。社交网络也被用来研究组织之间如何相互作用，描述了许多非正式的联系，以及不同组织的个别雇员之间的联系和联系。研究发现，组织内网络对组织承诺、组织认同、人际公民行为有显著影响。<br />
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===Social capital 社会资本===<br />
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[[Social capital]] is a form of [[Capital (economics)|economic]] and [[cultural capital]] in which social networks are central, [[Stock and flow|transactions]] are marked by [[Reciprocity (social psychology)|reciprocity]], [[Trust (social sciences)|trust]], and [[cooperation]], and [[Market (economics)|market]] [[Agent (economics)|agents]] produce [[goods and services]] not mainly for themselves, but for a [[common good]]. [[Social capital]] is split into three dimensions: the structural, the relational and the cognitive dimension. The structural dimension describes how partners interact with each other and which specific partners meet in a social network. Also The structural dimension of social capital indicates the level of ties among organizations.<ref>(Claridge, 2018).</ref>. This dimension is highly connected to the relational dimension which refers to trustworthiness, norms, expectations and idenfications of the bonds between partners.The relational dimension explains the nature of these ties which is mainly illustrated by the level of trust accorded to the network of organizations. <ref>(Claridge, 2018).</ref> The cognitive dimension analyses the extent to which organizations share common goals and objectives as a result of their ties and interactions. <ref>(Claridge, 2018).</ref><br />
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Social capital is a form of economic and cultural capital in which social networks are central, transactions are marked by reciprocity, trust, and cooperation, and market agents produce goods and services not mainly for themselves, but for a common good. Social capital is split into three dimensions: the structural, the relational and the cognitive dimension. The structural dimension describes how partners interact with each other and which specific partners meet in a social network. Also The structural dimension of social capital indicates the level of ties among organizations.. This dimension is highly connected to the relational dimension which refers to trustworthiness, norms, expectations and idenfications of the bonds between partners.The relational dimension explains the nature of these ties which is mainly illustrated by the level of trust accorded to the network of organizations. The cognitive dimension analyses the extent to which organizations share common goals and objectives as a result of their ties and interactions. <br />
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社会资本是一种经济和文化资本，其中社会网络是中心，交易以互惠、信任和合作为特征，市场代理人生产的商品和服务不主要是为他们自己，而是为了共同的利益。社会资本分为三个维度: 结构维度、关系维度和认知维度。结构维度描述了合作伙伴之间如何相互作用，以及哪些特定的合作伙伴在社交网络中相遇。社会资本的结构维度反映了组织之间的关系水平。这个维度与关系维度高度相关，关系维度指的是伴侣之间关系的可信度、规范、期望和相同性。 关系维度解释了这些联系的本质，主要表现在对组织网络的信任程度上。认知维度分析组织因其联系和相互作用而在多大程度上共享共同的目标和目的。<br />
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[[Social capital]] is a sociological concept about the value of [[social relation]]s and the role of cooperation and confidence to achieve positive outcomes. The term refers to the value one can get from their social ties. For example, newly arrived immigrants can make use of their social ties to established migrants to acquire jobs they may otherwise have trouble getting (e.g., because of unfamiliarity with the local language). A positive relationship exists between social capital and the intensity of social network use.<ref>{{cite journal|last=Sebastián|first=Valenzuela|author2=Namsu Park |author3=Kerk F. Kee |title=Is There Social Capital in a Social Network Site? Facebook Use and College Students' Life Satisfaction, Trust, and Participation|journal=Journal of Computer-Mediated Communication|year=2009|volume=14|issue=4|pages=875–901|doi=10.1111/j.1083-6101.2009.01474.x|doi-access=free}}</ref><ref>{{cite journal | title=Social Connectivity in America: Changes in Adult Friendship Network Size from 2002 to 2007 |author1=Wang, Hua |author2=Barry Wellman |lastauthoramp=yes | journal=American Behavioral Scientist | year=2010 | volume=53 | issue=8 | pages=1148–1169 | doi=10.1177/0002764209356247|url=https://semanticscholar.org/paper/402068f74efb35c093e7fd8c0f8268521219e242 }}</ref> In a dynamic framework, higher activity in a network feeds into higher social capital which itself encourages more activity.<ref name="GaudeulGiannetti2013">{{cite journal|last1=Gaudeul|first1=Alexia|last2=Giannetti|first2=Caterina|title=The role of reciprocation in social network formation, with an application to LiveJournal|journal=Social Networks|volume=35|issue=3|year=2013|pages=317–330|issn=03788733|doi=10.1016/j.socnet.2013.03.003}}</ref><br />
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Social capital is a sociological concept about the value of social relations and the role of cooperation and confidence to achieve positive outcomes. The term refers to the value one can get from their social ties. For example, newly arrived immigrants can make use of their social ties to established migrants to acquire jobs they may otherwise have trouble getting (e.g., because of unfamiliarity with the local language). A positive relationship exists between social capital and the intensity of social network use. In a dynamic framework, higher activity in a network feeds into higher social capital which itself encourages more activity.<br />
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社会资本是一个关于社会关系的价值和合作与信心的作用的社会学概念，以实现积极的成果。这个术语指的是一个人可以从他们的社会关系中获得的价值。例如，新来的移民可以利用他们与老移民的社会关系，获得他们在其他情况下可能难以获得的工作(例如，因为不熟悉当地语言)。社会资本与社会网络使用强度存在正相关关系。在一个动态的框架中，网络中更高的活动会产生更高的社会资本，而社会资本本身又会鼓励更多的活动。<br />
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=== Advertising ===<br />
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=== Advertising ===<br />
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广告<br />
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This particular cluster focuses on brand-image and promotional strategy effectiveness, taking into account the impact of customer participation on sales and brand-image. This is gauged through techniques such as sentiment analysis which rely on mathematical areas of study such as data mining and analytics. This area of research produces vast numbers of commercial applications as the main goal of any study is to understand consumer behaviour and drive sales.<br />
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This particular cluster focuses on brand-image and promotional strategy effectiveness, taking into account the impact of customer participation on sales and brand-image. This is gauged through techniques such as sentiment analysis which rely on mathematical areas of study such as data mining and analytics. This area of research produces vast numbers of commercial applications as the main goal of any study is to understand consumer behaviour and drive sales.<br />
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这个集群侧重于品牌形象和促销策略的有效性，同时考虑到顾客参与对销售和品牌形象的影响。这是通过诸如情感分析等技术来衡量的，这些技术依赖于数学领域的研究，如数据挖掘和分析。这一研究领域产生了大量的商业应用，因为任何研究的主要目标都是了解消费者行为并推动销售。<br />
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===Network position and benefits===<br />
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===Network position and benefits===<br />
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网络地位和利益<br />
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In many [[Formal organizations|organizations]], members tend to focus their activities inside their own groups, which stifles creativity and restricts opportunities. A player whose network bridges structural holes has an advantage in detecting and developing rewarding opportunities.<ref name="Burt 2004"/> Such a player can mobilize social capital by acting as a "broker" of information between two clusters that otherwise would not have been in contact, thus providing access to new ideas, opinions and opportunities. British philosopher and political economist [[John Stuart Mill]], writes, "it is hardly possible to overrate the value ... of placing human beings in contact with persons dissimilar to themselves.... Such communication [is] one of the primary sources of progress."<ref>{{cite book|last=Mill|first=John|title=Principles of Political Economy|year=1909|publisher=William J Ashley|location=Library of Economics and Liberty}}</ref> Thus, a player with a network rich in structural holes can add value to an organization through new ideas and opportunities. This in turn, helps an individual's career development and advancement.<br />
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In many organizations, members tend to focus their activities inside their own groups, which stifles creativity and restricts opportunities. A player whose network bridges structural holes has an advantage in detecting and developing rewarding opportunities. Thus, a player with a network rich in structural holes can add value to an organization through new ideas and opportunities. This in turn, helps an individual's career development and advancement.<br />
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在许多组织中，成员倾向于把他们的活动集中在他们自己的团队中，这抑制了创造力和限制了机会。一个玩家的网络桥梁结构洞有优势，在检测和发展奖励机会。因此，一个拥有丰富结构性漏洞的网络的玩家可以通过新的想法和机会为组织增加价值。这反过来又有助于个人的职业发展和晋升。<br />
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A social capital broker also reaps control benefits of being the facilitator of information flow between contacts. In the case of consulting firm Eden McCallum, the founders were able to advance their careers by bridging their connections with former big three consulting firm consultants and mid-size industry firms.<ref>{{cite journal|last=Gardner|first=Heidi|author2=Eccles, Robert |title=Eden McCallum: A Network Based Consulting Firm|journal=Harvard Business School Review|year=2011}}</ref> By bridging structural holes and mobilizing social capital, players can advance their careers by executing new opportunities between contacts.<br />
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A social capital broker also reaps control benefits of being the facilitator of information flow between contacts. In the case of consulting firm Eden McCallum, the founders were able to advance their careers by bridging their connections with former big three consulting firm consultants and mid-size industry firms. By bridging structural holes and mobilizing social capital, players can advance their careers by executing new opportunities between contacts.<br />
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社会资本经纪人还可以从促进联系人之间的信息流动中获得控制权利。在咨询公司 Eden McCallum 的案例中，创始人能够通过与前三大咨询公司顾问和中等规模的行业公司建立联系来提升自己的职业生涯。通过弥合结构漏洞和调动社会资本，玩家可以通过在联系人之间执行新的机会来提升自己的职业生涯。<br />
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There has been research that both substantiates and refutes the benefits of information brokerage. A study of high tech Chinese firms by Zhixing Xiao found that the control benefits of structural holes are "dissonant to the dominant firm-wide spirit of cooperation and the information benefits cannot materialize due to the communal sharing values" of such organizations.<ref>Xiao, Zhixing; Tsui, Anne (2007). "When Brokers May Not Work: The Cultural Contingency of [http://iniciarsesionentrar.com/facebook/ Social] Capital in Chinese High-tech Firms". ''Administrative Science Quarterly''.</ref> However, this study only analyzed Chinese firms, which tend to have strong communal sharing values. Information and control benefits of structural holes are still valuable in firms that are not quite as inclusive and cooperative on the firm-wide level. In 2004, Ronald Burt studied 673 managers who ran the supply chain for one of America's largest electronics companies. He found that managers who often discussed issues with other groups were better paid, received more positive job evaluations and were more likely to be promoted.<ref name="Burt 2004"/> Thus, bridging structural holes can be beneficial to an organization, and in turn, to an individual's career.<br />
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There has been research that both substantiates and refutes the benefits of information brokerage. A study of high tech Chinese firms by Zhixing Xiao found that the control benefits of structural holes are "dissonant to the dominant firm-wide spirit of cooperation and the information benefits cannot materialize due to the communal sharing values" of such organizations. However, this study only analyzed Chinese firms, which tend to have strong communal sharing values. Information and control benefits of structural holes are still valuable in firms that are not quite as inclusive and cooperative on the firm-wide level. In 2004, Ronald Burt studied 673 managers who ran the supply chain for one of America's largest electronics companies. He found that managers who often discussed issues with other groups were better paid, received more positive job evaluations and were more likely to be promoted. Thus, bridging structural holes can be beneficial to an organization, and in turn, to an individual's career.<br />
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有研究证实和反驳了信息经纪的好处。知行晓对中国高科技企业的一项研究发现，结构性漏洞的控制效益“与占主导地位的企业合作精神不协调，由于这类组织的共享价值观，信息效益不能实现”。然而，这项研究只分析了中国企业，这些企业往往有很强的公共共享价值观。结构性漏洞的信息和控制利益对于那些在整个公司层面上不那么具有包容性和合作性的公司来说仍然是有价值的。2004年，罗纳德 · 伯特研究了673名管理者，他们为美国最大的电子公司之一管理供应链。他发现，那些经常与其他团体讨论问题的管理人员收入更高，得到的工作评价更积极，也更有可能获得晋升。因此，架起结构性漏洞对组织有益，反过来对个人的职业生涯也有益。<br />
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===Social media===<br />
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社交媒体<br />
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[[Computer network]]s combined with social networking software produces a new medium for social interaction.<ref>{{Cite book|title=The International Encyclopedia of Media Effects|last=Amichai-Hamburger|first=Yair|last2=Hayat|first2=Tsahi|date=2017|publisher=John Wiley & Sons, Inc.|isbn=9781118783764|language=en|doi=10.1002/9781118783764.wbieme0170}}</ref> A relationship over a computerized [[social networking service]] can be characterized by context, direction, and strength. The content of a relation refers to the resource that is exchanged. In a [[computer mediated communication]] context, social pairs exchange different kinds of information, including sending a data file or a computer program as well as providing emotional support or arranging a meeting. With the rise of [[electronic commerce]], information exchanged may also correspond to exchanges of money, goods or services in the "real" world.<ref>{{cite journal | title=Studying Online Social Networks | last1=Garton|first1=Laura|first2=Caroline |last2=Haythornthwaite|author2-link=Caroline Haythornthwaite|first3=Barry|last3=Wellman|author3-link=Barry Wellman | journal=Journal of Computer-Mediated Communication |date=23 June 2006 | volume=3 | issue=1 |pages = 0| doi=10.1111/j.1083-6101.1997.tb00062.x}}</ref> [[Social network analysis]] methods have become essential to examining these types of computer mediated communication.<br />
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Computer networks combined with social networking software produces a new medium for social interaction. A relationship over a computerized social networking service can be characterized by context, direction, and strength. The content of a relation refers to the resource that is exchanged. In a computer mediated communication context, social pairs exchange different kinds of information, including sending a data file or a computer program as well as providing emotional support or arranging a meeting. With the rise of electronic commerce, information exchanged may also correspond to exchanges of money, goods or services in the "real" world. Social network analysis methods have become essential to examining these types of computer mediated communication.<br />
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计算机网络与社交网络软件的结合为社交互动提供了一种新的媒介。通过计算机化的社交网络服务建立起来的关系可以是拥有属性、方向和力量。关系的内容指的是交换的资源。在计算机中介的交流环境中，社会对交换不同种类的信息，包括发送数据文件或计算机程序，以及提供情感支持或安排会议。随着电子商务的兴起，交换的信息也可能对应于“现实”世界中的货币、商品或服务的交换。社会网络分析方法已经成为检验这些类型的计算机中介交流的必要手段。<br />
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In addition, the sheer size and the volatile nature of [[social media]] has given rise to new network metrics. A key concern with networks extracted from social media is the lack of robustness of network metrics given missing data.<ref>{{cite journal | last1 = Wei | first1 = Wei | last2 = Joseph | first2 = Kenneth | last3 = Liu | first3 = Huan | last4 = Carley | first4 = Kathleen M. | year = 2016 | title = Exploring Characteristics of Suspended Users and Network Stability on Twitter | url = | journal = Social Network Analysis and Mining | volume = 6 | issue = | page = 51 | doi = 10.1007/s13278-016-0358-5 }}</ref><br />
<br />
In addition, the sheer size and the volatile nature of social media has given rise to new network metrics. A key concern with networks extracted from social media is the lack of robustness of network metrics given missing data.<br />
<br />
此外，社交媒体的庞大规模和不稳定性也催生了新的网络指标。从社交媒体中提取网络的一个关键问题是，在缺失数据的情况下，网络指标缺乏健壮性。<br />
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==See also==<br />
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==See also==<br />
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参见<br />
<br />
{{cmn|colwidth=22em|<br />
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{{cmn|colwidth=22em|<br />
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22em | <br />
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* [[Bibliography of sociology]]<br />
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* [[Business networking]]<br />
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* [[Collective network]]<br />
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* [[International Network for Social Network Analysis]]<br />
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* [[Network society]]<br />
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* [[Network theory]]<br />
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*[[Network science]]<br />
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* [[Semiotics of social networking]]<br />
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* [[Scientific collaboration network]]<br />
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* [[Social network analysis]]<br />
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* [[Social network (sociolinguistics)]]<br />
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* [[Social networking service]]<br />
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* [[Social web]]<br />
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* [[Structural fold]]<br />
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}}<br />
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}}<br />
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}}<br />
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==References==<br />
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==References==<br />
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参考资料<br />
<br />
{{reflist}}<br />
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==Further reading==<br />
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==Further reading==<br />
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进一步阅读<br />
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* {{cite journal |last1=Aneja |first1=Nagender |last2=Gambhir |first2=Sapna |title=Ad-hoc Social Network: A Comprehensive Survey |url=http://www.ijser.org/researchpaper%5CAd-hoc-Social-Network-A-Comprehensive-Survey.pdf |journal=International Journal of Scientific & Engineering Research |volume=4 |issue=8 |date=August 2013 |pages=156–160|issn=2229-5518}}<br />
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* {{cite book |last=Barabási |first=Albert-László |year=2003 |title=Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life |publisher=Plum |isbn=978-0-452-28439-5 |url-access=registration |url=https://archive.org/details/linkedhoweveryth00bara }}<br />
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* {{cite book |last=Barnett |first=George A. |year=2011 |title=Encyclopedia of Social Networks |publisher=Sage |isbn=978-1-4129-7911-5}}<br />
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* {{cite book |last=Estrada |first=E |year=2011 |title=The Structure of Complex Networks: Theory and Applications |publisher=Oxford University Press |isbn=978-0-199-59175-6}}<br />
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* {{cite book |last=Ferguson |first=Niall |year=2018 |title=The Square and the Tower: Networks and Power, from the Freemasons to Facebook |isbn=978-0735222915 |publisher=Penguin Press}}<br />
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* {{cite book |last=Freeman |first=Linton C. |year=2004 |title=The Development of Social Network Analysis: A Study in the Sociology of Science |publisher=Empirical Press |isbn=978-1-59457-714-7}}<br />
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* {{cite book |last=Kadushin |first=Charles |year=2012 |title=Understanding Social Networks: Theories, Concepts, and Findings |location= |publisher=Oxford University Press |isbn=978-0-19-537946-4}}<br />
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* {{cite book |last1=Mauro |first1=Rios |last2=Petrella |first2=Carlos |year=2014 |title=La Quimera de las Redes Sociales |trans-title=The Chimera of Social Networks |language=Spanish |publisher=Bubok España |isbn=978-9974-99-637-3}}<br />
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* {{cite book |last1=Rainie |first1=Lee |last2=Wellman |first2=Barry |year=2012 |title=Networked: The New Social Operating System |url=https://archive.org/details/networkednewsoci0000rain |url-access=registration |location=Cambridge, Mass. |publisher=MIT Press |isbn=978-0262017190}}<br />
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* {{cite book |last=Scott |first=John |year=1991 |title=Social Network Analysis: A Handbook |publisher=Sage |isbn=978-0-7619-6338-7}}<br />
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* {{cite book |last1=Wasserman |first1=Stanley |last2=Faust |first2=Katherine |year=1994 |title=Social Network Analysis: Methods and Applications |publisher=Cambridge University Press |isbn=978-0-521-38269-4 |series=Structural Analysis in the Social Sciences }}<br />
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* {{cite book |last1=Wellman |first1=Barry |last2=Berkowitz |first2=S. D. |year=1988 |title=Social Structures: A Network Approach |series=Structural Analysis in the Social Sciences |publisher=Cambridge University Press |isbn=978-0-521-24441-1}}<br />
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==External links==<br />
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==External links==<br />
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外部链接<br />
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{{Commons category|Social networks}}<br />
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===Organizations===<br />
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===Organizations===<br />
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组织<br />
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* [http://www.insna.org/ International Network for Social Network Analysis]<br />
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===Peer-reviewed journals===<br />
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===Peer-reviewed journals===<br />
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同行评审期刊<br />
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* ''[http://www.sciencedirect.com/science/journal/03788733 Social Networks]''<br />
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* ''[http://journals.cambridge.org/action/displaySpecialPage?pageId=3656 Network Science]''<br />
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* ''[http://www.cmu.edu/joss/content/articles/volindex.html Journal of Social Structure]''<br />
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* ''[http://www.tandfonline.com/toc/gmas20/current Journal of Mathematical Sociology]''<br />
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* ''[https://www.springer.com/computer/database+management+%26+information+retrieval/journal/13278 Social Network Analysis and Mining (SNAM)]''<br />
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* {{cite journal |title=INSNA - ''Connections'' Journal |journal=Connections Bulletin of the International Network for Social Network Analysis |location=Toronto |publisher=International Network for Social Network Analysis |url=http://www.insna.org/pubs/connections/ |issn=0226-1766 |url-status=dead |archiveurl=https://web.archive.org/web/20130718063641/http://www.insna.org/pubs/connections/ |archivedate=2013-07-18 }}<br />
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===Textbooks and educational resources===<br />
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===Textbooks and educational resources===<br />
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课本及教育资源<br />
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* ''[http://www.cs.cornell.edu/home/kleinber/networks-book/ Networks, Crowds, and Markets]'' (2010) by D. Easley & J. Kleinberg<br />
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* ''[http://faculty.ucr.edu/~hanneman/nettext/ Introduction to Social Networks Methods]'' (2005) by R. Hanneman & M. Riddle<br />
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* [http://www.analytictech.com/networks/ Social Network Analysis Instructional Web Site] by S. Borgatti<br />
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* ''[https://web.archive.org/web/20160304063126/http://www.novagob.org/file/view/153259/ebook-gratuito-sobre-redes-sociales-virtuales Guide for virtual social networks for public administrations]'' (2015) by Mauro D. Ríos (in Spanish)<br />
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===Data sets===<br />
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===Data sets===<br />
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数据集<br />
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* [http://pajek.imfm.si/doku.php?id=data:urls:index Pajek's list of lists of datasets]<br />
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* [http://networkdata.ics.uci.edu/index.html UC Irvine Network Data Repository]<br />
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* [http://snap.stanford.edu/data/ Stanford Large Network Dataset Collection]<br />
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* [http://www-personal.umich.edu/~mejn/netdata/ M.E.J. Newman datasets]<br />
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* [http://vlado.fmf.uni-lj.si/pub/networks/data/ Pajek datasets]<br />
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* [http://wiki.gephi.org/index.php?title=Datasets#Social_networks Gephi datasets]<br />
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* [http://konect.uni-koblenz.de/networks KONECT – Koblenz network collection]<br />
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* [http://www.stats.ox.ac.uk/~snijders/siena/ RSiena datasets]<br />
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==编者推荐==<br />
集智俱乐部：[https://swarma.org/?p=18646 社会网络分析的四个维度：研究方法、应用和软件工具综述 | 网络科学论文速递25篇]<br />
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[http://arxiv.org/abs/2002.09485 The Four Dimensions of Social Network Analysis: An Overview of Research Methods, Applications, and Software Tools]：近年来，基于社交网络的应用呈指数级增长，其中一个方面的原因是，这个应用领域提供了一个特别肥沃的地方，可以测试和开发最先进的计算技术，从 Web 中提取有价值的信息。这项工作的主要贡献有三个方面：(1) 我们对社会网络分析 (SNA) 的最新发展水平进行了文献综述；(2) 我们提出了一套基于 SNA 四个基本特征（或维度）的新度量标准；(3) 最后，我们对一套流行的 SNA 工具和框架进行了定量分析。我们还进行了一项科学计量学研究，以检测该领域中最活跃的研究领域和应用领域。本工作提出了四个不同维度的定义，即(a)模式、(b)知识发现、(c)信息融合、(d)集成、可伸缩性和可视化，用于定义一组新的度量（称为度），以评估 SNA 的不同软件工具和框架（根据之前的度量对一组 20 个 SNA-软件工具进行分析和排序）。这些维度连同定义的程度，可以评价和衡量社会网络技术的成熟度，寻求对它们的定量评估，从而揭示这一活跃领域的挑战和未来发展趋势。<br />
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{{Social networking}}<br />
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{{Online social networking}}<br />
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{{DEFAULTSORT:Social Network}}<br />
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[[Category:Social networks| ]]<br />
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[[Category:Communication theory]]<br />
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Category:Communication theory<br />
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范畴: 传播理论<br />
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[[Category:Community building]]<br />
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Category:Community building<br />
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类别: 社区建设<br />
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[[Category:Complex systems theory]]<br />
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Category:Complex systems theory<br />
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范畴: 复杂系统理论<br />
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[[Category:Network theory]]<br />
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范畴: 网络理论<br />
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[[Category:Organizational theory]]<br />
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Category:Organizational theory<br />
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类别: 组织行为学<br />
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[[Category:Self-organization]]<br />
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Category:Self-organization<br />
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类别: 自我组织<br />
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[[Category:Social information processing]]<br />
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Category:Social information processing<br />
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类别: 社会信息处理<br />
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[[Category:Social systems]]<br />
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类别: 社会系统<br />
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[[Category:Sociological terminology]]<br />
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Category:Sociological terminology<br />
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类别: 社会学术语<br />
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[[Category:Sociological theories]]<br />
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Category:Sociological theories<br />
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范畴: 社会学理论<br />
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Category:Systems theory<br />
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范畴: 系统论<br />
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<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Social network]]. Its edit history can be viewed at [[社会网络/edithistory]]</small></noinclude><br />
<br />
[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%B2%8D%E5%BE%B7%E6%B8%A9%E6%95%88%E5%BA%94&diff=12059鲍德温效应2020-08-11T12:03:27Z<p>Dorr：</p>
<hr />
<div>已由Dorr初步翻译<br />
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此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。<br />
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{{short description|Effect of learned behavior on evolution}}<br />
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{{Distinguish|Baldwin's rules}}<br />
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[[File:Lamarck Compared to Darwin, Baldwin, Waddington.svg|thumb|upright=2.5|The Baldwin effect compared to [[Lamarckism|Lamarck's theory of evolution]], [[Darwinian evolution]], and Waddington's [[genetic assimilation]]. All the theories offer explanations of how organisms respond to a changed environment with adaptive inherited change.]]<br />
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Lamarck's theory of evolution, Darwinian evolution, and Waddington's genetic assimilation. All the theories offer explanations of how organisms respond to a changed environment with adaptive inherited change.]]<br />
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拉马克的进化论、达尔文进化论和华丁顿的遗传同化论。[参考译文]所有的理论都解释了有机体是如何通过适应性遗传变化来应对环境变化的<br />
--~英文wiki无此段<br />
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In [[evolutionary biology]], the '''Baldwin effect''' describes the effect of learned behavior on evolution. In brief, [[James Mark Baldwin]] and others suggested during the [[eclipse of Darwinism]] in the late 19th century that an organism's ''ability to learn'' new behaviors (e.g. to acclimatise to a new stressor) will affect its reproductive success and will therefore have an effect on the genetic makeup of its species through [[natural selection]]. Though this process appears similar to [[Lamarckism|Lamarckian evolution]], Lamarck proposed that living things ''inherited'' their parents' acquired characteristics. The Baldwin effect has been independently proposed several times, and today it is generally recognized as part of the [[Modern synthesis (20th century)|modern synthesis]].<br />
<br />
In evolutionary biology, the Baldwin effect describes the effect of learned behavior on evolution. In brief, James Mark Baldwin and others suggested during the eclipse of Darwinism in the late 19th century that an organism's ability to learn new behaviors (e.g. to acclimatise to a new stressor) will affect its reproductive success and will therefore have an effect on the genetic makeup of its species through natural selection. Though this process appears similar to Lamarckian evolution, Lamarck proposed that living things inherited their parents' acquired characteristics. The Baldwin effect has been independently proposed several times, and today it is generally recognized as part of the modern synthesis.<br />
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在'''<font color="#ff8000">演化生物学 Evolutionary Biology</font>'''中，'''<font color="#ff8000">鲍德温效应 Baldwin Effect</font>'''描述了习得行为对进化的影响。简而言之，在19世纪末'''<font color="#ff8000">达尔文主义 Darwinism</font>'''的衰落期间，'''詹姆斯·马克·鲍德温 James Mark Baldwin'''及其他一些人提出，生物体学习新行为的能力（如为了适应新的压力）会影响其繁殖，进而会通过'''<font color="#ff8000">自然选择 Natural Selection</font>'''影响其物种的基因组成。这个过程似乎与'''<font color="#ff8000">拉马克主义 Lamarckism</font>'''中相似，但拉马克认为生物继承了其父母后天获得的特性。鲍德温效应已独立提出过多次，如今普遍认为其为'''<font color="#ff8000">现代进化综论 Modern Evolutionary Synthesis</font>'''的一部分。<br />
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=="A New Factor in Evolution" “进化的新因素”==<br />
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The effect, then unnamed, was put forward in 1896 in a paper "A New Factor in Evolution" by [[United States|American]] [[psychology|psychologist]] [[James Mark Baldwin]]. The paper proposed a mechanism for specific selection for general learning ability. As [[Robert J. Richards|Robert Richards]] explains:<ref name=Richards1987>{{cite book | author=Richards, Robert J. | year=1987 | title=Darwin and the Emergence of Evolutionary Theories of Mind and Behavior| publisher=The University of Chicago Press |location=Chicago | isbn=978-0-226-71199-7 | page=399| author-link=Robert J. Richards }}</ref><br />
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The effect, then unnamed, was put forward in 1896 in a paper "A New Factor in Evolution" by American psychologist James Mark Baldwin. The paper proposed a mechanism for specific selection for general learning ability. As Robert Richards explains:<br />
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1896年，美国心理学家詹姆斯·马克·鲍德温在一篇名为《进化的新因素》的论文中提出了这一效应（1987年又发表了第二篇论文）。该文提出了一种针对一般学习能力的具体选择机制。正如'''罗伯特·理查兹 Robert Richards'''解释的那样:<br />
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块引号<br />
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If animals entered a new environment—or their old environment rapidly changed—those<br />
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If animals entered a new environment—or their old environment rapidly changed—those<br />
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that could flexibly respond by learning new behaviors or by ontogenetically adapting would<br />
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that could flexibly respond by learning new behaviors or by ontogenetically adapting would<br />
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be naturally preserved. This saved remnant would, over several generations, have the<br />
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be naturally preserved. This saved remnant would, over several generations, have the<br />
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opportunity to exhibit spontaneously congenital variations similar to their acquired traits and<br />
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opportunity to exhibit spontaneously congenital variations similar to their acquired traits and<br />
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have these variations naturally selected. It would look as though the acquired traits had sunk<br />
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have these variations naturally selected. It would look as though the acquired traits had sunk<br />
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into the hereditary substance in a Lamarckian fashion, but the process would really be [[Neo-Darwinism|neo-Darwinian]].<br />
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into the hereditary substance in a Lamarckian fashion, but the process would really be neo-Darwinian.<br />
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如果动物处于新环境或其原来所处的环境急剧变化，那些能够通过学习新行为或自身基因适应来灵活作出反应的动物会自然地留存下来。这些幸存者，经过几代之后，有机会自发地表现出与他们获得的特征相似的先天变异，并让其经自然选择而留存。看上去获得性特征以拉马克的方式留在了遗传物质上，但这个过程实际上是'''<font color="#ff8000">新达尔文主义 Neo-Darwinism</font>'''的。<br />
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Selected offspring would tend to have an increased capacity for learning new skills rather than being confined to genetically coded, relatively fixed abilities. In effect, it places emphasis on the fact that the sustained behavior of a species or group can shape the evolution of that species. The "Baldwin effect" is better understood in [[evolutionary developmental biology]] literature as a scenario in which a character or [[biological trait|trait]] change occurring in an [[organism]] as a result of its interaction with its environment [[genetic assimilation|becomes gradually assimilated]] into its developmental genetic or epigenetic repertoire (Simpson, 1953; Newman, 2002). In the words of [[Daniel Dennett]],<ref name=dennett03/><br />
<br />
Selected offspring would tend to have an increased capacity for learning new skills rather than being confined to genetically coded, relatively fixed abilities. In effect, it places emphasis on the fact that the sustained behavior of a species or group can shape the evolution of that species. The "Baldwin effect" is better understood in evolutionary developmental biology literature as a scenario in which a character or trait change occurring in an organism as a result of its interaction with its environment becomes gradually assimilated into its developmental genetic or epigenetic repertoire (Simpson, 1953; Newman, 2002). In the words of Daniel Dennett,<br />
<br />
被选择留存的后代往往具有更强的学习新技能的能力，而不仅局限于基因编码的相对固定的能力。实际上，它强调以下事实：一个物种或群体的持续行为可以影响其进化。在'''<font color="#ff8000">演化发育生物学 Evolutionary developmental biology</font>'''的文献中，“鲍德温效应”得到了更好的理解——生物体由于与其环境的相互作用而发生的的特征或性状变化逐渐被同化到其发育遗传或表观遗传库中。用'''丹尼尔·丹尼特 Daniel Dennett'''的话来说,<br />
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块引号<br />
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Thanks to the Baldwin effect, species can be said to pretest the efficacy of particular different designs by phenotypic (individual) exploration of the space of nearby possibilities. If a particularly winning setting is thereby discovered, this discovery will ''create'' a new selection pressure: organisms that are closer in the adaptive landscape to that discovery will have a clear advantage over those more distant.<br />
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Thanks to the Baldwin effect, species can be said to pretest the efficacy of particular different designs by phenotypic (individual) exploration of the space of nearby possibilities. If a particularly winning setting is thereby discovered, this discovery will create a new selection pressure: organisms that are closer in the adaptive landscape to that discovery will have a clear advantage over those more distant.<br />
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由于鲍德温效应，可以说物种通过表型（个体）探索附近可能性的空间，预先测试了特定不同设计的功效。若由此发现一个特别适宜的环境，该发现将导致新的选择压力: 在适应环境中，更接近此发现的生物体将较其他生物体有明显的优势。<br />
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</blockquote><br />
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/ blockquote<br />
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An update to the Baldwin effect was developed by [[Jean Piaget]], [[Paul Alfred Weiss|Paul Weiss]], and [[Conrad Waddington]] in the 1960s–1970s. This new version included an explicit role for the social in shaping subsequent natural change in humans (both evolutionary and developmental), with reference to alterations of selection pressures.<ref>{{cite journal | author = Burman J. T. | year = 2013 | title = Updating the Baldwin Effect: The biological levels behind Piaget's new theory | url = | journal = New Ideas in Psychology | volume = 31 | issue = 3| pages = 363–373 | doi = 10.1016/j.newideapsych.2012.07.003 }}</ref><br />
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An update to the Baldwin effect was developed by Jean Piaget, Paul Weiss, and Conrad Waddington in the 1960s–1970s. This new version included an explicit role for the social in shaping subsequent natural change in humans (both evolutionary and developmental), with reference to alterations of selection pressures.<br />
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1960-1970年代，'''让·皮亚杰 Jean Piaget'''、'''保罗·维斯 Paul Weiss'''和'''康拉德·华丁顿 Conrad Waddington'''修正了鲍德温效应。该新版本明确了社会在塑造人类随后的自然变化（包括进化和发展）中的作用，并提到了选择压力的改变。<br />
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As is to be expected from [[Stigler's law]], subsequent research shows that Baldwin was not the first to identify the process; [[Douglas Spalding]] mentioned it in 1873.<ref>Noble, R.; [[Denis Noble|Noble, D.]] (2017) Was the Watchmaker Blind? Or Was She One-Eyed? ''Biology'' 2017, 6(4), 47; doi:10.3390/biology6040047, quoting Bateson, P. The adaptability driver: Links between behaviour and evolution. Biol. Theory 2006, 1, 342–345.</ref><br />
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As is to be expected from Stigler's law, subsequent research shows that Baldwin was not the first to identify the process; Douglas Spalding mentioned it in 1873.<br />
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正如'''<font color="#ff8000">斯蒂格勒定律 Stigler's Law</font>'''所指，随后的研究表明，鲍德温并不是第一个发现这个过程的人; '''道格拉斯·斯波尔丁 Douglas Spalding'''在1873年提到过这个过程。<br />
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==Examples 举例==<br />
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Suppose a species is threatened by a new [[predator]] and there is a behavior that makes it more difficult for the predator to kill individuals of the species. Individuals who learn the behavior more quickly will obviously be at an advantage. As time goes on, the ability to learn the behavior will improve (by genetic selection), and at some point it will seem to be an [[instinct]]. Baldwin gives the following case involving cooperation: "Animals may be kept alive let us say in a given environment by social cooperation only; these transmit this social type of variation to posterity; thus social adaptation sets the direction of physical phylogeny and physical heredity is determined in part by this factor" (Baldwin, 1896, p.&nbsp;553).<br />
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Suppose a species is threatened by a new predator and there is a behavior that makes it more difficult for the predator to kill individuals of the species. Individuals who learn the behavior more quickly will obviously be at an advantage. As time goes on, the ability to learn the behavior will improve (by genetic selection), and at some point it will seem to be an instinct. Baldwin gives the following case involving cooperation: "Animals may be kept alive let us say in a given environment by social cooperation only; these transmit this social type of variation to posterity; thus social adaptation sets the direction of physical phylogeny and physical heredity is determined in part by this factor" (Baldwin, 1896, p.&nbsp;553).<br />
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假设一个物种受到一个新的捕食者的威胁，并且有一种行为使得捕食者更加难以杀死该物种的个体。更快学会这种行为的个体显然会处于有利地位。随着时间的推移，学习行为的能力将得到提高(通过基因选择) ，在某种程度上，这似乎是一种本能。鲍德温给出了以下涉及合作的案例: “动物可以保持活着，让我们说，在一个特定的环境中，只有通过社会合作; 这些传递这种社会类型的变异给子孙后代; 因此，社会适应确定了物理系统发育和物理遗传的方向，部分由这个因素决定”(鲍德温，1896年，第553页)。<br />
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The appearance of lactose tolerance (i.e., [[lactase persistence]]) in human populations with a long tradition of raising domesticated animals for milk production has been suggested as another example.<ref>[https://epdf.pub/evolution-and-learning-the-baldwin-effect-reconsidered.html Evolution and Learning: The Baldwin Effect Reconsidered]</ref> This argument holds that a feedback loop operates whereby a dairy culture increases the selective advantage from this genetic trait, while the average population genotype increases the collective rewards of a dairy culture.<br />
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The appearance of lactose tolerance (i.e., lactase persistence) in human populations with a long tradition of raising domesticated animals for milk production has been suggested as another example. This argument holds that a feedback loop operates whereby a dairy culture increases the selective advantage from this genetic trait, while the average population genotype increases the collective rewards of a dairy culture.<br />
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另一个例子是，在人类中出现了乳糖耐受性(即乳糖分解酶持续性) ，而人类在饲养家畜产奶方面有着悠久的传统。这种观点认为，乳制品文化通过一个反馈循环增加了这种遗传特性的选择优势，而普通群体的基因型增加了乳制品文化的集体回报。<br />
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==Controversy and acceptance 争议和接受==<br />
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Initially Baldwin's ideas were not incompatible with the prevailing, but uncertain, ideas about the mechanism of transmission of hereditary information and at least two other biologists put forward very similar ideas in 1896.<ref>{{Cite journal |last=Morgan |first=C. L. |authorlink=C. Lloyd Morgan |year=1896 |title=On modification and variation |journal=Science|volume=4|issue=99 |pages=733–740 |doi=10.1126/science.4.99.733|pmid=17735249 |url=https://zenodo.org/record/1448227 }}</ref><ref>{{Cite journal |last=Osborne |first=H. F. |authorlink=Henry Fairfield Osborne |year=1896 |title=Ontogenic and phylogenic variation |journal=Science |volume=4|issue=100 |pages=786–789 |doi=10.1126/science.4.100.786|pmid=17734840 |url=https://zenodo.org/record/1448151 }}</ref> In 1901, [[Maurice Maeterlinck]] referred to behavioural adaptations to prevailing climates in different species of bees as ‘what had merely been an idea, therefore, and opposed to instinct, has thus by slow degrees become an instinctive habit’.<ref>{{Cite book |title=The Life of the Bee|last=Materlinck |first=Maurice |authorlink=Maurice Maeterlinck |publisher=Dodd, Mead and Co. |year=1901 |location=New York |pages=Chapter VII section 102}}</ref> The Baldwin effect theory subsequently became more controversial, with scholars being split between "Baldwin boosters" and "Baldwin skeptics".<ref name="depew03">Depew, David J. (2003), "Baldwin Boosters, Baldwin Skeptics" in: {{cite book |title=Evolution and learning: The Baldwin effect reconsidered |last=Weber |first=Bruce H. |first2=David J. |last2=Depew |year=2003 |publisher=MIT Press |location=Cambridge, MA |isbn=978-0-262-23229-6 |pages=3–31 |url=https://books.google.com/books?id=yBtRzBilw1MC&pg=PR9&dq=baldwin+effect#v=onepage&q&f=false }}</ref> The theory was first called the "Baldwin effect" by [[George Gaylord Simpson]] in 1953.<ref name="depew03" /> Simpson "admitted that the idea was theoretically consistent, that is, not inconsistent with the [[Modern synthesis (20th century)|modern synthesis]]",<ref name="depew03" /> but he doubted that the phenomenon occurred very often, or if so, could be proven to occur. In his discussion of the reception of the Baldwin-effect theory Simpson points out that the theory appears to provide a reconciliation between a neo-Darwinian and a [[Lamarckism|neo-Lamarckian]] approach and that "Mendelism and later genetic theory so conclusively ruled out the extreme neo-Lamarckian position that reconciliation came to seem unnecessary".<ref>{{cite journal |title=The Baldwin effect |authorlink=George Gaylord Simpson |author=Simpson, George Gaylord |year=1953 |journal=Evolution |volume=7 | issue=2 |pages=110–117 |doi=10.2307/2405746|jstor=2405746 }}</ref> In 1942, the evolutionary biologist [[Julian Huxley]] promoted the Baldwin effect as part of the modern synthesis, saying the concept had been unduly neglected by evolutionists.<ref name=Julian1942>{{cite book | authorlink=Julian Huxley |author=Huxley, Julian |year=1942 |location=London |title=Evolution: The Modern Synthesis |publisher=George Allen & Unwin Ltd|title-link=Evolution: The Modern Synthesis }}</ref><br />
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Initially Baldwin's ideas were not incompatible with the prevailing, but uncertain, ideas about the mechanism of transmission of hereditary information and at least two other biologists put forward very similar ideas in 1896. In 1901, Maurice Maeterlinck referred to behavioural adaptations to prevailing climates in different species of bees as ‘what had merely been an idea, therefore, and opposed to instinct, has thus by slow degrees become an instinctive habit’. The Baldwin effect theory subsequently became more controversial, with scholars being split between "Baldwin boosters" and "Baldwin skeptics". The theory was first called the "Baldwin effect" by George Gaylord Simpson in 1953. In 1942, the evolutionary biologist Julian Huxley promoted the Baldwin effect as part of the modern synthesis, saying the concept had been unduly neglected by evolutionists.<br />
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最初，鲍德温的观点与广泛流传但未加验证的关于遗传信息传递机制的观点并不矛盾，且在1896年至少有两位生物学家提出了非常相似的观点。1901年，'''莫里斯·梅特林克 Maurice Maeterlinck'''提到不同种类的蜜蜂对当时气候的行为适应性，称其之前仅仅是一种想法，因此，与本能相反，慢慢地变成了一种本能的习惯。鲍德温效应理论随后变得更具争议性，学者们分为“鲍德温支持者”和“鲍德温怀疑论者”。1953年'''乔治·盖洛德·辛普森 George Gaylord Simpson'''首次命名该理论为“鲍德温效应”。1942年，进化生物学家'''朱利安·赫胥黎 Julian Huxley'''将鲍德温效应作为现代进化综论的一部分加以推广，称进化论者过度忽视了这一概念。<br />
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In the 1960s, the evolutionary biologist [[Ernst Mayr]] contended that the Baldwin effect theory was untenable because<br />
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In the 1960s, the evolutionary biologist Ernst Mayr contended that the Baldwin effect theory was untenable because<br />
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在20世纪60年代，进化生物学家'''恩斯特·迈尔'''称鲍德温效应理论是站不住脚的，因为<br />
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# the argument is stated in terms of the individual genotype, whereas what is really exposed to the selection pressure is a phenotypically and genetically variable population;<br />
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the argument is stated in terms of the individual genotype, whereas what is really exposed to the selection pressure is a phenotypically and genetically variable population;<br />
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这个论点是从个体基因型的角度来阐述的，而真正暴露在选择压力下的是表型和遗传可变的种群;<br />
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# it is not sufficiently emphasized that the degree of modification of the phenotype is in itself genetically controlled;<br />
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it is not sufficiently emphasized that the degree of modification of the phenotype is in itself genetically controlled;<br />
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没有充分强调表型的改变程度本身是受遗传控制的;<br />
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# it is assumed that phenotypic rigidity is selectively superior to phenotypic flexibility.<ref name=Mayr1963>{{cite book| authorlink=Ernst Mayr |author=Mayr, Ernst |year=1963 |location=Cambridge, MA |title=Animal Species and Evolution| url=https://archive.org/details/animalspeciesevo00mayr | url-access=registration |publisher=Harvard University Press|isbn=978-0-674-03750-2}}</ref><br />
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it is assumed that phenotypic rigidity is selectively superior to phenotypic flexibility.<br />
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假定表型刚性有选择性优于表型柔性。<br />
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In 1987 [[Geoffrey Hinton]] and Steven Nowlan demonstrated by computer simulation that learning can accelerate evolution, and they associated this with the Baldwin effect.<ref>{{cite journal |title=How learning can guide evolution |last=Hinton |first=Geoffrey E. |author-link=Geoffrey Hinton |last2=Nowlan |first2=Steven J. |year=1987 |journal=Complex Systems |volume=1 |pages=495–502}}</ref><ref>{{cite journal |title=When learning guides evolution|authorlink=John Maynard Smith |author=Maynard Smith, John |year=1987 |journal=Nature |volume=329 |issue=6142 |pages=761–762 |doi=10.1038/329761a0|pmid=3670381 }}</ref><ref name=Puentedura2003>{{cite book| author=Puentedura, Ruben R. |year=2003|location=Cambridge, MA |chapter=The Baldwin effect in the age of computation |title=Evolution and Learning: The Baldwin Effect Reconsidered |editor-last=Weber |editor-first=Bruce H. |editor-last2=Depew |editor-first2=David J.|publisher=MIT press |pages=219–234}}</ref><br />
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In 1987 Geoffrey Hinton and Steven Nowlan demonstrated by computer simulation that learning can accelerate evolution, and they associated this with the Baldwin effect.<br />
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1987年，'''杰弗里·辛顿 Geoffrey Hinton''' 和 '''史蒂芬·诺兰 Steven Nowlan''' 通过计算机模拟证明了学习可以加速进化，他们将此与鲍德温效应联系起来。<br />
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Paul Griffiths<ref name=Griffiths2003>{{cite book| author=Griffiths, Paul E. |year=2003|location=Cambridge, MA|chapter=Beyond the Baldwin effect: James Mark Baldwin’s ‘social heredity,’ epigenetic inheritance, and niche construction|title=Evolution and Learning: The Baldwin Effect Reconsidered|publisher=MIT press|pages=193–215|editor-last=Weber |editor-first=Bruce H. |editor-last2=Depew |editor-first2=David J.}}</ref> suggests two reasons for the continuing interest in the Baldwin effect. The first is the role ''mind'' is understood to play in the effect. The second is the connection between ''development'' and evolution in the effect. Baldwin’s account of how neurophysiological and conscious mental factors may contribute to the effect<ref>{{cite journal |title=Heredity and instinct|last=Baldwin |first=J. Mark |author-link=James Mark Baldwin|year=1896 |journal=Science |volume=3 |issue=64|pages=438–441, 558–561|doi=10.1126/science.3.64.438 |pmid=17780356|url=https://zenodo.org/record/1448082 }}</ref><ref>{{cite journal |title=Consciousness and evolution |last=Baldwin |first=J. Mark |author-link=James Mark Baldwin|year=1896 |journal=Psychological Review |volume=3 |issue=3|pages=300–309 |doi=10.1037/h0063996}}</ref><ref>{{cite journal |title=A new factor in evolution |last=Baldwin |first=J. Mark |author-link=James Mark Baldwin|year=1896 |journal=The American Naturalist |volume=30 |issue=354|pages=441–451, 536–553 |doi=10.1086/276408|url=https://semanticscholar.org/paper/0c1386d88cb54eb17c6a2745cd4bf15dbb3f2b09 }}</ref> brings into focus the question of the possible survival value of consciousness.<ref name=Lindahl2001>{{cite book| author=Lindahl, B. I. B. |year=2001 |location=Amsterdam |chapter=Consciousness, behavioural patterns and the direction of biological evolution: implications for the mind–brain problem |title=Dimensions of Conscious Experience |publisher=John Benjamins |pages=73–99 |editor-last=Pylkkänen |editor-first=Paavo |editor-last2=Vadén |editor-first2=Tere |isbn=978-90-272-5157-2}}</ref><br />
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Paul Griffiths suggests two reasons for the continuing interest in the Baldwin effect. The first is the role mind is understood to play in the effect. The second is the connection between development and evolution in the effect. Baldwin’s account of how neurophysiological and conscious mental factors may contribute to the effect brings into focus the question of the possible survival value of consciousness.<br />
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'''保罗·格里菲斯 Paul Griffiths'''指出了人们对鲍德温效应持续兴趣的两个原因。首先，我们可以理解心智在这种效果中所扮演的角色。二是其中发展与演变的关系。鲍德温关于神经生理学和意识心理因素如何促成这种效应的解释，使人们更加关注意识可能的存在价值问题。<br />
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[[File:Carpodacus mexicanus -Madison, Wisconsin, USA-8.jpg|thumb|The [[house finch]]'s colonisation of North America has provided empirical evidence of the Baldwin effect.<ref name=Badyaev/>]]<br />
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The [[house finch's colonisation of North America has provided empirical evidence of the Baldwin effect.]]<br />
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[图一：家朱雀在北美的定殖提供了鲍德温效应的经验证明。]<br />
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Still, observes David Depew, "it is striking that a rather diverse lot of contemporary evolutionary theorists, most of whom regard themselves as supporters of the Modern Synthesis, have of late become 'Baldwin boosters{{'"}}<ref name="depew03" /> These Baldwin boosters<br />
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Still, observes David Depew, "it is striking that a rather diverse lot of contemporary evolutionary theorists, most of whom regard themselves as supporters of the Modern Synthesis, have of late become 'Baldwin boosters These Baldwin boosters<br />
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块引号<br />
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are typically [[evolutionary psychologist]]s who are searching for scenarios in which a population can get itself by behavioral trial and error onto a "hard to find" part of the fitness landscape in which human brain, language, and mind can rapidly coevolve. They are searching for what [[Daniel Dennett]], himself a Baldwin booster, calls an "evolutionary crane," an instrument to do some heavy lifting fast.<ref name="depew03" /><br />
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are typically evolutionary psychologists who are searching for scenarios in which a population can get itself by behavioral trial and error onto a "hard to find" part of the fitness landscape in which human brain, language, and mind can rapidly coevolve. They are searching for what Daniel Dennett, himself a Baldwin booster, calls an "evolutionary crane," an instrument to do some heavy lifting fast.<br />
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不过，'''戴维•迪皮尤 David Depew'''评论道: “令人惊讶的是，很多当代进化论理论家（其中大多数人自认为是现代进化综论的支持者），最近都成为了‘鲍德温支持者’，这些人是典型的致力于寻找“很难找到的”，群体可以通过行为试错达到人脑、语言和思维快速共同进化的适应性图景之场景的'''<font color="#ff8000">进化心理学 Evolutionary Psychology</font>'''家。他们正在寻找被丹尼尔·丹尼特（鲍德温的支持者）所称的“进化起重机”（快速升高重物的工具）。<br />
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/ blockquote<br />
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According to [[Daniel Dennett]], recent work has rendered the Baldwin effect "no longer a controversial wrinkle in orthodox Darwinism".<ref name="dennett03">[[Daniel Dennett|Dennett, Daniel]] (2003), "The Baldwin Effect: a Crane, not a Skyhook" in: {{cite book |title=Evolution and learning: The Baldwin effect reconsidered |last=Weber |first=Bruce H. |first2=David J. |last2=Depew |year=2003 |publisher=MIT Press |location=Cambridge, MA |isbn=978-0-262-23229-6 |pages=69–106 |url=https://books.google.com/books?id=yBtRzBilw1MC&pg=PR9&dq=baldwin+effect#v=onepage&q&f=false }}</ref> Potential genetic mechanisms underlying the Baldwin effect have been proposed for the evolution of natural (genetically-determinant) antibodies.<ref>{{Cite journal|last=Anderson|first=Russell|date=1996|title=How the adaptive antibodies facilitate the evolution of natural antibodies|url=|journal=Immunology and Cell Biology|volume=74 | issue = 2 |pages=286–291|doi=10.1038/icb.1996.50|pmid=8799730}}</ref> In 2009, empirical evidence for the Baldwin effect was provided from the colonisation of North America by the [[house finch]].<ref name=Badyaev>{{cite journal |last1=Badyaev |first1=Alexander V. |title=Evolutionary significance of phenotypic accommodation in novel environments: an empirical test of the Baldwin effect |journal=Phil. Trans. R. Soc. Lond. B |date=March 2009 |volume=364 |issue=1520 |doi=10.1098/rstb.2008.0285 |pmid=19324617 |pages=1125–1141|pmc=2666683 }}</ref><br />
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According to Daniel Dennett, recent work has rendered the Baldwin effect "no longer a controversial wrinkle in orthodox Darwinism". Potential genetic mechanisms underlying the Baldwin effect have been proposed for the evolution of natural (genetically-determinant) antibodies. In 2009, empirical evidence for the Baldwin effect was provided from the colonisation of North America by the house finch.<br />
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丹尼尔•丹尼特表示，最近的研究工作已使鲍德温效应“在正统达尔文主义中不再是争议性问题”。科学家已提出鲍德温效应背后潜在的遗传机制用于自然（基因决定）抗体的进化。2009年，'''家朱雀 House Finch'''在北美的定殖为鲍德温效应提供了经验证明。<br />
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==Comparison with genetic assimilation 与遗传同化的区别==<br />
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{{further|Genetic assimilation}}<br />
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The Baldwin effect has been confused with, and sometimes conflated with, a different evolutionary theory also based on [[phenotypic plasticity]], [[C. H. Waddington]]'s [[genetic assimilation]]. The Baldwin effect includes genetic accommodation, of which one type is genetic assimilation.<ref>{{cite journal | last1=Crispo | first1=Erika | year=2007 | title=The Baldwin effect and genetic assimilation: revisiting two mechanisms of evolutionary change mediated by phenotypic plasticity | journal=Evolution | volume=61 | issue=11 | pages=2469–2479 | doi=10.1111/j.1558-5646.2007.00203.x| pmid=17714500 }}</ref><!--more needed--><br />
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The Baldwin effect has been confused with, and sometimes conflated with, a different evolutionary theory also based on phenotypic plasticity, C. H. Waddington's genetic assimilation. The Baldwin effect includes genetic accommodation, of which one type is genetic assimilation.<!--more needed--><br />
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鲍德温效应与另一种同样基于'''<font color="#ff8000">表型可塑性 Henotypic Plasticity</font>'''的进化理论——C.H.华丁顿的'''<font color="#ff8000">遗传同化 Genetic Assimilation</font>'''相混淆，有时甚至混为一谈。实际上，鲍德温效应包括遗传调节，其中一种类型是遗传同化。<br />
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==See also 参见==<br />
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* {{annotated link|Evolvability}}<br />
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* {{annotated link|Lamarckism}}<br />
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* {{annotated link|Maternal effect}}<br />
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* {{annotated link|Meme}}<br />
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* {{annotated link|Pangenesis}}<br />
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* {{annotated link|Weismann barrier}}<br />
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==Notes 注释==<br />
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{{reflist|30em}}<br />
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== References 参考资料==<br />
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* {{cite journal | last = Bateson |first=Patrick | year = 2004 | title = The Active Role of Behaviour in Evolution | url = | journal = Biology and Philosophy | volume = 19 | issue = 2| pages = 283–298 | doi=10.1023/b:biph.0000024468.12161.83}}<br />
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* {{cite journal | last1 = Baldwin | first1 = J. Mark | authorlink = James Mark Baldwin | year = 1896 | title = A New Factor in Evolution | url = http://www.brocku.ca/MeadProject/Baldwin/Baldwin_1896_h.html | journal = The American Naturalist | volume = 30 | issue = 354| pages = 441–451 | doi=10.1086/276408}}<br />
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* {{cite journal | last1 = Baldwin | first1 = J. Mark | authorlink = James Mark Baldwin | year = 1897 | title = Organic Selection | journal = Science | volume = 5 | issue = 121| pages = 634–636 | doi=10.1126/science.5.121.634| pmid = 17781159 | url = https://zenodo.org/record/1448253 }}<br />
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* {{cite book |last1=Cochran |first1=Gregory |last2=Harpending |first2=Henry |date=2009 |title=The 10,000 Year Explosion: How Civilization Accelerated Human Evolution |publisher=Basic Books |isbn=978-0-465-00221-4|title-link=The 10,000 Year Explosion }}<br />
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* {{cite journal | last = Hall |first=Brian K. | year = 2001 | title = Organic Selection: Proximate Environmental Effects on the Evolution of Morphology and Behaviour | doi = 10.1023/a:1006773408919 | journal = Biology and Philosophy | volume = 16 | issue = 2| pages = 215–237 }}<br />
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* {{cite journal|author=Newman, Stuart A |year=2002 |title=Putting Genes in their place |url=http://www.nymc.edu/sanewman/PDFs/Putting%20Genes%20in%20their%20Place.pdf |journal=Journal of Biosciences |volume=27 |issue= 2|pages=97–104 |doi=10.1007/bf02703765 |pmid=11937679 |url-status=dead |archiveurl=https://web.archive.org/web/20081217013308/http://www.nymc.edu/sanewman/PDFs/Putting%20Genes%20in%20their%20Place.pdf |archivedate=2008-12-17 }}<br />
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* {{cite journal | last1 = Osborn | first1 = Henry F. | authorlink = Henry Fairfield Osborn | year = 1896 | title = Ontogenic and Phylogenic Variation| journal = Science | volume = 4 | issue = 100| pages = 786–789 | doi=10.1126/science.4.100.786| pmid = 17734840 | url = https://zenodo.org/record/1448151 }}<br />
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* [[Robert J. Richards|Richards, Robert J.]]; ''Darwin and the Emergence of Evolutionary Theories of Mind and Behavior'', Chicago: The University of Chicago Press, 1987.<br />
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* {{cite journal | last = Simpson |first=G. Gaylord |authorlink=George Gaylord Simpson | year = 1953 | title = The Baldwin effect | url = | journal = Evolution | volume = 7 | issue = 2| pages = 110–117 | doi=10.2307/2405746|jstor=2405746 }}<br />
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* {{cite journal |last1=Sznajder |first1=B. |last2=Sabelis |first2=M. W. |last3=Egas |first3=M. |title=How Adaptive Learning Affects Evolution: Reviewing Theory on the Baldwin Effect |journal=Evolutionary Biology |date=September 2012|volume=39|issue=3 |pages=301–310 |doi=10.1007/s11692-011-9155-2 |pmid=22923852 |pmc=3423563}}<!--a review paper, a recent secondary source--><br />
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==External links 外部链接==<br />
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* [http://www.ecs.soton.ac.uk/~harnad/Hypermail/Cognition.Sociobiology.98/0101.html Baldwinian evolution]<br />
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* [https://web.archive.org/web/20060810005608/http://www2.create.human.nagoya-u.ac.jp/baldwin/ Bibliography]<br />
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[[Category:Evolutionary biology]]<br />
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Category:Evolutionary biology<br />
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分类: 进化生物学<br />
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[[Category:Selection]]<br />
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Category:Selection<br />
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类别: 选择<br />
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<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Baldwin effect]]. Its edit history can be viewed at [[鲍德温效应/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%B2%8D%E5%BE%B7%E6%B8%A9%E6%95%88%E5%BA%94&diff=12058鲍德温效应2020-08-11T12:01:22Z<p>Dorr：</p>
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<div>已由Dorr初步翻译<br />
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此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。<br />
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{{short description|Effect of learned behavior on evolution}}<br />
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{{Distinguish|Baldwin's rules}}<br />
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[[File:Lamarck Compared to Darwin, Baldwin, Waddington.svg|thumb|upright=2.5|The Baldwin effect compared to [[Lamarckism|Lamarck's theory of evolution]], [[Darwinian evolution]], and Waddington's [[genetic assimilation]]. All the theories offer explanations of how organisms respond to a changed environment with adaptive inherited change.]]<br />
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Lamarck's theory of evolution, Darwinian evolution, and Waddington's genetic assimilation. All the theories offer explanations of how organisms respond to a changed environment with adaptive inherited change.]]<br />
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拉马克的进化论、达尔文进化论和华丁顿的遗传同化论。[参考译文]所有的理论都解释了有机体是如何通过适应性遗传变化来应对环境变化的<br />
--~英文wiki无此段<br />
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In [[evolutionary biology]], the '''Baldwin effect''' describes the effect of learned behavior on evolution. In brief, [[James Mark Baldwin]] and others suggested during the [[eclipse of Darwinism]] in the late 19th century that an organism's ''ability to learn'' new behaviors (e.g. to acclimatise to a new stressor) will affect its reproductive success and will therefore have an effect on the genetic makeup of its species through [[natural selection]]. Though this process appears similar to [[Lamarckism|Lamarckian evolution]], Lamarck proposed that living things ''inherited'' their parents' acquired characteristics. The Baldwin effect has been independently proposed several times, and today it is generally recognized as part of the [[Modern synthesis (20th century)|modern synthesis]].<br />
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In evolutionary biology, the Baldwin effect describes the effect of learned behavior on evolution. In brief, James Mark Baldwin and others suggested during the eclipse of Darwinism in the late 19th century that an organism's ability to learn new behaviors (e.g. to acclimatise to a new stressor) will affect its reproductive success and will therefore have an effect on the genetic makeup of its species through natural selection. Though this process appears similar to Lamarckian evolution, Lamarck proposed that living things inherited their parents' acquired characteristics. The Baldwin effect has been independently proposed several times, and today it is generally recognized as part of the modern synthesis.<br />
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在'''<font color="#ff8000">演化生物学 Evolutionary Biology</font>'''中，'''<font color="#ff8000">鲍德温效应 Baldwin Effect</font>'''描述了习得行为对进化的影响。简而言之，在19世纪末'''<font color="#ff8000">达尔文主义 Darwinism</font>'''的衰落期间，'''詹姆斯·马克·鲍德温 James Mark Baldwin'''及其他一些人提出，生物体学习新行为的能力（如为了适应新的压力）会影响其繁殖，进而会通过'''<font color="#ff8000">自然选择 Natural Selection</font>'''影响其物种的基因组成。这个过程似乎与'''<font color="#ff8000">拉马克主义 Lamarckism</font>'''中相似，但拉马克认为生物继承了其父母后天获得的特性。鲍德温效应已独立提出过多次，如今普遍认为其为'''<font color="#ff8000">现代进化综论 Modern Evolutionary Synthesis</font>'''的一部分。<br />
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=="A New Factor in Evolution" “进化的新因素”==<br />
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The effect, then unnamed, was put forward in 1896 in a paper "A New Factor in Evolution" by [[United States|American]] [[psychology|psychologist]] [[James Mark Baldwin]]. The paper proposed a mechanism for specific selection for general learning ability. As [[Robert J. Richards|Robert Richards]] explains:<ref name=Richards1987>{{cite book | author=Richards, Robert J. | year=1987 | title=Darwin and the Emergence of Evolutionary Theories of Mind and Behavior| publisher=The University of Chicago Press |location=Chicago | isbn=978-0-226-71199-7 | page=399| author-link=Robert J. Richards }}</ref><br />
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The effect, then unnamed, was put forward in 1896 in a paper "A New Factor in Evolution" by American psychologist James Mark Baldwin. The paper proposed a mechanism for specific selection for general learning ability. As Robert Richards explains:<br />
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1896年，美国心理学家詹姆斯·马克·鲍德温在一篇名为《进化的新因素》的论文中提出了这一效应（1987年又发表了第二篇论文）。该文提出了一种针对一般学习能力的具体选择机制。正如'''罗伯特·理查兹 Robert Richards'''解释的那样:<br />
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If animals entered a new environment—or their old environment rapidly changed—those<br />
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If animals entered a new environment—or their old environment rapidly changed—those<br />
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that could flexibly respond by learning new behaviors or by ontogenetically adapting would<br />
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that could flexibly respond by learning new behaviors or by ontogenetically adapting would<br />
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be naturally preserved. This saved remnant would, over several generations, have the<br />
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be naturally preserved. This saved remnant would, over several generations, have the<br />
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opportunity to exhibit spontaneously congenital variations similar to their acquired traits and<br />
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opportunity to exhibit spontaneously congenital variations similar to their acquired traits and<br />
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have these variations naturally selected. It would look as though the acquired traits had sunk<br />
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have these variations naturally selected. It would look as though the acquired traits had sunk<br />
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into the hereditary substance in a Lamarckian fashion, but the process would really be [[Neo-Darwinism|neo-Darwinian]].<br />
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into the hereditary substance in a Lamarckian fashion, but the process would really be neo-Darwinian.<br />
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如果动物处于新环境或其原来所处的环境急剧变化，那些能够通过学习新行为或自身基因适应来灵活作出反应的动物会自然地留存下来。这些幸存者，经过几代之后，有机会自发地表现出与他们获得的特征相似的先天变异，并让其经自然选择而留存。看上去获得性特征以拉马克的方式留在了遗传物质上，但这个过程实际上是'''<font color="#ff8000">新达尔文主义 Neo-Darwinism</font>'''的。<br />
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Selected offspring would tend to have an increased capacity for learning new skills rather than being confined to genetically coded, relatively fixed abilities. In effect, it places emphasis on the fact that the sustained behavior of a species or group can shape the evolution of that species. The "Baldwin effect" is better understood in [[evolutionary developmental biology]] literature as a scenario in which a character or [[biological trait|trait]] change occurring in an [[organism]] as a result of its interaction with its environment [[genetic assimilation|becomes gradually assimilated]] into its developmental genetic or epigenetic repertoire (Simpson, 1953; Newman, 2002). In the words of [[Daniel Dennett]],<ref name=dennett03/><br />
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Selected offspring would tend to have an increased capacity for learning new skills rather than being confined to genetically coded, relatively fixed abilities. In effect, it places emphasis on the fact that the sustained behavior of a species or group can shape the evolution of that species. The "Baldwin effect" is better understood in evolutionary developmental biology literature as a scenario in which a character or trait change occurring in an organism as a result of its interaction with its environment becomes gradually assimilated into its developmental genetic or epigenetic repertoire (Simpson, 1953; Newman, 2002). In the words of Daniel Dennett,<br />
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被选择留存的后代往往具有更强的学习新技能的能力，而不仅局限于基因编码的相对固定的能力。实际上，它强调以下事实：一个物种或群体的持续行为可以影响其进化。在'''<font color="#ff8000">演化发育生物学 Evolutionary developmental biology</font>'''的文献中，“鲍德温效应”得到了更好的理解——生物体由于与其环境的相互作用而发生的的特征或性状变化逐渐被同化到其发育遗传或表观遗传库中。用'''丹尼尔·丹尼特 Daniel Dennett'''的话来说,<br />
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Thanks to the Baldwin effect, species can be said to pretest the efficacy of particular different designs by phenotypic (individual) exploration of the space of nearby possibilities. If a particularly winning setting is thereby discovered, this discovery will ''create'' a new selection pressure: organisms that are closer in the adaptive landscape to that discovery will have a clear advantage over those more distant.<br />
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Thanks to the Baldwin effect, species can be said to pretest the efficacy of particular different designs by phenotypic (individual) exploration of the space of nearby possibilities. If a particularly winning setting is thereby discovered, this discovery will create a new selection pressure: organisms that are closer in the adaptive landscape to that discovery will have a clear advantage over those more distant.<br />
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由于鲍德温效应，可以说物种通过表型（个体）探索附近可能性的空间，预先测试了特定不同设计的功效。若由此发现一个特别适宜的环境，该发现将导致新的选择压力: 在适应环境中，更接近此发现的生物体将较其他生物体有明显的优势。<br />
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An update to the Baldwin effect was developed by [[Jean Piaget]], [[Paul Alfred Weiss|Paul Weiss]], and [[Conrad Waddington]] in the 1960s–1970s. This new version included an explicit role for the social in shaping subsequent natural change in humans (both evolutionary and developmental), with reference to alterations of selection pressures.<ref>{{cite journal | author = Burman J. T. | year = 2013 | title = Updating the Baldwin Effect: The biological levels behind Piaget's new theory | url = | journal = New Ideas in Psychology | volume = 31 | issue = 3| pages = 363–373 | doi = 10.1016/j.newideapsych.2012.07.003 }}</ref><br />
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An update to the Baldwin effect was developed by Jean Piaget, Paul Weiss, and Conrad Waddington in the 1960s–1970s. This new version included an explicit role for the social in shaping subsequent natural change in humans (both evolutionary and developmental), with reference to alterations of selection pressures.<br />
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1960-1970年代，'''让·皮亚杰 Jean Piaget'''、'''保罗·维斯 Paul Weiss'''和'''康拉德·华丁顿 Conrad Waddington'''修正了鲍德温效应。该新版本明确了社会在塑造人类随后的自然变化（包括进化和发展）中的作用，并提到了选择压力的改变。<br />
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As is to be expected from [[Stigler's law]], subsequent research shows that Baldwin was not the first to identify the process; [[Douglas Spalding]] mentioned it in 1873.<ref>Noble, R.; [[Denis Noble|Noble, D.]] (2017) Was the Watchmaker Blind? Or Was She One-Eyed? ''Biology'' 2017, 6(4), 47; doi:10.3390/biology6040047, quoting Bateson, P. The adaptability driver: Links between behaviour and evolution. Biol. Theory 2006, 1, 342–345.</ref><br />
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As is to be expected from Stigler's law, subsequent research shows that Baldwin was not the first to identify the process; Douglas Spalding mentioned it in 1873.<br />
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正如'''<font color="#ff8000">斯蒂格勒定律 Stigler's Law</font>'''所指，随后的研究表明，鲍德温并不是第一个发现这个过程的人; '''<font color="#ff8000">道格拉斯·斯波尔丁 Douglas Spalding</font>'''在1873年提到过这个过程。<br />
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==Examples 举例==<br />
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Suppose a species is threatened by a new [[predator]] and there is a behavior that makes it more difficult for the predator to kill individuals of the species. Individuals who learn the behavior more quickly will obviously be at an advantage. As time goes on, the ability to learn the behavior will improve (by genetic selection), and at some point it will seem to be an [[instinct]]. Baldwin gives the following case involving cooperation: "Animals may be kept alive let us say in a given environment by social cooperation only; these transmit this social type of variation to posterity; thus social adaptation sets the direction of physical phylogeny and physical heredity is determined in part by this factor" (Baldwin, 1896, p.&nbsp;553).<br />
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Suppose a species is threatened by a new predator and there is a behavior that makes it more difficult for the predator to kill individuals of the species. Individuals who learn the behavior more quickly will obviously be at an advantage. As time goes on, the ability to learn the behavior will improve (by genetic selection), and at some point it will seem to be an instinct. Baldwin gives the following case involving cooperation: "Animals may be kept alive let us say in a given environment by social cooperation only; these transmit this social type of variation to posterity; thus social adaptation sets the direction of physical phylogeny and physical heredity is determined in part by this factor" (Baldwin, 1896, p.&nbsp;553).<br />
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假设一个物种受到一个新的捕食者的威胁，并且有一种行为使得捕食者更加难以杀死该物种的个体。更快学会这种行为的个体显然会处于有利地位。随着时间的推移，学习行为的能力将得到提高(通过基因选择) ，在某种程度上，这似乎是一种本能。鲍德温给出了以下涉及合作的案例: “动物可以保持活着，让我们说，在一个特定的环境中，只有通过社会合作; 这些传递这种社会类型的变异给子孙后代; 因此，社会适应确定了物理系统发育和物理遗传的方向，部分由这个因素决定”(鲍德温，1896年，第553页)。<br />
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The appearance of lactose tolerance (i.e., [[lactase persistence]]) in human populations with a long tradition of raising domesticated animals for milk production has been suggested as another example.<ref>[https://epdf.pub/evolution-and-learning-the-baldwin-effect-reconsidered.html Evolution and Learning: The Baldwin Effect Reconsidered]</ref> This argument holds that a feedback loop operates whereby a dairy culture increases the selective advantage from this genetic trait, while the average population genotype increases the collective rewards of a dairy culture.<br />
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The appearance of lactose tolerance (i.e., lactase persistence) in human populations with a long tradition of raising domesticated animals for milk production has been suggested as another example. This argument holds that a feedback loop operates whereby a dairy culture increases the selective advantage from this genetic trait, while the average population genotype increases the collective rewards of a dairy culture.<br />
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另一个例子是，在人类中出现了乳糖耐受性(即乳糖分解酶持续性) ，而人类在饲养家畜产奶方面有着悠久的传统。这种观点认为，乳制品文化通过一个反馈循环增加了这种遗传特性的选择优势，而普通群体的基因型增加了乳制品文化的集体回报。<br />
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==Controversy and acceptance 争议和接受==<br />
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Initially Baldwin's ideas were not incompatible with the prevailing, but uncertain, ideas about the mechanism of transmission of hereditary information and at least two other biologists put forward very similar ideas in 1896.<ref>{{Cite journal |last=Morgan |first=C. L. |authorlink=C. Lloyd Morgan |year=1896 |title=On modification and variation |journal=Science|volume=4|issue=99 |pages=733–740 |doi=10.1126/science.4.99.733|pmid=17735249 |url=https://zenodo.org/record/1448227 }}</ref><ref>{{Cite journal |last=Osborne |first=H. F. |authorlink=Henry Fairfield Osborne |year=1896 |title=Ontogenic and phylogenic variation |journal=Science |volume=4|issue=100 |pages=786–789 |doi=10.1126/science.4.100.786|pmid=17734840 |url=https://zenodo.org/record/1448151 }}</ref> In 1901, [[Maurice Maeterlinck]] referred to behavioural adaptations to prevailing climates in different species of bees as ‘what had merely been an idea, therefore, and opposed to instinct, has thus by slow degrees become an instinctive habit’.<ref>{{Cite book |title=The Life of the Bee|last=Materlinck |first=Maurice |authorlink=Maurice Maeterlinck |publisher=Dodd, Mead and Co. |year=1901 |location=New York |pages=Chapter VII section 102}}</ref> The Baldwin effect theory subsequently became more controversial, with scholars being split between "Baldwin boosters" and "Baldwin skeptics".<ref name="depew03">Depew, David J. (2003), "Baldwin Boosters, Baldwin Skeptics" in: {{cite book |title=Evolution and learning: The Baldwin effect reconsidered |last=Weber |first=Bruce H. |first2=David J. |last2=Depew |year=2003 |publisher=MIT Press |location=Cambridge, MA |isbn=978-0-262-23229-6 |pages=3–31 |url=https://books.google.com/books?id=yBtRzBilw1MC&pg=PR9&dq=baldwin+effect#v=onepage&q&f=false }}</ref> The theory was first called the "Baldwin effect" by [[George Gaylord Simpson]] in 1953.<ref name="depew03" /> Simpson "admitted that the idea was theoretically consistent, that is, not inconsistent with the [[Modern synthesis (20th century)|modern synthesis]]",<ref name="depew03" /> but he doubted that the phenomenon occurred very often, or if so, could be proven to occur. In his discussion of the reception of the Baldwin-effect theory Simpson points out that the theory appears to provide a reconciliation between a neo-Darwinian and a [[Lamarckism|neo-Lamarckian]] approach and that "Mendelism and later genetic theory so conclusively ruled out the extreme neo-Lamarckian position that reconciliation came to seem unnecessary".<ref>{{cite journal |title=The Baldwin effect |authorlink=George Gaylord Simpson |author=Simpson, George Gaylord |year=1953 |journal=Evolution |volume=7 | issue=2 |pages=110–117 |doi=10.2307/2405746|jstor=2405746 }}</ref> In 1942, the evolutionary biologist [[Julian Huxley]] promoted the Baldwin effect as part of the modern synthesis, saying the concept had been unduly neglected by evolutionists.<ref name=Julian1942>{{cite book | authorlink=Julian Huxley |author=Huxley, Julian |year=1942 |location=London |title=Evolution: The Modern Synthesis |publisher=George Allen & Unwin Ltd|title-link=Evolution: The Modern Synthesis }}</ref><br />
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Initially Baldwin's ideas were not incompatible with the prevailing, but uncertain, ideas about the mechanism of transmission of hereditary information and at least two other biologists put forward very similar ideas in 1896. In 1901, Maurice Maeterlinck referred to behavioural adaptations to prevailing climates in different species of bees as ‘what had merely been an idea, therefore, and opposed to instinct, has thus by slow degrees become an instinctive habit’. The Baldwin effect theory subsequently became more controversial, with scholars being split between "Baldwin boosters" and "Baldwin skeptics". The theory was first called the "Baldwin effect" by George Gaylord Simpson in 1953. In 1942, the evolutionary biologist Julian Huxley promoted the Baldwin effect as part of the modern synthesis, saying the concept had been unduly neglected by evolutionists.<br />
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最初，鲍德温的观点与广泛流传但未加验证的关于遗传信息传递机制的观点并不矛盾，且在1896年至少有两位生物学家提出了非常相似的观点。1901年，'''<font color="#ff8000">莫里斯·梅特林克 Maurice Maeterlinck</font>'''提到不同种类的蜜蜂对当时气候的行为适应性，称其之前仅仅是一种想法，因此，与本能相反，慢慢地变成了一种本能的习惯。鲍德温效应理论随后变得更具争议性，学者们分为“鲍德温支持者”和“鲍德温怀疑论者”。1953年'''乔治·盖洛德·辛普森 George Gaylord Simpson'''首次命名该理论为“鲍德温效应”。1942年，进化生物学家'''朱利安·赫胥黎 Julian Huxley'''将鲍德温效应作为现代进化综论的一部分加以推广，称进化论者过度忽视了这一概念。<br />
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In the 1960s, the evolutionary biologist [[Ernst Mayr]] contended that the Baldwin effect theory was untenable because<br />
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In the 1960s, the evolutionary biologist Ernst Mayr contended that the Baldwin effect theory was untenable because<br />
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在20世纪60年代，进化生物学家'''恩斯特·迈尔'''称鲍德温效应理论是站不住脚的，因为<br />
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# the argument is stated in terms of the individual genotype, whereas what is really exposed to the selection pressure is a phenotypically and genetically variable population;<br />
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the argument is stated in terms of the individual genotype, whereas what is really exposed to the selection pressure is a phenotypically and genetically variable population;<br />
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这个论点是从个体基因型的角度来阐述的，而真正暴露在选择压力下的是表型和遗传可变的种群;<br />
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# it is not sufficiently emphasized that the degree of modification of the phenotype is in itself genetically controlled;<br />
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it is not sufficiently emphasized that the degree of modification of the phenotype is in itself genetically controlled;<br />
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没有充分强调表型的改变程度本身是受遗传控制的;<br />
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# it is assumed that phenotypic rigidity is selectively superior to phenotypic flexibility.<ref name=Mayr1963>{{cite book| authorlink=Ernst Mayr |author=Mayr, Ernst |year=1963 |location=Cambridge, MA |title=Animal Species and Evolution| url=https://archive.org/details/animalspeciesevo00mayr | url-access=registration |publisher=Harvard University Press|isbn=978-0-674-03750-2}}</ref><br />
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it is assumed that phenotypic rigidity is selectively superior to phenotypic flexibility.<br />
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假定表型刚性有选择性优于表型柔性。<br />
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In 1987 [[Geoffrey Hinton]] and Steven Nowlan demonstrated by computer simulation that learning can accelerate evolution, and they associated this with the Baldwin effect.<ref>{{cite journal |title=How learning can guide evolution |last=Hinton |first=Geoffrey E. |author-link=Geoffrey Hinton |last2=Nowlan |first2=Steven J. |year=1987 |journal=Complex Systems |volume=1 |pages=495–502}}</ref><ref>{{cite journal |title=When learning guides evolution|authorlink=John Maynard Smith |author=Maynard Smith, John |year=1987 |journal=Nature |volume=329 |issue=6142 |pages=761–762 |doi=10.1038/329761a0|pmid=3670381 }}</ref><ref name=Puentedura2003>{{cite book| author=Puentedura, Ruben R. |year=2003|location=Cambridge, MA |chapter=The Baldwin effect in the age of computation |title=Evolution and Learning: The Baldwin Effect Reconsidered |editor-last=Weber |editor-first=Bruce H. |editor-last2=Depew |editor-first2=David J.|publisher=MIT press |pages=219–234}}</ref><br />
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In 1987 Geoffrey Hinton and Steven Nowlan demonstrated by computer simulation that learning can accelerate evolution, and they associated this with the Baldwin effect.<br />
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1987年，'''杰弗里·辛顿 Geoffrey Hinton''' 和 '''史蒂芬·诺兰 Steven Nowlan''' 通过计算机模拟证明了学习可以加速进化，他们将此与鲍德温效应联系起来。<br />
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Paul Griffiths<ref name=Griffiths2003>{{cite book| author=Griffiths, Paul E. |year=2003|location=Cambridge, MA|chapter=Beyond the Baldwin effect: James Mark Baldwin’s ‘social heredity,’ epigenetic inheritance, and niche construction|title=Evolution and Learning: The Baldwin Effect Reconsidered|publisher=MIT press|pages=193–215|editor-last=Weber |editor-first=Bruce H. |editor-last2=Depew |editor-first2=David J.}}</ref> suggests two reasons for the continuing interest in the Baldwin effect. The first is the role ''mind'' is understood to play in the effect. The second is the connection between ''development'' and evolution in the effect. Baldwin’s account of how neurophysiological and conscious mental factors may contribute to the effect<ref>{{cite journal |title=Heredity and instinct|last=Baldwin |first=J. Mark |author-link=James Mark Baldwin|year=1896 |journal=Science |volume=3 |issue=64|pages=438–441, 558–561|doi=10.1126/science.3.64.438 |pmid=17780356|url=https://zenodo.org/record/1448082 }}</ref><ref>{{cite journal |title=Consciousness and evolution |last=Baldwin |first=J. Mark |author-link=James Mark Baldwin|year=1896 |journal=Psychological Review |volume=3 |issue=3|pages=300–309 |doi=10.1037/h0063996}}</ref><ref>{{cite journal |title=A new factor in evolution |last=Baldwin |first=J. Mark |author-link=James Mark Baldwin|year=1896 |journal=The American Naturalist |volume=30 |issue=354|pages=441–451, 536–553 |doi=10.1086/276408|url=https://semanticscholar.org/paper/0c1386d88cb54eb17c6a2745cd4bf15dbb3f2b09 }}</ref> brings into focus the question of the possible survival value of consciousness.<ref name=Lindahl2001>{{cite book| author=Lindahl, B. I. B. |year=2001 |location=Amsterdam |chapter=Consciousness, behavioural patterns and the direction of biological evolution: implications for the mind–brain problem |title=Dimensions of Conscious Experience |publisher=John Benjamins |pages=73–99 |editor-last=Pylkkänen |editor-first=Paavo |editor-last2=Vadén |editor-first2=Tere |isbn=978-90-272-5157-2}}</ref><br />
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Paul Griffiths suggests two reasons for the continuing interest in the Baldwin effect. The first is the role mind is understood to play in the effect. The second is the connection between development and evolution in the effect. Baldwin’s account of how neurophysiological and conscious mental factors may contribute to the effect brings into focus the question of the possible survival value of consciousness.<br />
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'''保罗·格里菲斯 Paul Griffiths'''指出了人们对鲍德温效应持续兴趣的两个原因。首先，我们可以理解心智在这种效果中所扮演的角色。二是其中发展与演变的关系。鲍德温关于神经生理学和意识心理因素如何促成这种效应的解释，使人们更加关注意识可能的存在价值问题。<br />
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[[File:Carpodacus mexicanus -Madison, Wisconsin, USA-8.jpg|thumb|The [[house finch]]'s colonisation of North America has provided empirical evidence of the Baldwin effect.<ref name=Badyaev/>]]<br />
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The [[house finch's colonisation of North America has provided empirical evidence of the Baldwin effect.]]<br />
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[图一：家朱雀在北美的定殖提供了鲍德温效应的经验证明。]<br />
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Still, observes David Depew, "it is striking that a rather diverse lot of contemporary evolutionary theorists, most of whom regard themselves as supporters of the Modern Synthesis, have of late become 'Baldwin boosters{{'"}}<ref name="depew03" /> These Baldwin boosters<br />
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Still, observes David Depew, "it is striking that a rather diverse lot of contemporary evolutionary theorists, most of whom regard themselves as supporters of the Modern Synthesis, have of late become 'Baldwin boosters These Baldwin boosters<br />
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are typically [[evolutionary psychologist]]s who are searching for scenarios in which a population can get itself by behavioral trial and error onto a "hard to find" part of the fitness landscape in which human brain, language, and mind can rapidly coevolve. They are searching for what [[Daniel Dennett]], himself a Baldwin booster, calls an "evolutionary crane," an instrument to do some heavy lifting fast.<ref name="depew03" /><br />
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are typically evolutionary psychologists who are searching for scenarios in which a population can get itself by behavioral trial and error onto a "hard to find" part of the fitness landscape in which human brain, language, and mind can rapidly coevolve. They are searching for what Daniel Dennett, himself a Baldwin booster, calls an "evolutionary crane," an instrument to do some heavy lifting fast.<br />
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不过，'''戴维•迪皮尤 David Depew'''评论道: “令人惊讶的是，很多当代进化论理论家（其中大多数人自认为是现代进化综论的支持者），最近都成为了‘鲍德温支持者’，这些人是典型的致力于寻找“很难找到的”，群体可以通过行为试错达到人脑、语言和思维快速共同进化的适应性图景之场景的'''<font color="#ff8000">进化心理学 Evolutionary Psychology</font>'''家。他们正在寻找被丹尼尔·丹尼特（鲍德温的支持者）所称的“进化起重机”（快速升高重物的工具）。<br />
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According to [[Daniel Dennett]], recent work has rendered the Baldwin effect "no longer a controversial wrinkle in orthodox Darwinism".<ref name="dennett03">[[Daniel Dennett|Dennett, Daniel]] (2003), "The Baldwin Effect: a Crane, not a Skyhook" in: {{cite book |title=Evolution and learning: The Baldwin effect reconsidered |last=Weber |first=Bruce H. |first2=David J. |last2=Depew |year=2003 |publisher=MIT Press |location=Cambridge, MA |isbn=978-0-262-23229-6 |pages=69–106 |url=https://books.google.com/books?id=yBtRzBilw1MC&pg=PR9&dq=baldwin+effect#v=onepage&q&f=false }}</ref> Potential genetic mechanisms underlying the Baldwin effect have been proposed for the evolution of natural (genetically-determinant) antibodies.<ref>{{Cite journal|last=Anderson|first=Russell|date=1996|title=How the adaptive antibodies facilitate the evolution of natural antibodies|url=|journal=Immunology and Cell Biology|volume=74 | issue = 2 |pages=286–291|doi=10.1038/icb.1996.50|pmid=8799730}}</ref> In 2009, empirical evidence for the Baldwin effect was provided from the colonisation of North America by the [[house finch]].<ref name=Badyaev>{{cite journal |last1=Badyaev |first1=Alexander V. |title=Evolutionary significance of phenotypic accommodation in novel environments: an empirical test of the Baldwin effect |journal=Phil. Trans. R. Soc. Lond. B |date=March 2009 |volume=364 |issue=1520 |doi=10.1098/rstb.2008.0285 |pmid=19324617 |pages=1125–1141|pmc=2666683 }}</ref><br />
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According to Daniel Dennett, recent work has rendered the Baldwin effect "no longer a controversial wrinkle in orthodox Darwinism". Potential genetic mechanisms underlying the Baldwin effect have been proposed for the evolution of natural (genetically-determinant) antibodies. In 2009, empirical evidence for the Baldwin effect was provided from the colonisation of North America by the house finch.<br />
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丹尼尔•丹尼特表示，最近的研究工作已使鲍德温效应“在正统达尔文主义中不再是争议性问题”。科学家已提出鲍德温效应背后潜在的遗传机制用于自然（基因决定）抗体的进化。2009年，'''家朱雀 House Finch'''在北美的定殖为鲍德温效应提供了经验证明。<br />
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==Comparison with genetic assimilation 与遗传同化的区别==<br />
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{{further|Genetic assimilation}}<br />
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The Baldwin effect has been confused with, and sometimes conflated with, a different evolutionary theory also based on [[phenotypic plasticity]], [[C. H. Waddington]]'s [[genetic assimilation]]. The Baldwin effect includes genetic accommodation, of which one type is genetic assimilation.<ref>{{cite journal | last1=Crispo | first1=Erika | year=2007 | title=The Baldwin effect and genetic assimilation: revisiting two mechanisms of evolutionary change mediated by phenotypic plasticity | journal=Evolution | volume=61 | issue=11 | pages=2469–2479 | doi=10.1111/j.1558-5646.2007.00203.x| pmid=17714500 }}</ref><!--more needed--><br />
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The Baldwin effect has been confused with, and sometimes conflated with, a different evolutionary theory also based on phenotypic plasticity, C. H. Waddington's genetic assimilation. The Baldwin effect includes genetic accommodation, of which one type is genetic assimilation.<!--more needed--><br />
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鲍德温效应与另一种同样基于'''<font color="#ff8000">表型可塑性 Henotypic Plasticity</font>'''的进化理论——C.H.华丁顿的'''<font color="#ff8000">遗传同化 Genetic Assimilation</font>'''相混淆，有时甚至混为一谈。实际上，鲍德温效应包括遗传调节，其中一种类型是遗传同化。<br />
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==See also 参见==<br />
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* {{annotated link|Evolvability}}<br />
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* {{annotated link|Lamarckism}}<br />
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* {{annotated link|Maternal effect}}<br />
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* {{annotated link|Meme}}<br />
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* {{annotated link|Pangenesis}}<br />
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* {{annotated link|Weismann barrier}}<br />
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==Notes 注释==<br />
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{{reflist|30em}}<br />
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== References 参考资料==<br />
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* {{cite journal | last = Bateson |first=Patrick | year = 2004 | title = The Active Role of Behaviour in Evolution | url = | journal = Biology and Philosophy | volume = 19 | issue = 2| pages = 283–298 | doi=10.1023/b:biph.0000024468.12161.83}}<br />
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* {{cite journal | last1 = Baldwin | first1 = J. Mark | authorlink = James Mark Baldwin | year = 1896 | title = A New Factor in Evolution | url = http://www.brocku.ca/MeadProject/Baldwin/Baldwin_1896_h.html | journal = The American Naturalist | volume = 30 | issue = 354| pages = 441–451 | doi=10.1086/276408}}<br />
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* {{cite journal | last1 = Baldwin | first1 = J. Mark | authorlink = James Mark Baldwin | year = 1897 | title = Organic Selection | journal = Science | volume = 5 | issue = 121| pages = 634–636 | doi=10.1126/science.5.121.634| pmid = 17781159 | url = https://zenodo.org/record/1448253 }}<br />
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* {{cite book |last1=Cochran |first1=Gregory |last2=Harpending |first2=Henry |date=2009 |title=The 10,000 Year Explosion: How Civilization Accelerated Human Evolution |publisher=Basic Books |isbn=978-0-465-00221-4|title-link=The 10,000 Year Explosion }}<br />
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* {{cite journal | last = Hall |first=Brian K. | year = 2001 | title = Organic Selection: Proximate Environmental Effects on the Evolution of Morphology and Behaviour | doi = 10.1023/a:1006773408919 | journal = Biology and Philosophy | volume = 16 | issue = 2| pages = 215–237 }}<br />
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* {{cite journal|author=Newman, Stuart A |year=2002 |title=Putting Genes in their place |url=http://www.nymc.edu/sanewman/PDFs/Putting%20Genes%20in%20their%20Place.pdf |journal=Journal of Biosciences |volume=27 |issue= 2|pages=97–104 |doi=10.1007/bf02703765 |pmid=11937679 |url-status=dead |archiveurl=https://web.archive.org/web/20081217013308/http://www.nymc.edu/sanewman/PDFs/Putting%20Genes%20in%20their%20Place.pdf |archivedate=2008-12-17 }}<br />
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* {{cite journal | last1 = Osborn | first1 = Henry F. | authorlink = Henry Fairfield Osborn | year = 1896 | title = Ontogenic and Phylogenic Variation| journal = Science | volume = 4 | issue = 100| pages = 786–789 | doi=10.1126/science.4.100.786| pmid = 17734840 | url = https://zenodo.org/record/1448151 }}<br />
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* [[Robert J. Richards|Richards, Robert J.]]; ''Darwin and the Emergence of Evolutionary Theories of Mind and Behavior'', Chicago: The University of Chicago Press, 1987.<br />
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* {{cite journal | last = Simpson |first=G. Gaylord |authorlink=George Gaylord Simpson | year = 1953 | title = The Baldwin effect | url = | journal = Evolution | volume = 7 | issue = 2| pages = 110–117 | doi=10.2307/2405746|jstor=2405746 }}<br />
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* {{cite journal |last1=Sznajder |first1=B. |last2=Sabelis |first2=M. W. |last3=Egas |first3=M. |title=How Adaptive Learning Affects Evolution: Reviewing Theory on the Baldwin Effect |journal=Evolutionary Biology |date=September 2012|volume=39|issue=3 |pages=301–310 |doi=10.1007/s11692-011-9155-2 |pmid=22923852 |pmc=3423563}}<!--a review paper, a recent secondary source--><br />
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==External links 外部链接==<br />
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* [http://www.ecs.soton.ac.uk/~harnad/Hypermail/Cognition.Sociobiology.98/0101.html Baldwinian evolution]<br />
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* [https://web.archive.org/web/20060810005608/http://www2.create.human.nagoya-u.ac.jp/baldwin/ Bibliography]<br />
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[[Category:Evolutionary biology]]<br />
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Category:Evolutionary biology<br />
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分类: 进化生物学<br />
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[[Category:Selection]]<br />
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Category:Selection<br />
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类别: 选择<br />
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<small>This page was moved from [[wikipedia:en:Baldwin effect]]. Its edit history can be viewed at [[鲍德温效应/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%B2%8D%E5%BE%B7%E6%B8%A9%E6%95%88%E5%BA%94&diff=12057鲍德温效应2020-08-11T12:00:01Z<p>Dorr：</p>
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<div>已由Dorr初步翻译<br />
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{{short description|Effect of learned behavior on evolution}}<br />
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{{Distinguish|Baldwin's rules}}<br />
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[[File:Lamarck Compared to Darwin, Baldwin, Waddington.svg|thumb|upright=2.5|The Baldwin effect compared to [[Lamarckism|Lamarck's theory of evolution]], [[Darwinian evolution]], and Waddington's [[genetic assimilation]]. All the theories offer explanations of how organisms respond to a changed environment with adaptive inherited change.]]<br />
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Lamarck's theory of evolution, Darwinian evolution, and Waddington's genetic assimilation. All the theories offer explanations of how organisms respond to a changed environment with adaptive inherited change.]]<br />
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拉马克的进化论、达尔文进化论和华丁顿的遗传同化论。[参考译文]所有的理论都解释了有机体是如何通过适应性遗传变化来应对环境变化的<br />
--~英文wiki无此段<br />
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In [[evolutionary biology]], the '''Baldwin effect''' describes the effect of learned behavior on evolution. In brief, [[James Mark Baldwin]] and others suggested during the [[eclipse of Darwinism]] in the late 19th century that an organism's ''ability to learn'' new behaviors (e.g. to acclimatise to a new stressor) will affect its reproductive success and will therefore have an effect on the genetic makeup of its species through [[natural selection]]. Though this process appears similar to [[Lamarckism|Lamarckian evolution]], Lamarck proposed that living things ''inherited'' their parents' acquired characteristics. The Baldwin effect has been independently proposed several times, and today it is generally recognized as part of the [[Modern synthesis (20th century)|modern synthesis]].<br />
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In evolutionary biology, the Baldwin effect describes the effect of learned behavior on evolution. In brief, James Mark Baldwin and others suggested during the eclipse of Darwinism in the late 19th century that an organism's ability to learn new behaviors (e.g. to acclimatise to a new stressor) will affect its reproductive success and will therefore have an effect on the genetic makeup of its species through natural selection. Though this process appears similar to Lamarckian evolution, Lamarck proposed that living things inherited their parents' acquired characteristics. The Baldwin effect has been independently proposed several times, and today it is generally recognized as part of the modern synthesis.<br />
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在'''<font color="#ff8000">演化生物学 Evolutionary Biology</font>'''中，'''<font color="#ff8000">鲍德温效应 Baldwin Effect</font>'''描述了习得行为对进化的影响。简而言之，在19世纪末'''<font color="#ff8000">达尔文主义 Darwinism</font>'''的衰落期间，'''詹姆斯·马克·鲍德温 James Mark Baldwin'''及其他一些人提出，生物体学习新行为的能力（如为了适应新的压力）会影响其繁殖，进而会通过'''<font color="#ff8000">自然选择 Natural Selection,/font>'''影响其物种的基因组成。这个过程似乎与'''<font color="#ff8000">拉马克主义 Lamarckism</font>'''中相似，但拉马克认为生物继承了其父母后天获得的特性。鲍德温效应已独立提出过多次，如今普遍认为其为'''<font color="#ff8000">现代进化综论 Modern Evolutionary Synthesis</font>'''的一部分。<br />
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=="A New Factor in Evolution" “进化的新因素”==<br />
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The effect, then unnamed, was put forward in 1896 in a paper "A New Factor in Evolution" by [[United States|American]] [[psychology|psychologist]] [[James Mark Baldwin]]. The paper proposed a mechanism for specific selection for general learning ability. As [[Robert J. Richards|Robert Richards]] explains:<ref name=Richards1987>{{cite book | author=Richards, Robert J. | year=1987 | title=Darwin and the Emergence of Evolutionary Theories of Mind and Behavior| publisher=The University of Chicago Press |location=Chicago | isbn=978-0-226-71199-7 | page=399| author-link=Robert J. Richards }}</ref><br />
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The effect, then unnamed, was put forward in 1896 in a paper "A New Factor in Evolution" by American psychologist James Mark Baldwin. The paper proposed a mechanism for specific selection for general learning ability. As Robert Richards explains:<br />
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1896年，美国心理学家詹姆斯·马克·鲍德温在一篇名为《进化的新因素》的论文中提出了这一效应（1987年又发表了第二篇论文）。该文提出了一种针对一般学习能力的具体选择机制。正如'''罗伯特·理查兹 Robert Richards'''解释的那样:<br />
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If animals entered a new environment—or their old environment rapidly changed—those<br />
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If animals entered a new environment—or their old environment rapidly changed—those<br />
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that could flexibly respond by learning new behaviors or by ontogenetically adapting would<br />
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that could flexibly respond by learning new behaviors or by ontogenetically adapting would<br />
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be naturally preserved. This saved remnant would, over several generations, have the<br />
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be naturally preserved. This saved remnant would, over several generations, have the<br />
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opportunity to exhibit spontaneously congenital variations similar to their acquired traits and<br />
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opportunity to exhibit spontaneously congenital variations similar to their acquired traits and<br />
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have these variations naturally selected. It would look as though the acquired traits had sunk<br />
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have these variations naturally selected. It would look as though the acquired traits had sunk<br />
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into the hereditary substance in a Lamarckian fashion, but the process would really be [[Neo-Darwinism|neo-Darwinian]].<br />
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into the hereditary substance in a Lamarckian fashion, but the process would really be neo-Darwinian.<br />
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如果动物处于新环境或其原来所处的环境急剧变化，那些能够通过学习新行为或自身基因适应来灵活作出反应的动物会自然地留存下来。这些幸存者，经过几代之后，有机会自发地表现出与他们获得的特征相似的先天变异，并让其经自然选择而留存。看上去获得性特征以拉马克的方式留在了遗传物质上，但这个过程实际上是'''<font color="#ff8000">新达尔文主义 Neo-Darwinism</font>'''的。<br />
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Selected offspring would tend to have an increased capacity for learning new skills rather than being confined to genetically coded, relatively fixed abilities. In effect, it places emphasis on the fact that the sustained behavior of a species or group can shape the evolution of that species. The "Baldwin effect" is better understood in [[evolutionary developmental biology]] literature as a scenario in which a character or [[biological trait|trait]] change occurring in an [[organism]] as a result of its interaction with its environment [[genetic assimilation|becomes gradually assimilated]] into its developmental genetic or epigenetic repertoire (Simpson, 1953; Newman, 2002). In the words of [[Daniel Dennett]],<ref name=dennett03/><br />
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Selected offspring would tend to have an increased capacity for learning new skills rather than being confined to genetically coded, relatively fixed abilities. In effect, it places emphasis on the fact that the sustained behavior of a species or group can shape the evolution of that species. The "Baldwin effect" is better understood in evolutionary developmental biology literature as a scenario in which a character or trait change occurring in an organism as a result of its interaction with its environment becomes gradually assimilated into its developmental genetic or epigenetic repertoire (Simpson, 1953; Newman, 2002). In the words of Daniel Dennett,<br />
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被选择留存的后代往往具有更强的学习新技能的能力，而不仅局限于基因编码的相对固定的能力。实际上，它强调以下事实：一个物种或群体的持续行为可以影响其进化。在'''<font color="#ff8000">演化发育生物学 Evolutionary developmental biology</font>'''的文献中，“鲍德温效应”得到了更好的理解——生物体由于与其环境的相互作用而发生的的特征或性状变化逐渐被同化到其发育遗传或表观遗传库中。用'''丹尼尔·丹尼特 Daniel Dennett'''的话来说,<br />
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Thanks to the Baldwin effect, species can be said to pretest the efficacy of particular different designs by phenotypic (individual) exploration of the space of nearby possibilities. If a particularly winning setting is thereby discovered, this discovery will ''create'' a new selection pressure: organisms that are closer in the adaptive landscape to that discovery will have a clear advantage over those more distant.<br />
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Thanks to the Baldwin effect, species can be said to pretest the efficacy of particular different designs by phenotypic (individual) exploration of the space of nearby possibilities. If a particularly winning setting is thereby discovered, this discovery will create a new selection pressure: organisms that are closer in the adaptive landscape to that discovery will have a clear advantage over those more distant.<br />
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由于鲍德温效应，可以说物种通过表型（个体）探索附近可能性的空间，预先测试了特定不同设计的功效。若由此发现一个特别适宜的环境，该发现将导致新的选择压力: 在适应环境中，更接近此发现的生物体将较其他生物体有明显的优势。<br />
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An update to the Baldwin effect was developed by [[Jean Piaget]], [[Paul Alfred Weiss|Paul Weiss]], and [[Conrad Waddington]] in the 1960s–1970s. This new version included an explicit role for the social in shaping subsequent natural change in humans (both evolutionary and developmental), with reference to alterations of selection pressures.<ref>{{cite journal | author = Burman J. T. | year = 2013 | title = Updating the Baldwin Effect: The biological levels behind Piaget's new theory | url = | journal = New Ideas in Psychology | volume = 31 | issue = 3| pages = 363–373 | doi = 10.1016/j.newideapsych.2012.07.003 }}</ref><br />
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An update to the Baldwin effect was developed by Jean Piaget, Paul Weiss, and Conrad Waddington in the 1960s–1970s. This new version included an explicit role for the social in shaping subsequent natural change in humans (both evolutionary and developmental), with reference to alterations of selection pressures.<br />
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1960-1970年代，'''让·皮亚杰 Jean Piaget'''、'''保罗·维斯 Paul Weiss'''和'''康拉德·华丁顿 Conrad Waddington'''修正了鲍德温效应。该新版本明确了社会在塑造人类随后的自然变化（包括进化和发展）中的作用，并提到了选择压力的改变。<br />
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As is to be expected from [[Stigler's law]], subsequent research shows that Baldwin was not the first to identify the process; [[Douglas Spalding]] mentioned it in 1873.<ref>Noble, R.; [[Denis Noble|Noble, D.]] (2017) Was the Watchmaker Blind? Or Was She One-Eyed? ''Biology'' 2017, 6(4), 47; doi:10.3390/biology6040047, quoting Bateson, P. The adaptability driver: Links between behaviour and evolution. Biol. Theory 2006, 1, 342–345.</ref><br />
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As is to be expected from Stigler's law, subsequent research shows that Baldwin was not the first to identify the process; Douglas Spalding mentioned it in 1873.<br />
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正如'''<font color="#ff8000">斯蒂格勒定律 Stigler's Law</font>'''所指，随后的研究表明，鲍德温并不是第一个发现这个过程的人; '''<font color="#ff8000">道格拉斯·斯波尔丁 Douglas Spalding</font>'''在1873年提到过这个过程。<br />
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==Examples 举例==<br />
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Suppose a species is threatened by a new [[predator]] and there is a behavior that makes it more difficult for the predator to kill individuals of the species. Individuals who learn the behavior more quickly will obviously be at an advantage. As time goes on, the ability to learn the behavior will improve (by genetic selection), and at some point it will seem to be an [[instinct]]. Baldwin gives the following case involving cooperation: "Animals may be kept alive let us say in a given environment by social cooperation only; these transmit this social type of variation to posterity; thus social adaptation sets the direction of physical phylogeny and physical heredity is determined in part by this factor" (Baldwin, 1896, p.&nbsp;553).<br />
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Suppose a species is threatened by a new predator and there is a behavior that makes it more difficult for the predator to kill individuals of the species. Individuals who learn the behavior more quickly will obviously be at an advantage. As time goes on, the ability to learn the behavior will improve (by genetic selection), and at some point it will seem to be an instinct. Baldwin gives the following case involving cooperation: "Animals may be kept alive let us say in a given environment by social cooperation only; these transmit this social type of variation to posterity; thus social adaptation sets the direction of physical phylogeny and physical heredity is determined in part by this factor" (Baldwin, 1896, p.&nbsp;553).<br />
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假设一个物种受到一个新的捕食者的威胁，并且有一种行为使得捕食者更加难以杀死该物种的个体。更快学会这种行为的个体显然会处于有利地位。随着时间的推移，学习行为的能力将得到提高(通过基因选择) ，在某种程度上，这似乎是一种本能。鲍德温给出了以下涉及合作的案例: “动物可以保持活着，让我们说，在一个特定的环境中，只有通过社会合作; 这些传递这种社会类型的变异给子孙后代; 因此，社会适应确定了物理系统发育和物理遗传的方向，部分由这个因素决定”(鲍德温，1896年，第553页)。<br />
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The appearance of lactose tolerance (i.e., [[lactase persistence]]) in human populations with a long tradition of raising domesticated animals for milk production has been suggested as another example.<ref>[https://epdf.pub/evolution-and-learning-the-baldwin-effect-reconsidered.html Evolution and Learning: The Baldwin Effect Reconsidered]</ref> This argument holds that a feedback loop operates whereby a dairy culture increases the selective advantage from this genetic trait, while the average population genotype increases the collective rewards of a dairy culture.<br />
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The appearance of lactose tolerance (i.e., lactase persistence) in human populations with a long tradition of raising domesticated animals for milk production has been suggested as another example. This argument holds that a feedback loop operates whereby a dairy culture increases the selective advantage from this genetic trait, while the average population genotype increases the collective rewards of a dairy culture.<br />
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另一个例子是，在人类中出现了乳糖耐受性(即乳糖分解酶持续性) ，而人类在饲养家畜产奶方面有着悠久的传统。这种观点认为，乳制品文化通过一个反馈循环增加了这种遗传特性的选择优势，而普通群体的基因型增加了乳制品文化的集体回报。<br />
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==Controversy and acceptance 争议和接受==<br />
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Initially Baldwin's ideas were not incompatible with the prevailing, but uncertain, ideas about the mechanism of transmission of hereditary information and at least two other biologists put forward very similar ideas in 1896.<ref>{{Cite journal |last=Morgan |first=C. L. |authorlink=C. Lloyd Morgan |year=1896 |title=On modification and variation |journal=Science|volume=4|issue=99 |pages=733–740 |doi=10.1126/science.4.99.733|pmid=17735249 |url=https://zenodo.org/record/1448227 }}</ref><ref>{{Cite journal |last=Osborne |first=H. F. |authorlink=Henry Fairfield Osborne |year=1896 |title=Ontogenic and phylogenic variation |journal=Science |volume=4|issue=100 |pages=786–789 |doi=10.1126/science.4.100.786|pmid=17734840 |url=https://zenodo.org/record/1448151 }}</ref> In 1901, [[Maurice Maeterlinck]] referred to behavioural adaptations to prevailing climates in different species of bees as ‘what had merely been an idea, therefore, and opposed to instinct, has thus by slow degrees become an instinctive habit’.<ref>{{Cite book |title=The Life of the Bee|last=Materlinck |first=Maurice |authorlink=Maurice Maeterlinck |publisher=Dodd, Mead and Co. |year=1901 |location=New York |pages=Chapter VII section 102}}</ref> The Baldwin effect theory subsequently became more controversial, with scholars being split between "Baldwin boosters" and "Baldwin skeptics".<ref name="depew03">Depew, David J. (2003), "Baldwin Boosters, Baldwin Skeptics" in: {{cite book |title=Evolution and learning: The Baldwin effect reconsidered |last=Weber |first=Bruce H. |first2=David J. |last2=Depew |year=2003 |publisher=MIT Press |location=Cambridge, MA |isbn=978-0-262-23229-6 |pages=3–31 |url=https://books.google.com/books?id=yBtRzBilw1MC&pg=PR9&dq=baldwin+effect#v=onepage&q&f=false }}</ref> The theory was first called the "Baldwin effect" by [[George Gaylord Simpson]] in 1953.<ref name="depew03" /> Simpson "admitted that the idea was theoretically consistent, that is, not inconsistent with the [[Modern synthesis (20th century)|modern synthesis]]",<ref name="depew03" /> but he doubted that the phenomenon occurred very often, or if so, could be proven to occur. In his discussion of the reception of the Baldwin-effect theory Simpson points out that the theory appears to provide a reconciliation between a neo-Darwinian and a [[Lamarckism|neo-Lamarckian]] approach and that "Mendelism and later genetic theory so conclusively ruled out the extreme neo-Lamarckian position that reconciliation came to seem unnecessary".<ref>{{cite journal |title=The Baldwin effect |authorlink=George Gaylord Simpson |author=Simpson, George Gaylord |year=1953 |journal=Evolution |volume=7 | issue=2 |pages=110–117 |doi=10.2307/2405746|jstor=2405746 }}</ref> In 1942, the evolutionary biologist [[Julian Huxley]] promoted the Baldwin effect as part of the modern synthesis, saying the concept had been unduly neglected by evolutionists.<ref name=Julian1942>{{cite book | authorlink=Julian Huxley |author=Huxley, Julian |year=1942 |location=London |title=Evolution: The Modern Synthesis |publisher=George Allen & Unwin Ltd|title-link=Evolution: The Modern Synthesis }}</ref><br />
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Initially Baldwin's ideas were not incompatible with the prevailing, but uncertain, ideas about the mechanism of transmission of hereditary information and at least two other biologists put forward very similar ideas in 1896. In 1901, Maurice Maeterlinck referred to behavioural adaptations to prevailing climates in different species of bees as ‘what had merely been an idea, therefore, and opposed to instinct, has thus by slow degrees become an instinctive habit’. The Baldwin effect theory subsequently became more controversial, with scholars being split between "Baldwin boosters" and "Baldwin skeptics". The theory was first called the "Baldwin effect" by George Gaylord Simpson in 1953. In 1942, the evolutionary biologist Julian Huxley promoted the Baldwin effect as part of the modern synthesis, saying the concept had been unduly neglected by evolutionists.<br />
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最初，鲍德温的观点与广泛流传但未加验证的关于遗传信息传递机制的观点并不矛盾，且在1896年至少有两位生物学家提出了非常相似的观点。1901年，'''<font color="#ff8000">莫里斯·梅特林克 Maurice Maeterlinck</font>'''提到不同种类的蜜蜂对当时气候的行为适应性，称其之前仅仅是一种想法，因此，与本能相反，慢慢地变成了一种本能的习惯。鲍德温效应理论随后变得更具争议性，学者们分为“鲍德温支持者”和“鲍德温怀疑论者”。1953年'''乔治·盖洛德·辛普森 George Gaylord Simpson'''首次命名该理论为“鲍德温效应”。1942年，进化生物学家'''朱利安·赫胥黎 Julian Huxley'''将鲍德温效应作为现代进化综论的一部分加以推广，称进化论者过度忽视了这一概念。<br />
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In the 1960s, the evolutionary biologist [[Ernst Mayr]] contended that the Baldwin effect theory was untenable because<br />
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In the 1960s, the evolutionary biologist Ernst Mayr contended that the Baldwin effect theory was untenable because<br />
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在20世纪60年代，进化生物学家'''恩斯特·迈尔'''称鲍德温效应理论是站不住脚的，因为<br />
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# the argument is stated in terms of the individual genotype, whereas what is really exposed to the selection pressure is a phenotypically and genetically variable population;<br />
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the argument is stated in terms of the individual genotype, whereas what is really exposed to the selection pressure is a phenotypically and genetically variable population;<br />
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这个论点是从个体基因型的角度来阐述的，而真正暴露在选择压力下的是表型和遗传可变的种群;<br />
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# it is not sufficiently emphasized that the degree of modification of the phenotype is in itself genetically controlled;<br />
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it is not sufficiently emphasized that the degree of modification of the phenotype is in itself genetically controlled;<br />
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没有充分强调表型的改变程度本身是受遗传控制的;<br />
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# it is assumed that phenotypic rigidity is selectively superior to phenotypic flexibility.<ref name=Mayr1963>{{cite book| authorlink=Ernst Mayr |author=Mayr, Ernst |year=1963 |location=Cambridge, MA |title=Animal Species and Evolution| url=https://archive.org/details/animalspeciesevo00mayr | url-access=registration |publisher=Harvard University Press|isbn=978-0-674-03750-2}}</ref><br />
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it is assumed that phenotypic rigidity is selectively superior to phenotypic flexibility.<br />
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假定表型刚性有选择性优于表型柔性。<br />
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In 1987 [[Geoffrey Hinton]] and Steven Nowlan demonstrated by computer simulation that learning can accelerate evolution, and they associated this with the Baldwin effect.<ref>{{cite journal |title=How learning can guide evolution |last=Hinton |first=Geoffrey E. |author-link=Geoffrey Hinton |last2=Nowlan |first2=Steven J. |year=1987 |journal=Complex Systems |volume=1 |pages=495–502}}</ref><ref>{{cite journal |title=When learning guides evolution|authorlink=John Maynard Smith |author=Maynard Smith, John |year=1987 |journal=Nature |volume=329 |issue=6142 |pages=761–762 |doi=10.1038/329761a0|pmid=3670381 }}</ref><ref name=Puentedura2003>{{cite book| author=Puentedura, Ruben R. |year=2003|location=Cambridge, MA |chapter=The Baldwin effect in the age of computation |title=Evolution and Learning: The Baldwin Effect Reconsidered |editor-last=Weber |editor-first=Bruce H. |editor-last2=Depew |editor-first2=David J.|publisher=MIT press |pages=219–234}}</ref><br />
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In 1987 Geoffrey Hinton and Steven Nowlan demonstrated by computer simulation that learning can accelerate evolution, and they associated this with the Baldwin effect.<br />
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1987年，'''杰弗里·辛顿 Geoffrey Hinton''' 和 '''史蒂芬·诺兰 Steven Nowlan''' 通过计算机模拟证明了学习可以加速进化，他们将此与鲍德温效应联系起来。<br />
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Paul Griffiths<ref name=Griffiths2003>{{cite book| author=Griffiths, Paul E. |year=2003|location=Cambridge, MA|chapter=Beyond the Baldwin effect: James Mark Baldwin’s ‘social heredity,’ epigenetic inheritance, and niche construction|title=Evolution and Learning: The Baldwin Effect Reconsidered|publisher=MIT press|pages=193–215|editor-last=Weber |editor-first=Bruce H. |editor-last2=Depew |editor-first2=David J.}}</ref> suggests two reasons for the continuing interest in the Baldwin effect. The first is the role ''mind'' is understood to play in the effect. The second is the connection between ''development'' and evolution in the effect. Baldwin’s account of how neurophysiological and conscious mental factors may contribute to the effect<ref>{{cite journal |title=Heredity and instinct|last=Baldwin |first=J. Mark |author-link=James Mark Baldwin|year=1896 |journal=Science |volume=3 |issue=64|pages=438–441, 558–561|doi=10.1126/science.3.64.438 |pmid=17780356|url=https://zenodo.org/record/1448082 }}</ref><ref>{{cite journal |title=Consciousness and evolution |last=Baldwin |first=J. Mark |author-link=James Mark Baldwin|year=1896 |journal=Psychological Review |volume=3 |issue=3|pages=300–309 |doi=10.1037/h0063996}}</ref><ref>{{cite journal |title=A new factor in evolution |last=Baldwin |first=J. Mark |author-link=James Mark Baldwin|year=1896 |journal=The American Naturalist |volume=30 |issue=354|pages=441–451, 536–553 |doi=10.1086/276408|url=https://semanticscholar.org/paper/0c1386d88cb54eb17c6a2745cd4bf15dbb3f2b09 }}</ref> brings into focus the question of the possible survival value of consciousness.<ref name=Lindahl2001>{{cite book| author=Lindahl, B. I. B. |year=2001 |location=Amsterdam |chapter=Consciousness, behavioural patterns and the direction of biological evolution: implications for the mind–brain problem |title=Dimensions of Conscious Experience |publisher=John Benjamins |pages=73–99 |editor-last=Pylkkänen |editor-first=Paavo |editor-last2=Vadén |editor-first2=Tere |isbn=978-90-272-5157-2}}</ref><br />
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Paul Griffiths suggests two reasons for the continuing interest in the Baldwin effect. The first is the role mind is understood to play in the effect. The second is the connection between development and evolution in the effect. Baldwin’s account of how neurophysiological and conscious mental factors may contribute to the effect brings into focus the question of the possible survival value of consciousness.<br />
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'''保罗·格里菲斯 Paul Griffiths'''指出了人们对鲍德温效应持续兴趣的两个原因。首先，我们可以理解心智在这种效果中所扮演的角色。二是其中发展与演变的关系。鲍德温关于神经生理学和意识心理因素如何促成这种效应的解释，使人们更加关注意识可能的存在价值问题。<br />
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[[File:Carpodacus mexicanus -Madison, Wisconsin, USA-8.jpg|thumb|The [[house finch]]'s colonisation of North America has provided empirical evidence of the Baldwin effect.<ref name=Badyaev/>]]<br />
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The [[house finch's colonisation of North America has provided empirical evidence of the Baldwin effect.]]<br />
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[图一：家朱雀在北美的定殖提供了鲍德温效应的经验证明。]<br />
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Still, observes David Depew, "it is striking that a rather diverse lot of contemporary evolutionary theorists, most of whom regard themselves as supporters of the Modern Synthesis, have of late become 'Baldwin boosters{{'"}}<ref name="depew03" /> These Baldwin boosters<br />
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Still, observes David Depew, "it is striking that a rather diverse lot of contemporary evolutionary theorists, most of whom regard themselves as supporters of the Modern Synthesis, have of late become 'Baldwin boosters These Baldwin boosters<br />
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块引号<br />
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are typically [[evolutionary psychologist]]s who are searching for scenarios in which a population can get itself by behavioral trial and error onto a "hard to find" part of the fitness landscape in which human brain, language, and mind can rapidly coevolve. They are searching for what [[Daniel Dennett]], himself a Baldwin booster, calls an "evolutionary crane," an instrument to do some heavy lifting fast.<ref name="depew03" /><br />
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are typically evolutionary psychologists who are searching for scenarios in which a population can get itself by behavioral trial and error onto a "hard to find" part of the fitness landscape in which human brain, language, and mind can rapidly coevolve. They are searching for what Daniel Dennett, himself a Baldwin booster, calls an "evolutionary crane," an instrument to do some heavy lifting fast.<br />
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不过，'''戴维•迪皮尤 David Depew'''评论道: “令人惊讶的是，很多当代进化论理论家（其中大多数人自认为是现代进化综论的支持者），最近都成为了‘鲍德温支持者’，这些人是典型的致力于寻找“很难找到的”，群体可以通过行为试错达到人脑、语言和思维快速共同进化的适应性图景之场景的'''<font color="#ff8000">进化心理学 Evolutionary Psychology</font>'''家。他们正在寻找被丹尼尔·丹尼特（鲍德温的支持者）所称的“进化起重机”（快速升高重物的工具）。<br />
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/ blockquote<br />
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According to [[Daniel Dennett]], recent work has rendered the Baldwin effect "no longer a controversial wrinkle in orthodox Darwinism".<ref name="dennett03">[[Daniel Dennett|Dennett, Daniel]] (2003), "The Baldwin Effect: a Crane, not a Skyhook" in: {{cite book |title=Evolution and learning: The Baldwin effect reconsidered |last=Weber |first=Bruce H. |first2=David J. |last2=Depew |year=2003 |publisher=MIT Press |location=Cambridge, MA |isbn=978-0-262-23229-6 |pages=69–106 |url=https://books.google.com/books?id=yBtRzBilw1MC&pg=PR9&dq=baldwin+effect#v=onepage&q&f=false }}</ref> Potential genetic mechanisms underlying the Baldwin effect have been proposed for the evolution of natural (genetically-determinant) antibodies.<ref>{{Cite journal|last=Anderson|first=Russell|date=1996|title=How the adaptive antibodies facilitate the evolution of natural antibodies|url=|journal=Immunology and Cell Biology|volume=74 | issue = 2 |pages=286–291|doi=10.1038/icb.1996.50|pmid=8799730}}</ref> In 2009, empirical evidence for the Baldwin effect was provided from the colonisation of North America by the [[house finch]].<ref name=Badyaev>{{cite journal |last1=Badyaev |first1=Alexander V. |title=Evolutionary significance of phenotypic accommodation in novel environments: an empirical test of the Baldwin effect |journal=Phil. Trans. R. Soc. Lond. B |date=March 2009 |volume=364 |issue=1520 |doi=10.1098/rstb.2008.0285 |pmid=19324617 |pages=1125–1141|pmc=2666683 }}</ref><br />
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According to Daniel Dennett, recent work has rendered the Baldwin effect "no longer a controversial wrinkle in orthodox Darwinism". Potential genetic mechanisms underlying the Baldwin effect have been proposed for the evolution of natural (genetically-determinant) antibodies. In 2009, empirical evidence for the Baldwin effect was provided from the colonisation of North America by the house finch.<br />
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丹尼尔•丹尼特表示，最近的研究工作已使鲍德温效应“在正统达尔文主义中不再是争议性问题”。科学家已提出鲍德温效应背后潜在的遗传机制用于自然（基因决定）抗体的进化。2009年，'''家朱雀 House Finch'''在北美的定殖为鲍德温效应提供了经验证明。<br />
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==Comparison with genetic assimilation 与遗传同化的区别==<br />
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{{further|Genetic assimilation}}<br />
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The Baldwin effect has been confused with, and sometimes conflated with, a different evolutionary theory also based on [[phenotypic plasticity]], [[C. H. Waddington]]'s [[genetic assimilation]]. The Baldwin effect includes genetic accommodation, of which one type is genetic assimilation.<ref>{{cite journal | last1=Crispo | first1=Erika | year=2007 | title=The Baldwin effect and genetic assimilation: revisiting two mechanisms of evolutionary change mediated by phenotypic plasticity | journal=Evolution | volume=61 | issue=11 | pages=2469–2479 | doi=10.1111/j.1558-5646.2007.00203.x| pmid=17714500 }}</ref><!--more needed--><br />
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The Baldwin effect has been confused with, and sometimes conflated with, a different evolutionary theory also based on phenotypic plasticity, C. H. Waddington's genetic assimilation. The Baldwin effect includes genetic accommodation, of which one type is genetic assimilation.<!--more needed--><br />
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鲍德温效应与另一种同样基于'''<font color="#ff8000">表型可塑性 Henotypic Plasticity</font>'''的进化理论——C.H.华丁顿的'''<font color="#ff8000">遗传同化 Genetic Assimilation</font>'''相混淆，有时甚至混为一谈。实际上，鲍德温效应包括遗传调节，其中一种类型是遗传同化。<br />
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==See also 参见==<br />
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* {{annotated link|Evolvability}}<br />
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* {{annotated link|Lamarckism}}<br />
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* {{annotated link|Maternal effect}}<br />
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* {{annotated link|Meme}}<br />
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* {{annotated link|Pangenesis}}<br />
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* {{annotated link|Weismann barrier}}<br />
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==Notes 注释==<br />
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{{reflist|30em}}<br />
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== References 参考资料==<br />
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* {{cite journal | last = Bateson |first=Patrick | year = 2004 | title = The Active Role of Behaviour in Evolution | url = | journal = Biology and Philosophy | volume = 19 | issue = 2| pages = 283–298 | doi=10.1023/b:biph.0000024468.12161.83}}<br />
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* {{cite journal | last1 = Baldwin | first1 = J. Mark | authorlink = James Mark Baldwin | year = 1896 | title = A New Factor in Evolution | url = http://www.brocku.ca/MeadProject/Baldwin/Baldwin_1896_h.html | journal = The American Naturalist | volume = 30 | issue = 354| pages = 441–451 | doi=10.1086/276408}}<br />
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* {{cite journal | last1 = Baldwin | first1 = J. Mark | authorlink = James Mark Baldwin | year = 1897 | title = Organic Selection | journal = Science | volume = 5 | issue = 121| pages = 634–636 | doi=10.1126/science.5.121.634| pmid = 17781159 | url = https://zenodo.org/record/1448253 }}<br />
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* {{cite book |last1=Cochran |first1=Gregory |last2=Harpending |first2=Henry |date=2009 |title=The 10,000 Year Explosion: How Civilization Accelerated Human Evolution |publisher=Basic Books |isbn=978-0-465-00221-4|title-link=The 10,000 Year Explosion }}<br />
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* {{cite journal | last = Hall |first=Brian K. | year = 2001 | title = Organic Selection: Proximate Environmental Effects on the Evolution of Morphology and Behaviour | doi = 10.1023/a:1006773408919 | journal = Biology and Philosophy | volume = 16 | issue = 2| pages = 215–237 }}<br />
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* {{cite journal|author=Newman, Stuart A |year=2002 |title=Putting Genes in their place |url=http://www.nymc.edu/sanewman/PDFs/Putting%20Genes%20in%20their%20Place.pdf |journal=Journal of Biosciences |volume=27 |issue= 2|pages=97–104 |doi=10.1007/bf02703765 |pmid=11937679 |url-status=dead |archiveurl=https://web.archive.org/web/20081217013308/http://www.nymc.edu/sanewman/PDFs/Putting%20Genes%20in%20their%20Place.pdf |archivedate=2008-12-17 }}<br />
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* {{cite journal | last1 = Osborn | first1 = Henry F. | authorlink = Henry Fairfield Osborn | year = 1896 | title = Ontogenic and Phylogenic Variation| journal = Science | volume = 4 | issue = 100| pages = 786–789 | doi=10.1126/science.4.100.786| pmid = 17734840 | url = https://zenodo.org/record/1448151 }}<br />
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* [[Robert J. Richards|Richards, Robert J.]]; ''Darwin and the Emergence of Evolutionary Theories of Mind and Behavior'', Chicago: The University of Chicago Press, 1987.<br />
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* {{cite journal | last = Simpson |first=G. Gaylord |authorlink=George Gaylord Simpson | year = 1953 | title = The Baldwin effect | url = | journal = Evolution | volume = 7 | issue = 2| pages = 110–117 | doi=10.2307/2405746|jstor=2405746 }}<br />
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* {{cite journal |last1=Sznajder |first1=B. |last2=Sabelis |first2=M. W. |last3=Egas |first3=M. |title=How Adaptive Learning Affects Evolution: Reviewing Theory on the Baldwin Effect |journal=Evolutionary Biology |date=September 2012|volume=39|issue=3 |pages=301–310 |doi=10.1007/s11692-011-9155-2 |pmid=22923852 |pmc=3423563}}<!--a review paper, a recent secondary source--><br />
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==External links 外部链接==<br />
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* [http://www.ecs.soton.ac.uk/~harnad/Hypermail/Cognition.Sociobiology.98/0101.html Baldwinian evolution]<br />
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* [https://web.archive.org/web/20060810005608/http://www2.create.human.nagoya-u.ac.jp/baldwin/ Bibliography]<br />
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[[Category:Evolutionary biology]]<br />
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Category:Evolutionary biology<br />
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分类: 进化生物学<br />
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[[Category:Selection]]<br />
<br />
Category:Selection<br />
<br />
类别: 选择<br />
<br />
<noinclude><br />
<br />
<small>This page was moved from [[wikipedia:en:Baldwin effect]]. Its edit history can be viewed at [[鲍德温效应/edithistory]]</small></noinclude><br />
<br />
[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%B8%9F%E7%BE%A4%E7%AE%97%E6%B3%95&diff=12056鸟群算法2020-08-11T11:53:56Z<p>Dorr：</p>
<hr />
<div>{{#seo:<br />
|keywords=鸟群算法,Craig Reynolds,计算机图形<br />
|description=涌现,分离,对齐,靠近<br />
}}<br />
已由Dorr进行初步翻译<br />
<br />
'''<font color="#ff8000">鸟群算法 Boids</font>'''是模拟鸟类群集行为的人工生命项目，由'''克雷格·雷诺兹 Craig Reynolds'''于1986年开发。该模型基于常用于计算机动画或计算机辅助设计的计算机三维几何。他关于这个主题的论文发表在1987年的 ACM SIGGRAPH（美国计算机协会计算机图形专业组组织的计算机图形学顶级年度会议）会议记录上。<ref>{{Cite book | last1=Reynolds| first1=Craig| title=Flocks, herds and schools: A distributed behavioral model.| year=1987 |journal=SIGGRAPH '87: Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques| publisher=Association for Computing Machinery | pages=25–34| isbn=978-0-89791-227-3| doi=10.1145/37401.37406| citeseerx=10.1.1.103.7187}}</ref>“ boid”是“ bird-oid object”的缩写，指一个类鸟对象。<ref>{{Cite journal| last1=Banks| first1=Alec| last2=Vincent| first2=Jonathan| last3=Anyakoha| first3=Chukwudi | title=A review of particle swarm optimization. Part I: background and development|date=July 2007| journal=Natural Computing| volume=6| issue=4| pages=467–484| doi=10.1007/s11047-007-9049-5| citeseerx=10.1.1.605.5879}}</ref>恰巧，“boid”也是纽约都市方言中“bird”的发音。<br />
<br />
<br />
==算法介绍==<br />
[[File:Rule_separation.gif|thumb|200px|分离]]<br />
[[File:Rule_cohesion.gif|thumb|200px|对齐]]<br />
[[File:Rule_alignment.gif|thumb|200px|靠近]]<br />
<br />
与大多数人工生命模拟一样，Boids 是[[涌现 Emergent]]行为的一个例子; 也就是说，Boids 的复杂性来自于遵循一系列简单规则的个体代理（这里是 Boids）的相互作用。在最简单的Bolds 世界中适用的规则如下，其描述了鸟群中的个体如何根据周边同伴的位置和速度移动:<br />
<br />
* '''<font color="#ff8000">分离 Separation</font>''': 移动以避开群体拥挤处<br />
<br />
* '''<font color="#ff8000">对齐 Alignment</font>''': 转向群体的平均航向<br />
<br />
* '''<font color="#ff8000">靠近 Cohesion</font>''': 朝群体的平均位置（质心）移动<br />
<br />
<br />
每个 boid 个体都可以得知整体的几何参数，但群体要求其只对其周围某个小邻近范围作出反应。该邻近范围由一个距离（从该个体的中心算起）和一个角度（从其飞行方向算起）决定。此范围外的同伴不予考虑。该临近范围可以认为是一个有限知觉的模型（就像浑水中的鱼一样）；但更恰当的想法可能是，其定义了鸟群影响了个体转向的区域范围。<br />
<br />
可以添加更复杂的规则，如避障和寻找目标。<br />
<br />
<br />
该基本模型自雷诺兹提出以来，已用多种方法扩展。例如，Delgado-Mata <ref>{{cite journal|first1= Carlos | last1= Delgado-Mata|first2= Jesus Ibanez | last2= Martinez|first3= Simon | last3= Bee|first4= Rocio | last4= Ruiz-Rodarte|first5= Ruth | last5= Aylett|title= On the use of Virtual Animals with Artificial Fear in Virtual Environments<br />
|journal= New Generation Computing|volume=25 |issue=2 |pages= 145–169|date=2007|doi= 10.1007/s00354-007-0009-5}}</ref>等人扩展基本模型以加入恐惧的影响。利过一种可用自由膨胀气体中的粒子模拟的信息素，动物间用嗅觉来传递情感。'''哈特曼 Hartman'''和'''贝内斯 Benes'''<ref>{{cite journal|first1= Christopher | last1= Hartman|first1= Christopher | last1= Hartman|first2= Bedr̆ich |last2= Benes̆|title= Autonomous boids|journal= Computer Animation and Virtual Worlds|volume=17 |issue=3–4 |pages= 199–206|date=July 2006|doi= 10.1002/cav.123}}</ref>为定位过程引入了一种补充力量，他们称之为领导权的更迭。这条决定了某只鸟成为领导者及试图逃跑的机会。<br />
<br />
<br />
Boids 的运动可以表现为混乱（分裂的群体和狂野的行为）或有序。意想不到的行为，比如群体分散和避开障碍后的团聚可以被认为是应急的。<br />
<br />
<br />
Boids 框架通常用于计算机图形学，提供鸟群和其他生物（如鱼群）的逼真表现。例如，在1998年的电子游戏《'''半条命 Half-Life'''》中，游戏结束时Xen中出现的类似鸟类的飞行生物就使用了该框架（游戏文件中命名为“ boid”）。<br />
<br />
<br />
Boids 模型可用于'''<font color="#ff8000">[[集群机器人]] Swarm Robotics</font>'''中简单的无人地面车辆（UGV）<ref>{{cite conference |title= Design and analysis of Group Escape Behavior for distributed autonomous mobile robots |first1= Hongkyu |last1= Min |first2= Zhidong |last2= Wang |year= 2011 |conference= IEEE International Conference on Robotics and Automation (ICRA)|doi= 10.1109/ICRA.2011.5980123 }}</ref> 或'''<font color="#ff8000">微型飞行器 Micro Aerial Vehicles</font>'''（MAV）<ref>{{cite conference |title= Swarms of micro aerial vehicles stabilized under a visual relative localization |first1= Martin |last1= Saska |first2= Vakula |last2= Jan |first3= Preucil |last3= Libor |year= 2014 |conference= IEEE International Conference on Robotics and Automation (ICRA)|doi= 10.1109/ICRA.2014.6907374 }}</ref>群体的直接控制和稳定。为了异质 UAV-UGV 群体的稳定性，Saska 等人将该模型用于板载相对定位。<ref>{{cite conference |title= Coordination and Navigation of Heterogeneous UAVs-UGVs Teams Localized by a Hawk-Eye Approach |first1= Martin |last1= Saska |first2= Vonasek |last2= Vojtech |first3= Krajnik |last3= Tomas |first4= Preucil |last4= Libor |year= 2012 |conference= IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)|url=http://labe.felk.cvut.cz/~tkrajnik/ardrone/articles/formace.pdf}}</ref><br />
<br />
<br />
在当时提出时，雷诺兹的方法相比于传统的计算机动画电影技术是一个巨大的进步。第一部利用了此模型的动画片是《'''史丹利和史黛拉: 破冰 Stanley and Stella in: Breaking The Ice'''》（1987），之后是'''蒂姆·伯顿 Tim Burton'''的故事片《'''蝙蝠侠归来 Batman Returns'''》（1992），电脑合成的蝙蝠群和成群的企鹅行进穿过'''哥谭市 Gotham City'''的街道。<ref>{{cite journal | url=http://lrss.fri.uni-lj.si/people/ilbajec/papers/ilb_ab09.pdf | last1=Lebar Bajec |first1= Iztok |first2= Frank H. |last2= Heppner | date=2009 | title=Organized flight in birds | journal=Animal Behaviour | volume = 78 | issue = 4 | pages = 777–789 | doi = 10.1016/j.anbehav.2009.07.007 }}</ref><br />
<br />
<br />
Boids 模型已经被用于其他有趣应用。该系统已应用于互联网多频道广播电台的自动节目编排，.<ref>{{cite conference| title= DJ-boids: emergent collective behavior as multichannel radio station programming| first1 = Jesús | last1 = Ibáñez| first2 = Antonio F. | last2 = Gómez-Skarmeta| first3 = Josep | last3 = Blat| date = 2003| booktitle = Proceedings of the 8th international conference on Intelligent User Interfaces| pages = 248–250| doi= 10.1145/604045.604089}}</ref>以及可视化信息<ref>{{cite conference| title= Time-Varying Data Visualization Using Information Flocking Boids| last = Moere | first = A V| date= 2004| booktitle = Proceedings of the IEEE Symposium on Information Visualization| pages= 97–104| doi= 10.1109/INFVIS.2004.65}}</ref>和优化任务。<ref>{{cite journal|first1 = Zhihua | last1 = Cui|first2 = Zhongzhi | last2 = Shi|title= Boid particle swarm optimisation|journal= International Journal of Innovative Computing and Applications|volume=2 |issue=2 |pages= 77–85|date=2009|doi= 10.1504/IJICA.2009.031778}}</ref><br />
<br />
==参见==<br />
<br />
*[[群体智能 Swarm intelligence]] <br />
<br />
*[[集体细胞迁移 Collective cell migration]]<br />
<br />
==参考文献==<br />
<br />
{{Reflist}}<br />
<br />
<br />
<br />
==外部链接==<br />
<br />
* [http://www.red3d.com/cwr/boids/ Craig Reynolds' Boids page]<br />
<br />
* [http://www.vergenet.net/~conrad/boids/pseudocode.html Explanation of algorithm in pseudocode]<br />
<br />
* [https://gpolo.github.com/birdflocking JavaScript implementation]<br />
<br />
* [https://lufemas.github.io/boid-ai-implementation/ JavaScript implementation with Phaser Framework]<br />
<br />
* [https://web.archive.org/web/20080205014305/http://www.navgen.com/3d_boids/ 3D Boids Simulation using OpenGL, used by the BBC's Natural History Unit]<br />
<br />
* [https://black-square.github.io/BirdFlock/ Live In-Browser 3D Simulation of Bird Flocking Behavior in Unity3D] – Open Source implementation for Windows, Linux and Mac<br />
<br />
* [https://github.com/piskvorky/PredatorPrey UNIX+Windows open source implementation in C++, using OpenGL and simulation controls]<br />
<br />
* [https://github.com/tofti/javafx-boids A java implementation using the javafx API]<br />
<br />
==编者推荐==<br />
===集智相关课程===<br />
[[File:Auklet_flock_Shumagins_1986.jpg|300px|thumb|[https://swarma.org/?p=16504 张江：乌合之众还是群智涌现？鸟群知道答案]]]<br />
====[https://campus.swarma.org/course/1104 透过人工鸟群Boid模型学习List的使用]====<br />
<br />
讲师：[[张江]](北师大系统科学学院教授、博士生导师，集智俱乐部、集智学园创始人。)<br />
<br />
本课程通过数个案例教会大家如何去动手搭建一个多主体仿真模型，以及如何利用NetLogo去实现。从生命游戏到人工鸟群，从模拟经济系统到病毒沿网络的传播，通过循序渐进的案例，该课程带你逐步走入NetLogo多主体建模的神奇世界。<br />
<br />
<br />
<br />
===集智相关文章===<br />
====[https://swarma.org/?p=16504 张江：乌合之众还是群智涌现？鸟群知道答案]====<br />
<br />
生命如何起源，智能如何涌现，因果如何反转，复杂谜题，引人深思。2019年7月13日，集智俱乐部创始人[[张江]]教授在混沌大学授课，详细解读了复杂系统中的混沌、涌现与进化，此文整理自课堂内容。<br />
<br />
<br />
<br />
===其他===<br />
====[https://v.qq.com/x/page/z13491sn7fy.html 动画片《Stanley and Stella in: Breaking the Ice (1987)》]====<br />
<br />
Craig Reynolds 与 Symbolics Graphics 和 Whitney / Demos Production 的同事合作制作的利用 boids 模型的小短片，讲述了飞鸟与鱼相爱，鸟儿冲破冰面与鱼相见的故事。影片最早于87年在 SIGGRAPH 电子剧院展出。<br />
<br />
<br />
<br />
----<br />
本中文词条由[[用户:Dorr|Dorr]]参与编译，[[用户:木子二月鸟|木子二月鸟]]审校，[[用户:薄荷|薄荷]]编辑，欢迎在讨论页面留言。<br />
<br />
'''本词条内容源自wikipedia及公开资料，遵守 CC3.0协议。'''<br />
<br />
<br />
<br />
[[Category:人工生命]]<br />
[[Category:1986软件]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%B8%9F%E7%BE%A4%E7%AE%97%E6%B3%95&diff=12055鸟群算法2020-08-11T11:52:56Z<p>Dorr：</p>
<hr />
<div>{{#seo:<br />
|keywords=鸟群算法,Craig Reynolds,计算机图形<br />
|description=涌现,分离,对齐,靠近<br />
}}<br />
已由Dorr进行初步翻译<br />
<br />
'''<font color="#ff8000">鸟群算法 Boids</font>'''是模拟鸟类群集行为的人工生命项目，由'''克雷格·雷诺兹 Craig Reynolds'''于1986年开发。该模型基于常用于计算机动画或计算机辅助设计的计算机三维几何。他关于这个主题的论文发表在1987年的 ACM SIGGRAPH（美国计算机协会计算机图形专业组组织的计算机图形学顶级年度会议）会议记录上。<ref>{{Cite book | last1=Reynolds| first1=Craig| title=Flocks, herds and schools: A distributed behavioral model.| year=1987 |journal=SIGGRAPH '87: Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques| publisher=Association for Computing Machinery | pages=25–34| isbn=978-0-89791-227-3| doi=10.1145/37401.37406| citeseerx=10.1.1.103.7187}}</ref>“ boid”是“ bird-oid object”的缩写，指一个类鸟对象。<ref>{{Cite journal| last1=Banks| first1=Alec| last2=Vincent| first2=Jonathan| last3=Anyakoha| first3=Chukwudi | title=A review of particle swarm optimization. Part I: background and development|date=July 2007| journal=Natural Computing| volume=6| issue=4| pages=467–484| doi=10.1007/s11047-007-9049-5| citeseerx=10.1.1.605.5879}}</ref>恰巧，“boid”也是纽约都市方言中“bird”的发音。<br />
<br />
<br />
==算法介绍==<br />
[[File:Rule_separation.gif|thumb|200px|分离]]<br />
[[File:Rule_cohesion.gif|thumb|200px|对齐]]<br />
[[File:Rule_alignment.gif|thumb|200px|靠近]]<br />
<br />
与大多数人工生命模拟一样，Boids 是[[涌现 Emergent]]行为的一个例子; 也就是说，Boids 的复杂性来自于遵循一系列简单规则的个体代理（这里是 Boids）的相互作用。在最简单的Bolds 世界中适用的规则如下，其描述了鸟群中的个体如何根据周边同伴的位置和速度移动:<br />
<br />
* '''<font color="#ff8000">分离 Separation</font>''': 移动以避开群体拥挤处<br />
<br />
* '''<font color="#ff8000">对齐 Alignment</font>''': 转向群体的平均航向<br />
<br />
* '''<font color="#ff8000">靠近 Cohesion</font>''': 朝群体的平均位置（质心）移动<br />
<br />
<br />
每个 boid 个体都可以得知整体的几何参数，但群体要求其只对其周围某个小邻近范围作出反应。该邻近范围由一个距离（从该个体的中心算起）和一个角度（从其飞行方向算起）决定。此范围外的同伴不予考虑。该临近范围可以认为是一个有限知觉的模型（就像浑水中的鱼一样）；但更恰当的想法可能是，其定义了鸟群影响了个体转向的区域范围。<br />
<br />
可以添加更复杂的规则，如避障和寻找目标。<br />
<br />
<br />
该基本模型自雷诺兹提出以来，已用多种方法扩展。例如，Delgado-Mata <ref>{{cite journal|first1= Carlos | last1= Delgado-Mata|first2= Jesus Ibanez | last2= Martinez|first3= Simon | last3= Bee|first4= Rocio | last4= Ruiz-Rodarte|first5= Ruth | last5= Aylett|title= On the use of Virtual Animals with Artificial Fear in Virtual Environments<br />
|journal= New Generation Computing|volume=25 |issue=2 |pages= 145–169|date=2007|doi= 10.1007/s00354-007-0009-5}}</ref>等人扩展基本模型以加入恐惧的影响。利过一种可用自由膨胀气体中的粒子模拟的信息素，动物间用嗅觉来传递情感。'''哈特曼 Hartman'''和'''贝内斯 Benes'''<ref>{{cite journal|first1= Christopher | last1= Hartman|first1= Christopher | last1= Hartman|first2= Bedr̆ich |last2= Benes̆|title= Autonomous boids|journal= Computer Animation and Virtual Worlds|volume=17 |issue=3–4 |pages= 199–206|date=July 2006|doi= 10.1002/cav.123}}</ref>为定位过程引入了一种补充力量，他们称之为领导权的更迭。这条决定了某只鸟成为领导者及试图逃跑的机会。<br />
<br />
<br />
Boids 的运动可以表现为混乱（分裂的群体和狂野的行为）或有序。意想不到的行为，比如群体分散和避开障碍后的团聚可以被认为是应急的。<br />
<br />
<br />
Boids 框架通常用于计算机图形学，提供鸟群和其他生物（如鱼群）的逼真表现。例如，在1998年的电子游戏《'''<font color="#ff8000">半条命 Half-Life</font>'''》中，游戏结束时Xen中出现的类似鸟类的飞行生物就使用了该框架（游戏文件中命名为“ boid”）。<br />
<br />
<br />
Boids 模型可用于'''<font color="#ff8000">[[集群机器人]] Swarm Robotics</font>'''中简单的无人地面车辆（UGV）<ref>{{cite conference |title= Design and analysis of Group Escape Behavior for distributed autonomous mobile robots |first1= Hongkyu |last1= Min |first2= Zhidong |last2= Wang |year= 2011 |conference= IEEE International Conference on Robotics and Automation (ICRA)|doi= 10.1109/ICRA.2011.5980123 }}</ref> 或'''<font color="#ff8000">微型飞行器 Micro Aerial Vehicles</font>'''（MAV）<ref>{{cite conference |title= Swarms of micro aerial vehicles stabilized under a visual relative localization |first1= Martin |last1= Saska |first2= Vakula |last2= Jan |first3= Preucil |last3= Libor |year= 2014 |conference= IEEE International Conference on Robotics and Automation (ICRA)|doi= 10.1109/ICRA.2014.6907374 }}</ref>群体的直接控制和稳定。为了异质 UAV-UGV 群体的稳定性，Saska 等人将该模型用于板载相对定位。<ref>{{cite conference |title= Coordination and Navigation of Heterogeneous UAVs-UGVs Teams Localized by a Hawk-Eye Approach |first1= Martin |last1= Saska |first2= Vonasek |last2= Vojtech |first3= Krajnik |last3= Tomas |first4= Preucil |last4= Libor |year= 2012 |conference= IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)|url=http://labe.felk.cvut.cz/~tkrajnik/ardrone/articles/formace.pdf}}</ref><br />
<br />
<br />
在当时提出时，雷诺兹的方法相比于传统的计算机动画电影技术是一个巨大的进步。第一部利用了此模型的动画片是《'''史丹利和史黛拉: 破冰 Stanley and Stella in: Breaking The Ice'''》（1987），之后是'''蒂姆·伯顿 Tim Burton'''的故事片《'''蝙蝠侠归来 Batman Returns'''》（1992），电脑合成的蝙蝠群和成群的企鹅行进穿过'''哥谭市 Gotham City'''的街道。<ref>{{cite journal | url=http://lrss.fri.uni-lj.si/people/ilbajec/papers/ilb_ab09.pdf | last1=Lebar Bajec |first1= Iztok |first2= Frank H. |last2= Heppner | date=2009 | title=Organized flight in birds | journal=Animal Behaviour | volume = 78 | issue = 4 | pages = 777–789 | doi = 10.1016/j.anbehav.2009.07.007 }}</ref><br />
<br />
<br />
Boids 模型已经被用于其他有趣应用。该系统已应用于互联网多频道广播电台的自动节目编排，.<ref>{{cite conference| title= DJ-boids: emergent collective behavior as multichannel radio station programming| first1 = Jesús | last1 = Ibáñez| first2 = Antonio F. | last2 = Gómez-Skarmeta| first3 = Josep | last3 = Blat| date = 2003| booktitle = Proceedings of the 8th international conference on Intelligent User Interfaces| pages = 248–250| doi= 10.1145/604045.604089}}</ref>以及可视化信息<ref>{{cite conference| title= Time-Varying Data Visualization Using Information Flocking Boids| last = Moere | first = A V| date= 2004| booktitle = Proceedings of the IEEE Symposium on Information Visualization| pages= 97–104| doi= 10.1109/INFVIS.2004.65}}</ref>和优化任务。<ref>{{cite journal|first1 = Zhihua | last1 = Cui|first2 = Zhongzhi | last2 = Shi|title= Boid particle swarm optimisation|journal= International Journal of Innovative Computing and Applications|volume=2 |issue=2 |pages= 77–85|date=2009|doi= 10.1504/IJICA.2009.031778}}</ref><br />
<br />
==参见==<br />
<br />
*[[群体智能 Swarm intelligence]] <br />
<br />
*[[集体细胞迁移 Collective cell migration]]<br />
<br />
==参考文献==<br />
<br />
{{Reflist}}<br />
<br />
<br />
<br />
==外部链接==<br />
<br />
* [http://www.red3d.com/cwr/boids/ Craig Reynolds' Boids page]<br />
<br />
* [http://www.vergenet.net/~conrad/boids/pseudocode.html Explanation of algorithm in pseudocode]<br />
<br />
* [https://gpolo.github.com/birdflocking JavaScript implementation]<br />
<br />
* [https://lufemas.github.io/boid-ai-implementation/ JavaScript implementation with Phaser Framework]<br />
<br />
* [https://web.archive.org/web/20080205014305/http://www.navgen.com/3d_boids/ 3D Boids Simulation using OpenGL, used by the BBC's Natural History Unit]<br />
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* [https://black-square.github.io/BirdFlock/ Live In-Browser 3D Simulation of Bird Flocking Behavior in Unity3D] – Open Source implementation for Windows, Linux and Mac<br />
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* [https://github.com/piskvorky/PredatorPrey UNIX+Windows open source implementation in C++, using OpenGL and simulation controls]<br />
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* [https://github.com/tofti/javafx-boids A java implementation using the javafx API]<br />
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==编者推荐==<br />
===集智相关课程===<br />
[[File:Auklet_flock_Shumagins_1986.jpg|300px|thumb|[https://swarma.org/?p=16504 张江：乌合之众还是群智涌现？鸟群知道答案]]]<br />
====[https://campus.swarma.org/course/1104 透过人工鸟群Boid模型学习List的使用]====<br />
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讲师：[[张江]](北师大系统科学学院教授、博士生导师，集智俱乐部、集智学园创始人。)<br />
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本课程通过数个案例教会大家如何去动手搭建一个多主体仿真模型，以及如何利用NetLogo去实现。从生命游戏到人工鸟群，从模拟经济系统到病毒沿网络的传播，通过循序渐进的案例，该课程带你逐步走入NetLogo多主体建模的神奇世界。<br />
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===集智相关文章===<br />
====[https://swarma.org/?p=16504 张江：乌合之众还是群智涌现？鸟群知道答案]====<br />
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生命如何起源，智能如何涌现，因果如何反转，复杂谜题，引人深思。2019年7月13日，集智俱乐部创始人[[张江]]教授在混沌大学授课，详细解读了复杂系统中的混沌、涌现与进化，此文整理自课堂内容。<br />
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===其他===<br />
====[https://v.qq.com/x/page/z13491sn7fy.html 动画片《Stanley and Stella in: Breaking the Ice (1987)》]====<br />
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Craig Reynolds 与 Symbolics Graphics 和 Whitney / Demos Production 的同事合作制作的利用 boids 模型的小短片，讲述了飞鸟与鱼相爱，鸟儿冲破冰面与鱼相见的故事。影片最早于87年在 SIGGRAPH 电子剧院展出。<br />
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本中文词条由[[用户:Dorr|Dorr]]参与编译，[[用户:木子二月鸟|木子二月鸟]]审校，[[用户:薄荷|薄荷]]编辑，欢迎在讨论页面留言。<br />
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'''本词条内容源自wikipedia及公开资料，遵守 CC3.0协议。'''<br />
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[[Category:人工生命]]<br />
[[Category:1986软件]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E5%9F%BA%E4%BA%8E%E4%B8%BB%E4%BD%93%E7%9A%84%E8%AE%A1%E7%AE%97%E7%BB%8F%E6%B5%8E%E5%AD%A6&diff=12054基于主体的计算经济学2020-08-11T11:47:57Z<p>Dorr：</p>
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<div>已由Dorr进行初步翻译<br />
此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。<br />
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{{Use dmy dates|date=February 2020}}<br />
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'''Agent-based computational economics''' ('''ACE''') is the area of [[computational economics]] that studies economic processes, including whole [[economy|economies]], as [[dynamic system]]s of interacting [[Agent (economics)|agents]]. As such, it falls in the [[paradigm]] of [[complex adaptive system]]s.<ref>• [[W. Brian Arthur]], 1994. "[https://ocw.tudelft.nl/wp-content/uploads/ElFarolArtur1994.pdf Inductive Reasoning and Bounded Rationality]," ''American Economic Review'', 84(2), pp. [http://www-personal.umich.edu/~samoore/bit885f2011/arthur-inductive.pdf 406-411] {{Webarchive|url=https://web.archive.org/web/20130521145936/http://www-personal.umich.edu/~samoore/bit885f2011/arthur-inductive.pdf |date=21 May 2013 }}.<br/>&nbsp;&nbsp; • [[Leigh Tesfatsion]], 2003. "Agent-based Computational Economics: Modeling Economies as Complex Adaptive Systems," ''Information Sciences'', 149(4), pp. [http://copper.math.buffalo.edu/urgewiki/uploads/Literature/Tesfatsion2002.pdf 262-268] {{webarchive|url=https://web.archive.org/web/20120426000037/http://copper.math.buffalo.edu/urgewiki/uploads/Literature/Tesfatsion2002.pdf |date=26 April 2012 }}.</ref> In corresponding [[agent-based model]]s, the "[[agent (economics)|agents]]" are "computational objects modeled as interacting according to rules" over space and time, not real people. The rules are formulated to model behavior and social interactions based on incentives and information.<ref>Scott E. Page (2008). "agent-based models," ''[[The New Palgrave Dictionary of Economics]]'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_A000218&edition=current&q=agent-based%20computational%20modeling&topicid=&result_number=1 Abstract].</ref> Such rules could also be the result of optimization, realized through use of AI methods (such as [[Q-learning]] and other reinforcement learning techniques).<ref>Richard S. Sutton and Andrew G. Barto, Reinforcement Learning: An Introduction, The MIT Press, Cambridge, MA, 1998 [http://www.cs.ualberta.ca/~sutton/book/ebook/the-book.html] {{Webarchive|url=https://web.archive.org/web/20090904194934/http://www.cs.ualberta.ca/~sutton/book/ebook/the-book.html |date=4 September 2009 }}</ref><br />
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Agent-based computational economics (ACE) is the area of computational economics that studies economic processes, including whole economies, as dynamic systems of interacting agents. As such, it falls in the paradigm of complex adaptive systems. In corresponding agent-based models, the "agents" are "computational objects modeled as interacting according to rules" over space and time, not real people. The rules are formulated to model behavior and social interactions based on incentives and information. Such rules could also be the result of optimization, realized through use of AI methods (such as Q-learning and other reinforcement learning techniques).<br />
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'''<font color="#ff8000">基于主体的计算经济学 Agent-based Computational Economics (ACE)</font>'''是[[计算经济学]]的一个领域，将整个经济视作不同主体相互作用的动态系统研究经济过程。因此，它属于复杂适应系统'''<font color="#ff8000">范式 Paradigm</font>'''。在相应的'''<font color="#ff8000">智能体基模型 Agent-based Model(ABM)</font>'''中，“主体”指时空中“根据规则建模的交互计算对象”，而非真实的人。这些规则是为了建立基于激励和信息的行为和社会互动模型而制定的。这些规则也可能是优化的结果，通过人工智能方法（如'''<font color="#ff8000">Q学习 Q-learning</font>'''及其他强化学习技术）来实现。<br />
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The theoretical assumption of [[mathematical optimization]] by agents in [[equilibrium (economics)|equilibrium]] is replaced by the less restrictive postulate of agents with [[bounded rationality]] ''adapting'' to market forces.<ref>• [[John H. Holland]] and John H. Miller (1991). "Artificial Adaptive Agents in Economic Theory," ''American Economic Review'', 81(2), pp. [http://www.santafe.edu/media/workingpapers/91-05-025.pdf 365-370] {{Webarchive|url=https://web.archive.org/web/20110105015853/http://www.santafe.edu/media/workingpapers/91-05-025.pdf |date=5 January 2011 }} p. 366.<br/>&nbsp;&nbsp; • [[Thomas C. Schelling]] (1978 [2006]). ''Micromotives and Macrobehavior'', Norton. [http://books.wwnorton.com/books/978-0-393-32946-9/ Description] {{Webarchive|url=https://web.archive.org/web/20171102093240/http://books.wwnorton.com/books/978-0-393-32946-9/ |date=2 November 2017 }}, [https://books.google.com/books?id=DenWKRgqzWMC&printsec=find&pg=PA1=#v=onepage&q&f=false preview].<br/>&nbsp;&nbsp; •<br />
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The theoretical assumption of mathematical optimization by agents in equilibrium is replaced by the less restrictive postulate of agents with bounded rationality adapting to market forces.&nbsp;&nbsp; • Thomas C. Schelling (1978 [2006]). Micromotives and Macrobehavior, Norton. [http://books.wwnorton.com/books/978-0-393-32946-9/ Description] , [https://books.google.com/books?id=DenWKRgqzWMC&printsec=find&pg=PA1=#v=onepage&q&f=false preview].<br/>&nbsp;&nbsp; •<br />
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•托马斯 · c · 谢林(Thomas c. Schelling，1978[2006])。微动机和宏行为，诺顿。[ http://books.wwnorton.com/books/978-0-393-32946-9/ 描述] ，[ https://books.google.com/books?id=denwkrgqzwmc&printsec=find&pg=pa1=#v=onepage&q&f=false 预览] . br / •<br />
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[[Thomas J. Sargent]], 1994. ''Bounded Rationality in Macroeconomics'', Oxford. [http://www.oup.com/us/catalog/general/subject/Economics/MacroeconomicTheory/?view=usa&ci=9780198288695 Description] and chapter-preview 1st-page [https://www.questia.com/library/book/bounded-rationality-in-macroeconomics-thomas-j-sargent-by-thomas-j-sargent.jsp links.]</ref> ACE models apply [[numerical methods]] of analysis to [[Computer simulation|computer-based simulations]] of complex dynamic problems for which more conventional methods, such as theorem formulation, may not find ready use.<ref>• Kenneth L. Judd, 2006. "Computationally Intensive Analyses in Economics," ''Handbook of Computational Economics'', v. 2, ch. 17, Introduction, p. 883. [Pp. [https://books.google.com/books?id=6ITfRkNmKQcC&pg=PA881 881-] 893. Pre-pub [http://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF].<br />
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Thomas J. Sargent, 1994. Bounded Rationality in Macroeconomics, Oxford. [http://www.oup.com/us/catalog/general/subject/Economics/MacroeconomicTheory/?view=usa&ci=9780198288695 Description] and chapter-preview 1st-page [https://www.questia.com/library/book/bounded-rationality-in-macroeconomics-thomas-j-sargent-by-thomas-j-sargent.jsp links.]</ref> ACE models apply numerical methods of analysis to computer-based simulations of complex dynamic problems for which more conventional methods, such as theorem formulation, may not find ready use.<ref>• Kenneth L. Judd, 2006. "Computationally Intensive Analyses in Economics," Handbook of Computational Economics, v. 2, ch. 17, Introduction, p. 883. [Pp. [https://books.google.com/books?id=6ITfRkNmKQcC&pg=PA881 881-] 893. Pre-pub [http://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF].<br />
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托马斯·萨金特，1994年。牛津大学宏观经济学有限理性。[ http://www.oup.com/us/catalog/general/subject/economics/macroeconomictheory/?view=usa&ci=9780198288695说明]和第一章预览页面[ https://www.questia.com/library/book/bounded-rationality-in-macroeconomics-thomas-j-sargent-by-thomas-j-sargent.jsp 链接]。] / ref <br />
。 文献• Kenneth l. Judd，2006。《经济学中的计算密集分析》 ，《计算经济学手册，第2卷，第5页。17，Introduction，p. 883.[ Pp.Https://books.google.com/books?id=6itfrknmkqcc&pg=pa881881-893.预订酒吧[ http://www2.econ.iastate.edu/tesfatsi/judd.finalrev.PDF PDF ]。<br />
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<br/>&nbsp;&nbsp; • _____, 1998. ''Numerical Methods in Economics'', MIT Press. Links to [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=3257 description] {{webarchive|url=https://web.archive.org/web/20120211061602/http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=3257 |date=11 February 2012 }} and [https://books.google.com/books?id=9Wxk_z9HskAC&pg=PR7 chapter previews].</ref> Starting from initial conditions specified by the modeler, the computational economy evolves over time as its constituent agents repeatedly interact with each other, including learning from interactions. In these respects, ACE has been characterized as a bottom-up culture-dish approach to the study of [[economic systems]].<ref>• Leigh Tesfatsion (2002). "Agent-Based Computational Economics: Growing Economies from the Bottom Up," ''Artificial Life'', 8(1), pp.55-82. [http://www.mitpressjournals.org/doi/abs/10.1162/106454602753694765 Abstract] and pre-pub [http://www.econ.brown.edu/fac/Peter_Howitt/SummerSchool/Agent.pdf PDF] {{webarchive|url=https://web.archive.org/web/20130514143904/http://www.econ.brown.edu/fac/Peter_Howitt/SummerSchool/Agent.pdf |date=14 May 2013 }}.<br/>&nbsp;&nbsp; • _____ (1997). "How Economists Can Get Alife," in W. B. Arthur, S. Durlauf, and D. Lane, eds., ''The Economy as an Evolving Complex System, II'', pp. 533-564. Addison-Wesley. Pre-pub [http://ageconsearch.umn.edu/bitstream/18196/1/er37.pdf PDF].</ref><br />
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<br/>&nbsp;&nbsp; • _____, 1998. Numerical Methods in Economics, MIT Press. Links to [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=3257 description] and [https://books.google.com/books?id=9Wxk_z9HskAC&pg=PR7 chapter previews].</ref> Starting from initial conditions specified by the modeler, the computational economy evolves over time as its constituent agents repeatedly interact with each other, including learning from interactions. In these respects, ACE has been characterized as a bottom-up culture-dish approach to the study of economic systems.<br />
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<br/>&nbsp;&nbsp; • _____, 1998.经济学数值方法，麻省理工学院出版社。链接到[ http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=3257说明]和[ https://books.google.com/books?id=9wxk_z9hskac&pg=pr7章预览]。<br />
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'''<font color="#ff8000">经济均衡 Economic Equilibrium</font>'''中主体'''<font color="#ff8000">最优化 Mathematical Optimization</font>'''的理论假设被限制较少的适应市场力量的'''<font color="#ff8000">有限理性 Bounded Rationality</font>'''主体假设所取代。ACE 模型将数值分析方法应用于复杂动力学问题的计算机模拟，对其更传统的方法，如定理公式可能无法方便地使用。从建模者指定的初始条件开始，计算经济随着时间的推移而发展，其组成主体不断地相互作用，包括从交互中学习。在这些方面，ACE 被认为是研究'''<font color="#ff8000">经济体系 Economic System</font>'''的一种自下而上的文化方法。<br />
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ACE has a similarity to, and overlap with, [[game theory]] as an agent-based method for modeling social interactions.<ref name="COMP&GT">• [[Joseph Y. Halpern]] (2008). "computer science and game theory," ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_C000566&edition=current&q=&topicid=&result_number=1 Abstract].<br/>&nbsp;&nbsp; • Yoav Shoham (2008). "Computer Science and Game Theory," ''Communications of the ACM'', 51(8), pp.<br />
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ACE has a similarity to, and overlap with, game theory as an agent-based method for modeling social interactions.&nbsp;&nbsp; • Yoav Shoham (2008). "Computer Science and Game Theory," Communications of the ACM, 51(8), pp.<br />
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• Yoav Shoham (2008)。《计算机科学与博弈论》 ，美国计算机学会通讯，51(8)页。<br />
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[http://www.robotics.stanford.edu/~shoham/www%20papers/CSGT-CACM-Shoham.pdf 75-79] {{Webarchive|url=https://web.archive.org/web/20120426005917/http://www.robotics.stanford.edu/~shoham/www%20papers/CSGT-CACM-Shoham.pdf |date=26 April 2012 }}.<br/>&nbsp;&nbsp; • [[Alvin E. Roth]] (2002). "The Economist as Engineer: Game Theory, Experimentation, and Computation as Tools for Design Economics," ''Econometrica'', 70(4), pp. [https://web.archive.org/web/20040414102216/http://kuznets.fas.harvard.edu/~aroth/papers/engineer.pdf 1341–1378].</ref> But practitioners have also noted differences from standard methods, for example in ACE events modeled being driven solely by initial conditions, whether or not equilibria exist or are computationally tractable, and in the modeling facilitation of agent autonomy and learning.<ref>Tesfatsion, Leigh (2006), "Agent-Based Computational Economics: A Constructive Approach to Economic Theory," ch. 16, ''Handbook of Computational Economics'', v. 2, part 2, ACE study of economic system. [http://www.sciencedirect.com/science/article/pii/S1574002105020162 Abstract] and pre-pub [http://econ2.econ.iastate.edu/tesfatsi/hbintlt.pdf PDF].</ref><br />
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[http://www.robotics.stanford.edu/~shoham/www%20papers/CSGT-CACM-Shoham.pdf 75-79] .<br/>&nbsp;&nbsp; • Alvin E. Roth (2002). "The Economist as Engineer: Game Theory, Experimentation, and Computation as Tools for Design Economics," Econometrica, 70(4), pp. [https://web.archive.org/web/20040414102216/http://kuznets.fas.harvard.edu/~aroth/papers/engineer.pdf 1341–1378].</ref> But practitioners have also noted differences from standard methods, for example in ACE events modeled being driven solely by initial conditions, whether or not equilibria exist or are computationally tractable, and in the modeling facilitation of agent autonomy and learning.<br />
<br />
[http://www.robotics.stanford.edu/~shoham/www%20papers/CSGT-CACM-Shoham.pdf 75-79] .<br/>&nbsp;&nbsp; • Alvin E. Roth (2002).《作为工程师的经济学家: 博弈论、实验和计算作为设计经济学的工具》 ，《经济学》 ，70(4) ，页。[ https://web.archive.org/web/20040414102216/http://kuznets.fas.harvard.edu/~aroth/papers/engineer.pdf 1341-1378]. <br />
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ACE 与博弈论（作为基于主体的社会互动建模方法时）有相似之处和重叠部分。但实践者也注意到了与标准方法的区别，例如，在仅由初始条件驱动的ACE事件建模中，均衡是否存在或者在计算上是否易于处理，以及在建模过程中促进主体自主性和学习方面。<br />
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The method has benefited from continuing improvements in modeling techniques of [[computer science]] and increased computer capabilities. The ultimate scientific objective of the method is to "test theoretical findings against real-world data in ways that permit empirically supported theories to cumulate over time, with each researcher’s work building appropriately on the work that has gone before."<ref>• Leigh Tesfatsion (2006). "Agent-Based Computational Economics: A Constructive Approach to Economic Theory," ch. 16, ''Handbook of Computational Economics'', v. 2, [pp. 831-880] sect. 5. [http://www.sciencedirect.com/science/article/pii/S1574002105020162 Abstract] and pre-pub [http://econ2.econ.iastate.edu/tesfatsi/hbintlt.pdf PDF].<br/>&nbsp;&nbsp; • [[Kenneth L. Judd]] (2006). "Computationally Intensive Analyses in Economics," ''Handbook of Computational Economics'', v. 2, ch. 17, pp. [https://books.google.com/books?id=6ITfRkNmKQcC&pg=PA881 881-] 893. Pre-pub [http://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF].<br/>&nbsp;&nbsp; • Leigh Tesfatsion and Kenneth L. Judd, ed. (2006). ''Handbook of Computational Economics'', v. 2. [http://www.elsevier.com/wps/find/bookdescription.cws_home/660847/description#description Description] {{Webarchive|url=https://web.archive.org/web/20120306100156/http://www.elsevier.com/wps/find/bookdescription.cws_home/660847/description#description |date=6 March 2012 }} & and chapter-preview<br />
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The method has benefited from continuing improvements in modeling techniques of computer science and increased computer capabilities. The ultimate scientific objective of the method is to "test theoretical findings against real-world data in ways that permit empirically supported theories to cumulate over time, with each researcher’s work building appropriately on the work that has gone before."&nbsp;&nbsp; • Kenneth L. Judd (2006). "Computationally Intensive Analyses in Economics," Handbook of Computational Economics, v. 2, ch. 17, pp. [https://books.google.com/books?id=6ITfRkNmKQcC&pg=PA881 881-] 893. Pre-pub [http://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF].<br/>&nbsp;&nbsp; • Leigh Tesfatsion and Kenneth L. Judd, ed. (2006). Handbook of Computational Economics, v. 2. [http://www.elsevier.com/wps/find/bookdescription.cws_home/660847/description#description Description] & and chapter-preview<br />
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• Kenneth l. Judd (2006)。《经济学中的计算密集分析》 ，计算经济学手册，第2卷，第5页。17页。Https://books.google.com/books?id=6itfrknmkqcc&pg=pa881881-893.Pre-pub [http://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF].<br/>&nbsp;&nbsp; • Leigh Tesfatsion and Kenneth L. Judd, ed.(2006).计算经济学手册第2节。[ http://www.elsevier.com/wps/find/bookdescription.cws_home/660847/Description#Description 说明]和章节预览<br />
<br />
[http://www.sciencedirect.com/science?_ob=PublicationURL&_hubEid=1-s2.0-S1574002105X02003&_cid=273377&_pubType=HS&_auth=y&_acct=C000228598&_version=1&_urlVersion=0&_userid=10&md5=e4757b4f65755ed6340a11fee9615200 links.]</ref> The subject has been applied to research areas like [[asset pricing]],<ref name=arthuretal>B. Arthur, J. Holland, B. LeBaron, R. Palmer, P. Taylor (1997), 'Asset pricing under endogenous expectations in an artificial stock market,' in ''The Economy as an Evolving Complex System II'', B. Arthur, S. Durlauf, and D. Lane, eds., Addison Wesley.</ref> [[competition]] and [[collaboration]],<ref>[[Robert Axelrod]] (1997). ''The Complexity of Cooperation: Agent-Based Models of Competition and Collaboration'', Princeton. [http://press.princeton.edu/titles/6144.html Description], [http://press.princeton.edu/titles/6144.html#TOC contents], and [https://books.google.com/books?id=J0dgRGMdjmQC&printsec=find&pg=PR11#v=onepage&q&f=false preview].</ref> [[transaction cost]]s,<ref>Tomas B. Klosa and [[Bart Nooteboom]], 2001. "Agent-based Computational Transaction Cost Economics," ''Journal of Economic Dynamics and Control'' 25(3–4), pp. 503–52. [http://www.sciencedirect.com/science/article/pii/S0165188900000348 Abstract.]</ref> [[market structure]] and [[industrial organization]] and dynamics,<ref>• Roberto Leombruni and Matteo Richiardi, ed. (2004), ''Industry and Labor Dynamics: The Agent-Based Computational Economics Approach.'' World Scientific Publishing {{ISBN|981-256-100-5}}. [http://www.worldscibooks.com/economics/5706.html Description] {{Webarchive|url=https://web.archive.org/web/20100727221149/http://www.worldscibooks.com/economics/5706.html |date=27 July 2010 }} and chapter-preview [https://books.google.com/books?id=P5O7A5D55nQC&printsec=fond&pg=PR5#v=onepage&q&f=false links].<br/>&nbsp;&nbsp; • [[Joshua M. Epstein]] (2006). "Growing Adaptive Organizations: An Agent-Based Computational Approach," in ''Generative Social Science: Studies in Agent-Based Computational Modeling'', pp. [https://books.google.com/books?id=543OS3qdxBYC&pg=PA309 309-] 344. [http://press.princeton.edu/titles/8277.html Description] {{Webarchive|url=https://web.archive.org/web/20120126180655/http://press.princeton.edu/titles/8277.html |date=26 January 2012 }} and [http://www.santafe.edu/research/working-papers/abstract/99895b6465e8b87656612f8e3570b34c/ abstract].</ref> [[welfare economics]],<ref>[[Robert Axtell]] (2005). "The Complexity of Exchange," ''Economic Journal'', 115(504, Features), pp. [http://econfaculty.gmu.edu/pboettke/workshop/archives/f05/Axtell.pdf F193-F210].</ref> and [[mechanism design]],<ref>• ''The New Palgrave Dictionary of Economics'' (2008), 2nd Edition: <br/>&nbsp;&nbsp;&nbsp;&nbsp; [[Roger B. Myerson]] "mechanism design." [http://www.dictionaryofeconomics.com/article?id=pde2008_M000132&edition=current&q=mechanism%20design&topicid=&result_number=3 Abstract.] <br/>&nbsp;&nbsp;&nbsp;&nbsp; _____. "revelation principle." [http://www.dictionaryofeconomics.com/article?id=pde2008_R000137&edition=current&q=moral&topicid=&result_number=1 Abstract.]<br/>&nbsp;&nbsp;&nbsp;&nbsp; Tuomas Sandholm. "computing in mechanism design." [http://www.dictionaryofeconomics.com/article?id=pde2008_C000563&edition=&field=keyword&q=algorithmic%20mechanism%20design&topicid=&result_number=1 Abstract.]<br/>&nbsp;&nbsp; • [[Noam Nisan]] and Amir Ronen (2001). "Algorithmic Mechanism Design," ''Games and Economic Behavior'', 35(1-2), pp. [https://www.cs.cmu.edu/~sandholm/cs15-892F09/Algorithmic%20mechanism%20design.pdf 166–196].<br/>&nbsp;&nbsp; • [[Noam Nisan]] ''et al''., ed. (2007). ''Algorithmic Game Theory'', Cambridge University Press. [http://www.cup.cam.ac.uk/asia/catalogue/catalogue.asp?isbn=9780521872829 Description] {{Webarchive|url=https://web.archive.org/web/20120505140924/http://www.cup.cam.ac.uk/asia/catalogue/catalogue.asp?isbn=9780521872829 |date=5 May 2012 }}.</ref> [[Information economics|information and uncertainty]],<ref>Tuomas W. Sandholm and Victor R. Lesser (2001). "Leveled Commitment Contracts and Strategic Breach," ''Games and Economic Behavior'', 35(1-2), pp. [https://www.cs.cmu.edu/afs/.cs.cmu.edu/Web/People/sandholm/leveled.geb.pdf 212-270].</ref> [[macroeconomics]],<ref>• [[David Colander]], [[Peter Howitt]], Alan Kirman, [[Axel Leijonhufvud]], and [[Perry Mehrling]], 2008. "Beyond DSGE Models: Toward an Empirically Based Macroeconomics," ''American Economic Review'', 98(2), pp. [https://www.jstor.org/pss/29730026 236]-240. Pre-pub [http://www.brown.edu/Departments/Economics/Faculty/Peter_Howitt/publication/complex%20macro6.pdf PDF].<br/>&nbsp;&nbsp; • [[Thomas J. Sargent]] (1994). ''Bounded Rationality in Macroeconomics'', Oxford. [http://www.oup.com/us/catalog/general/subject/Economics/MacroeconomicTheory/?view=usa&ci=9780198288695 Description] and chapter-preview 1st-page [https://www.questia.com/library/book/bounded-rationality-in-macroeconomics-thomas-j-sargent-by-thomas-j-sargent.jsp links].<br/>&nbsp;&nbsp; • M. Oeffner (2009). '[http://www.opus-bayern.de/uni-wuerzburg/volltexte/2009/3927/pdf/OeffnerDissohneAnhang.pdf Agent-based Keynesian Macroeconomics]'. PhD thesis, Faculty of Economics, University of Würzburg.</ref> and [[Marxist economics]].<ref>A. F. Cottrell, P. Cockshott, G. J. Michaelson, I. P. Wright, V. Yakovenko<br />
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[http://www.sciencedirect.com/science?_ob=PublicationURL&_hubEid=1-s2.0-S1574002105X02003&_cid=273377&_pubType=HS&_auth=y&_acct=C000228598&_version=1&_urlVersion=0&_userid=10&md5=e4757b4f65755ed6340a11fee9615200 links.]</ref> The subject has been applied to research areas like asset pricing, competition and collaboration, transaction costs, market structure and industrial organization and dynamics, welfare economics, and mechanism design, information and uncertainty, macroeconomics, and Marxist economics.<ref>A. F. Cottrell, P. Cockshott, G. J. Michaelson, I. P. Wright, V. Yakovenko<br />
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[ http://www.sciencedirect.com/science?_ob=publicationurl&_hubeid=1-s2.0-s1574002105x02003&_cid=273377&_pubtype=hs&_auth=y&_acct=c000228598&_version=1&_urlversion=0&_userid=10&md5=e4757b4f65755ed6340a11fee9615200连结。参考文献<br />
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参考文献 a。科特雷尔，p. Cockshott，g. j. Michaelson，i. p. Wright，v. Yakovenko<br />
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(2009), ''Classical Econophysics''. Routledge, {{ISBN|978-0-415-47848-9}}.</ref><ref>Leigh Tesfatsion (2006), "Agent-Based Computational Economics: A Constructive Approach to Economic Theory," ch. 16, ''Handbook of Computational Economics'', v. 2, part 2, ACE study of economic system. [http://www.sciencedirect.com/science/article/pii/S1574002105020162 Abstract] and pre-pub [http://econ2.econ.iastate.edu/tesfatsi/hbintlt.pdf PDF].</ref><br />
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(2009), Classical Econophysics. Routledge, .</ref><br />
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(2009) ，古典经济物理学。路特利奇，参考文献<br />
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该方法得益于计算机科学建模技术的不断改进和计算机能力的提高。其最终科学目标是“用实际数据来检验理论发现——以允许经验支持的理论随着时间的推移而累积的方式（每位研究者的工作恰当建立在之前工作的基础之上）。”其现已应用于如资产定价、竞争与合作、'''<font color="#ff8000">交易成本 Transaction Cost</font>'''、市场形式与'''<font color="#ff8000">产业组织 Industrial Organization</font>'''与动态、'''<font color="#ff8000">福利经济学 Welfare Economics</font>'''、'''<font color="#ff8000">机制设计 Mechanism Design</font>'''、信息与不确定性、'''<font color="#ff8000">宏观经济学 Macroeconomics</font>'''和'''<font color="#ff8000">马克思主义政治经济学 Marxist economics</font>'''等研究领域。<br />
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==Overview 概述==<br />
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The "[[Agent (economics)|agents]]" in ACE models can represent individuals (e.g. people), social groupings (e.g. firms), biological entities (e.g. growing crops), and/or physical systems (e.g. transport systems). The ACE modeler provides the initial configuration of a computational economic system comprising multiple interacting agents. The modeler then steps back to observe the development of the system over time without further intervention. In particular, system events should be driven by agent interactions without external imposition of equilibrium conditions.<ref>[http://www.socsci.aau.dk/ae2006/ Summary of methods] {{Webarchive|url=https://web.archive.org/web/20070526105550/http://www.socsci.aau.dk/ae2006/ |date=26 May 2007 }}: ''Department of Economics, Politics and Public Administration, Aalborg University, Denmark'' website.</ref> Issues include those common to [[experimental economics]] in general<ref>[[Vernon L. Smith]], 2008. "experimental economics," ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_E000277&q=experimental%20&topicid=&result_number=2 Abstract].</ref> and development of a common framework for empirical validation and resolving open questions in agent-based modeling.<ref>Giorgio Fagiolo, Alessio Moneta, and Paul Windrum, 2007. "A Critical Guide to Empirical Validation of Agent-Based Models in Economics: Methodologies, Procedures, and Open Problems," ''Computational Economics'', 30, pp. [https://doi.org/10.1007%2Fs10614-007-9104-4 195]–226.</ref><br />
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The "agents" in ACE models can represent individuals (e.g. people), social groupings (e.g. firms), biological entities (e.g. growing crops), and/or physical systems (e.g. transport systems). The ACE modeler provides the initial configuration of a computational economic system comprising multiple interacting agents. The modeler then steps back to observe the development of the system over time without further intervention. In particular, system events should be driven by agent interactions without external imposition of equilibrium conditions. Issues include those common to experimental economics in general and development of a common framework for empirical validation and resolving open questions in agent-based modeling.<br />
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ACE 模型中的“主体”可以代表个体（如人）、社会群体（如公司）、生物实体（如生长中的庄稼）和/或物理系统（如运输系统）。ACE 模型提供了由多个相互作用的主体组成的计算经济系统的初始配置。然后，建模人员返回来观察系统随着时间的推移而发展的情况，而不需要进一步的干预。注意，系统活动应由主体间的相互作用驱动，而不需要外部强加的平衡条件。研究问题包括'''<font color="#ff8000">实验经济学 Experimental economics</font>'''通用的问题，以及开发用于经验验证的通用框架和解决基于主体的建模中尚未解决的问题。<br />
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ACE is an officially designated special interest group (SIG) of the Society for Computational Economics.<ref>[http://comp-econ.org/ Society for Computational Economics] website.</ref> Researchers at the [[Santa Fe Institute]] have contributed to the development of ACE.<br />
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ACE is an officially designated special interest group (SIG) of the Society for Computational Economics. Researchers at the Santa Fe Institute have contributed to the development of ACE.<br />
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ACE 是美国计算经济学学会的一个官方指定的特殊利益集团 (SIG) 。[[圣菲研究所 Santa Fe Institute]]的研究人员为 ACE 的发展做出了贡献。<br />
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==Example: finance 举例：金融==<br />
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One area where ACE methodology has frequently been applied is asset pricing. [[W. Brian Arthur]], Eric Baum, [[William A. Brock (economist)|William Brock]], Cars Hommes, and Blake LeBaron, among others, have developed computational models in which many agents choose from a set of possible forecasting strategies in order to predict stock prices, which affects their asset demands and thus affects stock prices. These models assume that agents are more likely to choose forecasting strategies which have recently been successful. The success of any strategy will depend on market conditions and also on the set of strategies that are currently being used. These models frequently find that large booms and busts in asset prices may occur as agents switch across forecasting strategies.<ref name=arthuretal/><ref>W. Brock and C. Hommes (1997), 'A rational route to randomness.' ''Econometrica'' 65 (5), pp. 1059-1095.</ref><ref>C. Hommes (2008), 'Interacting agents in finance,' in ''The New Palgrave Dictionary of Economics''.</ref> More recently, Brock, Hommes, and Wagener (2009) have used a model of this type to argue that the introduction of new hedging instruments may destabilize the market,<ref>{{cite journal |first=W. |last=Brock |first2=C. |last2=Hommes |first3=F. |last3=Wagener |year=2009 |title=More hedging instruments may destabilize markets |journal=Journal of Economic Dynamics and Control |volume=33 |issue=11 |pages=1912–1928 |doi=10.1016/j.jedc.2009.05.004 |url=http://cendef.uva.nl/binaries/content/assets/subsites/amsterdam-school-of-economics/amsterdam-school-of-economics-research-institute/cendef/working-papers-2006/brohomwag.pdf?1417181713282 }}</ref> and some papers have suggested that ACE might be a useful methodology for understanding the recent [[financial crisis]].<ref>M. Buchanan (2009), '[http://pagesperso-orange.fr/mark.buchanan/nature_economic_modelling.pdf Meltdown modelling. Could agent-based computer models prevent another financial crisis?].' Nature, Vol. 460, No. 7256. (5 August 2009), pp. 680-682.</ref><ref>J.D. Farmer, D. Foley (2009), 'The economy needs agent-based modelling.' Nature, Vol. 460, No. 7256. (5 August 2009), pp. 685-686.</ref><ref>M. Holcombe, S. Coakley, M.Kiran, S. Chin, C. Greenough, D.Worth, S.Cincotti, M.Raberto, A. Teglio, C. Deissenberg, S. van der Hoog, H. Dawid, S. Gemkow, P. Harting, M. Neugart. Large-scale Modeling of Economic Systems, Complex Systems, 22(2), 175-191, 2013</ref><br />
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One area where ACE methodology has frequently been applied is asset pricing. W. Brian Arthur, Eric Baum, William Brock, Cars Hommes, and Blake LeBaron, among others, have developed computational models in which many agents choose from a set of possible forecasting strategies in order to predict stock prices, which affects their asset demands and thus affects stock prices. These models assume that agents are more likely to choose forecasting strategies which have recently been successful. The success of any strategy will depend on market conditions and also on the set of strategies that are currently being used. These models frequently find that large booms and busts in asset prices may occur as agents switch across forecasting strategies. More recently, Brock, Hommes, and Wagener (2009) have used a model of this type to argue that the introduction of new hedging instruments may destabilize the market, and some papers have suggested that ACE might be a useful methodology for understanding the recent financial crisis.<br />
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ACE 方法已广泛应用的领域之一是资产定价。'''布莱恩·亚瑟 W. Brian Arthur'''，'''埃里克·鲍姆 Eric Baum'''，'''威廉·布洛克 William Brock'''，Cars Hommes，'''布莱克·勒巴朗 Blake LeBaron'''等人已开发出一个计算模型，其中许多主体从一组可能的预测策略中选择以预测股票价格，这影响了他们的资产需求，进而影响股票价格。这些模型假设主体更有可能选择最近成功的预测策略。任何策略的成功取决于市场条件，也取决于目前正在使用的一组策略。这些模型经常表明，随着主体改换预测策略，资产价格可能会出现大起大落。最近，布洛克、Hommes 和'''瓦格纳 Wagener''' （2009）使用了这类模型来论证引入新的对冲工具可能会破坏市场稳定，一些论文提出，ACE 可能是理解最近金融危机的有用方法。<br />
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==See also 参见==<br />
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* [[ACEGES]] 基于主体的全球能源系统的计算经济学<br />
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* [[Agent-based social simulation]] 基于主体的社会仿真<br />
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* [[Artificial economics]] 人工经济学<br />
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* [[Computational economics]] 计算经济学<br />
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* [[Econophysics]] 经济物理学<br />
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* [[Macroeconomic model]] 宏观经济模型<br />
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* [[Multi-agent system]] 多智能体系统<br />
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* [[Statistical finance]] 统计金融<br />
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==References 参考资料==<br />
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{{Reflist}}<br />
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==编者推荐==<br />
[https://campus.swarma.org/ 集智学园]课程：[https://campus.swarma.org/course/714 计算社会经济学的内容与方法]<br />
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本课程系统性地介绍计算社会经济学的主要研究内容和分析方法，阐述大规模社会经济数据（包括卫星遥感数据、手机通讯数据、社交媒体数据等）在解决全球贫困问题、刻画区域经济结构、推断个体财富和应急抢险救灾中的具体应用。<br />
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[[Category:Computational economics]]<br />
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Category:Computational economics<br />
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类别: 计算经济学<br />
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[[Category:Monte Carlo methods in finance]]<br />
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Category:Monte Carlo methods in finance<br />
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类别: 金融中的蒙特卡罗方法<br />
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[[Category:Computational fields of study]]<br />
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Category:Computational fields of study<br />
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类别: 研究的计算领域<br />
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<small>This page was moved from [[wikipedia:en:Agent-based computational economics]]. Its edit history can be viewed at [[基于主体的计算经济学/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E7%A4%BE%E4%BC%9A%E7%BD%91%E7%BB%9C&diff=11911社会网络2020-08-10T15:08:50Z<p>Dorr：</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Social structure made up of a set of social actors}}<br />
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{{about|the theoretical concept as used in the social and behavioral sciences|social networking sites|Social networking service|the 2010 movie|The Social Network|other uses|Social network (disambiguation)}}<br />
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A '''social network ''' is a [[social structure]] made up of a set of [[social]] actors (such as individuals or organizations), sets of [[Dyad (sociology)|dyadic]] ties, and other [[Social relation|social interactions]] between actors. The social network perspective provides a set of methods for analyzing the structure of whole social entities as well as a variety of theories explaining the patterns observed in these structures.<ref name="WF94CH1">{{cite book|last1=Wasserman |first1=Stanley | author-link1= Stanley Wasserman|last2=Faust |first2=Katherine |year=1994 |title=Social Network Analysis: Methods and Applications |isbn=9780521387071 |chapter=Social Network Analysis in the Social and Behavioral Sciences |pages=1–27 |publisher=Cambridge University Press}}</ref> The study of these structures uses [[social network analysis]] to identify local and global patterns, locate influential entities, and examine network dynamics.<br />
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A social network is a social structure made up of a set of social actors (such as individuals or organizations), sets of dyadic ties, and other social interactions between actors. The social network perspective provides a set of methods for analyzing the structure of whole social entities as well as a variety of theories explaining the patterns observed in these structures. The study of these structures uses social network analysis to identify local and global patterns, locate influential entities, and examine network dynamics.<br />
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'''<font color="#FFD700">社会网络 Social Network</font>'''是一种由一组社会行为者（个人或组织）、一组二元关系以及行为者之间的其他社会互动组成的社会结构。社会网络视角为分析整个社会实体的结构提供了一套方法，也为解释在这些结构中观察到的模式提供了各种理论。对这些结构的研究使用'''<font color="#FFD700">社会网络分析 Social Network Analysis</font>'''以确定本地及全球模式，定位有影响力的实体，并审查网络动态。<br />
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Social networks and the analysis of them is an inherently [[Interdisciplinarity|interdisciplinary]] academic field which emerged from [[social psychology]], [[sociology]], [[statistics]], and [[graph theory]]. [[Georg Simmel]] authored early structural theories in sociology emphasizing the dynamics of triads and "web of group affiliations".<ref>{{cite book|last1=Scott |first1=W. Richard |last2=Davis |first2=Gerald F. |author-link1= William Richard Scott|author-link2= Gerald F. Davis|title=Organizations and Organizing |chapter=Networks In and Around Organizations |year=2003 |isbn=978-0-13-195893-7 |publisher=Pearson Prentice Hall}}</ref> [[Jacob Moreno]] is credited with developing the first [[sociogram]]s in the 1930s to study interpersonal relationships. These approaches were mathematically formalized in the 1950s and theories and methods of social networks became pervasive in the [[Social science|social and behavioral sciences]] by the 1980s.<ref name="WF94CH1"/><ref name="Freeman History">{{cite book|last=Freeman |first=Linton |year=2004 |publisher=Empirical Press |isbn=978-1-59457-714-7 |title=The Development of Social Network Analysis: A Study in the Sociology of Science}}</ref> [[Social network analysis]] is now one of the major paradigms in contemporary sociology, and is also employed in a number of other social and formal sciences. Together with other [[complex network]]s, it forms part of the nascent field of [[network science]].<ref>{{cite journal|journal=Science |year=2009 |volume=323 |number=5916 |pages=892–895 |doi=10.1126/science.1165821 |pmid=19213908 |title=Network Analysis in the Social Sciences |first1=Stephen P. |last1=Borgatti |first2=Ajay |last2=Mehra |first3=Daniel J. |last3=Brass |first4=Giuseppe |last4=Labianca|bibcode=2009Sci...323..892B |citeseerx=10.1.1.536.5568 }}</ref><ref name="EK">{{cite book|title=Networks, Crowds, and Markets: Reasoning about a Highly Connected World |first1=David |last1=Easley |first2=Jon |last2=Kleinberg |publisher=Cambridge University Press |year=2010 |chapter=Overview |pages=1–20 |isbn=978-0-521-19533-1}}</ref><br />
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Social networks and the analysis of them is an inherently interdisciplinary academic field which emerged from social psychology, sociology, statistics, and graph theory. Georg Simmel authored early structural theories in sociology emphasizing the dynamics of triads and "web of group affiliations". Jacob Moreno is credited with developing the first sociograms in the 1930s to study interpersonal relationships. These approaches were mathematically formalized in the 1950s and theories and methods of social networks became pervasive in the social and behavioral sciences by the 1980s. Social network analysis is now one of the major paradigms in contemporary sociology, and is also employed in a number of other social and formal sciences. Together with other complex networks, it forms part of the nascent field of network science.<br />
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社会网络及其分析是社会心理学、社会学、统计学和图论中内在的跨学科学术领域。'''格奥尔格·齐美尔 Georg Simmel'''是早期社会学结构主义理论的作者，他强调三合会和“群体关系网”的动态性。'''雅各布·莫雷诺 Jacob Moreno'''被认为是在20世纪30年代发展了第一份'''<font color="#FFD700">社交关系图 Sociogram</font>'''以研究人际关系的人。这些方法在20世纪50年代得到数学化，到20世纪80年代，社会网络的理论和方法在社会和行为科学中变得普遍。社会网络分析现在是当代社会学的主要范式之一，也被用于许多其他社会科学及形式科学。它与其他复杂网络一起构成了网络科学新兴领域的一部分。<br />
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==Overview 概览==<br />
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[[File:Barabasi Albert model.gif|thumb|left|Evolution graph of a social network: [[Barabási–Albert model|Barabási model]].图一：一个社会网络的演化图：巴拉巴西模型。]]<br />
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The social network is a [[Scientific theory|theoretical]] [[Construct (philosophy of science)|construct]] useful in the [[social sciences]] to study relationships between individuals, [[social groups|groups]], [[formal organizations|organizations]], or even entire [[society|societies]] ([[social unit]]s, see [[Differentiation (sociology)|differentiation]]). The term is used to describe a [[social structure]] determined by such [[social interactions|interactions]]. The ties through which any given social unit connects represent the convergence of the various social contacts of that unit. This theoretical approach is, necessarily, relational. An [[axiom]] of the social network approach to understanding [[social interaction]] is that social phenomena should be primarily conceived and investigated through the properties of relations between and within units, instead of the properties of these units themselves. Thus, one common criticism of social network theory is that [[Agency (sociology)|individual agency]] is often ignored<ref name="jscott">Scott, John P. (2000). ''Social Network Analysis: A Handbook'' (2nd edition). Thousand Oaks, CA: Sage Publications.</ref> although this may not be the case in practice (see [[agent-based model]]ing). Precisely because many different types of relations, singular or in combination, form these network configurations, [[Network science|network analytics]] are useful to a broad range of research enterprises. In social science, these fields of study include, but are not limited to [[anthropology]], [[biology]], [[communication studies]], [[economics]], [[geography]], [[information science]], [[organizational studies]], [[social psychology]], [[sociology]], and [[sociolinguistics]].<br />
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The social network is a theoretical construct useful in the social sciences to study relationships between individuals, groups, organizations, or even entire societies (social units, see differentiation). The term is used to describe a social structure determined by such interactions. The ties through which any given social unit connects represent the convergence of the various social contacts of that unit. This theoretical approach is, necessarily, relational. An axiom of the social network approach to understanding social interaction is that social phenomena should be primarily conceived and investigated through the properties of relations between and within units, instead of the properties of these units themselves. Thus, one common criticism of social network theory is that individual agency is often ignored although this may not be the case in practice (see agent-based modeling). Precisely because many different types of relations, singular or in combination, form these network configurations, network analytics are useful to a broad range of research enterprises. In social science, these fields of study include, but are not limited to anthropology, biology, communication studies, economics, geography, information science, organizational studies, social psychology, sociology, and sociolinguistics.<br />
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社会网络是社会科学中研究个人、团体、组织甚至整个社会（社会单位，见分化）之间关系的理论构造。这个术语用来描述由这种相互作用决定的社会结构。任何一个特定的社会单元之间的联系都代表着这个单元各种社会联系的聚合。这种理论方法必然是相关的。理解社会互动的社会网络方法的一个公理是，社会现象应该主要通过单元之间和单元内部关系的性质来构思和研究，而不是这些单元本身的性质。因此，社会网络理论总被诟病的一点是其常忽视个体代理，而实践中可能并非如此（见基于主体的建模）。正是因为许多不同类型的关系（单独或组合形式）形成这些网络配置，网络分析在广泛的研究中有用。在社会科学中，这些研究领域包括但不限于'''<font color="#FFD700">人类学 Anthropology</font>'''、生物学、'''<font color="#FFD700">传播学 Communication Studies</font>'''、经济学、地理学、信息科学、组织学、社会心理学、社会学和'''<font color="#FFD700">社会语言学 Sociolinguistics</font>'''。<br />
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==History 历史==<br />
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In the late 1890s, both [[Émile Durkheim]] and [[Ferdinand Tönnies]] foreshadowed the idea of social networks in their theories and research of [[social group]]s. Tönnies argued that social groups can exist as personal and direct social ties that either link individuals who share values and belief (''[[Gemeinschaft]]'', German, commonly translated as "[[community]]") or impersonal, formal, and instrumental social links (''[[Gesellschaft]]'', German, commonly translated as "[[society]]").<ref>Tönnies, Ferdinand (1887). ''Gemeinschaft und Gesellschaft'', Leipzig: Fues's Verlag. (Translated, 1957 by Charles Price Loomis as ''Community and Society'', East Lansing: Michigan State University Press.)</ref> Durkheim gave a non-individualistic explanation of social facts, arguing that social phenomena arise when interacting individuals constitute a reality that can no longer be accounted for in terms of the properties of individual actors.<ref>Durkheim, Emile (1893). ''De la division du travail social: étude sur l'organisation des sociétés supérieures'', Paris: F. Alcan. (Translated, 1964, by Lewis A. Coser as ''The Division of Labor in Society'', New York: Free Press.)</ref> [[Georg Simmel]], writing at the turn of the twentieth century, pointed to the nature of networks and the effect of network size on interaction and examined the likelihood of interaction in loosely knit networks rather than groups.<ref>Simmel, Georg (1908). ''Soziologie'', Leipzig: Duncker & Humblot.</ref><br />
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In the late 1890s, both Émile Durkheim and Ferdinand Tönnies foreshadowed the idea of social networks in their theories and research of social groups. Tönnies argued that social groups can exist as personal and direct social ties that either link individuals who share values and belief (Gemeinschaft, German, commonly translated as "community") or impersonal, formal, and instrumental social links (Gesellschaft, German, commonly translated as "society"). Durkheim gave a non-individualistic explanation of social facts, arguing that social phenomena arise when interacting individuals constitute a reality that can no longer be accounted for in terms of the properties of individual actors. Georg Simmel, writing at the turn of the twentieth century, pointed to the nature of networks and the effect of network size on interaction and examined the likelihood of interaction in loosely knit networks rather than groups.<br />
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在19世纪90年代晚期，'''埃米尔·涂尔干 Émile Durkheim'''和'''斐迪南·滕尼斯 Ferdinand Tönnies'''在他们关于社会群体的理论和研究中都预示了社会网络的概念。滕尼斯认为，社会群体可以作为个人和直接的社会关系存在，这种关系或者将具有共同价值观和信仰的个人（德语 Gemeinschaft，通常翻译为“社区”）联系在一起，或者将非个人的、正式的和工具性的社会关系（德语 Gesellschaft，通常翻译为“社会”）联系在一起。涂尔干对社会事实给出了非个人主义的解释，认为当相互作用的个体构成一种再也不能用个体行为者的特性来解释的现实时，社会现象就产生了。格奥尔格·齐美尔在20世纪之交的著作中指出了网络的本质和网络规模对互动的影响，并研究了在松散的网络而非群体中互动的可能性。<br />
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[[File:Moreno Sociogram 2nd Grade.png|thumb|Moreno's sociogram of a 2nd grade class 图二：莫雷诺对一个二年级班级的社会关系图]]<br />
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Major developments in the field can be seen in the 1930s by several groups in psychology, anthropology, and mathematics working independently.<ref name=jscott /><ref>For a historical overview of the development of social network analysis, see: {{cite book|last1=Carrington|first1=Peter J.|last2=Scott|first2=John|chapter=Introduction|title=The Sage Handbook of Social Network Analysis| publisher=Sage|year=2011|isbn=978-1-84787-395-8|page=1|chapter-url=https://books.google.com/books?id=2chSmLzClXgC&pg=PA1}}</ref><ref>See also the diagram in {{cite book|author=Scott, John|title=Social Network Analysis: A Handbook|publisher=Sage|year=2000|isbn=978-0-7619-6339-4|page=8|url=https://books.google.com/books?id=Ww3_bKcz6kgC&pg=PA8}}</ref> In [[psychology]], in the 1930s, [[Jacob L. Moreno]] began systematic recording and analysis of social interaction in small groups, especially classrooms and work groups (see [[sociometry]]). In [[anthropology]], the foundation for social network theory is the theoretical and [[ethnography|ethnographic]] work of [[Bronislaw Malinowski]],<ref>Malinowski, Bronislaw (1913). ''The Family Among the Australian Aborigines: A Sociological Study''. London: University of London Press.</ref> [[Radcliffe-Brown|Alfred Radcliffe-Brown]],<ref>Radcliffe-Brown, Alfred Reginald (1930) ''The social organization of Australian tribes''. Sydney, Australia: University of Sydney ''Oceania'' monographs, No.1.</ref><ref>{{cite journal | last1 = Radcliffe-Brown | first1 = A. R. | year = 1940 | title = On social structure | url = | journal = Journal of the Royal Anthropological Institute | volume = 70 | issue = 1| pages = 1–12 | doi=10.2307/2844197| jstor = 2844197 }}</ref> and [[Claude Lévi-Strauss]].<ref>Lévi-Strauss, Claude ([1947]1967). ''Les structures élémentaires de la parenté''. Paris: La Haye, Mouton et Co. (Translated, 1969 by J. H. Bell, J. R. von Sturmer, and R. Needham, 1969, as ''The Elementary Structures of Kinship'', Boston: Beacon Press.)</ref> A group of social anthropologists associated with [[Max Gluckman]] and the [[Manchester school (anthropology)|Manchester School]], including [[John Arundel Barnes|John A. Barnes]],<ref>Barnes, John (1954). "Class and Committees in a Norwegian Island Parish". ''Human Relations'', (7): 39–58.</ref> [[J. Clyde Mitchell]] and [[Elizabeth Bott Spillius]],<ref>{{cite journal | last1 = Freeman | first1 = Linton C. | last2 = Wellman | first2 = Barry | year = 1995 | title = A note on the ancestoral Toronto home of social network analysis | url = | journal = Connections | volume = 18 | issue = 2| pages = 15–19 }}</ref><ref>{{cite journal | last1 = Savage | first1 = Mike | year = 2008 | title = Elizabeth Bott and the formation of modern British sociology | url = | journal = The Sociological Review | volume = 56 | issue = 4| pages = 579–605 | doi=10.1111/j.1467-954x.2008.00806.x}}</ref> often are credited with performing some of the first fieldwork from which network analyses were performed, investigating community networks in southern Africa, India and the United Kingdom.<ref name=jscott /> Concomitantly, British anthropologist [[Siegfried Frederick Nadel|S. F. Nadel]] codified a theory of social structure that was influential in later network analysis.<ref>Nadel, S. F. 1957. ''The Theory of Social Structure''. London: Cohen and West.</ref> In [[sociology]], the early (1930s) work of [[Talcott Parsons]] set the stage for taking a relational approach to understanding social structure.<ref>Parsons, Talcott ([1937] 1949). ''The Structure of Social Action: A Study in Social Theory with Special Reference to a Group of European Writers''. New York: The Free Press.</ref><ref>Parsons, Talcott (1951). ''The Social System''. New York: The Free Press.</ref> Later, drawing upon Parsons' theory, the work of sociologist [[Peter Blau]] provides a strong impetus for analyzing the relational ties of social units with his work on [[social exchange theory]].<ref>Blau, Peter (1956). ''Bureaucracy in Modern Society''. New York: Random House, Inc.</ref><ref>Blau, Peter (1960). "A Theory of Social Integration". ''The American Journal of Sociology'', (65)6: 545–556, (May).</ref><ref>Blau, Peter (1964). ''Exchange and Power in Social Life''.</ref><br />
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Major developments in the field can be seen in the 1930s by several groups in psychology, anthropology, and mathematics working independently. In psychology, in the 1930s, Jacob L. Moreno began systematic recording and analysis of social interaction in small groups, especially classrooms and work groups (see sociometry). In anthropology, the foundation for social network theory is the theoretical and ethnographic work of Bronislaw Malinowski, Alfred Radcliffe-Brown, and Claude Lévi-Strauss. A group of social anthropologists associated with Max Gluckman and the Manchester School, including John A. Barnes, J. Clyde Mitchell and Elizabeth Bott Spillius, often are credited with performing some of the first fieldwork from which network analyses were performed, investigating community networks in southern Africa, India and the United Kingdom. In sociology, the early (1930s) work of Talcott Parsons set the stage for taking a relational approach to understanding social structure. Later, drawing upon Parsons' theory, the work of sociologist Peter Blau provides a strong impetus for analyzing the relational ties of social units with his work on social exchange theory.<br />
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20世纪30年代，心理学、人类学和数学领域的几个独立研究小组已经看到了这一领域的重大发展。在心理学方面，在20世纪30年代，雅各布·L·莫雷诺开始系统地记录和分析小团体中的社会互动，尤其是课堂和工作团体中的社会互动（见'''<font color="#FFD700">社会测量 Sociometry）</font>'''。在人类学中，社会网络理论的基础是'''布罗尼斯拉夫·马林诺夫斯基 Bronislaw Malinowski'''，'''阿尔弗雷德·拉德克利夫-布朗 Alfred Radcliffe-Brown'''和'''克洛德·列维-斯特劳斯 Claude Lévi-Strauss'''的理论和人种学著作。包括'''约翰·A·巴恩斯 John A. Barnes'''、'''J·克莱德·米切尔 J. Clyde Mitchell'''和'''伊丽莎白·博特·斯皮利厄斯 Elizabeth Bott Spillius'''在内的一群与'''马克斯·格拉克曼 Max Gluckman'''和'''曼彻斯特学派 Manchester School'''有关的社会人类学家，经常被认为是执行了一些最初的实地工作，从而进行了网络分析，调查了南非、印度和英国的社区网络。在社会学方面，'''塔尔科特·帕森斯 Talcott Parsons'''的早期工作（1930年代）为采用关系方法理解社会结构奠定了基础。后来，社会学家'''彼得·布劳 Peter Blau'''的'''<font color="#FFD700">社会交换论 Social Exchange Theory</font>'''为分析社会单位之间的关系提供了强大的动力。<br />
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By the 1970s, a growing number of scholars worked to combine the different tracks and traditions. One group consisted of sociologist [[Harrison White]] and his students at the [[Harvard Department of Social Relations|Harvard University Department of Social Relations]]. Also independently active in the Harvard Social Relations department at the time were [[Charles Tilly]], who focused on networks in political and community sociology and social movements, and [[Stanley Milgram]], who developed the "six degrees of separation" thesis.<ref>{{cite web |url=http://www.semioticon.com/semiotix/semiotix14/sem-14-05.html |title=The Networked Individual: A Profile of Barry Wellman |author=Bernie Hogan}}</ref> [[Mark Granovetter]]<ref name="Introduction for the French Reader">{{cite journal | last1 = Granovetter | first1 = Mark | year = 2007 | title = Introduction for the French Reader | url = | journal = Sociologica | volume = 2 | issue = | pages = 1–8 }}</ref> and [[Barry Wellman]]<ref>Wellman, Barry (1988). "Structural analysis: From method and metaphor to theory and substance". pp. 19–61 in B. Wellman and S. D. Berkowitz (eds.) ''Social Structures: A Network Approach'', Cambridge, UK: Cambridge University Press.</ref> are among the former students of White who elaborated and championed the analysis of social networks.<ref name="Introduction for the French Reader"/><ref>Mullins, Nicholas. ''Theories and Theory Groups in Contemporary American Sociology''. New York: Harper and Row, 1973.</ref><ref>Tilly, Charles, ed. ''An Urban World''. Boston: Little Brown, 1974.</ref><ref>Wellman, Barry. 1988. "Structural Analysis: From Method and Metaphor to Theory and Substance". pp. 19–61 in ''Social Structures: A Network Approach'', edited by Barry Wellman and S. D. Berkowitz. Cambridge: Cambridge University Press.</ref><br />
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By the 1970s, a growing number of scholars worked to combine the different tracks and traditions. One group consisted of sociologist Harrison White and his students at the Harvard University Department of Social Relations. Also independently active in the Harvard Social Relations department at the time were Charles Tilly, who focused on networks in political and community sociology and social movements, and Stanley Milgram, who developed the "six degrees of separation" thesis. Mark Granovetter and Barry Wellman are among the former students of White who elaborated and championed the analysis of social networks.<br />
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到了20世纪70年代，越来越多的学者致力于将不同的轨迹和传统结合起来。其中一组由社会学家'''哈里森·怀特 Harrison White'''和他在哈佛大学社会关系系的学生组成。当时在哈佛大学社会关系系独立活动的还有专注于政治和社区社会学和社会运动的网络的'''查尔斯·堤利 Charles Tilly'''，还有发表了六度分隔理论论文的'''斯坦利·米尔格拉姆 Stanley Milgram'''。'''马克·格兰诺维特 Mark Granovetter'''和'''巴里·威尔曼 Barry Wellman'''是怀特以前的学生，他们阐述并支持对社交网络的分析。<br />
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Beginning in the late 1990s, social network analysis experienced work by sociologists, political scientists, and physicists such as [[Duncan J. Watts]], [[Albert-László Barabási]], [[Peter Bearman]], [[Nicholas A. Christakis]], [[James H. Fowler]], and others, developing and applying new models and methods to emerging data available about online social networks, as well as "digital traces" regarding face-to-face networks.<br />
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Beginning in the late 1990s, social network analysis experienced work by sociologists, political scientists, and physicists such as Duncan J. Watts, Albert-László Barabási, Peter Bearman, Nicholas A. Christakis, James H. Fowler, and others, developing and applying new models and methods to emerging data available about online social networks, as well as "digital traces" regarding face-to-face networks.<br />
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从20世纪90年代末开始，社会网络分析经历了社会学家、政治学家和物理学家的工作，如'''[[邓肯·瓦茨 Duncan J. Watts]]'''、'''[[艾伯特-拉斯洛·巴拉巴西 Albert-László Barabási]]'''、Peter Bearman、Nicholas A. Christakis、James H. Fowler等人，开发和应用新的模型和方法来获得有关在线社会网络的新兴数据，以及有关面对面网络的“数字痕迹”。<br />
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==Levels of analysis 分析水平==<br />
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[[File:Network self-organization stages.png|thumb|right|Self-organization of a network, based on Nagler, Levina, & Timme, (2011) 图三：基于Nagler, Levina和Timme的自组织网络，(2011)<ref>{{cite journal|author1=Nagler, Jan|author2=Anna Levina|author3=Marc Timme|year=2011|title=Impact of single links in competitive percolation|journal=Nature Physics|volume=7|issue=3|pages=265–270|doi=10.1038/nphys1860|arxiv=1103.0922|bibcode=2011NatPh...7..265N}}</ref>]]<br />
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[[File:Social Network Diagram (large).svg|right|thumb|Centrality]]<br />
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Centrality<br />
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中心性<br />
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In general, social networks are [[self-organization|self-organizing]], [[emergence|emergent]], and [[social complexity|complex]], such that a globally coherent pattern appears from the local interaction of the elements that make up the system.<ref>Newman, Mark, Albert-László Barabási and Duncan J. Watts (2006). ''The Structure and Dynamics of Networks'' (Princeton Studies in Complexity). Oxford: Princeton University Press.</ref><ref>{{cite journal | last1 = Wellman | first1 = Barry | year = 2008 | title = Review: The development of social network analysis: A study in the sociology of science | url = | journal = Contemporary Sociology | volume = 37 | issue = 3| pages = 221–222 | doi=10.1177/009430610803700308}}</ref> These patterns become more apparent as network size increases. However, a global network analysis<ref>{{cite book|last=Faust|first=Stanley Wasserman; Katherine|title=Social network analysis : methods and applications|year=1998|publisher=Cambridge Univ. Press|location=Cambridge [u.a.]|isbn=978-0521382694|edition=Reprint.}}</ref> of, for example, all [[interpersonal relationships]] in the world is not feasible and is likely to contain so much [[Information theory|information]] as to be uninformative. Practical limitations of computing power, ethics and participant recruitment and payment also limit the scope of a social network analysis.<ref name="Kadu12">Kadushin, C. (2012). ''Understanding social networks: Theories, concepts, and findings''. Oxford: Oxford University Press.</ref><ref>{{Cite journal| author=Granovetter, M.|title= Network sampling: Some first steps| year=1976 |pages=1287–1303 |volume=81| issue=6 |journal= American Journal of Sociology | doi=10.1086/226224}}</ref> The nuances of a local system may be lost in a large network analysis, hence the quality of information may be more important than its scale for understanding network properties. Thus, social networks are analyzed at the scale relevant to the researcher's theoretical question. Although [[level of analysis|levels of analysis]] are not necessarily [[Mutually exclusive events|mutually exclusive]], there are three general levels into which networks may fall: [[Microsociology|micro-level]], [[wikt:meso-|meso-level]], and [[Macrosociology|macro-level]].<br />
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In general, social networks are self-organizing, emergent, and complex, such that a globally coherent pattern appears from the local interaction of the elements that make up the system. These patterns become more apparent as network size increases. However, a global network analysis of, for example, all interpersonal relationships in the world is not feasible and is likely to contain so much information as to be uninformative. Practical limitations of computing power, ethics and participant recruitment and payment also limit the scope of a social network analysis. The nuances of a local system may be lost in a large network analysis, hence the quality of information may be more important than its scale for understanding network properties. Thus, social networks are analyzed at the scale relevant to the researcher's theoretical question. Although levels of analysis are not necessarily mutually exclusive, there are three general levels into which networks may fall: micro-level, meso-level, and macro-level.<br />
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一般来说，社会网络是自组织的、涌现的和复杂的，这样，一个全局一致的模式就会从组成系统的元素的局部交互中显现出来。随着网络规模的增大，这些模式变得更加明显。然而，一个全球网络分析（如世界上所有的人际关系）是不可行的，它可能包含太多的信息，以至于相当于没有提供信息。计算能力的实际限制、道德规范以及参与者的招募和支付也限制了社会网络分析的范围。本地系统的细微差别在大型网络分析中可能会丢失，因此对于理解网络属性来说，信息的质量可能比其规模更重要。因此，社会网络被分析在与研究者的理论问题相关的尺度上。虽然分析层次不一定相互排斥，但网络可以分为三个一般层次: 微观、中观和宏观。<br />
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===Micro level 微观层面===<br />
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At the micro-level, social network research typically begins with an individual, [[Snowball sampling|snowballing]] as social relationships are traced, or may begin with a small group of individuals in a particular social context.<br />
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At the micro-level, social network research typically begins with an individual, snowballing as social relationships are traced, or may begin with a small group of individuals in a particular social context.<br />
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在微观层面上，社会网络研究通常从个人开始，随着社会关系的追踪而像滚雪球一样扩大，或者可能从特定社会背景下的一小群个体开始。<br />
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'''Dyadic level''': A [[Dyad (sociology)|dyad]] is a social relationship between two individuals. Network research on dyads may concentrate on [[Structural functionalism|structure]] of the relationship (e.g. multiplexity, strength), [[social equality]], and tendencies toward [[Reciprocity (social and political philosophy)|reciprocity/mutuality]].<br />
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Dyadic level: A dyad is a social relationship between two individuals. Network research on dyads may concentrate on structure of the relationship (e.g. multiplexity, strength), social equality, and tendencies toward reciprocity/mutuality.<br />
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二元层面：二元是两个个体之间的社会关系。网络对二元关系的研究可以集中在关系的结构上（如多样性、力量)、社会平等以及互惠互利的倾向。<br />
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'''Triadic level''': Add one individual to a dyad, and you have a [[Triadic relation|triad]]. Research at this level may concentrate on factors such as [[Independence number|balance]] and [[Vertex-transitive graph|transitivity]], as well as [[social equality]] and tendencies toward [[Reciprocity (social and political philosophy)|reciprocity/mutuality]].<ref name="Kadu12"/> In the [[balance theory]] of [[Fritz Heider]] the triad is the key to social dynamics. The discord in a rivalrous [[love triangle]] is an example of an unbalanced triad, likely to change to a balanced triad by a change in one of the relations. The dynamics of social friendships in society has been modeled by balancing triads. The study is carried forward with the theory of [[signed graph]]s.<br />
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Triadic level: Add one individual to a dyad, and you have a triad. Research at this level may concentrate on factors such as balance and transitivity, as well as social equality and tendencies toward reciprocity/mutuality. In the balance theory of Fritz Heider the triad is the key to social dynamics. The discord in a rivalrous love triangle is an example of an unbalanced triad, likely to change to a balanced triad by a change in one of the relations. The dynamics of social friendships in society has been modeled by balancing triads. The study is carried forward with the theory of signed graphs.<br />
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三元层面：再加一个个体到二元组中，就得到一个三元组。这一层次的研究可能集中在诸如平衡和传递性等因素，以及社会平等和互惠互利的倾向。在'''弗里茨 · 海德 Fritz Heider'''的'''平衡理论 Balance Theory'''中，三元组是社会动力学的关键。在敌对的三角恋中的不和谐是不平衡的三角关系的一个例子，很可能通过其中一种关系的改变而变成平衡的三角关系。社会中社会友谊的动态模型是通过平衡三角关系建立起来的。该研究利用了符号图理论。<br />
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'''Actor level''': The smallest unit of analysis in a social network is an individual in their social setting, i.e., an "actor" or "ego". Egonetwork analysis focuses on network characteristics such as size, relationship strength, density, [[centrality]], [[wikt:prestige|prestige]] and roles such as [[isolates|isolates, liaisons]], and [[Bridge (interpersonal)|bridges]].<ref name="Jone11"/> Such analyses, are most commonly used in the fields of [[psychology]] or [[Social psychology (sociology)|social psychology]], [[ethnographic]] [[kinship]] analysis or other [[genealogy|genealogical]] studies of relationships between individuals.<br />
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Actor level: The smallest unit of analysis in a social network is an individual in their social setting, i.e., an "actor" or "ego". Egonetwork analysis focuses on network characteristics such as size, relationship strength, density, centrality, prestige and roles such as isolates, liaisons, and bridges. Such analyses, are most commonly used in the fields of psychology or social psychology, ethnographic kinship analysis or other genealogical studies of relationships between individuals.<br />
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行为者层面：社会网络中最小的分析单位是社会环境中的个体，即“行为者”或“自我”。自我网络分析主要关注网络特征，例如大小、关系强度、密度、中心性、声望和隔离、联络和桥梁等角色。这种分析最常用于心理学或社会心理学、人种学亲属关系分析或其他个体关系的系谱研究领域。<br />
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'''Subset level''': [[Subset]] levels of network research problems begin at the micro-level, but may cross over into the meso-level of analysis. Subset level research may focus on [[Distance (graph theory)|distance]] and reachability, [[cliques]], [[Cohesion (social policy)|cohesive]] subgroups, or other [[Group action (sociology)|group actions]] or [[Group behaviour|behavior]].<ref>{{cite book | title="Graph Theoretical Approaches to Social Network Analysis". in Computational Complexity: Theory, Techniques, and Applications (Robert A. Meyers, ed.) | publisher=Springer | author=de Nooy, Wouter |year=2012 | pages=2864–2877 | isbn=978-1-4614-1800-9|doi=10.1007/978-1-4614-1800-9_176| chapter=Social Network Analysis, Graph Theoretical Approaches to }}</ref><br />
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Subset level: Subset levels of network research problems begin at the micro-level, but may cross over into the meso-level of analysis. Subset level research may focus on distance and reachability, cliques, cohesive subgroups, or other group actions or behavior.<br />
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子集层面：网络研究问题的子集级别开始于微观级别，但可能跨越到中观级别的分析。子集级别的研究可能集中在距离和可达性、派系、凝聚子群或其他群体行为或行为。<br />
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===Meso level 中观层面===<br />
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In general, meso-level theories begin with a [[Sample population|population]] size that falls between the micro- and macro-levels. However, meso-level may also refer to analyses that are specifically designed to reveal connections between micro- and macro-levels. Meso-level networks are low density and may exhibit causal processes distinct from interpersonal micro-level networks.<ref>{{cite journal | last1 = Hedström | first1 = Peter | last2 = Sandell | first2 = Rickard | last3 = Stern | first3 = Charlotta | year = 2000 | title = Mesolevel Networks and the Diffusion of Social Movements: The Case of the Swedish Social Democratic Party | url = http://www.nuffield.ox.ac.uk/users/hedstrom/ajs3.pdf | journal = American Journal of Sociology | volume = 106 | issue = 1| pages = 145–172 | doi = 10.1086/303109 }}</ref><br />
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In general, meso-level theories begin with a population size that falls between the micro- and macro-levels. However, meso-level may also refer to analyses that are specifically designed to reveal connections between micro- and macro-levels. Meso-level networks are low density and may exhibit causal processes distinct from interpersonal micro-level networks.<br />
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一般来说，中观层面的理论始于介于微观和宏观层面之间的规模。然而，中观层面也可以指专门为揭示微观和宏观层面之间的联系而设计的分析。中观层次的网络是密度低，可能表现出不同于人际微观层面网络的因果过程。<br />
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[[File:Social Red.jpg|thumb|right|Social network diagram, meso-level 图四：中观层而社会网络图]]<br />
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'''Organizations''': Formal [[organizations]] are [[social group]]s that distribute tasks for a collective [[goal]].<ref name="Riketta, M. 2007">{{cite journal | last1 = Riketta | first1 = M. | last2 = Nienber | first2 = S. | year = 2007 | title = Multiple identities and work motivation: The role of perceived compatibility between nested organizational units | url = | journal = British Journal of Management | volume = 18 | issue = | pages = S61–77 | doi=10.1111/j.1467-8551.2007.00526.x}}</ref> Network research on organizations may focus on either intra-organizational or inter-organizational ties in terms of [[Formal organization|formal]] or [[Informal organization|informal]] relationships. Intra-organizational networks themselves often contain multiple levels of analysis, especially in larger organizations with multiple branches, franchises or semi-autonomous departments. In these cases, research is often conducted at a work group level and organization level, focusing on the interplay between the two structures.<ref name="Riketta, M. 2007"/> Experiments with networked groups online have documented ways to optimize group-level coordination through diverse interventions, including the addition of autonomous agents to the groups.<ref>{{Cite journal|last=Shirado|first=Hirokazu|last2=Christakis|first2=Nicholas A|title=Locally noisy autonomous agents improve global human coordination in network experiments|journal=Nature|volume=545|issue=7654|pages=370–374|doi=10.1038/nature22332|pmid=28516927|pmc=5912653|bibcode=2017Natur.545..370S|year=2017}}</ref><br />
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Organizations: Formal organizations are social groups that distribute tasks for a collective goal. Network research on organizations may focus on either intra-organizational or inter-organizational ties in terms of formal or informal relationships. Intra-organizational networks themselves often contain multiple levels of analysis, especially in larger organizations with multiple branches, franchises or semi-autonomous departments. In these cases, research is often conducted at a work group level and organization level, focusing on the interplay between the two structures.<br />
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组织: 正规组织是为共同目标分配任务的社会团体。关于组织的网络研究可以侧重于正式或非正式关系方面的组织内或组织间联系。组织内网络本身往往包含多层次的分析，特别是在具有多个分支机构、特许权或半自治部门的较大组织中。在这些情况下，研究通常在工作组和组织层面进行，重点放在两个结构之间的相互作用。<br />
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'''Randomly distributed networks''': [[Exponential random graph models]] of social networks became state-of-the-art methods of social network analysis in the 1980s. This framework has the capacity to represent social-structural effects commonly observed in many human social networks, including general [[Degree (graph theory)|degree]]-based structural effects commonly observed in many human social networks as well as [[Reciprocity (social and political philosophy)|reciprocity]] and [[Transitive set|transitivity]], and at the node-level, [[homophily]] and [[Attribute-value system|attribute]]-based activity and popularity effects, as derived from explicit hypotheses about [[Dependency graph|dependencies]] among network ties. [[Parameter]]s are given in terms of the prevalence of small [[Induced subgraph|subgraph]] configurations in the network and can be interpreted as describing the combinations of local social processes from which a given network emerges. These probability models for networks on a given set of actors allow generalization beyond the restrictive dyadic independence assumption of micro-networks, allowing models to be built from theoretical structural foundations of social behavior.<ref>{{cite journal | last1 = Cranmer | first1 = Skyler J. | last2 = Desmarais | first2 = Bruce A. | year = 2011 | title = Inferential Network Analysis with Exponential Random Graph Models | url = | journal = Political Analysis | volume = 19 | issue = 1| pages = 66–86 | doi=10.1093/pan/mpq037| citeseerx = 10.1.1.623.751 }}</ref><br />
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Randomly distributed networks: Exponential random graph models of social networks became state-of-the-art methods of social network analysis in the 1980s. This framework has the capacity to represent social-structural effects commonly observed in many human social networks, including general degree-based structural effects commonly observed in many human social networks as well as reciprocity and transitivity, and at the node-level, homophily and attribute-based activity and popularity effects, as derived from explicit hypotheses about dependencies among network ties. Parameters are given in terms of the prevalence of small subgraph configurations in the network and can be interpreted as describing the combinations of local social processes from which a given network emerges. These probability models for networks on a given set of actors allow generalization beyond the restrictive dyadic independence assumption of micro-networks, allowing models to be built from theoretical structural foundations of social behavior.<br />
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随机分布的网络: 社会网络的'''<font color="#FFD700">指数随机图模型 Exponential random graph models</font>'''在20世纪80年代成为最先进的社会网络分析方法。这个框架有能力表示在许多人类社会网络中普遍观察到的社会结构效应，包括在许多人类社会网络中普遍观察到的基于程度的一般性结构效应以及互惠性和传递性，以及在节点一级、同相性和基于属性的活动和流行性效应，这些效应源于关于网络关系之间依赖性的明确假设。参数是根据网络中小型子图配置的流行程度给出的，可以解释为描述一个给定网络出现的局部社会过程的组合。这些网络的概率模型在给定的参与者集合上允许超越微型网络的限制性并元独立性假设的泛化，允许模型从社会行为的理论结构基础上建立。<br />
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[[File:Scale-free network sample.png|thumb|right|Examples of a random network and a scale-free network. Each graph has 32 nodes and 32 links. Note the "hubs" (shaded) in the scale-free diagram (on the right).图五：一个随机网络和一个无尺度网络的例子。每个图有32个顶点和32条边。注意无标度图中的“集线器”（阴影部分）（右侧）。]]<br />
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'''Scale-free networks''': A [[scale-free network]] is a [[complex network|network]] whose [[degree distribution]] follows a [[power law]], at least [[asymptotic]]ally. In [[network theory]] a scale-free ideal network is a [[random network]] with a [[degree distribution]] that unravels the size distribution of social groups.<ref>{{cite journal |author1=Moreira, André A.|author2=Demétrius R. Paula|author3=Raimundo N. Costa Filho|author4=José S. Andrade, Jr. |title=Competitive cluster growth in complex networks |year=2006 |doi=10.1103/PhysRevE.73.065101 |pmid=16906890 |journal=Physical Review E |volume=73 |issue=6 |pages=065101 |arxiv=cond-mat/0603272 |bibcode=2006PhRvE..73f5101M }}</ref> Specific characteristics of scale-free networks vary with the theories and analytical tools used to create them, however, in general, scale-free networks have some common characteristics. One notable characteristic in a scale-free network is the relative commonness of [[Vertex (graph theory)|vertices]] with a [[Maximum degree|degree]] that greatly exceeds the average. The highest-degree nodes are often called "hubs", and may serve specific purposes in their networks, although this depends greatly on the social context. Another general characteristic of scale-free networks is the [[clustering coefficient]] distribution, which decreases as the node degree increases. This distribution also follows a [[power law]].<ref>Barabási, Albert-László (2003). ''Linked: how everything is connected to everything else and what it means for business, science, and everyday life''. New York: Plum.</ref> The [[Barabási–Albert model|Barabási]] model of network evolution shown above is an example of a scale-free network.<br />
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Scale-free networks: A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. In network theory a scale-free ideal network is a random network with a degree distribution that unravels the size distribution of social groups. Specific characteristics of scale-free networks vary with the theories and analytical tools used to create them, however, in general, scale-free networks have some common characteristics. One notable characteristic in a scale-free network is the relative commonness of vertices with a degree that greatly exceeds the average. The highest-degree nodes are often called "hubs", and may serve specific purposes in their networks, although this depends greatly on the social context. Another general characteristic of scale-free networks is the clustering coefficient distribution, which decreases as the node degree increases. This distribution also follows a power law. The Barabási model of network evolution shown above is an example of a scale-free network.<br />
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'''<font color="#FFD700">无标度网络 A Scale-free Network</font>''': 无标度网络网络是一个'''<font color="#FFD700">度分布 Degree Distribution</font>'''遵循幂律的网络，至少是渐近的。在网络理论中，无标度理想网络是一个具有度分布的随机网络，它揭示了社会群体的规模分布。无标度网络的具体特征随创建无标度网络的理论和分析工具的不同而不同，然而，一般来说，无标度网络具有一些共同的特征。无尺度网络的一个显著特征是，度大大超过平均值的顶点具有相对的共性。最高度的节点通常被称为“枢纽” ，并且可能在其网络中服务于特定的目的，尽管这在很大程度上取决于社会环境。无标度网络的另一个普遍特征是集聚系数分布，它随着节点度的增加而减少。这个分布也遵循一个幂定律。上面显示的网络演化的 barab si 模型就是无尺度网络的一个例子。<br />
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===Macro level 宏观层面===<br />
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Rather than tracing interpersonal interactions, macro-level analyses generally trace the outcomes of interactions, such as [[economic]] or other [[resource]] [[Transfer function|transfer]] interactions over a large [[Sample population|population]].<br />
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Rather than tracing interpersonal interactions, macro-level analyses generally trace the outcomes of interactions, such as economic or other resource transfer interactions over a large population.<br />
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宏观层面的分析不是追踪人与人之间的相互作用，而是通常追踪相互作用的结果，例如经济或其他资源转移在一大群体中的相互作用。<br />
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[[File:Diagram of a social network.jpg|thumb|right|Diagram: section of a large-scale social network]]<br />
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Diagram: section of a large-scale social network<br />
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图: 一个大型社交网络的一部分<br />
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'''Large-scale networks''': [[Large-scale macroeconometric model|Large-scale network]] is a term somewhat synonymous with "macro-level" as used, primarily, in [[social science|social]] and [[Behavioural sciences|behavioral]] sciences, in [[economics]]. Originally, the term was used extensively in the [[computer sciences]] (see [[Network mapping#Large-scale mapping project|large-scale network mapping]]).<br />
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Large-scale networks: Large-scale network is a term somewhat synonymous with "macro-level" as used, primarily, in social and behavioral sciences, in economics. Originally, the term was used extensively in the computer sciences (see large-scale network mapping).<br />
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大规模网络: 大规模网络是一个与“宏观层面”同义的术语，主要用于社会和行为科学，经济学。最初，这个术语在计算机科学中被广泛使用(见大比例尺网络地图)。<br />
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'''Complex networks''': Most larger social networks display features of [[social complexity]], which involves substantial non-trivial features of [[network topology]], with patterns of complex connections between elements that are neither purely regular nor purely random (see, [[complexity science]], [[dynamical system]] and [[chaos theory]]), as do [[biological]], and [[Computer network|technological networks]]. Such [[complex network]] features include a heavy tail in the [[degree distribution]], a high [[clustering coefficient]], [[assortativity]] or disassortativity among vertices, [[community structure]] (see [[stochastic block model]]), and [[hierarchy|hierarchical structure]]. In the case of [[Agency (philosophy)|agency-directed]] networks these features also include [[Reciprocity in network|reciprocity]], triad significance profile (TSP, see [[network motif]]), and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as [[lattice graph|lattices]] and [[random graph]]s, do not show these features.<ref>{{cite journal|author=Strogatz, Steven H.|year=2001|title=Exploring complex networks|journal=Nature|volume=410|pages=268–276|doi=10.1038/35065725|pmid=11258382|issue=6825|bibcode=2001Natur.410..268S|doi-access=free}}</ref><br />
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Complex networks: Most larger social networks display features of social complexity, which involves substantial non-trivial features of network topology, with patterns of complex connections between elements that are neither purely regular nor purely random (see, complexity science, dynamical system and chaos theory), as do biological, and technological networks. Such complex network features include a heavy tail in the degree distribution, a high clustering coefficient, assortativity or disassortativity among vertices, community structure (see stochastic block model), and hierarchical structure. In the case of agency-directed networks these features also include reciprocity, triad significance profile (TSP, see network motif), and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as lattices and random graphs, do not show these features.<br />
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复杂网络: 大多数较大的社会网络呈现出社会复杂性的特征，包括网络拓扑的大量非琐碎特征，以及元素之间的复杂连接模式，这些元素既不是纯规则的也不是纯随机的(见复杂性科学、动力系统和混沌理论) ，生物学和技术网络也是如此。这些复杂的网络特征包括程度分布的重尾、高集聚系数、顶点之间的协调性或不协调性、社区结构(见随机块模型)和层次结构。在代理导向网络的情况下，这些特征还包括互惠性、三重显著性特征(TSP，见网络主题)和其他特征。相比之下，许多过去研究过的网络数学模型，如格和随机图，并没有表现出这些特征。<br />
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==Theoretical links 理论联系==<br />
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===Imported theories 输入的理论===<br />
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Various theoretical frameworks have been imported for the use of social network analysis. The most prominent of these are [[Graph theory]], [[Balance theory]], Social comparison theory, and more recently, the [[Social identity approach]].<ref>{{cite book |last1=Kilduff|first1=M.|last2=Tsai|first2=W. |year=2003 |title= Social networks and organisations |publisher=Sage Publications}}</ref><br />
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Various theoretical frameworks have been imported for the use of social network analysis. The most prominent of these are Graph theory, Balance theory, Social comparison theory, and more recently, the Social identity approach.<br />
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为了使用社会网络分析，已经引入了各种理论框架。其中最突出的是图论、平衡论、社会比较论，以及最近的社会认同方法。<br />
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===Indigenous theories 本土理论===<br />
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Few complete theories have been produced from social network analysis. Two that have are [[role theory|structural role theory]] and [[heterophily|heterophily theory]].<br />
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Few complete theories have been produced from social network analysis. Two that have are structural role theory and heterophily theory.<br />
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很少有完整的理论产生于社会网络分析。结构角色理论和异质性理论。<br />
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The basis of Heterophily Theory was the finding in one study that more numerous weak ties can be important in seeking information and innovation, as cliques have a tendency to have more homogeneous opinions as well as share many common traits. This homophilic tendency was the reason for the members of the cliques to be attracted together in the first place. However, being similar, each member of the clique would also know more or less what the other members knew. To find new information or insights, members of the clique will have to look beyond the clique to its other friends and acquaintances. This is what Granovetter called "the strength of weak ties".<ref>{{Cite journal | author=Granovetter, M. |year=1973 |title= The strength of weak ties |journal= American Journal of Sociology |volume=78 |issue=6 |pages= 1360–1380 |doi=10.1086/225469}}</ref><br />
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The basis of Heterophily Theory was the finding in one study that more numerous weak ties can be important in seeking information and innovation, as cliques have a tendency to have more homogeneous opinions as well as share many common traits. This homophilic tendency was the reason for the members of the cliques to be attracted together in the first place. However, being similar, each member of the clique would also know more or less what the other members knew. To find new information or insights, members of the clique will have to look beyond the clique to its other friends and acquaintances. This is what Granovetter called "the strength of weak ties".<br />
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异质性理论的基础是在一项研究中发现，更多的弱关系可以在寻求信息和创新方面发挥重要作用，因为小团体倾向于有更同质化的观点，也有许多共同特征。这种亲同性倾向是小团体成员被吸引到一起的首要原因。然而，由于相似，小圈子里的每一个成员或多或少都知道其他成员所知道的事情。为了找到新的信息或见解，小圈子里的成员将不得不超越小圈子，关注其他的朋友和熟人。这就是格兰诺维特所说的“弱关系的力量”。<br />
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==Structural holes 结构性漏洞==<br />
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In the context of networks, [[social capital]] exists where people have an advantage because of their location in a network. Contacts in a network provide information, opportunities and perspectives that can be beneficial to the central player in the network. Most social structures tend to be characterized by dense clusters of strong connections.<ref name="Burt 2004">{{cite journal|last=Burt|first=Ronald|title=Structural Holes and Good Ideas|journal=American Journal of Sociology|year=2004|doi=10.1086/421787|volume=110|issue=2|pages=349–399|citeseerx=10.1.1.388.2251}}</ref> Information within these clusters tends to be rather homogeneous and redundant. Non-redundant information is most often obtained through contacts in different clusters.<ref name="Burt 1992">{{cite book|last=Burt|first=Ronald|title=Structural Holes: The Social Structure of Competition|year=1992|publisher=Harvard University Press|location=Cambridge, MA}}</ref> When two separate clusters possess non-redundant information, there is said to be a structural hole between them.<ref name="Burt 1992"/> Thus, a network that bridges [[structural holes]] will provide network benefits that are in some degree additive, rather than overlapping. An ideal network structure has a vine and cluster structure, providing access to many different clusters and structural holes.<ref name="Burt 1992"/><br />
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In the context of networks, social capital exists where people have an advantage because of their location in a network. Contacts in a network provide information, opportunities and perspectives that can be beneficial to the central player in the network. Most social structures tend to be characterized by dense clusters of strong connections. Information within these clusters tends to be rather homogeneous and redundant. Non-redundant information is most often obtained through contacts in different clusters. When two separate clusters possess non-redundant information, there is said to be a structural hole between them. Thus, a network that bridges structural holes will provide network benefits that are in some degree additive, rather than overlapping. An ideal network structure has a vine and cluster structure, providing access to many different clusters and structural holes.<br />
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在网络的背景下，社会资本存在于人们因为他们在网络中的位置而具有优势的地方。网络中的联系人提供的信息、机会和观点可能有利于网络中的核心参与者。大多数社会结构倾向于拥有属性密集的强连接。这些集群中的信息往往是相当均匀和冗余的。非冗余信息通常是通过不同集群中的联系人获得的。当两个独立的簇拥有非冗余信息时，我们称它们之间存在一个结构空洞。因此，一个桥接结构孔的网络将在某种程度上提供附加的网络效益，而不是重叠的。理想的网络结构具有蔓生结构和集群结构，提供对许多不同集群和结构洞的访问。<br />
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Networks rich in structural holes are a form of social capital in that they offer [[Information theory|information]] benefits. The main player in a network that bridges structural holes is able to access information from diverse sources and clusters.<ref name="Burt 1992"/> For example, in [[Business networking|business networks]], this is beneficial to an individual's career because he is more likely to hear of job openings and opportunities if his network spans a wide range of contacts in different industries/sectors. This concept is similar to Mark Granovetter's theory of [[Interpersonal ties|weak ties]], which rests on the basis that having a broad range of contacts is most effective for job attainment.<br />
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Networks rich in structural holes are a form of social capital in that they offer information benefits. The main player in a network that bridges structural holes is able to access information from diverse sources and clusters. For example, in business networks, this is beneficial to an individual's career because he is more likely to hear of job openings and opportunities if his network spans a wide range of contacts in different industries/sectors. This concept is similar to Mark Granovetter's theory of weak ties, which rests on the basis that having a broad range of contacts is most effective for job attainment.<br />
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富含结构性漏洞的网络是社会资本的一种形式，因为它们提供信息利益。桥接结构漏洞的网络中的主要参与者能够访问来自不同来源和集群的信息。例如，在商业网络中，这对个人的职业生涯是有益的，因为如果他的关系网涵盖不同行业 / 部门的广泛联系，他更有可能听到职位空缺和机会。这个概念类似于马克·格兰诺维特的弱关系理论，该理论的基础是拥有广泛的联系对于获得工作是最有效的。<br />
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==Research clusters 研究集群==<br />
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===Communication 沟通===<br />
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[[Communication Studies]] are often considered a part of both the social sciences and the humanities, drawing heavily on fields such as [[sociology]], [[psychology]], [[anthropology]], [[information science]], [[biology]], [[political science]], and [[economics]] as well as [[rhetoric]], [[literary studies]], and [[semiotics]]. Many communication concepts describe the transfer of information from one source to another, and can thus be conceived of in terms of a network.<br />
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Communication Studies are often considered a part of both the social sciences and the humanities, drawing heavily on fields such as sociology, psychology, anthropology, information science, biology, political science, and economics as well as rhetoric, literary studies, and semiotics. Many communication concepts describe the transfer of information from one source to another, and can thus be conceived of in terms of a network.<br />
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传播学通常被认为是社会科学和人文科学的一部分，主要集中在社会学、心理学、人类学、信息科学、生物学、政治学、经济学以及修辞学、文学研究和符号学等领域。许多通信概念描述了信息从一个来源到另一个来源的转移，因此可以从网络的角度来考虑。<br />
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===Community 社区===<br />
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In J.A. Barnes' day, a "[[community]]" referred to a specific geographic location and studies of community ties had to do with who talked, associated, traded, and attended church with whom. Today, however, there are extended "online" communities developed through [[telecommunications]] devices and [[social network services]]. Such devices and services require extensive and ongoing maintenance and analysis, often using [[network science]] methods. [[Community development]] studies, today, also make extensive use of such methods.<br />
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In J.A. Barnes' day, a "community" referred to a specific geographic location and studies of community ties had to do with who talked, associated, traded, and attended church with whom. Today, however, there are extended "online" communities developed through telecommunications devices and social network services. Such devices and services require extensive and ongoing maintenance and analysis, often using network science methods. Community development studies, today, also make extensive use of such methods.<br />
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在耶路撒冷。巴恩斯日，一个“社区”指的是一个特定的地理位置和社区关系的研究与谁交谈，联系，贸易和参加教会与谁。然而，今天，通过电信设备和社交网络服务，有了扩展的“在线”社区。这样的设备和服务需要广泛和持续的维护和分析，通常使用网络科学的方法。社区发展研究，今天，也广泛使用这些方法。<br />
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===Complex networks 复杂网络===<br />
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[[Complex networks]] require methods specific to modelling and interpreting [[social complexity]] and [[complex adaptive system]]s, including techniques of [[dynamic network analysis]].<br />
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Complex networks require methods specific to modelling and interpreting social complexity and complex adaptive systems, including techniques of dynamic network analysis.<br />
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复杂网络需要建模和解释社会复杂性和复杂适应系统的特定方法，包括动态网络分析技术。<br />
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Mechanisms such as [[Dual-phase evolution#social networks|Dual-phase evolution]] explain how temporal changes in connectivity contribute to the formation of structure in social networks.<br />
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Mechanisms such as Dual-phase evolution explain how temporal changes in connectivity contribute to the formation of structure in social networks.<br />
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双阶段进化等机制解释了连接性的时间变化如何促进社会网络结构的形成。<br />
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===Criminal networks 犯罪网络===<br />
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In [[criminology]] and [[urban sociology]], much attention has been paid to the social networks among criminal actors. For example, Andrew Papachristos<ref>{{cite journal |last=Papachristos |first=Andrew |year=2009 |title=Murder by Structure: Dominance Relations and the Social Structure of Gang Homicide |journal=American Journal of Sociology |volume=115 |issue=1 |pages=74–128 |doi=10.2139/ssrn.855304 |url=http://www.papachristos.org/Publications_2_files/ajs_final_version.pdf |accessdate=29 March 2013 |url-status=dead |archiveurl=https://web.archive.org/web/20140407094725/http://www.papachristos.org/Publications_2_files/ajs_final_version.pdf |archivedate=7 April 2014 }}</ref> has studied gang murders as a series of exchanges between gangs. Murders can be seen to diffuse outwards from a single source, because weaker gangs cannot afford to kill members of stronger gangs in retaliation, but must commit other violent acts to maintain their reputation for strength.<br />
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In criminology and urban sociology, much attention has been paid to the social networks among criminal actors. For example, Andrew Papachristos has studied gang murders as a series of exchanges between gangs. Murders can be seen to diffuse outwards from a single source, because weaker gangs cannot afford to kill members of stronger gangs in retaliation, but must commit other violent acts to maintain their reputation for strength.<br />
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在犯罪学和城市社会学中，犯罪行为人之间的社会网络问题受到了广泛的关注。例如，安德鲁 · 帕帕克里斯托斯将帮派谋杀研究为帮派之间的一系列交流。谋杀可以被视为从单一来源向外扩散，因为较弱的帮派无法承担杀死较强帮派成员进行报复的代价，而必须采取其他暴力行动来维护他们强大的声誉。<br />
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===Diffusion of innovations 创新产品渗透理论===<br />
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[[Diffusion of innovations|Diffusion of ideas and innovations]] studies focus on the spread and use of ideas from one actor to another or one [[culture]] and another. This line of research seeks to explain why some become "early adopters" of ideas and innovations, and links social network structure with facilitating or impeding the spread of an innovation.<br />
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Diffusion of ideas and innovations studies focus on the spread and use of ideas from one actor to another or one culture and another. This line of research seeks to explain why some become "early adopters" of ideas and innovations, and links social network structure with facilitating or impeding the spread of an innovation.<br />
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思想和创新的传播研究的重点是思想从一个行动者传播到另一个行动者或一种文化和另一种文化。这一系列的研究试图解释为什么有些人成为创意和创新的“早期接受者” ，并将社交网络结构与促进或阻碍创新的传播联系起来。<br />
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===Demography 人口统计===<br />
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In [[demography]], the study of social networks has led to new sampling methods for estimating and reaching populations that are hard to enumerate (for example, homeless people or intravenous drug users.) For example, respondent driven sampling is a network-based sampling technique that relies on respondents to a survey recommending further respondents.<br />
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In demography, the study of social networks has led to new sampling methods for estimating and reaching populations that are hard to enumerate (for example, homeless people or intravenous drug users.) For example, respondent driven sampling is a network-based sampling technique that relies on respondents to a survey recommending further respondents.<br />
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在人口统计学方面，对社会网络的研究导致了新的抽样方法，用于估计和覆盖难以统计的人群(例如，无家可归者或静脉注射毒品者)例如，受访者驱动的抽样是一种基于网络的抽样技术，依赖于调查的受访者推荐更多的受访者。<br />
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===Economic sociology 经济社会学===<br />
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The field of [[sociology]] focuses almost entirely on networks of outcomes of social interactions. More narrowly, [[economic sociology]] considers behavioral interactions of individuals and groups through [[social capital]] and social "markets". Sociologists, such as Mark Granovetter, have developed core principles about the interactions of social structure, information, ability to punish or reward, and trust that frequently recur in their analyses of political, economic and other institutions. Granovetter examines how social structures and social networks can affect economic outcomes like hiring, price, productivity and innovation and describes sociologists' contributions to analyzing the impact of social structure and networks on the economy.<ref>{{cite journal |last=Granovetter |first=Mark |year=2005 |title=The Impact of Social Structure on Economic Outcomes |journal=The Journal of Economic Perspectives |jstor=4134991 |volume=19 |issue=1 |pages=33–50|doi=10.1257/0895330053147958 }}</ref><br />
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The field of sociology focuses almost entirely on networks of outcomes of social interactions. More narrowly, economic sociology considers behavioral interactions of individuals and groups through social capital and social "markets". Sociologists, such as Mark Granovetter, have developed core principles about the interactions of social structure, information, ability to punish or reward, and trust that frequently recur in their analyses of political, economic and other institutions. Granovetter examines how social structures and social networks can affect economic outcomes like hiring, price, productivity and innovation and describes sociologists' contributions to analyzing the impact of social structure and networks on the economy.<br />
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社会学领域几乎完全关注社会互动的结果网络。更狭义地说，经济社会学通过社会资本和社会“市场”考虑个人和群体的行为互动。社会学家，如马克·格兰诺维特，已经发展出关于社会结构、信息、惩罚或奖励能力和信任相互作用的核心原则，这些原则在他们对政治、经济和其他制度的分析中经常出现。格兰诺维特研究了社会结构和社会网络如何影响经济结果，如雇佣、价格、生产力和创新，并描述了社会学家对分析社会结构和网络对经济的影响的贡献。<br />
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===Health care 医疗保健===<br />
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Analysis of social networks is increasingly incorporated into [[health care analytics]], not only in [[epidemiology|epidemiological]] studies but also in models of [[Health Communication|patient communication]] and education, disease prevention, mental health diagnosis and treatment, and in the study of health care organizations and [[health care systems|systems]].<ref>Levy, Judith and Bernice Pescosolido (2002). ''Social Networks and Health''. Boston, MA: JAI Press.</ref><br />
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Analysis of social networks is increasingly incorporated into health care analytics, not only in epidemiological studies but also in models of patient communication and education, disease prevention, mental health diagnosis and treatment, and in the study of health care organizations and systems.<br />
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社会网络分析不仅在流行病学研究中，而且在病人沟通和教育模型、疾病预防、心理健康诊断和治疗以及卫生保健组织和系统的研究中，越来越多地被纳入卫生保健分析中。<br />
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===Human ecology 人类生态学===<br />
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[[Human ecology]] is an [[interdisciplinary]] and [[transdisciplinary]] study of the relationship between [[human]]s and their [[natural environment|natural]], [[Social environment|social]], and [[built environment]]s. The scientific philosophy of human ecology has a diffuse history with connections to [[geography]], [[sociology]], [[psychology]], [[anthropology]], [[zoology]], and natural [[ecology]].<ref>Crona, Beatrice and Klaus Hubacek (eds.) (2010). [http://www.ecologyandsociety.org/issues/view.php?sf=48 "Special Issue: Social network analysis in natural resource governance"]. ''Ecology and Society'', 48.</ref><ref>Ernstson, Henrich (2010). "Reading list: Using social network analysis (SNA) in social-ecological studies". [http://rs.resalliance.org/2010/11/03/reading-list-using-social-network-analysis-sna-in-social-ecological-studies/ ''Resilience Science'']</ref><br />
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Human ecology is an interdisciplinary and transdisciplinary study of the relationship between humans and their natural, social, and built environments. The scientific philosophy of human ecology has a diffuse history with connections to geography, sociology, psychology, anthropology, zoology, and natural ecology.<br />
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人类生态学是一门跨学科、跨学科的学科，研究人类与自然、社会和人造环境之间的关系。人类生态学的科学哲学与地理学、社会学、心理学、人类学、动物学和自然生态学有着密切的联系。<br />
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===Language and linguistics 语言与语言学===<br />
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Studies of [[language]] and [[linguistics]], particularly [[evolutionary linguistics]], focus on the development of [[Morphology (linguistics)|linguistic forms]] and transfer of changes, [[Phonology|sounds]] or words, from one language system to another through networks of social interaction. Social networks are also important in [[language shift]], as groups of people add and/or abandon languages to their repertoire.<br />
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Studies of language and linguistics, particularly evolutionary linguistics, focus on the development of linguistic forms and transfer of changes, sounds or words, from one language system to another through networks of social interaction. Social networks are also important in language shift, as groups of people add and/or abandon languages to their repertoire.<br />
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语言学和语言学的研究，特别是演化语言学，集中在语言形式的发展和变化，声音或词汇的转移，从一个语言系统到另一个通过社会互动的网络。社交网络在语言转换中也很重要，因为一群人增加或者放弃了他们的语言。<br />
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===Literary networks 文学网络===<br />
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In the study of literary systems, network analysis has been applied by Anheier, Gerhards and Romo,<ref>{{cite journal | last1 = Anheier | first1 = H. K. | last2 = Romo | first2 = F. P. | year = 1995 | title = Forms of capital and social structure of fields: examining Bourdieu's social topography | url = | journal = American Journal of Sociology | volume = 100 | issue = 4| pages = 859–903 | doi=10.1086/230603}}</ref> De Nooy,<ref>{{cite journal | last1 = De Nooy | first1 = W | year = 2003| title = Fields and networks: Correspondence analysis and social network analysis in the framework of Field Theory | url = | journal = Poetics | volume = 31 | issue = 5–6| pages = 305–327 | doi = 10.1016/S0304-422X(03)00035-4 }}</ref> and Senekal,<ref>{{cite journal | last1 = Senekal | first1 = B. A. | year = 2012 | title = Die Afrikaanse literêre sisteem: ʼn Eksperimentele benadering met behulp van Sosiale-netwerk-analise (SNA) | url = | journal = LitNet Akademies | volume = 9 | issue = | page = 3 }}</ref> to study various aspects of how literature functions. The basic premise is that polysystem theory, which has been around since the writings of [[Even-Zohar]], can be integrated with network theory and the relationships between different actors in the literary network, e.g. writers, critics, publishers, literary histories, etc., can be mapped using [[Computer graphics (computer science)|visualization]] from SNA.<br />
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In the study of literary systems, network analysis has been applied by Anheier, Gerhards and Romo, De Nooy, and Senekal, to study various aspects of how literature functions. The basic premise is that polysystem theory, which has been around since the writings of Even-Zohar, can be integrated with network theory and the relationships between different actors in the literary network, e.g. writers, critics, publishers, literary histories, etc., can be mapped using visualization from SNA.<br />
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在文学体系的研究中，网络分析被阿海尔、格尔哈兹和罗姆、努伊和 Senekal 应用于研究文学功能的各个方面。其基本前提是多元系统理论可以与网络理论以及文学网络中不同行为者之间的关系相结合。作家、评论家、出版商、文学史等等，都可以通过使用 SNA 的可视化来绘制地图。<br />
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===Organizational studies 组织研究===<br />
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Research studies of [[Formal organization|formal]] or [[informal organization]] [[Social relation|relationships]], [[organizational communication]], [[economics]], [[economic sociology]], and other [[resource]] [[Transfer function|transfers]]. Social networks have also been used to examine how organizations interact with each other, characterizing the many [[Interlocking directorate|informal connections]] that link executives together, as well as associations and connections between individual employees at different organizations.<ref>{{cite journal | last1 = Podolny | first1 = J. M. | last2 = Baron | first2 = J. N. | year = 1997 | title = Resources and relationships: Social networks and mobility in the workplace | url = | journal = American Sociological Review | volume = 62 | issue = 5| pages = 673–693 | doi=10.2307/2657354| jstor = 2657354 | citeseerx = 10.1.1.114.6822 }}</ref> Intra-organizational networks have been found to affect [[organizational commitment]],<ref>{{cite journal | last1 = Lee | first1 = J. | last2 = Kim | first2 = S. | year = 2011 | title = Exploring the role of social networks in affective organizational commitment: Network centrality, strength of ties, and structural holes | url = | journal = The American Review of Public Administration | volume = 41 | issue = 2| pages = 205–223 | doi=10.1177/0275074010373803}}</ref> [[organizational identification]],<ref name="Jone11">{{cite journal | last1 = Jones | first1 = C. | last2 = Volpe | first2 = E.H. | year = 2011 | title = Organizational identification: Extending our understanding of social identities through social networks | url = | journal = Journal of Organizational Behavior | volume = 32 | issue = 3| pages = 413–434 | doi=10.1002/job.694}}</ref> [[Organizational citizenship behavior|interpersonal citizenship behaviour]].<ref>{{cite journal | last1 = Bowler | first1 = W. M. | last2 = Brass | first2 = D. J. | year = 2011 | title = Relational correlates of interpersonal citizenship behaviour: A social network perspective | doi = 10.1037/0021-9010.91.1.70 | pmid = 16435939 | journal = Journal of Applied Psychology | volume = 91 | issue = 1| pages = 70–82 | citeseerx = 10.1.1.516.8746 }}</ref><br />
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Research studies of formal or informal organization relationships, organizational communication, economics, economic sociology, and other resource transfers. Social networks have also been used to examine how organizations interact with each other, characterizing the many informal connections that link executives together, as well as associations and connections between individual employees at different organizations. Intra-organizational networks have been found to affect organizational commitment, organizational identification, interpersonal citizenship behaviour.<br />
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研究正式或非正式的组织关系，组织沟通，经济，经济社会学和其他资源转移。社交网络也被用来研究组织之间如何相互作用，描述了许多非正式的联系，以及不同组织的个别雇员之间的联系和联系。研究发现，组织内网络对组织承诺、组织认同、人际公民行为有显著影响。<br />
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===Social capital 社会资本===<br />
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[[Social capital]] is a form of [[Capital (economics)|economic]] and [[cultural capital]] in which social networks are central, [[Stock and flow|transactions]] are marked by [[Reciprocity (social psychology)|reciprocity]], [[Trust (social sciences)|trust]], and [[cooperation]], and [[Market (economics)|market]] [[Agent (economics)|agents]] produce [[goods and services]] not mainly for themselves, but for a [[common good]]. [[Social capital]] is split into three dimensions: the structural, the relational and the cognitive dimension. The structural dimension describes how partners interact with each other and which specific partners meet in a social network. Also The structural dimension of social capital indicates the level of ties among organizations.<ref>(Claridge, 2018).</ref>. This dimension is highly connected to the relational dimension which refers to trustworthiness, norms, expectations and idenfications of the bonds between partners.The relational dimension explains the nature of these ties which is mainly illustrated by the level of trust accorded to the network of organizations. <ref>(Claridge, 2018).</ref> The cognitive dimension analyses the extent to which organizations share common goals and objectives as a result of their ties and interactions. <ref>(Claridge, 2018).</ref><br />
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Social capital is a form of economic and cultural capital in which social networks are central, transactions are marked by reciprocity, trust, and cooperation, and market agents produce goods and services not mainly for themselves, but for a common good. Social capital is split into three dimensions: the structural, the relational and the cognitive dimension. The structural dimension describes how partners interact with each other and which specific partners meet in a social network. Also The structural dimension of social capital indicates the level of ties among organizations.. This dimension is highly connected to the relational dimension which refers to trustworthiness, norms, expectations and idenfications of the bonds between partners.The relational dimension explains the nature of these ties which is mainly illustrated by the level of trust accorded to the network of organizations. The cognitive dimension analyses the extent to which organizations share common goals and objectives as a result of their ties and interactions. <br />
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社会资本是一种经济和文化资本，其中社会网络是中心，交易以互惠、信任和合作为特征，市场代理人生产的商品和服务不主要是为他们自己，而是为了共同的利益。社会资本分为三个维度: 结构维度、关系维度和认知维度。结构维度描述了合作伙伴之间如何相互作用，以及哪些特定的合作伙伴在社交网络中相遇。社会资本的结构维度反映了组织之间的关系水平。这个维度与关系维度高度相关，关系维度指的是伴侣之间关系的可信度、规范、期望和相同性。 关系维度解释了这些联系的本质，主要表现在对组织网络的信任程度上。认知维度分析组织因其联系和相互作用而在多大程度上共享共同的目标和目的。<br />
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[[Social capital]] is a sociological concept about the value of [[social relation]]s and the role of cooperation and confidence to achieve positive outcomes. The term refers to the value one can get from their social ties. For example, newly arrived immigrants can make use of their social ties to established migrants to acquire jobs they may otherwise have trouble getting (e.g., because of unfamiliarity with the local language). A positive relationship exists between social capital and the intensity of social network use.<ref>{{cite journal|last=Sebastián|first=Valenzuela|author2=Namsu Park |author3=Kerk F. Kee |title=Is There Social Capital in a Social Network Site? Facebook Use and College Students' Life Satisfaction, Trust, and Participation|journal=Journal of Computer-Mediated Communication|year=2009|volume=14|issue=4|pages=875–901|doi=10.1111/j.1083-6101.2009.01474.x|doi-access=free}}</ref><ref>{{cite journal | title=Social Connectivity in America: Changes in Adult Friendship Network Size from 2002 to 2007 |author1=Wang, Hua |author2=Barry Wellman |lastauthoramp=yes | journal=American Behavioral Scientist | year=2010 | volume=53 | issue=8 | pages=1148–1169 | doi=10.1177/0002764209356247|url=https://semanticscholar.org/paper/402068f74efb35c093e7fd8c0f8268521219e242 }}</ref> In a dynamic framework, higher activity in a network feeds into higher social capital which itself encourages more activity.<ref name="GaudeulGiannetti2013">{{cite journal|last1=Gaudeul|first1=Alexia|last2=Giannetti|first2=Caterina|title=The role of reciprocation in social network formation, with an application to LiveJournal|journal=Social Networks|volume=35|issue=3|year=2013|pages=317–330|issn=03788733|doi=10.1016/j.socnet.2013.03.003}}</ref><br />
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Social capital is a sociological concept about the value of social relations and the role of cooperation and confidence to achieve positive outcomes. The term refers to the value one can get from their social ties. For example, newly arrived immigrants can make use of their social ties to established migrants to acquire jobs they may otherwise have trouble getting (e.g., because of unfamiliarity with the local language). A positive relationship exists between social capital and the intensity of social network use. In a dynamic framework, higher activity in a network feeds into higher social capital which itself encourages more activity.<br />
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社会资本是一个关于社会关系的价值和合作与信心的作用的社会学概念，以实现积极的成果。这个术语指的是一个人可以从他们的社会关系中获得的价值。例如，新来的移民可以利用他们与老移民的社会关系，获得他们在其他情况下可能难以获得的工作(例如，因为不熟悉当地语言)。社会资本与社会网络使用强度存在正相关关系。在一个动态的框架中，网络中更高的活动会产生更高的社会资本，而社会资本本身又会鼓励更多的活动。<br />
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=== Advertising ===<br />
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=== Advertising ===<br />
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广告<br />
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This particular cluster focuses on brand-image and promotional strategy effectiveness, taking into account the impact of customer participation on sales and brand-image. This is gauged through techniques such as sentiment analysis which rely on mathematical areas of study such as data mining and analytics. This area of research produces vast numbers of commercial applications as the main goal of any study is to understand consumer behaviour and drive sales.<br />
<br />
This particular cluster focuses on brand-image and promotional strategy effectiveness, taking into account the impact of customer participation on sales and brand-image. This is gauged through techniques such as sentiment analysis which rely on mathematical areas of study such as data mining and analytics. This area of research produces vast numbers of commercial applications as the main goal of any study is to understand consumer behaviour and drive sales.<br />
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这个集群侧重于品牌形象和促销策略的有效性，同时考虑到顾客参与对销售和品牌形象的影响。这是通过诸如情感分析等技术来衡量的，这些技术依赖于数学领域的研究，如数据挖掘和分析。这一研究领域产生了大量的商业应用，因为任何研究的主要目标都是了解消费者行为并推动销售。<br />
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===Network position and benefits===<br />
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===Network position and benefits===<br />
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网络地位和利益<br />
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In many [[Formal organizations|organizations]], members tend to focus their activities inside their own groups, which stifles creativity and restricts opportunities. A player whose network bridges structural holes has an advantage in detecting and developing rewarding opportunities.<ref name="Burt 2004"/> Such a player can mobilize social capital by acting as a "broker" of information between two clusters that otherwise would not have been in contact, thus providing access to new ideas, opinions and opportunities. British philosopher and political economist [[John Stuart Mill]], writes, "it is hardly possible to overrate the value ... of placing human beings in contact with persons dissimilar to themselves.... Such communication [is] one of the primary sources of progress."<ref>{{cite book|last=Mill|first=John|title=Principles of Political Economy|year=1909|publisher=William J Ashley|location=Library of Economics and Liberty}}</ref> Thus, a player with a network rich in structural holes can add value to an organization through new ideas and opportunities. This in turn, helps an individual's career development and advancement.<br />
<br />
In many organizations, members tend to focus their activities inside their own groups, which stifles creativity and restricts opportunities. A player whose network bridges structural holes has an advantage in detecting and developing rewarding opportunities. Thus, a player with a network rich in structural holes can add value to an organization through new ideas and opportunities. This in turn, helps an individual's career development and advancement.<br />
<br />
在许多组织中，成员倾向于把他们的活动集中在他们自己的团队中，这抑制了创造力和限制了机会。一个玩家的网络桥梁结构洞有优势，在检测和发展奖励机会。因此，一个拥有丰富结构性漏洞的网络的玩家可以通过新的想法和机会为组织增加价值。这反过来又有助于个人的职业发展和晋升。<br />
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A social capital broker also reaps control benefits of being the facilitator of information flow between contacts. In the case of consulting firm Eden McCallum, the founders were able to advance their careers by bridging their connections with former big three consulting firm consultants and mid-size industry firms.<ref>{{cite journal|last=Gardner|first=Heidi|author2=Eccles, Robert |title=Eden McCallum: A Network Based Consulting Firm|journal=Harvard Business School Review|year=2011}}</ref> By bridging structural holes and mobilizing social capital, players can advance their careers by executing new opportunities between contacts.<br />
<br />
A social capital broker also reaps control benefits of being the facilitator of information flow between contacts. In the case of consulting firm Eden McCallum, the founders were able to advance their careers by bridging their connections with former big three consulting firm consultants and mid-size industry firms. By bridging structural holes and mobilizing social capital, players can advance their careers by executing new opportunities between contacts.<br />
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社会资本经纪人还可以从促进联系人之间的信息流动中获得控制权利。在咨询公司 Eden McCallum 的案例中，创始人能够通过与前三大咨询公司顾问和中等规模的行业公司建立联系来提升自己的职业生涯。通过弥合结构漏洞和调动社会资本，玩家可以通过在联系人之间执行新的机会来提升自己的职业生涯。<br />
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There has been research that both substantiates and refutes the benefits of information brokerage. A study of high tech Chinese firms by Zhixing Xiao found that the control benefits of structural holes are "dissonant to the dominant firm-wide spirit of cooperation and the information benefits cannot materialize due to the communal sharing values" of such organizations.<ref>Xiao, Zhixing; Tsui, Anne (2007). "When Brokers May Not Work: The Cultural Contingency of [http://iniciarsesionentrar.com/facebook/ Social] Capital in Chinese High-tech Firms". ''Administrative Science Quarterly''.</ref> However, this study only analyzed Chinese firms, which tend to have strong communal sharing values. Information and control benefits of structural holes are still valuable in firms that are not quite as inclusive and cooperative on the firm-wide level. In 2004, Ronald Burt studied 673 managers who ran the supply chain for one of America's largest electronics companies. He found that managers who often discussed issues with other groups were better paid, received more positive job evaluations and were more likely to be promoted.<ref name="Burt 2004"/> Thus, bridging structural holes can be beneficial to an organization, and in turn, to an individual's career.<br />
<br />
There has been research that both substantiates and refutes the benefits of information brokerage. A study of high tech Chinese firms by Zhixing Xiao found that the control benefits of structural holes are "dissonant to the dominant firm-wide spirit of cooperation and the information benefits cannot materialize due to the communal sharing values" of such organizations. However, this study only analyzed Chinese firms, which tend to have strong communal sharing values. Information and control benefits of structural holes are still valuable in firms that are not quite as inclusive and cooperative on the firm-wide level. In 2004, Ronald Burt studied 673 managers who ran the supply chain for one of America's largest electronics companies. He found that managers who often discussed issues with other groups were better paid, received more positive job evaluations and were more likely to be promoted. Thus, bridging structural holes can be beneficial to an organization, and in turn, to an individual's career.<br />
<br />
有研究证实和反驳了信息经纪的好处。知行晓对中国高科技企业的一项研究发现，结构性漏洞的控制效益“与占主导地位的企业合作精神不协调，由于这类组织的共享价值观，信息效益不能实现”。然而，这项研究只分析了中国企业，这些企业往往有很强的公共共享价值观。结构性漏洞的信息和控制利益对于那些在整个公司层面上不那么具有包容性和合作性的公司来说仍然是有价值的。2004年，罗纳德 · 伯特研究了673名管理者，他们为美国最大的电子公司之一管理供应链。他发现，那些经常与其他团体讨论问题的管理人员收入更高，得到的工作评价更积极，也更有可能获得晋升。因此，架起结构性漏洞对组织有益，反过来对个人的职业生涯也有益。<br />
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===Social media===<br />
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===Social media===<br />
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社交媒体<br />
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{{main article|Social media}}<br />
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[[Computer network]]s combined with social networking software produces a new medium for social interaction.<ref>{{Cite book|title=The International Encyclopedia of Media Effects|last=Amichai-Hamburger|first=Yair|last2=Hayat|first2=Tsahi|date=2017|publisher=John Wiley & Sons, Inc.|isbn=9781118783764|language=en|doi=10.1002/9781118783764.wbieme0170}}</ref> A relationship over a computerized [[social networking service]] can be characterized by context, direction, and strength. The content of a relation refers to the resource that is exchanged. In a [[computer mediated communication]] context, social pairs exchange different kinds of information, including sending a data file or a computer program as well as providing emotional support or arranging a meeting. With the rise of [[electronic commerce]], information exchanged may also correspond to exchanges of money, goods or services in the "real" world.<ref>{{cite journal | title=Studying Online Social Networks | last1=Garton|first1=Laura|first2=Caroline |last2=Haythornthwaite|author2-link=Caroline Haythornthwaite|first3=Barry|last3=Wellman|author3-link=Barry Wellman | journal=Journal of Computer-Mediated Communication |date=23 June 2006 | volume=3 | issue=1 |pages = 0| doi=10.1111/j.1083-6101.1997.tb00062.x}}</ref> [[Social network analysis]] methods have become essential to examining these types of computer mediated communication.<br />
<br />
Computer networks combined with social networking software produces a new medium for social interaction. A relationship over a computerized social networking service can be characterized by context, direction, and strength. The content of a relation refers to the resource that is exchanged. In a computer mediated communication context, social pairs exchange different kinds of information, including sending a data file or a computer program as well as providing emotional support or arranging a meeting. With the rise of electronic commerce, information exchanged may also correspond to exchanges of money, goods or services in the "real" world. Social network analysis methods have become essential to examining these types of computer mediated communication.<br />
<br />
计算机网络与社交网络软件的结合为社交互动提供了一种新的媒介。通过计算机化的社交网络服务建立起来的关系可以是拥有属性、方向和力量。关系的内容指的是交换的资源。在计算机中介的交流环境中，社会对交换不同种类的信息，包括发送数据文件或计算机程序，以及提供情感支持或安排会议。随着电子商务的兴起，交换的信息也可能对应于“现实”世界中的货币、商品或服务的交换。社会网络分析方法已经成为检验这些类型的计算机中介交流的必要手段。<br />
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In addition, the sheer size and the volatile nature of [[social media]] has given rise to new network metrics. A key concern with networks extracted from social media is the lack of robustness of network metrics given missing data.<ref>{{cite journal | last1 = Wei | first1 = Wei | last2 = Joseph | first2 = Kenneth | last3 = Liu | first3 = Huan | last4 = Carley | first4 = Kathleen M. | year = 2016 | title = Exploring Characteristics of Suspended Users and Network Stability on Twitter | url = | journal = Social Network Analysis and Mining | volume = 6 | issue = | page = 51 | doi = 10.1007/s13278-016-0358-5 }}</ref><br />
<br />
In addition, the sheer size and the volatile nature of social media has given rise to new network metrics. A key concern with networks extracted from social media is the lack of robustness of network metrics given missing data.<br />
<br />
此外，社交媒体的庞大规模和不稳定性也催生了新的网络指标。从社交媒体中提取网络的一个关键问题是，在缺失数据的情况下，网络指标缺乏健壮性。<br />
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==See also==<br />
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==See also==<br />
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参见<br />
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* [[Bibliography of sociology]]<br />
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* [[Business networking]]<br />
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* [[Collective network]]<br />
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* [[International Network for Social Network Analysis]]<br />
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* [[Network society]]<br />
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* [[Network theory]]<br />
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*[[Network science]]<br />
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* [[Semiotics of social networking]]<br />
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* [[Scientific collaboration network]]<br />
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* [[Social network analysis]]<br />
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* [[Social network (sociolinguistics)]]<br />
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* [[Social networking service]]<br />
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* [[Social web]]<br />
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* [[Structural fold]]<br />
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==References==<br />
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==References==<br />
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参考资料<br />
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{{reflist}}<br />
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==Further reading==<br />
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==Further reading==<br />
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进一步阅读<br />
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* {{cite journal |last1=Aneja |first1=Nagender |last2=Gambhir |first2=Sapna |title=Ad-hoc Social Network: A Comprehensive Survey |url=http://www.ijser.org/researchpaper%5CAd-hoc-Social-Network-A-Comprehensive-Survey.pdf |journal=International Journal of Scientific & Engineering Research |volume=4 |issue=8 |date=August 2013 |pages=156–160|issn=2229-5518}}<br />
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* {{cite book |last=Barabási |first=Albert-László |year=2003 |title=Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life |publisher=Plum |isbn=978-0-452-28439-5 |url-access=registration |url=https://archive.org/details/linkedhoweveryth00bara }}<br />
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* {{cite book |last=Barnett |first=George A. |year=2011 |title=Encyclopedia of Social Networks |publisher=Sage |isbn=978-1-4129-7911-5}}<br />
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* {{cite book |last=Estrada |first=E |year=2011 |title=The Structure of Complex Networks: Theory and Applications |publisher=Oxford University Press |isbn=978-0-199-59175-6}}<br />
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* {{cite book |last=Ferguson |first=Niall |year=2018 |title=The Square and the Tower: Networks and Power, from the Freemasons to Facebook |isbn=978-0735222915 |publisher=Penguin Press}}<br />
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* {{cite book |last=Freeman |first=Linton C. |year=2004 |title=The Development of Social Network Analysis: A Study in the Sociology of Science |publisher=Empirical Press |isbn=978-1-59457-714-7}}<br />
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* {{cite book |last=Kadushin |first=Charles |year=2012 |title=Understanding Social Networks: Theories, Concepts, and Findings |location= |publisher=Oxford University Press |isbn=978-0-19-537946-4}}<br />
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* {{cite book |last1=Mauro |first1=Rios |last2=Petrella |first2=Carlos |year=2014 |title=La Quimera de las Redes Sociales |trans-title=The Chimera of Social Networks |language=Spanish |publisher=Bubok España |isbn=978-9974-99-637-3}}<br />
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* {{cite book |last1=Rainie |first1=Lee |last2=Wellman |first2=Barry |year=2012 |title=Networked: The New Social Operating System |url=https://archive.org/details/networkednewsoci0000rain |url-access=registration |location=Cambridge, Mass. |publisher=MIT Press |isbn=978-0262017190}}<br />
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* {{cite book |last=Scott |first=John |year=1991 |title=Social Network Analysis: A Handbook |publisher=Sage |isbn=978-0-7619-6338-7}}<br />
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* {{cite book |last1=Wasserman |first1=Stanley |last2=Faust |first2=Katherine |year=1994 |title=Social Network Analysis: Methods and Applications |publisher=Cambridge University Press |isbn=978-0-521-38269-4 |series=Structural Analysis in the Social Sciences }}<br />
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* {{cite book |last1=Wellman |first1=Barry |last2=Berkowitz |first2=S. D. |year=1988 |title=Social Structures: A Network Approach |series=Structural Analysis in the Social Sciences |publisher=Cambridge University Press |isbn=978-0-521-24441-1}}<br />
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==External links==<br />
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==External links==<br />
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外部链接<br />
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{{Commons category|Social networks}}<br />
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===Organizations===<br />
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组织<br />
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* [http://www.insna.org/ International Network for Social Network Analysis]<br />
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===Peer-reviewed journals===<br />
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===Peer-reviewed journals===<br />
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同行评审期刊<br />
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* ''[http://www.sciencedirect.com/science/journal/03788733 Social Networks]''<br />
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* ''[http://journals.cambridge.org/action/displaySpecialPage?pageId=3656 Network Science]''<br />
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* ''[http://www.cmu.edu/joss/content/articles/volindex.html Journal of Social Structure]''<br />
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* ''[http://www.tandfonline.com/toc/gmas20/current Journal of Mathematical Sociology]''<br />
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* ''[https://www.springer.com/computer/database+management+%26+information+retrieval/journal/13278 Social Network Analysis and Mining (SNAM)]''<br />
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* {{cite journal |title=INSNA - ''Connections'' Journal |journal=Connections Bulletin of the International Network for Social Network Analysis |location=Toronto |publisher=International Network for Social Network Analysis |url=http://www.insna.org/pubs/connections/ |issn=0226-1766 |url-status=dead |archiveurl=https://web.archive.org/web/20130718063641/http://www.insna.org/pubs/connections/ |archivedate=2013-07-18 }}<br />
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===Textbooks and educational resources===<br />
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===Textbooks and educational resources===<br />
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课本及教育资源<br />
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* ''[http://www.cs.cornell.edu/home/kleinber/networks-book/ Networks, Crowds, and Markets]'' (2010) by D. Easley & J. Kleinberg<br />
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* ''[http://faculty.ucr.edu/~hanneman/nettext/ Introduction to Social Networks Methods]'' (2005) by R. Hanneman & M. Riddle<br />
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* [http://www.analytictech.com/networks/ Social Network Analysis Instructional Web Site] by S. Borgatti<br />
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* ''[https://web.archive.org/web/20160304063126/http://www.novagob.org/file/view/153259/ebook-gratuito-sobre-redes-sociales-virtuales Guide for virtual social networks for public administrations]'' (2015) by Mauro D. Ríos (in Spanish)<br />
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===Data sets===<br />
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===Data sets===<br />
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数据集<br />
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* [http://pajek.imfm.si/doku.php?id=data:urls:index Pajek's list of lists of datasets]<br />
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* [http://networkdata.ics.uci.edu/index.html UC Irvine Network Data Repository]<br />
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* [http://snap.stanford.edu/data/ Stanford Large Network Dataset Collection]<br />
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* [http://www-personal.umich.edu/~mejn/netdata/ M.E.J. Newman datasets]<br />
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* [http://vlado.fmf.uni-lj.si/pub/networks/data/ Pajek datasets]<br />
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* [http://wiki.gephi.org/index.php?title=Datasets#Social_networks Gephi datasets]<br />
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* [http://konect.uni-koblenz.de/networks KONECT – Koblenz network collection]<br />
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* [http://www.stats.ox.ac.uk/~snijders/siena/ RSiena datasets]<br />
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==编者推荐==<br />
集智俱乐部：[https://swarma.org/?p=18646 社会网络分析的四个维度：研究方法、应用和软件工具综述 | 网络科学论文速递25篇]<br />
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[http://arxiv.org/abs/2002.09485 The Four Dimensions of Social Network Analysis: An Overview of Research Methods, Applications, and Software Tools]：近年来，基于社交网络的应用呈指数级增长，其中一个方面的原因是，这个应用领域提供了一个特别肥沃的地方，可以测试和开发最先进的计算技术，从 Web 中提取有价值的信息。这项工作的主要贡献有三个方面：(1) 我们对社会网络分析 (SNA) 的最新发展水平进行了文献综述；(2) 我们提出了一套基于 SNA 四个基本特征（或维度）的新度量标准；(3) 最后，我们对一套流行的 SNA 工具和框架进行了定量分析。我们还进行了一项科学计量学研究，以检测该领域中最活跃的研究领域和应用领域。本工作提出了四个不同维度的定义，即(a)模式、(b)知识发现、(c)信息融合、(d)集成、可伸缩性和可视化，用于定义一组新的度量（称为度），以评估 SNA 的不同软件工具和框架（根据之前的度量对一组 20 个 SNA-软件工具进行分析和排序）。这些维度连同定义的程度，可以评价和衡量社会网络技术的成熟度，寻求对它们的定量评估，从而揭示这一活跃领域的挑战和未来发展趋势。<br />
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[[Category:Social networks| ]]<br />
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[[Category:Communication theory]]<br />
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范畴: 复杂系统理论<br />
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范畴: 网络理论<br />
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类别: 社会系统<br />
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[[Category:Sociological terminology]]<br />
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类别: 社会学术语<br />
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[[Category:Sociological theories]]<br />
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范畴: 系统论<br />
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<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Social network]]. Its edit history can be viewed at [[社会网络/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E7%A4%BE%E4%BC%9A%E7%BD%91%E7%BB%9C&diff=11880社会网络2020-08-09T15:51:09Z<p>Dorr：</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Social structure made up of a set of social actors}}<br />
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{{about|the theoretical concept as used in the social and behavioral sciences|social networking sites|Social networking service|the 2010 movie|The Social Network|other uses|Social network (disambiguation)}}<br />
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{{Sociology}}<br />
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{{Network Science}}<br />
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A '''social network ''' is a [[social structure]] made up of a set of [[social]] actors (such as individuals or organizations), sets of [[Dyad (sociology)|dyadic]] ties, and other [[Social relation|social interactions]] between actors. The social network perspective provides a set of methods for analyzing the structure of whole social entities as well as a variety of theories explaining the patterns observed in these structures.<ref name="WF94CH1">{{cite book|last1=Wasserman |first1=Stanley | author-link1= Stanley Wasserman|last2=Faust |first2=Katherine |year=1994 |title=Social Network Analysis: Methods and Applications |isbn=9780521387071 |chapter=Social Network Analysis in the Social and Behavioral Sciences |pages=1–27 |publisher=Cambridge University Press}}</ref> The study of these structures uses [[social network analysis]] to identify local and global patterns, locate influential entities, and examine network dynamics.<br />
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A social network is a social structure made up of a set of social actors (such as individuals or organizations), sets of dyadic ties, and other social interactions between actors. The social network perspective provides a set of methods for analyzing the structure of whole social entities as well as a variety of theories explaining the patterns observed in these structures. The study of these structures uses social network analysis to identify local and global patterns, locate influential entities, and examine network dynamics.<br />
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'''<font color="#FFD700">社会网络 Social Network</font>'''是一种由一组社会行为者（个人或组织）、一组二元关系以及行为者之间的其他社会互动组成的社会结构。社会网络视角为分析整个社会实体的结构提供了一套方法，也为解释在这些结构中观察到的模式提供了各种理论。对这些结构的研究使用'''<font color="#FFD700">社会网络分析 Social Network Analysis</font>'''以确定本地及全球模式，定位有影响力的实体，并审查网络动态。<br />
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Social networks and the analysis of them is an inherently [[Interdisciplinarity|interdisciplinary]] academic field which emerged from [[social psychology]], [[sociology]], [[statistics]], and [[graph theory]]. [[Georg Simmel]] authored early structural theories in sociology emphasizing the dynamics of triads and "web of group affiliations".<ref>{{cite book|last1=Scott |first1=W. Richard |last2=Davis |first2=Gerald F. |author-link1= William Richard Scott|author-link2= Gerald F. Davis|title=Organizations and Organizing |chapter=Networks In and Around Organizations |year=2003 |isbn=978-0-13-195893-7 |publisher=Pearson Prentice Hall}}</ref> [[Jacob Moreno]] is credited with developing the first [[sociogram]]s in the 1930s to study interpersonal relationships. These approaches were mathematically formalized in the 1950s and theories and methods of social networks became pervasive in the [[Social science|social and behavioral sciences]] by the 1980s.<ref name="WF94CH1"/><ref name="Freeman History">{{cite book|last=Freeman |first=Linton |year=2004 |publisher=Empirical Press |isbn=978-1-59457-714-7 |title=The Development of Social Network Analysis: A Study in the Sociology of Science}}</ref> [[Social network analysis]] is now one of the major paradigms in contemporary sociology, and is also employed in a number of other social and formal sciences. Together with other [[complex network]]s, it forms part of the nascent field of [[network science]].<ref>{{cite journal|journal=Science |year=2009 |volume=323 |number=5916 |pages=892–895 |doi=10.1126/science.1165821 |pmid=19213908 |title=Network Analysis in the Social Sciences |first1=Stephen P. |last1=Borgatti |first2=Ajay |last2=Mehra |first3=Daniel J. |last3=Brass |first4=Giuseppe |last4=Labianca|bibcode=2009Sci...323..892B |citeseerx=10.1.1.536.5568 }}</ref><ref name="EK">{{cite book|title=Networks, Crowds, and Markets: Reasoning about a Highly Connected World |first1=David |last1=Easley |first2=Jon |last2=Kleinberg |publisher=Cambridge University Press |year=2010 |chapter=Overview |pages=1–20 |isbn=978-0-521-19533-1}}</ref><br />
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Social networks and the analysis of them is an inherently interdisciplinary academic field which emerged from social psychology, sociology, statistics, and graph theory. Georg Simmel authored early structural theories in sociology emphasizing the dynamics of triads and "web of group affiliations". Jacob Moreno is credited with developing the first sociograms in the 1930s to study interpersonal relationships. These approaches were mathematically formalized in the 1950s and theories and methods of social networks became pervasive in the social and behavioral sciences by the 1980s. Social network analysis is now one of the major paradigms in contemporary sociology, and is also employed in a number of other social and formal sciences. Together with other complex networks, it forms part of the nascent field of network science.<br />
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社会网络及其分析是社会心理学、社会学、统计学和图论中内在的跨学科学术领域。'''格奥尔格·齐美尔 Georg Simmel'''是早期社会学结构主义理论的作者，他强调三合会和“群体关系网”的动态性。'''雅各布·莫雷诺 Jacob Moreno'''被认为是在20世纪30年代发展了第一份'''<font color="#FFD700">社交关系图 Sociogram</font>'''以研究人际关系的人。这些方法在20世纪50年代得到数学化，到20世纪80年代，社会网络的理论和方法在社会和行为科学中变得普遍。社会网络分析现在是当代社会学的主要范式之一，也被用于许多其他社会科学及形式科学。它与其他复杂网络一起构成了网络科学新兴领域的一部分。<br />
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==Overview 概览==<br />
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[[File:Barabasi Albert model.gif|thumb|left|Evolution graph of a social network: [[Barabási–Albert model|Barabási model]].图一：一个社会网络的演化图：巴拉巴西模型。]]<br />
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Barabási model.]]<br />
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Barabási model.]]<br />
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The social network is a [[Scientific theory|theoretical]] [[Construct (philosophy of science)|construct]] useful in the [[social sciences]] to study relationships between individuals, [[social groups|groups]], [[formal organizations|organizations]], or even entire [[society|societies]] ([[social unit]]s, see [[Differentiation (sociology)|differentiation]]). The term is used to describe a [[social structure]] determined by such [[social interactions|interactions]]. The ties through which any given social unit connects represent the convergence of the various social contacts of that unit. This theoretical approach is, necessarily, relational. An [[axiom]] of the social network approach to understanding [[social interaction]] is that social phenomena should be primarily conceived and investigated through the properties of relations between and within units, instead of the properties of these units themselves. Thus, one common criticism of social network theory is that [[Agency (sociology)|individual agency]] is often ignored<ref name="jscott">Scott, John P. (2000). ''Social Network Analysis: A Handbook'' (2nd edition). Thousand Oaks, CA: Sage Publications.</ref> although this may not be the case in practice (see [[agent-based model]]ing). Precisely because many different types of relations, singular or in combination, form these network configurations, [[Network science|network analytics]] are useful to a broad range of research enterprises. In social science, these fields of study include, but are not limited to [[anthropology]], [[biology]], [[communication studies]], [[economics]], [[geography]], [[information science]], [[organizational studies]], [[social psychology]], [[sociology]], and [[sociolinguistics]].<br />
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The social network is a theoretical construct useful in the social sciences to study relationships between individuals, groups, organizations, or even entire societies (social units, see differentiation). The term is used to describe a social structure determined by such interactions. The ties through which any given social unit connects represent the convergence of the various social contacts of that unit. This theoretical approach is, necessarily, relational. An axiom of the social network approach to understanding social interaction is that social phenomena should be primarily conceived and investigated through the properties of relations between and within units, instead of the properties of these units themselves. Thus, one common criticism of social network theory is that individual agency is often ignored although this may not be the case in practice (see agent-based modeling). Precisely because many different types of relations, singular or in combination, form these network configurations, network analytics are useful to a broad range of research enterprises. In social science, these fields of study include, but are not limited to anthropology, biology, communication studies, economics, geography, information science, organizational studies, social psychology, sociology, and sociolinguistics.<br />
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社会网络是社会科学中研究个人、团体、组织甚至整个社会（社会单位，见分化）之间关系的理论构造。这个术语用来描述由这种相互作用决定的社会结构。任何一个特定的社会单元之间的联系都代表着这个单元各种社会联系的聚合。这种理论方法必然是相关的。理解社会互动的社会网络方法的一个公理是，社会现象应该主要通过单元之间和单元内部关系的性质来构思和研究，而不是这些单元本身的性质。因此，社会网络理论总被诟病的一点是其常忽视个体代理，而实践中可能并非如此（见基于主体的建模）。正是因为许多不同类型的关系（单独或组合形式）形成这些网络配置，网络分析在广泛的研究中有用。在社会科学中，这些研究领域包括但不限于'''<font color="#FFD700">人类学 Anthropology</font>'''、生物学、'''<font color="#FFD700">传播学 Communication Studies</font>'''、经济学、地理学、信息科学、组织学、社会心理学、社会学和'''<font color="#FFD700">社会语言学 Sociolinguistics</font>'''。<br />
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==History 历史==<br />
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In the late 1890s, both [[Émile Durkheim]] and [[Ferdinand Tönnies]] foreshadowed the idea of social networks in their theories and research of [[social group]]s. Tönnies argued that social groups can exist as personal and direct social ties that either link individuals who share values and belief (''[[Gemeinschaft]]'', German, commonly translated as "[[community]]") or impersonal, formal, and instrumental social links (''[[Gesellschaft]]'', German, commonly translated as "[[society]]").<ref>Tönnies, Ferdinand (1887). ''Gemeinschaft und Gesellschaft'', Leipzig: Fues's Verlag. (Translated, 1957 by Charles Price Loomis as ''Community and Society'', East Lansing: Michigan State University Press.)</ref> Durkheim gave a non-individualistic explanation of social facts, arguing that social phenomena arise when interacting individuals constitute a reality that can no longer be accounted for in terms of the properties of individual actors.<ref>Durkheim, Emile (1893). ''De la division du travail social: étude sur l'organisation des sociétés supérieures'', Paris: F. Alcan. (Translated, 1964, by Lewis A. Coser as ''The Division of Labor in Society'', New York: Free Press.)</ref> [[Georg Simmel]], writing at the turn of the twentieth century, pointed to the nature of networks and the effect of network size on interaction and examined the likelihood of interaction in loosely knit networks rather than groups.<ref>Simmel, Georg (1908). ''Soziologie'', Leipzig: Duncker & Humblot.</ref><br />
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In the late 1890s, both Émile Durkheim and Ferdinand Tönnies foreshadowed the idea of social networks in their theories and research of social groups. Tönnies argued that social groups can exist as personal and direct social ties that either link individuals who share values and belief (Gemeinschaft, German, commonly translated as "community") or impersonal, formal, and instrumental social links (Gesellschaft, German, commonly translated as "society"). Durkheim gave a non-individualistic explanation of social facts, arguing that social phenomena arise when interacting individuals constitute a reality that can no longer be accounted for in terms of the properties of individual actors. Georg Simmel, writing at the turn of the twentieth century, pointed to the nature of networks and the effect of network size on interaction and examined the likelihood of interaction in loosely knit networks rather than groups.<br />
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在19世纪90年代晚期，'''埃米尔·涂尔干 Émile Durkheim'''和'''斐迪南·滕尼斯 Ferdinand Tönnies'''在他们关于社会群体的理论和研究中都预示了社会网络的概念。滕尼斯认为，社会群体可以作为个人和直接的社会关系存在，这种关系或者将具有共同价值观和信仰的个人（德语 Gemeinschaft，通常翻译为“社区”）联系在一起，或者将非个人的、正式的和工具性的社会关系（德语 Gesellschaft，通常翻译为“社会”）联系在一起。涂尔干对社会事实给出了非个人主义的解释，认为当相互作用的个体构成一种再也不能用个体行为者的特性来解释的现实时，社会现象就产生了。格奥尔格·齐美尔在20世纪之交的著作中指出了网络的本质和网络规模对互动的影响，并研究了在松散的网络而非群体中互动的可能性。<br />
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[[File:Moreno Sociogram 2nd Grade.png|thumb|Moreno's sociogram of a 2nd grade class 图二：莫雷诺对一个二年级班级的社会关系图]]<br />
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Major developments in the field can be seen in the 1930s by several groups in psychology, anthropology, and mathematics working independently.<ref name=jscott /><ref>For a historical overview of the development of social network analysis, see: {{cite book|last1=Carrington|first1=Peter J.|last2=Scott|first2=John|chapter=Introduction|title=The Sage Handbook of Social Network Analysis| publisher=Sage|year=2011|isbn=978-1-84787-395-8|page=1|chapter-url=https://books.google.com/books?id=2chSmLzClXgC&pg=PA1}}</ref><ref>See also the diagram in {{cite book|author=Scott, John|title=Social Network Analysis: A Handbook|publisher=Sage|year=2000|isbn=978-0-7619-6339-4|page=8|url=https://books.google.com/books?id=Ww3_bKcz6kgC&pg=PA8}}</ref> In [[psychology]], in the 1930s, [[Jacob L. Moreno]] began systematic recording and analysis of social interaction in small groups, especially classrooms and work groups (see [[sociometry]]). In [[anthropology]], the foundation for social network theory is the theoretical and [[ethnography|ethnographic]] work of [[Bronislaw Malinowski]],<ref>Malinowski, Bronislaw (1913). ''The Family Among the Australian Aborigines: A Sociological Study''. London: University of London Press.</ref> [[Radcliffe-Brown|Alfred Radcliffe-Brown]],<ref>Radcliffe-Brown, Alfred Reginald (1930) ''The social organization of Australian tribes''. Sydney, Australia: University of Sydney ''Oceania'' monographs, No.1.</ref><ref>{{cite journal | last1 = Radcliffe-Brown | first1 = A. R. | year = 1940 | title = On social structure | url = | journal = Journal of the Royal Anthropological Institute | volume = 70 | issue = 1| pages = 1–12 | doi=10.2307/2844197| jstor = 2844197 }}</ref> and [[Claude Lévi-Strauss]].<ref>Lévi-Strauss, Claude ([1947]1967). ''Les structures élémentaires de la parenté''. Paris: La Haye, Mouton et Co. (Translated, 1969 by J. H. Bell, J. R. von Sturmer, and R. Needham, 1969, as ''The Elementary Structures of Kinship'', Boston: Beacon Press.)</ref> A group of social anthropologists associated with [[Max Gluckman]] and the [[Manchester school (anthropology)|Manchester School]], including [[John Arundel Barnes|John A. Barnes]],<ref>Barnes, John (1954). "Class and Committees in a Norwegian Island Parish". ''Human Relations'', (7): 39–58.</ref> [[J. Clyde Mitchell]] and [[Elizabeth Bott Spillius]],<ref>{{cite journal | last1 = Freeman | first1 = Linton C. | last2 = Wellman | first2 = Barry | year = 1995 | title = A note on the ancestoral Toronto home of social network analysis | url = | journal = Connections | volume = 18 | issue = 2| pages = 15–19 }}</ref><ref>{{cite journal | last1 = Savage | first1 = Mike | year = 2008 | title = Elizabeth Bott and the formation of modern British sociology | url = | journal = The Sociological Review | volume = 56 | issue = 4| pages = 579–605 | doi=10.1111/j.1467-954x.2008.00806.x}}</ref> often are credited with performing some of the first fieldwork from which network analyses were performed, investigating community networks in southern Africa, India and the United Kingdom.<ref name=jscott /> Concomitantly, British anthropologist [[Siegfried Frederick Nadel|S. F. Nadel]] codified a theory of social structure that was influential in later network analysis.<ref>Nadel, S. F. 1957. ''The Theory of Social Structure''. London: Cohen and West.</ref> In [[sociology]], the early (1930s) work of [[Talcott Parsons]] set the stage for taking a relational approach to understanding social structure.<ref>Parsons, Talcott ([1937] 1949). ''The Structure of Social Action: A Study in Social Theory with Special Reference to a Group of European Writers''. New York: The Free Press.</ref><ref>Parsons, Talcott (1951). ''The Social System''. New York: The Free Press.</ref> Later, drawing upon Parsons' theory, the work of sociologist [[Peter Blau]] provides a strong impetus for analyzing the relational ties of social units with his work on [[social exchange theory]].<ref>Blau, Peter (1956). ''Bureaucracy in Modern Society''. New York: Random House, Inc.</ref><ref>Blau, Peter (1960). "A Theory of Social Integration". ''The American Journal of Sociology'', (65)6: 545–556, (May).</ref><ref>Blau, Peter (1964). ''Exchange and Power in Social Life''.</ref><br />
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Major developments in the field can be seen in the 1930s by several groups in psychology, anthropology, and mathematics working independently. In psychology, in the 1930s, Jacob L. Moreno began systematic recording and analysis of social interaction in small groups, especially classrooms and work groups (see sociometry). In anthropology, the foundation for social network theory is the theoretical and ethnographic work of Bronislaw Malinowski, Alfred Radcliffe-Brown, and Claude Lévi-Strauss. A group of social anthropologists associated with Max Gluckman and the Manchester School, including John A. Barnes, J. Clyde Mitchell and Elizabeth Bott Spillius, often are credited with performing some of the first fieldwork from which network analyses were performed, investigating community networks in southern Africa, India and the United Kingdom. In sociology, the early (1930s) work of Talcott Parsons set the stage for taking a relational approach to understanding social structure. Later, drawing upon Parsons' theory, the work of sociologist Peter Blau provides a strong impetus for analyzing the relational ties of social units with his work on social exchange theory.<br />
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20世纪30年代，心理学、人类学和数学领域的几个独立研究小组已经看到了这一领域的重大发展。在心理学方面，在20世纪30年代，雅各布·L·莫雷诺开始系统地记录和分析小团体中的社会互动，尤其是课堂和工作团体中的社会互动（见'''<font color="#FFD700">社会测量 Sociometry）</font>'''。在人类学中，社会网络理论的基础是'''布罗尼斯拉夫·马林诺夫斯基 Bronislaw Malinowski'''，'''阿尔弗雷德·拉德克利夫-布朗 Alfred Radcliffe-Brown'''和'''克洛德·列维-斯特劳斯 Claude Lévi-Strauss'''的理论和人种学著作。包括约翰·A·巴恩斯 John A. Barnes、 j · 克莱德 · 米切尔(j. Clyde Mitchell)和伊丽莎白 · 博特 · 斯皮利厄斯(Elizabeth Bott Spillius)在内的一群与马克斯 · 格拉克曼(Max Gluckman)和曼彻斯特学院(Manchester School)有关的社会人类学家，经常被认为是进行了一些最早的野。在社会学方面，塔尔科特 · 帕森斯早期(1930年代)的著作为采用关系方法理解社会结构奠定了基础。后来，社会学家彼得 · 布劳的社会交换理论为分析社会单位之间的关系提供了强大的动力。<br />
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By the 1970s, a growing number of scholars worked to combine the different tracks and traditions. One group consisted of sociologist [[Harrison White]] and his students at the [[Harvard Department of Social Relations|Harvard University Department of Social Relations]]. Also independently active in the Harvard Social Relations department at the time were [[Charles Tilly]], who focused on networks in political and community sociology and social movements, and [[Stanley Milgram]], who developed the "six degrees of separation" thesis.<ref>{{cite web |url=http://www.semioticon.com/semiotix/semiotix14/sem-14-05.html |title=The Networked Individual: A Profile of Barry Wellman |author=Bernie Hogan}}</ref> [[Mark Granovetter]]<ref name="Introduction for the French Reader">{{cite journal | last1 = Granovetter | first1 = Mark | year = 2007 | title = Introduction for the French Reader | url = | journal = Sociologica | volume = 2 | issue = | pages = 1–8 }}</ref> and [[Barry Wellman]]<ref>Wellman, Barry (1988). "Structural analysis: From method and metaphor to theory and substance". pp. 19–61 in B. Wellman and S. D. Berkowitz (eds.) ''Social Structures: A Network Approach'', Cambridge, UK: Cambridge University Press.</ref> are among the former students of White who elaborated and championed the analysis of social networks.<ref name="Introduction for the French Reader"/><ref>Mullins, Nicholas. ''Theories and Theory Groups in Contemporary American Sociology''. New York: Harper and Row, 1973.</ref><ref>Tilly, Charles, ed. ''An Urban World''. Boston: Little Brown, 1974.</ref><ref>Wellman, Barry. 1988. "Structural Analysis: From Method and Metaphor to Theory and Substance". pp. 19–61 in ''Social Structures: A Network Approach'', edited by Barry Wellman and S. D. Berkowitz. Cambridge: Cambridge University Press.</ref><br />
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By the 1970s, a growing number of scholars worked to combine the different tracks and traditions. One group consisted of sociologist Harrison White and his students at the Harvard University Department of Social Relations. Also independently active in the Harvard Social Relations department at the time were Charles Tilly, who focused on networks in political and community sociology and social movements, and Stanley Milgram, who developed the "six degrees of separation" thesis. Mark Granovetter and Barry Wellman are among the former students of White who elaborated and championed the analysis of social networks.<br />
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到了20世纪70年代，越来越多的学者致力于将不同的轨迹和传统结合起来。其中一组由社会学家哈里森 · 怀特和他在哈佛大学社会关系系的学生组成。当时在哈佛大学社会关系系独立活动的还有 Charles Tilly，他专注于政治和社区社会学和社会运动的网络，还有 Stanley Milgram，他发表了六度分隔理论的论文。马克·格兰诺维特和 Barry Wellman 是 White 以前的学生，他们阐述并支持对社交网络的分析。<br />
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Beginning in the late 1990s, social network analysis experienced work by sociologists, political scientists, and physicists such as [[Duncan J. Watts]], [[Albert-László Barabási]], [[Peter Bearman]], [[Nicholas A. Christakis]], [[James H. Fowler]], and others, developing and applying new models and methods to emerging data available about online social networks, as well as "digital traces" regarding face-to-face networks.<br />
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Beginning in the late 1990s, social network analysis experienced work by sociologists, political scientists, and physicists such as Duncan J. Watts, Albert-László Barabási, Peter Bearman, Nicholas A. Christakis, James H. Fowler, and others, developing and applying new models and methods to emerging data available about online social networks, as well as "digital traces" regarding face-to-face networks.<br />
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从20世纪90年代末开始，社会网络分析经历了社会学家、政治学家和物理学家的工作，如邓肯 · j · 瓦茨、阿尔伯特 · l · 斯兹尔 · 巴拉布 · 西、彼得 · 贝尔曼、尼古拉斯 · a · 克里斯塔基斯、詹姆斯 · h · 福勒等人，开发和应用新的模型和方法来获得有关在线社会网络的新兴数据，以及有关面对面网络的“数字痕迹”。<br />
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==Levels of analysis==<br />
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==Levels of analysis==<br />
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分析水平<br />
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[[File:Network self-organization stages.png|thumb|right|Self-organization of a network, based on Nagler, Levina, & Timme, (2011)<ref>{{cite journal|author1=Nagler, Jan|author2=Anna Levina|author3=Marc Timme|year=2011|title=Impact of single links in competitive percolation|journal=Nature Physics|volume=7|issue=3|pages=265–270|doi=10.1038/nphys1860|arxiv=1103.0922|bibcode=2011NatPh...7..265N}}</ref>]]<br />
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Self-organization of a network, based on Nagler, Levina, & Timme, (2011)<br />
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一个网络的自我组织，基于 Nagler，Levina，& Timme，(2011)<br />
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[[File:Social Network Diagram (large).svg|right|thumb|Centrality]]<br />
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Centrality<br />
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中心性<br />
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In general, social networks are [[self-organization|self-organizing]], [[emergence|emergent]], and [[social complexity|complex]], such that a globally coherent pattern appears from the local interaction of the elements that make up the system.<ref>Newman, Mark, Albert-László Barabási and Duncan J. Watts (2006). ''The Structure and Dynamics of Networks'' (Princeton Studies in Complexity). Oxford: Princeton University Press.</ref><ref>{{cite journal | last1 = Wellman | first1 = Barry | year = 2008 | title = Review: The development of social network analysis: A study in the sociology of science | url = | journal = Contemporary Sociology | volume = 37 | issue = 3| pages = 221–222 | doi=10.1177/009430610803700308}}</ref> These patterns become more apparent as network size increases. However, a global network analysis<ref>{{cite book|last=Faust|first=Stanley Wasserman; Katherine|title=Social network analysis : methods and applications|year=1998|publisher=Cambridge Univ. Press|location=Cambridge [u.a.]|isbn=978-0521382694|edition=Reprint.}}</ref> of, for example, all [[interpersonal relationships]] in the world is not feasible and is likely to contain so much [[Information theory|information]] as to be uninformative. Practical limitations of computing power, ethics and participant recruitment and payment also limit the scope of a social network analysis.<ref name="Kadu12">Kadushin, C. (2012). ''Understanding social networks: Theories, concepts, and findings''. Oxford: Oxford University Press.</ref><ref>{{Cite journal| author=Granovetter, M.|title= Network sampling: Some first steps| year=1976 |pages=1287–1303 |volume=81| issue=6 |journal= American Journal of Sociology | doi=10.1086/226224}}</ref> The nuances of a local system may be lost in a large network analysis, hence the quality of information may be more important than its scale for understanding network properties. Thus, social networks are analyzed at the scale relevant to the researcher's theoretical question. Although [[level of analysis|levels of analysis]] are not necessarily [[Mutually exclusive events|mutually exclusive]], there are three general levels into which networks may fall: [[Microsociology|micro-level]], [[wikt:meso-|meso-level]], and [[Macrosociology|macro-level]].<br />
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In general, social networks are self-organizing, emergent, and complex, such that a globally coherent pattern appears from the local interaction of the elements that make up the system. These patterns become more apparent as network size increases. However, a global network analysis of, for example, all interpersonal relationships in the world is not feasible and is likely to contain so much information as to be uninformative. Practical limitations of computing power, ethics and participant recruitment and payment also limit the scope of a social network analysis. The nuances of a local system may be lost in a large network analysis, hence the quality of information may be more important than its scale for understanding network properties. Thus, social networks are analyzed at the scale relevant to the researcher's theoretical question. Although levels of analysis are not necessarily mutually exclusive, there are three general levels into which networks may fall: micro-level, meso-level, and macro-level.<br />
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一般来说，社会网络是自组织的、涌现的和复杂的，以至于从组成系统的元素的局部相互作用中出现了一个全局连贯的模式。随着网络规模的增大，这些模式变得更加明显。然而，一个全球网络分析，例如，世界上所有的人际关系是不可行的，可能包含太多的信息，以至于没有提供信息。计算能力的实际限制、道德规范以及参与者的招募和支付也限制了社会网络分析的范围。本地系统的细微差别在大型网络分析中可能会丢失，因此对于理解网络属性来说，信息的质量可能比其规模更重要。因此，社会网络被分析在与研究者的理论问题相关的尺度上。虽然分析层次不一定相互排斥，但网络可以分为三个一般层次: 微观层次、中观层次和宏观层次。<br />
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===Micro level===<br />
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===Micro level===<br />
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微观层面<br />
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At the micro-level, social network research typically begins with an individual, [[Snowball sampling|snowballing]] as social relationships are traced, or may begin with a small group of individuals in a particular social context.<br />
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At the micro-level, social network research typically begins with an individual, snowballing as social relationships are traced, or may begin with a small group of individuals in a particular social context.<br />
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在微观层面上，社会网络研究通常从个人开始，随着社会关系的追踪而像滚雪球一样扩大，或者可能从特定社会背景下的一小群个人开始。<br />
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'''Dyadic level''': A [[Dyad (sociology)|dyad]] is a social relationship between two individuals. Network research on dyads may concentrate on [[Structural functionalism|structure]] of the relationship (e.g. multiplexity, strength), [[social equality]], and tendencies toward [[Reciprocity (social and political philosophy)|reciprocity/mutuality]].<br />
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Dyadic level: A dyad is a social relationship between two individuals. Network research on dyads may concentrate on structure of the relationship (e.g. multiplexity, strength), social equality, and tendencies toward reciprocity/mutuality.<br />
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二元层次: 二元是两个人之间的社会关系。网络对二元关系的研究可以集中在关系的结构上。多样性、力量)、社会平等以及互惠互利的倾向。<br />
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'''Triadic level''': Add one individual to a dyad, and you have a [[Triadic relation|triad]]. Research at this level may concentrate on factors such as [[Independence number|balance]] and [[Vertex-transitive graph|transitivity]], as well as [[social equality]] and tendencies toward [[Reciprocity (social and political philosophy)|reciprocity/mutuality]].<ref name="Kadu12"/> In the [[balance theory]] of [[Fritz Heider]] the triad is the key to social dynamics. The discord in a rivalrous [[love triangle]] is an example of an unbalanced triad, likely to change to a balanced triad by a change in one of the relations. The dynamics of social friendships in society has been modeled by balancing triads. The study is carried forward with the theory of [[signed graph]]s.<br />
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Triadic level: Add one individual to a dyad, and you have a triad. Research at this level may concentrate on factors such as balance and transitivity, as well as social equality and tendencies toward reciprocity/mutuality. In the balance theory of Fritz Heider the triad is the key to social dynamics. The discord in a rivalrous love triangle is an example of an unbalanced triad, likely to change to a balanced triad by a change in one of the relations. The dynamics of social friendships in society has been modeled by balancing triads. The study is carried forward with the theory of signed graphs.<br />
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三元等级: 把一个人加到一个二元组中，你就有了一个三元组。这一层次的研究可能集中在诸如平衡和传递性等因素，以及社会平等和互惠 / 互惠的倾向。在弗里茨 · 海德的平衡理论中，三元组是社会动力学的关键。争强好胜的三角恋中的不和谐是不平衡的三角关系的一个例子，很可能通过其中一种关系的改变而变成平衡的三角关系。社会中社会友谊的动态模型是通过平衡三合会建立起来的。本文利用符号图理论进行了研究。<br />
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'''Actor level''': The smallest unit of analysis in a social network is an individual in their social setting, i.e., an "actor" or "ego". Egonetwork analysis focuses on network characteristics such as size, relationship strength, density, [[centrality]], [[wikt:prestige|prestige]] and roles such as [[isolates|isolates, liaisons]], and [[Bridge (interpersonal)|bridges]].<ref name="Jone11"/> Such analyses, are most commonly used in the fields of [[psychology]] or [[Social psychology (sociology)|social psychology]], [[ethnographic]] [[kinship]] analysis or other [[genealogy|genealogical]] studies of relationships between individuals.<br />
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Actor level: The smallest unit of analysis in a social network is an individual in their social setting, i.e., an "actor" or "ego". Egonetwork analysis focuses on network characteristics such as size, relationship strength, density, centrality, prestige and roles such as isolates, liaisons, and bridges. Such analyses, are most commonly used in the fields of psychology or social psychology, ethnographic kinship analysis or other genealogical studies of relationships between individuals.<br />
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行为者层面: 社会网络中最小的分析单位是社会环境中的个体，即“行为者”或“自我”。Egonwork 分析主要关注网络特征，例如大小、关系强度、密度、中心性、声望和角色，例如隔离、联络和桥梁。这种分析最常用于心理学或社会心理学、人种学亲属关系分析或其他个人关系的系谱研究领域。<br />
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'''Subset level''': [[Subset]] levels of network research problems begin at the micro-level, but may cross over into the meso-level of analysis. Subset level research may focus on [[Distance (graph theory)|distance]] and reachability, [[cliques]], [[Cohesion (social policy)|cohesive]] subgroups, or other [[Group action (sociology)|group actions]] or [[Group behaviour|behavior]].<ref>{{cite book | title="Graph Theoretical Approaches to Social Network Analysis". in Computational Complexity: Theory, Techniques, and Applications (Robert A. Meyers, ed.) | publisher=Springer | author=de Nooy, Wouter |year=2012 | pages=2864–2877 | isbn=978-1-4614-1800-9|doi=10.1007/978-1-4614-1800-9_176| chapter=Social Network Analysis, Graph Theoretical Approaches to }}</ref><br />
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Subset level: Subset levels of network research problems begin at the micro-level, but may cross over into the meso-level of analysis. Subset level research may focus on distance and reachability, cliques, cohesive subgroups, or other group actions or behavior.<br />
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子集级别: 网络研究问题的子集级别开始于微观级别，但可能跨越到中观级别的分析。子集级别的研究可能集中在距离和可达性、派系、凝聚子群或其他群体行为或行为。<br />
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===Meso level===<br />
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===Meso level===<br />
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中观水平<br />
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In general, meso-level theories begin with a [[Sample population|population]] size that falls between the micro- and macro-levels. However, meso-level may also refer to analyses that are specifically designed to reveal connections between micro- and macro-levels. Meso-level networks are low density and may exhibit causal processes distinct from interpersonal micro-level networks.<ref>{{cite journal | last1 = Hedström | first1 = Peter | last2 = Sandell | first2 = Rickard | last3 = Stern | first3 = Charlotta | year = 2000 | title = Mesolevel Networks and the Diffusion of Social Movements: The Case of the Swedish Social Democratic Party | url = http://www.nuffield.ox.ac.uk/users/hedstrom/ajs3.pdf | journal = American Journal of Sociology | volume = 106 | issue = 1| pages = 145–172 | doi = 10.1086/303109 }}</ref><br />
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In general, meso-level theories begin with a population size that falls between the micro- and macro-levels. However, meso-level may also refer to analyses that are specifically designed to reveal connections between micro- and macro-levels. Meso-level networks are low density and may exhibit causal processes distinct from interpersonal micro-level networks.<br />
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一般来说，中观层面的理论始于介于微观和宏观层面之间的人口规模。然而，中观层面也可以指专门为揭示微观和宏观层面之间的联系而设计的分析。中观层次的网络是低密度的，可能表现出不同于人际微观层次网络的因果过程。<br />
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[[File:Social Red.jpg|thumb|right|Social network diagram, meso-level]]<br />
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Social network diagram, meso-level<br />
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中层社会网络图<br />
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'''Organizations''': Formal [[organizations]] are [[social group]]s that distribute tasks for a collective [[goal]].<ref name="Riketta, M. 2007">{{cite journal | last1 = Riketta | first1 = M. | last2 = Nienber | first2 = S. | year = 2007 | title = Multiple identities and work motivation: The role of perceived compatibility between nested organizational units | url = | journal = British Journal of Management | volume = 18 | issue = | pages = S61–77 | doi=10.1111/j.1467-8551.2007.00526.x}}</ref> Network research on organizations may focus on either intra-organizational or inter-organizational ties in terms of [[Formal organization|formal]] or [[Informal organization|informal]] relationships. Intra-organizational networks themselves often contain multiple levels of analysis, especially in larger organizations with multiple branches, franchises or semi-autonomous departments. In these cases, research is often conducted at a work group level and organization level, focusing on the interplay between the two structures.<ref name="Riketta, M. 2007"/> Experiments with networked groups online have documented ways to optimize group-level coordination through diverse interventions, including the addition of autonomous agents to the groups.<ref>{{Cite journal|last=Shirado|first=Hirokazu|last2=Christakis|first2=Nicholas A|title=Locally noisy autonomous agents improve global human coordination in network experiments|journal=Nature|volume=545|issue=7654|pages=370–374|doi=10.1038/nature22332|pmid=28516927|pmc=5912653|bibcode=2017Natur.545..370S|year=2017}}</ref><br />
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Organizations: Formal organizations are social groups that distribute tasks for a collective goal. Network research on organizations may focus on either intra-organizational or inter-organizational ties in terms of formal or informal relationships. Intra-organizational networks themselves often contain multiple levels of analysis, especially in larger organizations with multiple branches, franchises or semi-autonomous departments. In these cases, research is often conducted at a work group level and organization level, focusing on the interplay between the two structures.<br />
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组织: 正式组织是为共同目标分配任务的社会团体。关于组织的网络研究可以侧重于正式或非正式关系方面的组织内或组织间联系。组织内网络本身往往包含多层次的分析，特别是在具有多个分支机构、特许经营权或半自治部门的较大组织中。在这些情况下，研究通常在工作组和组织层面进行，重点放在两个结构之间的相互作用。<br />
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'''Randomly distributed networks''': [[Exponential random graph models]] of social networks became state-of-the-art methods of social network analysis in the 1980s. This framework has the capacity to represent social-structural effects commonly observed in many human social networks, including general [[Degree (graph theory)|degree]]-based structural effects commonly observed in many human social networks as well as [[Reciprocity (social and political philosophy)|reciprocity]] and [[Transitive set|transitivity]], and at the node-level, [[homophily]] and [[Attribute-value system|attribute]]-based activity and popularity effects, as derived from explicit hypotheses about [[Dependency graph|dependencies]] among network ties. [[Parameter]]s are given in terms of the prevalence of small [[Induced subgraph|subgraph]] configurations in the network and can be interpreted as describing the combinations of local social processes from which a given network emerges. These probability models for networks on a given set of actors allow generalization beyond the restrictive dyadic independence assumption of micro-networks, allowing models to be built from theoretical structural foundations of social behavior.<ref>{{cite journal | last1 = Cranmer | first1 = Skyler J. | last2 = Desmarais | first2 = Bruce A. | year = 2011 | title = Inferential Network Analysis with Exponential Random Graph Models | url = | journal = Political Analysis | volume = 19 | issue = 1| pages = 66–86 | doi=10.1093/pan/mpq037| citeseerx = 10.1.1.623.751 }}</ref><br />
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Randomly distributed networks: Exponential random graph models of social networks became state-of-the-art methods of social network analysis in the 1980s. This framework has the capacity to represent social-structural effects commonly observed in many human social networks, including general degree-based structural effects commonly observed in many human social networks as well as reciprocity and transitivity, and at the node-level, homophily and attribute-based activity and popularity effects, as derived from explicit hypotheses about dependencies among network ties. Parameters are given in terms of the prevalence of small subgraph configurations in the network and can be interpreted as describing the combinations of local social processes from which a given network emerges. These probability models for networks on a given set of actors allow generalization beyond the restrictive dyadic independence assumption of micro-networks, allowing models to be built from theoretical structural foundations of social behavior.<br />
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随机分布的网络: 社会网络的指数随机图模型在20世纪80年代成为最先进的社会网络分析方法。这个框架有能力表示在许多人类社会网络中普遍观察到的社会结构效应，包括在许多人类社会网络中普遍观察到的基于程度的一般性结构效应以及互惠性和传递性，以及在节点一级、同相性和基于属性的活动和流行性效应，这些效应源于关于网络关系之间依赖性的明确假设。参数是根据网络中小型子图配置的流行程度给出的，可以解释为描述一个给定网络出现的局部社会过程的组合。这些网络的概率模型在给定的参与者集合上允许超越微型网络的限制性并元独立性假设的泛化，允许模型从社会行为的理论结构基础上建立。<br />
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[[File:Scale-free network sample.png|thumb|right|Examples of a random network and a scale-free network. Each graph has 32 nodes and 32 links. Note the "hubs" (shaded) in the scale-free diagram (on the right).]]<br />
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Examples of a random network and a scale-free network. Each graph has 32 nodes and 32 links. Note the "hubs" (shaded) in the scale-free diagram (on the right).<br />
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一个随机网络和一个无尺度网络的例子。每个图有32个节点和32个链接。注意无标度图中的“集线器”(阴影部分)(右侧)。<br />
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'''Scale-free networks''': A [[scale-free network]] is a [[complex network|network]] whose [[degree distribution]] follows a [[power law]], at least [[asymptotic]]ally. In [[network theory]] a scale-free ideal network is a [[random network]] with a [[degree distribution]] that unravels the size distribution of social groups.<ref>{{cite journal |author1=Moreira, André A.|author2=Demétrius R. Paula|author3=Raimundo N. Costa Filho|author4=José S. Andrade, Jr. |title=Competitive cluster growth in complex networks |year=2006 |doi=10.1103/PhysRevE.73.065101 |pmid=16906890 |journal=Physical Review E |volume=73 |issue=6 |pages=065101 |arxiv=cond-mat/0603272 |bibcode=2006PhRvE..73f5101M }}</ref> Specific characteristics of scale-free networks vary with the theories and analytical tools used to create them, however, in general, scale-free networks have some common characteristics. One notable characteristic in a scale-free network is the relative commonness of [[Vertex (graph theory)|vertices]] with a [[Maximum degree|degree]] that greatly exceeds the average. The highest-degree nodes are often called "hubs", and may serve specific purposes in their networks, although this depends greatly on the social context. Another general characteristic of scale-free networks is the [[clustering coefficient]] distribution, which decreases as the node degree increases. This distribution also follows a [[power law]].<ref>Barabási, Albert-László (2003). ''Linked: how everything is connected to everything else and what it means for business, science, and everyday life''. New York: Plum.</ref> The [[Barabási–Albert model|Barabási]] model of network evolution shown above is an example of a scale-free network.<br />
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Scale-free networks: A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. In network theory a scale-free ideal network is a random network with a degree distribution that unravels the size distribution of social groups. Specific characteristics of scale-free networks vary with the theories and analytical tools used to create them, however, in general, scale-free networks have some common characteristics. One notable characteristic in a scale-free network is the relative commonness of vertices with a degree that greatly exceeds the average. The highest-degree nodes are often called "hubs", and may serve specific purposes in their networks, although this depends greatly on the social context. Another general characteristic of scale-free networks is the clustering coefficient distribution, which decreases as the node degree increases. This distribution also follows a power law. The Barabási model of network evolution shown above is an example of a scale-free network.<br />
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无标度网络: 无尺度网络网络是一个度分布遵循幂律的网络，至少是渐近的。在网络理论中，无标度理想网络是一个具有度分布的随机网络，它揭示了社会群体的规模分布。无标度网络的具体特征随创建无标度网络的理论和分析工具的不同而不同，然而，一般来说，无标度网络具有一些共同的特征。无尺度网络的一个显著特征是，度大大超过平均值的顶点具有相对的共性。最高度的节点通常被称为“枢纽” ，并且可能在其网络中服务于特定的目的，尽管这在很大程度上取决于社会环境。无标度网络的另一个普遍特征是集聚系数分布，它随着节点度的增加而减少。这个分布也遵循一个幂定律。上面显示的网络演化的 barab si 模型就是无尺度网络的一个例子。<br />
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===Macro level===<br />
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===Macro level===<br />
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宏观层面<br />
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Rather than tracing interpersonal interactions, macro-level analyses generally trace the outcomes of interactions, such as [[economic]] or other [[resource]] [[Transfer function|transfer]] interactions over a large [[Sample population|population]].<br />
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Rather than tracing interpersonal interactions, macro-level analyses generally trace the outcomes of interactions, such as economic or other resource transfer interactions over a large population.<br />
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宏观层面的分析不是追踪人与人之间的相互作用，而是通常追踪相互作用的结果，例如经济或其他资源转移在一大群体中的相互作用。<br />
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[[File:Diagram of a social network.jpg|thumb|right|Diagram: section of a large-scale social network]]<br />
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Diagram: section of a large-scale social network<br />
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图: 一个大型社交网络的一部分<br />
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'''Large-scale networks''': [[Large-scale macroeconometric model|Large-scale network]] is a term somewhat synonymous with "macro-level" as used, primarily, in [[social science|social]] and [[Behavioural sciences|behavioral]] sciences, in [[economics]]. Originally, the term was used extensively in the [[computer sciences]] (see [[Network mapping#Large-scale mapping project|large-scale network mapping]]).<br />
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Large-scale networks: Large-scale network is a term somewhat synonymous with "macro-level" as used, primarily, in social and behavioral sciences, in economics. Originally, the term was used extensively in the computer sciences (see large-scale network mapping).<br />
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大规模网络: 大规模网络是一个与“宏观层面”同义的术语，主要用于社会和行为科学，经济学。最初，这个术语在计算机科学中被广泛使用(见大比例尺网络地图)。<br />
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'''Complex networks''': Most larger social networks display features of [[social complexity]], which involves substantial non-trivial features of [[network topology]], with patterns of complex connections between elements that are neither purely regular nor purely random (see, [[complexity science]], [[dynamical system]] and [[chaos theory]]), as do [[biological]], and [[Computer network|technological networks]]. Such [[complex network]] features include a heavy tail in the [[degree distribution]], a high [[clustering coefficient]], [[assortativity]] or disassortativity among vertices, [[community structure]] (see [[stochastic block model]]), and [[hierarchy|hierarchical structure]]. In the case of [[Agency (philosophy)|agency-directed]] networks these features also include [[Reciprocity in network|reciprocity]], triad significance profile (TSP, see [[network motif]]), and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as [[lattice graph|lattices]] and [[random graph]]s, do not show these features.<ref>{{cite journal|author=Strogatz, Steven H.|year=2001|title=Exploring complex networks|journal=Nature|volume=410|pages=268–276|doi=10.1038/35065725|pmid=11258382|issue=6825|bibcode=2001Natur.410..268S|doi-access=free}}</ref><br />
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Complex networks: Most larger social networks display features of social complexity, which involves substantial non-trivial features of network topology, with patterns of complex connections between elements that are neither purely regular nor purely random (see, complexity science, dynamical system and chaos theory), as do biological, and technological networks. Such complex network features include a heavy tail in the degree distribution, a high clustering coefficient, assortativity or disassortativity among vertices, community structure (see stochastic block model), and hierarchical structure. In the case of agency-directed networks these features also include reciprocity, triad significance profile (TSP, see network motif), and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as lattices and random graphs, do not show these features.<br />
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复杂网络: 大多数较大的社会网络呈现出社会复杂性的特征，包括网络拓扑的大量非琐碎特征，以及元素之间的复杂连接模式，这些元素既不是纯规则的也不是纯随机的(见复杂性科学、动力系统和混沌理论) ，生物学和技术网络也是如此。这些复杂的网络特征包括程度分布的重尾、高集聚系数、顶点之间的协调性或不协调性、社区结构(见随机块模型)和层次结构。在代理导向网络的情况下，这些特征还包括互惠性、三重显著性特征(TSP，见网络主题)和其他特征。相比之下，许多过去研究过的网络数学模型，如格和随机图，并没有表现出这些特征。<br />
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==Theoretical links==<br />
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==Theoretical links==<br />
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理论联系<br />
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===Imported theories===<br />
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===Imported theories===<br />
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输入的理论<br />
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Various theoretical frameworks have been imported for the use of social network analysis. The most prominent of these are [[Graph theory]], [[Balance theory]], Social comparison theory, and more recently, the [[Social identity approach]].<ref>{{cite book |last1=Kilduff|first1=M.|last2=Tsai|first2=W. |year=2003 |title= Social networks and organisations |publisher=Sage Publications}}</ref><br />
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Various theoretical frameworks have been imported for the use of social network analysis. The most prominent of these are Graph theory, Balance theory, Social comparison theory, and more recently, the Social identity approach.<br />
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为了使用社会网络分析，已经引入了各种理论框架。其中最突出的是图论、平衡论、社会比较论，以及最近的社会认同方法。<br />
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===Indigenous theories===<br />
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===Indigenous theories===<br />
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本土理论<br />
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Few complete theories have been produced from social network analysis. Two that have are [[role theory|structural role theory]] and [[heterophily|heterophily theory]].<br />
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Few complete theories have been produced from social network analysis. Two that have are structural role theory and heterophily theory.<br />
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很少有完整的理论产生于社会网络分析。结构角色理论和异质性理论。<br />
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The basis of Heterophily Theory was the finding in one study that more numerous weak ties can be important in seeking information and innovation, as cliques have a tendency to have more homogeneous opinions as well as share many common traits. This homophilic tendency was the reason for the members of the cliques to be attracted together in the first place. However, being similar, each member of the clique would also know more or less what the other members knew. To find new information or insights, members of the clique will have to look beyond the clique to its other friends and acquaintances. This is what Granovetter called "the strength of weak ties".<ref>{{Cite journal | author=Granovetter, M. |year=1973 |title= The strength of weak ties |journal= American Journal of Sociology |volume=78 |issue=6 |pages= 1360–1380 |doi=10.1086/225469}}</ref><br />
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The basis of Heterophily Theory was the finding in one study that more numerous weak ties can be important in seeking information and innovation, as cliques have a tendency to have more homogeneous opinions as well as share many common traits. This homophilic tendency was the reason for the members of the cliques to be attracted together in the first place. However, being similar, each member of the clique would also know more or less what the other members knew. To find new information or insights, members of the clique will have to look beyond the clique to its other friends and acquaintances. This is what Granovetter called "the strength of weak ties".<br />
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异质性理论的基础是在一项研究中发现，更多的弱关系可以在寻求信息和创新方面发挥重要作用，因为小团体倾向于有更同质化的观点，也有许多共同特征。这种亲同性倾向是小团体成员被吸引到一起的首要原因。然而，由于相似，小圈子里的每一个成员或多或少都知道其他成员所知道的事情。为了找到新的信息或见解，小圈子里的成员将不得不超越小圈子，关注其他的朋友和熟人。这就是格兰诺维特所说的“弱关系的力量”。<br />
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==Structural holes==<br />
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==Structural holes==<br />
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结构性漏洞<br />
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In the context of networks, [[social capital]] exists where people have an advantage because of their location in a network. Contacts in a network provide information, opportunities and perspectives that can be beneficial to the central player in the network. Most social structures tend to be characterized by dense clusters of strong connections.<ref name="Burt 2004">{{cite journal|last=Burt|first=Ronald|title=Structural Holes and Good Ideas|journal=American Journal of Sociology|year=2004|doi=10.1086/421787|volume=110|issue=2|pages=349–399|citeseerx=10.1.1.388.2251}}</ref> Information within these clusters tends to be rather homogeneous and redundant. Non-redundant information is most often obtained through contacts in different clusters.<ref name="Burt 1992">{{cite book|last=Burt|first=Ronald|title=Structural Holes: The Social Structure of Competition|year=1992|publisher=Harvard University Press|location=Cambridge, MA}}</ref> When two separate clusters possess non-redundant information, there is said to be a structural hole between them.<ref name="Burt 1992"/> Thus, a network that bridges [[structural holes]] will provide network benefits that are in some degree additive, rather than overlapping. An ideal network structure has a vine and cluster structure, providing access to many different clusters and structural holes.<ref name="Burt 1992"/><br />
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In the context of networks, social capital exists where people have an advantage because of their location in a network. Contacts in a network provide information, opportunities and perspectives that can be beneficial to the central player in the network. Most social structures tend to be characterized by dense clusters of strong connections. Information within these clusters tends to be rather homogeneous and redundant. Non-redundant information is most often obtained through contacts in different clusters. When two separate clusters possess non-redundant information, there is said to be a structural hole between them. Thus, a network that bridges structural holes will provide network benefits that are in some degree additive, rather than overlapping. An ideal network structure has a vine and cluster structure, providing access to many different clusters and structural holes.<br />
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在网络的背景下，社会资本存在于人们因为他们在网络中的位置而具有优势的地方。网络中的联系人提供的信息、机会和观点可能有利于网络中的核心参与者。大多数社会结构倾向于拥有属性密集的强连接。这些集群中的信息往往是相当均匀和冗余的。非冗余信息通常是通过不同集群中的联系人获得的。当两个独立的簇拥有非冗余信息时，我们称它们之间存在一个结构空洞。因此，一个桥接结构孔的网络将在某种程度上提供附加的网络效益，而不是重叠的。理想的网络结构具有蔓生结构和集群结构，提供对许多不同集群和结构洞的访问。<br />
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Networks rich in structural holes are a form of social capital in that they offer [[Information theory|information]] benefits. The main player in a network that bridges structural holes is able to access information from diverse sources and clusters.<ref name="Burt 1992"/> For example, in [[Business networking|business networks]], this is beneficial to an individual's career because he is more likely to hear of job openings and opportunities if his network spans a wide range of contacts in different industries/sectors. This concept is similar to Mark Granovetter's theory of [[Interpersonal ties|weak ties]], which rests on the basis that having a broad range of contacts is most effective for job attainment.<br />
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Networks rich in structural holes are a form of social capital in that they offer information benefits. The main player in a network that bridges structural holes is able to access information from diverse sources and clusters. For example, in business networks, this is beneficial to an individual's career because he is more likely to hear of job openings and opportunities if his network spans a wide range of contacts in different industries/sectors. This concept is similar to Mark Granovetter's theory of weak ties, which rests on the basis that having a broad range of contacts is most effective for job attainment.<br />
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富含结构性漏洞的网络是社会资本的一种形式，因为它们提供信息利益。桥接结构漏洞的网络中的主要参与者能够访问来自不同来源和集群的信息。例如，在商业网络中，这对个人的职业生涯是有益的，因为如果他的关系网涵盖不同行业 / 部门的广泛联系，他更有可能听到职位空缺和机会。这个概念类似于马克·格兰诺维特的弱关系理论，该理论的基础是拥有广泛的联系对于获得工作是最有效的。<br />
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==Research clusters==<br />
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==Research clusters==<br />
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研究集群<br />
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===Communication===<br />
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===Communication===<br />
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沟通<br />
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[[Communication Studies]] are often considered a part of both the social sciences and the humanities, drawing heavily on fields such as [[sociology]], [[psychology]], [[anthropology]], [[information science]], [[biology]], [[political science]], and [[economics]] as well as [[rhetoric]], [[literary studies]], and [[semiotics]]. Many communication concepts describe the transfer of information from one source to another, and can thus be conceived of in terms of a network.<br />
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Communication Studies are often considered a part of both the social sciences and the humanities, drawing heavily on fields such as sociology, psychology, anthropology, information science, biology, political science, and economics as well as rhetoric, literary studies, and semiotics. Many communication concepts describe the transfer of information from one source to another, and can thus be conceived of in terms of a network.<br />
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传播学通常被认为是社会科学和人文科学的一部分，主要集中在社会学、心理学、人类学、信息科学、生物学、政治学、经济学以及修辞学、文学研究和符号学等领域。许多通信概念描述了信息从一个来源到另一个来源的转移，因此可以从网络的角度来考虑。<br />
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===Community===<br />
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===Community===<br />
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社区<br />
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In J.A. Barnes' day, a "[[community]]" referred to a specific geographic location and studies of community ties had to do with who talked, associated, traded, and attended church with whom. Today, however, there are extended "online" communities developed through [[telecommunications]] devices and [[social network services]]. Such devices and services require extensive and ongoing maintenance and analysis, often using [[network science]] methods. [[Community development]] studies, today, also make extensive use of such methods.<br />
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In J.A. Barnes' day, a "community" referred to a specific geographic location and studies of community ties had to do with who talked, associated, traded, and attended church with whom. Today, however, there are extended "online" communities developed through telecommunications devices and social network services. Such devices and services require extensive and ongoing maintenance and analysis, often using network science methods. Community development studies, today, also make extensive use of such methods.<br />
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在耶路撒冷。巴恩斯日，一个“社区”指的是一个特定的地理位置和社区关系的研究与谁交谈，联系，贸易和参加教会与谁。然而，今天，通过电信设备和社交网络服务，有了扩展的“在线”社区。这样的设备和服务需要广泛和持续的维护和分析，通常使用网络科学的方法。社区发展研究，今天，也广泛使用这些方法。<br />
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===Complex networks===<br />
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===Complex networks===<br />
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复杂的网络<br />
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[[Complex networks]] require methods specific to modelling and interpreting [[social complexity]] and [[complex adaptive system]]s, including techniques of [[dynamic network analysis]].<br />
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Complex networks require methods specific to modelling and interpreting social complexity and complex adaptive systems, including techniques of dynamic network analysis.<br />
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复杂网络需要建模和解释社会复杂性和复杂适应系统的特定方法，包括动态网络分析技术。<br />
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Mechanisms such as [[Dual-phase evolution#social networks|Dual-phase evolution]] explain how temporal changes in connectivity contribute to the formation of structure in social networks.<br />
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Mechanisms such as Dual-phase evolution explain how temporal changes in connectivity contribute to the formation of structure in social networks.<br />
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双阶段进化等机制解释了连接性的时间变化如何促进社会网络结构的形成。<br />
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===Criminal networks===<br />
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===Criminal networks===<br />
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犯罪网络<br />
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In [[criminology]] and [[urban sociology]], much attention has been paid to the social networks among criminal actors. For example, Andrew Papachristos<ref>{{cite journal |last=Papachristos |first=Andrew |year=2009 |title=Murder by Structure: Dominance Relations and the Social Structure of Gang Homicide |journal=American Journal of Sociology |volume=115 |issue=1 |pages=74–128 |doi=10.2139/ssrn.855304 |url=http://www.papachristos.org/Publications_2_files/ajs_final_version.pdf |accessdate=29 March 2013 |url-status=dead |archiveurl=https://web.archive.org/web/20140407094725/http://www.papachristos.org/Publications_2_files/ajs_final_version.pdf |archivedate=7 April 2014 }}</ref> has studied gang murders as a series of exchanges between gangs. Murders can be seen to diffuse outwards from a single source, because weaker gangs cannot afford to kill members of stronger gangs in retaliation, but must commit other violent acts to maintain their reputation for strength.<br />
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In criminology and urban sociology, much attention has been paid to the social networks among criminal actors. For example, Andrew Papachristos has studied gang murders as a series of exchanges between gangs. Murders can be seen to diffuse outwards from a single source, because weaker gangs cannot afford to kill members of stronger gangs in retaliation, but must commit other violent acts to maintain their reputation for strength.<br />
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在犯罪学和城市社会学中，犯罪行为人之间的社会网络问题受到了广泛的关注。例如，安德鲁 · 帕帕克里斯托斯将帮派谋杀研究为帮派之间的一系列交流。谋杀可以被视为从单一来源向外扩散，因为较弱的帮派无法承担杀死较强帮派成员进行报复的代价，而必须采取其他暴力行动来维护他们强大的声誉。<br />
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===Diffusion of innovations===<br />
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===Diffusion of innovations===<br />
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创新产品渗透理论<br />
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[[Diffusion of innovations|Diffusion of ideas and innovations]] studies focus on the spread and use of ideas from one actor to another or one [[culture]] and another. This line of research seeks to explain why some become "early adopters" of ideas and innovations, and links social network structure with facilitating or impeding the spread of an innovation.<br />
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Diffusion of ideas and innovations studies focus on the spread and use of ideas from one actor to another or one culture and another. This line of research seeks to explain why some become "early adopters" of ideas and innovations, and links social network structure with facilitating or impeding the spread of an innovation.<br />
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思想和创新的传播研究的重点是思想从一个行动者传播到另一个行动者或一种文化和另一种文化。这一系列的研究试图解释为什么有些人成为创意和创新的“早期接受者” ，并将社交网络结构与促进或阻碍创新的传播联系起来。<br />
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===Demography===<br />
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===Demography===<br />
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人口统计<br />
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In [[demography]], the study of social networks has led to new sampling methods for estimating and reaching populations that are hard to enumerate (for example, homeless people or intravenous drug users.) For example, respondent driven sampling is a network-based sampling technique that relies on respondents to a survey recommending further respondents.<br />
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In demography, the study of social networks has led to new sampling methods for estimating and reaching populations that are hard to enumerate (for example, homeless people or intravenous drug users.) For example, respondent driven sampling is a network-based sampling technique that relies on respondents to a survey recommending further respondents.<br />
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在人口统计学方面，对社会网络的研究导致了新的抽样方法，用于估计和覆盖难以统计的人群(例如，无家可归者或静脉注射毒品者)例如，受访者驱动的抽样是一种基于网络的抽样技术，依赖于调查的受访者推荐更多的受访者。<br />
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===Economic sociology===<br />
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===Economic sociology===<br />
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经济社会学<br />
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The field of [[sociology]] focuses almost entirely on networks of outcomes of social interactions. More narrowly, [[economic sociology]] considers behavioral interactions of individuals and groups through [[social capital]] and social "markets". Sociologists, such as Mark Granovetter, have developed core principles about the interactions of social structure, information, ability to punish or reward, and trust that frequently recur in their analyses of political, economic and other institutions. Granovetter examines how social structures and social networks can affect economic outcomes like hiring, price, productivity and innovation and describes sociologists' contributions to analyzing the impact of social structure and networks on the economy.<ref>{{cite journal |last=Granovetter |first=Mark |year=2005 |title=The Impact of Social Structure on Economic Outcomes |journal=The Journal of Economic Perspectives |jstor=4134991 |volume=19 |issue=1 |pages=33–50|doi=10.1257/0895330053147958 }}</ref><br />
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The field of sociology focuses almost entirely on networks of outcomes of social interactions. More narrowly, economic sociology considers behavioral interactions of individuals and groups through social capital and social "markets". Sociologists, such as Mark Granovetter, have developed core principles about the interactions of social structure, information, ability to punish or reward, and trust that frequently recur in their analyses of political, economic and other institutions. Granovetter examines how social structures and social networks can affect economic outcomes like hiring, price, productivity and innovation and describes sociologists' contributions to analyzing the impact of social structure and networks on the economy.<br />
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社会学领域几乎完全关注社会互动的结果网络。更狭义地说，经济社会学通过社会资本和社会“市场”考虑个人和群体的行为互动。社会学家，如马克·格兰诺维特，已经发展出关于社会结构、信息、惩罚或奖励能力和信任相互作用的核心原则，这些原则在他们对政治、经济和其他制度的分析中经常出现。格兰诺维特研究了社会结构和社会网络如何影响经济结果，如雇佣、价格、生产力和创新，并描述了社会学家对分析社会结构和网络对经济的影响的贡献。<br />
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===Health care===<br />
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===Health care===<br />
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医疗保健<br />
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Analysis of social networks is increasingly incorporated into [[health care analytics]], not only in [[epidemiology|epidemiological]] studies but also in models of [[Health Communication|patient communication]] and education, disease prevention, mental health diagnosis and treatment, and in the study of health care organizations and [[health care systems|systems]].<ref>Levy, Judith and Bernice Pescosolido (2002). ''Social Networks and Health''. Boston, MA: JAI Press.</ref><br />
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Analysis of social networks is increasingly incorporated into health care analytics, not only in epidemiological studies but also in models of patient communication and education, disease prevention, mental health diagnosis and treatment, and in the study of health care organizations and systems.<br />
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社会网络分析不仅在流行病学研究中，而且在病人沟通和教育模型、疾病预防、心理健康诊断和治疗以及卫生保健组织和系统的研究中，越来越多地被纳入卫生保健分析中。<br />
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===Human ecology===<br />
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===Human ecology===<br />
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人类生态学<br />
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[[Human ecology]] is an [[interdisciplinary]] and [[transdisciplinary]] study of the relationship between [[human]]s and their [[natural environment|natural]], [[Social environment|social]], and [[built environment]]s. The scientific philosophy of human ecology has a diffuse history with connections to [[geography]], [[sociology]], [[psychology]], [[anthropology]], [[zoology]], and natural [[ecology]].<ref>Crona, Beatrice and Klaus Hubacek (eds.) (2010). [http://www.ecologyandsociety.org/issues/view.php?sf=48 "Special Issue: Social network analysis in natural resource governance"]. ''Ecology and Society'', 48.</ref><ref>Ernstson, Henrich (2010). "Reading list: Using social network analysis (SNA) in social-ecological studies". [http://rs.resalliance.org/2010/11/03/reading-list-using-social-network-analysis-sna-in-social-ecological-studies/ ''Resilience Science'']</ref><br />
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Human ecology is an interdisciplinary and transdisciplinary study of the relationship between humans and their natural, social, and built environments. The scientific philosophy of human ecology has a diffuse history with connections to geography, sociology, psychology, anthropology, zoology, and natural ecology.<br />
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人类生态学是一门跨学科、跨学科的学科，研究人类与自然、社会和人造环境之间的关系。人类生态学的科学哲学与地理学、社会学、心理学、人类学、动物学和自然生态学有着密切的联系。<br />
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===Language and linguistics===<br />
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===Language and linguistics===<br />
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语言与语言学<br />
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Studies of [[language]] and [[linguistics]], particularly [[evolutionary linguistics]], focus on the development of [[Morphology (linguistics)|linguistic forms]] and transfer of changes, [[Phonology|sounds]] or words, from one language system to another through networks of social interaction. Social networks are also important in [[language shift]], as groups of people add and/or abandon languages to their repertoire.<br />
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Studies of language and linguistics, particularly evolutionary linguistics, focus on the development of linguistic forms and transfer of changes, sounds or words, from one language system to another through networks of social interaction. Social networks are also important in language shift, as groups of people add and/or abandon languages to their repertoire.<br />
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语言学和语言学的研究，特别是演化语言学，集中在语言形式的发展和变化，声音或词汇的转移，从一个语言系统到另一个通过社会互动的网络。社交网络在语言转换中也很重要，因为一群人增加或者放弃了他们的语言。<br />
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===Literary networks===<br />
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===Literary networks===<br />
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文学网络<br />
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In the study of literary systems, network analysis has been applied by Anheier, Gerhards and Romo,<ref>{{cite journal | last1 = Anheier | first1 = H. K. | last2 = Romo | first2 = F. P. | year = 1995 | title = Forms of capital and social structure of fields: examining Bourdieu's social topography | url = | journal = American Journal of Sociology | volume = 100 | issue = 4| pages = 859–903 | doi=10.1086/230603}}</ref> De Nooy,<ref>{{cite journal | last1 = De Nooy | first1 = W | year = 2003| title = Fields and networks: Correspondence analysis and social network analysis in the framework of Field Theory | url = | journal = Poetics | volume = 31 | issue = 5–6| pages = 305–327 | doi = 10.1016/S0304-422X(03)00035-4 }}</ref> and Senekal,<ref>{{cite journal | last1 = Senekal | first1 = B. A. | year = 2012 | title = Die Afrikaanse literêre sisteem: ʼn Eksperimentele benadering met behulp van Sosiale-netwerk-analise (SNA) | url = | journal = LitNet Akademies | volume = 9 | issue = | page = 3 }}</ref> to study various aspects of how literature functions. The basic premise is that polysystem theory, which has been around since the writings of [[Even-Zohar]], can be integrated with network theory and the relationships between different actors in the literary network, e.g. writers, critics, publishers, literary histories, etc., can be mapped using [[Computer graphics (computer science)|visualization]] from SNA.<br />
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In the study of literary systems, network analysis has been applied by Anheier, Gerhards and Romo, De Nooy, and Senekal, to study various aspects of how literature functions. The basic premise is that polysystem theory, which has been around since the writings of Even-Zohar, can be integrated with network theory and the relationships between different actors in the literary network, e.g. writers, critics, publishers, literary histories, etc., can be mapped using visualization from SNA.<br />
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在文学体系的研究中，网络分析被阿海尔、格尔哈兹和罗姆、努伊和 Senekal 应用于研究文学功能的各个方面。其基本前提是多元系统理论可以与网络理论以及文学网络中不同行为者之间的关系相结合。作家、评论家、出版商、文学史等等，都可以通过使用 SNA 的可视化来绘制地图。<br />
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===Organizational studies===<br />
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===Organizational studies===<br />
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组织研究<br />
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Research studies of [[Formal organization|formal]] or [[informal organization]] [[Social relation|relationships]], [[organizational communication]], [[economics]], [[economic sociology]], and other [[resource]] [[Transfer function|transfers]]. Social networks have also been used to examine how organizations interact with each other, characterizing the many [[Interlocking directorate|informal connections]] that link executives together, as well as associations and connections between individual employees at different organizations.<ref>{{cite journal | last1 = Podolny | first1 = J. M. | last2 = Baron | first2 = J. N. | year = 1997 | title = Resources and relationships: Social networks and mobility in the workplace | url = | journal = American Sociological Review | volume = 62 | issue = 5| pages = 673–693 | doi=10.2307/2657354| jstor = 2657354 | citeseerx = 10.1.1.114.6822 }}</ref> Intra-organizational networks have been found to affect [[organizational commitment]],<ref>{{cite journal | last1 = Lee | first1 = J. | last2 = Kim | first2 = S. | year = 2011 | title = Exploring the role of social networks in affective organizational commitment: Network centrality, strength of ties, and structural holes | url = | journal = The American Review of Public Administration | volume = 41 | issue = 2| pages = 205–223 | doi=10.1177/0275074010373803}}</ref> [[organizational identification]],<ref name="Jone11">{{cite journal | last1 = Jones | first1 = C. | last2 = Volpe | first2 = E.H. | year = 2011 | title = Organizational identification: Extending our understanding of social identities through social networks | url = | journal = Journal of Organizational Behavior | volume = 32 | issue = 3| pages = 413–434 | doi=10.1002/job.694}}</ref> [[Organizational citizenship behavior|interpersonal citizenship behaviour]].<ref>{{cite journal | last1 = Bowler | first1 = W. M. | last2 = Brass | first2 = D. J. | year = 2011 | title = Relational correlates of interpersonal citizenship behaviour: A social network perspective | doi = 10.1037/0021-9010.91.1.70 | pmid = 16435939 | journal = Journal of Applied Psychology | volume = 91 | issue = 1| pages = 70–82 | citeseerx = 10.1.1.516.8746 }}</ref><br />
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Research studies of formal or informal organization relationships, organizational communication, economics, economic sociology, and other resource transfers. Social networks have also been used to examine how organizations interact with each other, characterizing the many informal connections that link executives together, as well as associations and connections between individual employees at different organizations. Intra-organizational networks have been found to affect organizational commitment, organizational identification, interpersonal citizenship behaviour.<br />
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研究正式或非正式的组织关系，组织沟通，经济，经济社会学和其他资源转移。社交网络也被用来研究组织之间如何相互作用，描述了许多非正式的联系，以及不同组织的个别雇员之间的联系和联系。研究发现，组织内网络对组织承诺、组织认同、人际公民行为有显著影响。<br />
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===Social capital===<br />
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===Social capital===<br />
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社会资本<br />
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[[Social capital]] is a form of [[Capital (economics)|economic]] and [[cultural capital]] in which social networks are central, [[Stock and flow|transactions]] are marked by [[Reciprocity (social psychology)|reciprocity]], [[Trust (social sciences)|trust]], and [[cooperation]], and [[Market (economics)|market]] [[Agent (economics)|agents]] produce [[goods and services]] not mainly for themselves, but for a [[common good]]. [[Social capital]] is split into three dimensions: the structural, the relational and the cognitive dimension. The structural dimension describes how partners interact with each other and which specific partners meet in a social network. Also The structural dimension of social capital indicates the level of ties among organizations.<ref>(Claridge, 2018).</ref>. This dimension is highly connected to the relational dimension which refers to trustworthiness, norms, expectations and idenfications of the bonds between partners.The relational dimension explains the nature of these ties which is mainly illustrated by the level of trust accorded to the network of organizations. <ref>(Claridge, 2018).</ref> The cognitive dimension analyses the extent to which organizations share common goals and objectives as a result of their ties and interactions. <ref>(Claridge, 2018).</ref><br />
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Social capital is a form of economic and cultural capital in which social networks are central, transactions are marked by reciprocity, trust, and cooperation, and market agents produce goods and services not mainly for themselves, but for a common good. Social capital is split into three dimensions: the structural, the relational and the cognitive dimension. The structural dimension describes how partners interact with each other and which specific partners meet in a social network. Also The structural dimension of social capital indicates the level of ties among organizations.. This dimension is highly connected to the relational dimension which refers to trustworthiness, norms, expectations and idenfications of the bonds between partners.The relational dimension explains the nature of these ties which is mainly illustrated by the level of trust accorded to the network of organizations. The cognitive dimension analyses the extent to which organizations share common goals and objectives as a result of their ties and interactions. <br />
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社会资本是一种经济和文化资本，其中社会网络是中心，交易以互惠、信任和合作为特征，市场代理人生产的商品和服务不主要是为他们自己，而是为了共同的利益。社会资本分为三个维度: 结构维度、关系维度和认知维度。结构维度描述了合作伙伴之间如何相互作用，以及哪些特定的合作伙伴在社交网络中相遇。社会资本的结构维度反映了组织之间的关系水平。这个维度与关系维度高度相关，关系维度指的是伴侣之间关系的可信度、规范、期望和相同性。 关系维度解释了这些联系的本质，主要表现在对组织网络的信任程度上。认知维度分析组织因其联系和相互作用而在多大程度上共享共同的目标和目的。<br />
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[[Social capital]] is a sociological concept about the value of [[social relation]]s and the role of cooperation and confidence to achieve positive outcomes. The term refers to the value one can get from their social ties. For example, newly arrived immigrants can make use of their social ties to established migrants to acquire jobs they may otherwise have trouble getting (e.g., because of unfamiliarity with the local language). A positive relationship exists between social capital and the intensity of social network use.<ref>{{cite journal|last=Sebastián|first=Valenzuela|author2=Namsu Park |author3=Kerk F. Kee |title=Is There Social Capital in a Social Network Site? Facebook Use and College Students' Life Satisfaction, Trust, and Participation|journal=Journal of Computer-Mediated Communication|year=2009|volume=14|issue=4|pages=875–901|doi=10.1111/j.1083-6101.2009.01474.x|doi-access=free}}</ref><ref>{{cite journal | title=Social Connectivity in America: Changes in Adult Friendship Network Size from 2002 to 2007 |author1=Wang, Hua |author2=Barry Wellman |lastauthoramp=yes | journal=American Behavioral Scientist | year=2010 | volume=53 | issue=8 | pages=1148–1169 | doi=10.1177/0002764209356247|url=https://semanticscholar.org/paper/402068f74efb35c093e7fd8c0f8268521219e242 }}</ref> In a dynamic framework, higher activity in a network feeds into higher social capital which itself encourages more activity.<ref name="GaudeulGiannetti2013">{{cite journal|last1=Gaudeul|first1=Alexia|last2=Giannetti|first2=Caterina|title=The role of reciprocation in social network formation, with an application to LiveJournal|journal=Social Networks|volume=35|issue=3|year=2013|pages=317–330|issn=03788733|doi=10.1016/j.socnet.2013.03.003}}</ref><br />
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Social capital is a sociological concept about the value of social relations and the role of cooperation and confidence to achieve positive outcomes. The term refers to the value one can get from their social ties. For example, newly arrived immigrants can make use of their social ties to established migrants to acquire jobs they may otherwise have trouble getting (e.g., because of unfamiliarity with the local language). A positive relationship exists between social capital and the intensity of social network use. In a dynamic framework, higher activity in a network feeds into higher social capital which itself encourages more activity.<br />
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社会资本是一个关于社会关系的价值和合作与信心的作用的社会学概念，以实现积极的成果。这个术语指的是一个人可以从他们的社会关系中获得的价值。例如，新来的移民可以利用他们与老移民的社会关系，获得他们在其他情况下可能难以获得的工作(例如，因为不熟悉当地语言)。社会资本与社会网络使用强度存在正相关关系。在一个动态的框架中，网络中更高的活动会产生更高的社会资本，而社会资本本身又会鼓励更多的活动。<br />
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=== Advertising ===<br />
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=== Advertising ===<br />
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广告<br />
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This particular cluster focuses on brand-image and promotional strategy effectiveness, taking into account the impact of customer participation on sales and brand-image. This is gauged through techniques such as sentiment analysis which rely on mathematical areas of study such as data mining and analytics. This area of research produces vast numbers of commercial applications as the main goal of any study is to understand consumer behaviour and drive sales.<br />
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This particular cluster focuses on brand-image and promotional strategy effectiveness, taking into account the impact of customer participation on sales and brand-image. This is gauged through techniques such as sentiment analysis which rely on mathematical areas of study such as data mining and analytics. This area of research produces vast numbers of commercial applications as the main goal of any study is to understand consumer behaviour and drive sales.<br />
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这个集群侧重于品牌形象和促销策略的有效性，同时考虑到顾客参与对销售和品牌形象的影响。这是通过诸如情感分析等技术来衡量的，这些技术依赖于数学领域的研究，如数据挖掘和分析。这一研究领域产生了大量的商业应用，因为任何研究的主要目标都是了解消费者行为并推动销售。<br />
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===Network position and benefits===<br />
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===Network position and benefits===<br />
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网络地位和利益<br />
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In many [[Formal organizations|organizations]], members tend to focus their activities inside their own groups, which stifles creativity and restricts opportunities. A player whose network bridges structural holes has an advantage in detecting and developing rewarding opportunities.<ref name="Burt 2004"/> Such a player can mobilize social capital by acting as a "broker" of information between two clusters that otherwise would not have been in contact, thus providing access to new ideas, opinions and opportunities. British philosopher and political economist [[John Stuart Mill]], writes, "it is hardly possible to overrate the value ... of placing human beings in contact with persons dissimilar to themselves.... Such communication [is] one of the primary sources of progress."<ref>{{cite book|last=Mill|first=John|title=Principles of Political Economy|year=1909|publisher=William J Ashley|location=Library of Economics and Liberty}}</ref> Thus, a player with a network rich in structural holes can add value to an organization through new ideas and opportunities. This in turn, helps an individual's career development and advancement.<br />
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In many organizations, members tend to focus their activities inside their own groups, which stifles creativity and restricts opportunities. A player whose network bridges structural holes has an advantage in detecting and developing rewarding opportunities. Thus, a player with a network rich in structural holes can add value to an organization through new ideas and opportunities. This in turn, helps an individual's career development and advancement.<br />
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在许多组织中，成员倾向于把他们的活动集中在他们自己的团队中，这抑制了创造力和限制了机会。一个玩家的网络桥梁结构洞有优势，在检测和发展奖励机会。因此，一个拥有丰富结构性漏洞的网络的玩家可以通过新的想法和机会为组织增加价值。这反过来又有助于个人的职业发展和晋升。<br />
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A social capital broker also reaps control benefits of being the facilitator of information flow between contacts. In the case of consulting firm Eden McCallum, the founders were able to advance their careers by bridging their connections with former big three consulting firm consultants and mid-size industry firms.<ref>{{cite journal|last=Gardner|first=Heidi|author2=Eccles, Robert |title=Eden McCallum: A Network Based Consulting Firm|journal=Harvard Business School Review|year=2011}}</ref> By bridging structural holes and mobilizing social capital, players can advance their careers by executing new opportunities between contacts.<br />
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A social capital broker also reaps control benefits of being the facilitator of information flow between contacts. In the case of consulting firm Eden McCallum, the founders were able to advance their careers by bridging their connections with former big three consulting firm consultants and mid-size industry firms. By bridging structural holes and mobilizing social capital, players can advance their careers by executing new opportunities between contacts.<br />
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社会资本经纪人还可以从促进联系人之间的信息流动中获得控制权利。在咨询公司 Eden McCallum 的案例中，创始人能够通过与前三大咨询公司顾问和中等规模的行业公司建立联系来提升自己的职业生涯。通过弥合结构漏洞和调动社会资本，玩家可以通过在联系人之间执行新的机会来提升自己的职业生涯。<br />
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There has been research that both substantiates and refutes the benefits of information brokerage. A study of high tech Chinese firms by Zhixing Xiao found that the control benefits of structural holes are "dissonant to the dominant firm-wide spirit of cooperation and the information benefits cannot materialize due to the communal sharing values" of such organizations.<ref>Xiao, Zhixing; Tsui, Anne (2007). "When Brokers May Not Work: The Cultural Contingency of [http://iniciarsesionentrar.com/facebook/ Social] Capital in Chinese High-tech Firms". ''Administrative Science Quarterly''.</ref> However, this study only analyzed Chinese firms, which tend to have strong communal sharing values. Information and control benefits of structural holes are still valuable in firms that are not quite as inclusive and cooperative on the firm-wide level. In 2004, Ronald Burt studied 673 managers who ran the supply chain for one of America's largest electronics companies. He found that managers who often discussed issues with other groups were better paid, received more positive job evaluations and were more likely to be promoted.<ref name="Burt 2004"/> Thus, bridging structural holes can be beneficial to an organization, and in turn, to an individual's career.<br />
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There has been research that both substantiates and refutes the benefits of information brokerage. A study of high tech Chinese firms by Zhixing Xiao found that the control benefits of structural holes are "dissonant to the dominant firm-wide spirit of cooperation and the information benefits cannot materialize due to the communal sharing values" of such organizations. However, this study only analyzed Chinese firms, which tend to have strong communal sharing values. Information and control benefits of structural holes are still valuable in firms that are not quite as inclusive and cooperative on the firm-wide level. In 2004, Ronald Burt studied 673 managers who ran the supply chain for one of America's largest electronics companies. He found that managers who often discussed issues with other groups were better paid, received more positive job evaluations and were more likely to be promoted. Thus, bridging structural holes can be beneficial to an organization, and in turn, to an individual's career.<br />
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有研究证实和反驳了信息经纪的好处。知行晓对中国高科技企业的一项研究发现，结构性漏洞的控制效益“与占主导地位的企业合作精神不协调，由于这类组织的共享价值观，信息效益不能实现”。然而，这项研究只分析了中国企业，这些企业往往有很强的公共共享价值观。结构性漏洞的信息和控制利益对于那些在整个公司层面上不那么具有包容性和合作性的公司来说仍然是有价值的。2004年，罗纳德 · 伯特研究了673名管理者，他们为美国最大的电子公司之一管理供应链。他发现，那些经常与其他团体讨论问题的管理人员收入更高，得到的工作评价更积极，也更有可能获得晋升。因此，架起结构性漏洞对组织有益，反过来对个人的职业生涯也有益。<br />
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===Social media===<br />
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===Social media===<br />
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社交媒体<br />
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{{main article|Social media}}<br />
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[[Computer network]]s combined with social networking software produces a new medium for social interaction.<ref>{{Cite book|title=The International Encyclopedia of Media Effects|last=Amichai-Hamburger|first=Yair|last2=Hayat|first2=Tsahi|date=2017|publisher=John Wiley & Sons, Inc.|isbn=9781118783764|language=en|doi=10.1002/9781118783764.wbieme0170}}</ref> A relationship over a computerized [[social networking service]] can be characterized by context, direction, and strength. The content of a relation refers to the resource that is exchanged. In a [[computer mediated communication]] context, social pairs exchange different kinds of information, including sending a data file or a computer program as well as providing emotional support or arranging a meeting. With the rise of [[electronic commerce]], information exchanged may also correspond to exchanges of money, goods or services in the "real" world.<ref>{{cite journal | title=Studying Online Social Networks | last1=Garton|first1=Laura|first2=Caroline |last2=Haythornthwaite|author2-link=Caroline Haythornthwaite|first3=Barry|last3=Wellman|author3-link=Barry Wellman | journal=Journal of Computer-Mediated Communication |date=23 June 2006 | volume=3 | issue=1 |pages = 0| doi=10.1111/j.1083-6101.1997.tb00062.x}}</ref> [[Social network analysis]] methods have become essential to examining these types of computer mediated communication.<br />
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Computer networks combined with social networking software produces a new medium for social interaction. A relationship over a computerized social networking service can be characterized by context, direction, and strength. The content of a relation refers to the resource that is exchanged. In a computer mediated communication context, social pairs exchange different kinds of information, including sending a data file or a computer program as well as providing emotional support or arranging a meeting. With the rise of electronic commerce, information exchanged may also correspond to exchanges of money, goods or services in the "real" world. Social network analysis methods have become essential to examining these types of computer mediated communication.<br />
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计算机网络与社交网络软件的结合为社交互动提供了一种新的媒介。通过计算机化的社交网络服务建立起来的关系可以是拥有属性、方向和力量。关系的内容指的是交换的资源。在计算机中介的交流环境中，社会对交换不同种类的信息，包括发送数据文件或计算机程序，以及提供情感支持或安排会议。随着电子商务的兴起，交换的信息也可能对应于“现实”世界中的货币、商品或服务的交换。社会网络分析方法已经成为检验这些类型的计算机中介交流的必要手段。<br />
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In addition, the sheer size and the volatile nature of [[social media]] has given rise to new network metrics. A key concern with networks extracted from social media is the lack of robustness of network metrics given missing data.<ref>{{cite journal | last1 = Wei | first1 = Wei | last2 = Joseph | first2 = Kenneth | last3 = Liu | first3 = Huan | last4 = Carley | first4 = Kathleen M. | year = 2016 | title = Exploring Characteristics of Suspended Users and Network Stability on Twitter | url = | journal = Social Network Analysis and Mining | volume = 6 | issue = | page = 51 | doi = 10.1007/s13278-016-0358-5 }}</ref><br />
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In addition, the sheer size and the volatile nature of social media has given rise to new network metrics. A key concern with networks extracted from social media is the lack of robustness of network metrics given missing data.<br />
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此外，社交媒体的庞大规模和不稳定性也催生了新的网络指标。从社交媒体中提取网络的一个关键问题是，在缺失数据的情况下，网络指标缺乏健壮性。<br />
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* {{cite journal |last1=Aneja |first1=Nagender |last2=Gambhir |first2=Sapna |title=Ad-hoc Social Network: A Comprehensive Survey |url=http://www.ijser.org/researchpaper%5CAd-hoc-Social-Network-A-Comprehensive-Survey.pdf |journal=International Journal of Scientific & Engineering Research |volume=4 |issue=8 |date=August 2013 |pages=156–160|issn=2229-5518}}<br />
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* {{cite book |last=Barabási |first=Albert-László |year=2003 |title=Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life |publisher=Plum |isbn=978-0-452-28439-5 |url-access=registration |url=https://archive.org/details/linkedhoweveryth00bara }}<br />
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* {{cite book |last=Barnett |first=George A. |year=2011 |title=Encyclopedia of Social Networks |publisher=Sage |isbn=978-1-4129-7911-5}}<br />
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* {{cite book |last=Estrada |first=E |year=2011 |title=The Structure of Complex Networks: Theory and Applications |publisher=Oxford University Press |isbn=978-0-199-59175-6}}<br />
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* {{cite book |last=Ferguson |first=Niall |year=2018 |title=The Square and the Tower: Networks and Power, from the Freemasons to Facebook |isbn=978-0735222915 |publisher=Penguin Press}}<br />
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* {{cite book |last=Freeman |first=Linton C. |year=2004 |title=The Development of Social Network Analysis: A Study in the Sociology of Science |publisher=Empirical Press |isbn=978-1-59457-714-7}}<br />
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* {{cite book |last=Kadushin |first=Charles |year=2012 |title=Understanding Social Networks: Theories, Concepts, and Findings |location= |publisher=Oxford University Press |isbn=978-0-19-537946-4}}<br />
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* {{cite book |last1=Mauro |first1=Rios |last2=Petrella |first2=Carlos |year=2014 |title=La Quimera de las Redes Sociales |trans-title=The Chimera of Social Networks |language=Spanish |publisher=Bubok España |isbn=978-9974-99-637-3}}<br />
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* {{cite book |last1=Rainie |first1=Lee |last2=Wellman |first2=Barry |year=2012 |title=Networked: The New Social Operating System |url=https://archive.org/details/networkednewsoci0000rain |url-access=registration |location=Cambridge, Mass. |publisher=MIT Press |isbn=978-0262017190}}<br />
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* {{cite book |last=Scott |first=John |year=1991 |title=Social Network Analysis: A Handbook |publisher=Sage |isbn=978-0-7619-6338-7}}<br />
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* {{cite book |last1=Wasserman |first1=Stanley |last2=Faust |first2=Katherine |year=1994 |title=Social Network Analysis: Methods and Applications |publisher=Cambridge University Press |isbn=978-0-521-38269-4 |series=Structural Analysis in the Social Sciences }}<br />
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* {{cite book |last1=Wellman |first1=Barry |last2=Berkowitz |first2=S. D. |year=1988 |title=Social Structures: A Network Approach |series=Structural Analysis in the Social Sciences |publisher=Cambridge University Press |isbn=978-0-521-24441-1}}<br />
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同行评审期刊<br />
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* ''[http://www.sciencedirect.com/science/journal/03788733 Social Networks]''<br />
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* ''[http://journals.cambridge.org/action/displaySpecialPage?pageId=3656 Network Science]''<br />
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* ''[http://www.cmu.edu/joss/content/articles/volindex.html Journal of Social Structure]''<br />
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* ''[http://www.tandfonline.com/toc/gmas20/current Journal of Mathematical Sociology]''<br />
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* ''[https://www.springer.com/computer/database+management+%26+information+retrieval/journal/13278 Social Network Analysis and Mining (SNAM)]''<br />
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* {{cite journal |title=INSNA - ''Connections'' Journal |journal=Connections Bulletin of the International Network for Social Network Analysis |location=Toronto |publisher=International Network for Social Network Analysis |url=http://www.insna.org/pubs/connections/ |issn=0226-1766 |url-status=dead |archiveurl=https://web.archive.org/web/20130718063641/http://www.insna.org/pubs/connections/ |archivedate=2013-07-18 }}<br />
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===Textbooks and educational resources===<br />
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===Textbooks and educational resources===<br />
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课本及教育资源<br />
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* ''[http://www.cs.cornell.edu/home/kleinber/networks-book/ Networks, Crowds, and Markets]'' (2010) by D. Easley & J. Kleinberg<br />
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* ''[http://faculty.ucr.edu/~hanneman/nettext/ Introduction to Social Networks Methods]'' (2005) by R. Hanneman & M. Riddle<br />
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* [http://www.analytictech.com/networks/ Social Network Analysis Instructional Web Site] by S. Borgatti<br />
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* ''[https://web.archive.org/web/20160304063126/http://www.novagob.org/file/view/153259/ebook-gratuito-sobre-redes-sociales-virtuales Guide for virtual social networks for public administrations]'' (2015) by Mauro D. Ríos (in Spanish)<br />
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===Data sets===<br />
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数据集<br />
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* [http://pajek.imfm.si/doku.php?id=data:urls:index Pajek's list of lists of datasets]<br />
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* [http://networkdata.ics.uci.edu/index.html UC Irvine Network Data Repository]<br />
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* [http://snap.stanford.edu/data/ Stanford Large Network Dataset Collection]<br />
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* [http://www-personal.umich.edu/~mejn/netdata/ M.E.J. Newman datasets]<br />
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* [http://vlado.fmf.uni-lj.si/pub/networks/data/ Pajek datasets]<br />
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* [http://wiki.gephi.org/index.php?title=Datasets#Social_networks Gephi datasets]<br />
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* [http://konect.uni-koblenz.de/networks KONECT – Koblenz network collection]<br />
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* [http://www.stats.ox.ac.uk/~snijders/siena/ RSiena datasets]<br />
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==编者推荐==<br />
集智俱乐部：[https://swarma.org/?p=18646 社会网络分析的四个维度：研究方法、应用和软件工具综述 | 网络科学论文速递25篇]<br />
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[http://arxiv.org/abs/2002.09485 The Four Dimensions of Social Network Analysis: An Overview of Research Methods, Applications, and Software Tools]：近年来，基于社交网络的应用呈指数级增长，其中一个方面的原因是，这个应用领域提供了一个特别肥沃的地方，可以测试和开发最先进的计算技术，从 Web 中提取有价值的信息。这项工作的主要贡献有三个方面：(1) 我们对社会网络分析 (SNA) 的最新发展水平进行了文献综述；(2) 我们提出了一套基于 SNA 四个基本特征（或维度）的新度量标准；(3) 最后，我们对一套流行的 SNA 工具和框架进行了定量分析。我们还进行了一项科学计量学研究，以检测该领域中最活跃的研究领域和应用领域。本工作提出了四个不同维度的定义，即(a)模式、(b)知识发现、(c)信息融合、(d)集成、可伸缩性和可视化，用于定义一组新的度量（称为度），以评估 SNA 的不同软件工具和框架（根据之前的度量对一组 20 个 SNA-软件工具进行分析和排序）。这些维度连同定义的程度，可以评价和衡量社会网络技术的成熟度，寻求对它们的定量评估，从而揭示这一活跃领域的挑战和未来发展趋势。<br />
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类别: 社会系统<br />
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范畴: 系统论<br />
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<small>This page was moved from [[wikipedia:en:Social network]]. Its edit history can be viewed at [[社会网络/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E7%A4%BE%E4%BC%9A%E7%BD%91%E7%BB%9C&diff=11878社会网络2020-08-09T15:03:51Z<p>Dorr：</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Social structure made up of a set of social actors}}<br />
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{{about|the theoretical concept as used in the social and behavioral sciences|social networking sites|Social networking service|the 2010 movie|The Social Network|other uses|Social network (disambiguation)}}<br />
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A '''social network ''' is a [[social structure]] made up of a set of [[social]] actors (such as individuals or organizations), sets of [[Dyad (sociology)|dyadic]] ties, and other [[Social relation|social interactions]] between actors. The social network perspective provides a set of methods for analyzing the structure of whole social entities as well as a variety of theories explaining the patterns observed in these structures.<ref name="WF94CH1">{{cite book|last1=Wasserman |first1=Stanley | author-link1= Stanley Wasserman|last2=Faust |first2=Katherine |year=1994 |title=Social Network Analysis: Methods and Applications |isbn=9780521387071 |chapter=Social Network Analysis in the Social and Behavioral Sciences |pages=1–27 |publisher=Cambridge University Press}}</ref> The study of these structures uses [[social network analysis]] to identify local and global patterns, locate influential entities, and examine network dynamics.<br />
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A social network is a social structure made up of a set of social actors (such as individuals or organizations), sets of dyadic ties, and other social interactions between actors. The social network perspective provides a set of methods for analyzing the structure of whole social entities as well as a variety of theories explaining the patterns observed in these structures. The study of these structures uses social network analysis to identify local and global patterns, locate influential entities, and examine network dynamics.<br />
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'''<font color="#FFD700">社会网络 Social Network</font>'''是一种由一组社会行为者（个人或组织）、一组二元关系以及行为者之间的其他社会互动组成的社会结构。社会网络视角为分析整个社会实体的结构提供了一套方法，也为解释在这些结构中观察到的模式提供了各种理论。对这些结构的研究使用'''<font color="#FFD700">社会网络分析 Social Network Analysis</font>'''以确定本地及全球模式，定位有影响力的实体，并审查网络动态。<br />
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Social networks and the analysis of them is an inherently [[Interdisciplinarity|interdisciplinary]] academic field which emerged from [[social psychology]], [[sociology]], [[statistics]], and [[graph theory]]. [[Georg Simmel]] authored early structural theories in sociology emphasizing the dynamics of triads and "web of group affiliations".<ref>{{cite book|last1=Scott |first1=W. Richard |last2=Davis |first2=Gerald F. |author-link1= William Richard Scott|author-link2= Gerald F. Davis|title=Organizations and Organizing |chapter=Networks In and Around Organizations |year=2003 |isbn=978-0-13-195893-7 |publisher=Pearson Prentice Hall}}</ref> [[Jacob Moreno]] is credited with developing the first [[sociogram]]s in the 1930s to study interpersonal relationships. These approaches were mathematically formalized in the 1950s and theories and methods of social networks became pervasive in the [[Social science|social and behavioral sciences]] by the 1980s.<ref name="WF94CH1"/><ref name="Freeman History">{{cite book|last=Freeman |first=Linton |year=2004 |publisher=Empirical Press |isbn=978-1-59457-714-7 |title=The Development of Social Network Analysis: A Study in the Sociology of Science}}</ref> [[Social network analysis]] is now one of the major paradigms in contemporary sociology, and is also employed in a number of other social and formal sciences. Together with other [[complex network]]s, it forms part of the nascent field of [[network science]].<ref>{{cite journal|journal=Science |year=2009 |volume=323 |number=5916 |pages=892–895 |doi=10.1126/science.1165821 |pmid=19213908 |title=Network Analysis in the Social Sciences |first1=Stephen P. |last1=Borgatti |first2=Ajay |last2=Mehra |first3=Daniel J. |last3=Brass |first4=Giuseppe |last4=Labianca|bibcode=2009Sci...323..892B |citeseerx=10.1.1.536.5568 }}</ref><ref name="EK">{{cite book|title=Networks, Crowds, and Markets: Reasoning about a Highly Connected World |first1=David |last1=Easley |first2=Jon |last2=Kleinberg |publisher=Cambridge University Press |year=2010 |chapter=Overview |pages=1–20 |isbn=978-0-521-19533-1}}</ref><br />
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Social networks and the analysis of them is an inherently interdisciplinary academic field which emerged from social psychology, sociology, statistics, and graph theory. Georg Simmel authored early structural theories in sociology emphasizing the dynamics of triads and "web of group affiliations". Jacob Moreno is credited with developing the first sociograms in the 1930s to study interpersonal relationships. These approaches were mathematically formalized in the 1950s and theories and methods of social networks became pervasive in the social and behavioral sciences by the 1980s. Social network analysis is now one of the major paradigms in contemporary sociology, and is also employed in a number of other social and formal sciences. Together with other complex networks, it forms part of the nascent field of network science.<br />
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社会网络及其分析是社会心理学、社会学、统计学和图论中内在的跨学科学术领域。'''格奥尔格·西梅尔 Georg Simmel'''是早期社会学结构主义理论的作者，他强调三合会和“群体关系网”的动态性。'''雅各布·莫雷诺 Jacob Moreno'''被认为是在20世纪30年代发展了第一份'''<font color="#FFD700">社交关系图 Sociogram</font>'''以研究人际关系的人。这些方法在20世纪50年代得到数学化，到20世纪80年代，社会网络的理论和方法在社会和行为科学中变得普遍。社会网络分析现在是当代社会学的主要范式之一，也被用于许多其他社会科学及形式科学。它与其他复杂网络一起构成了网络科学新兴领域的一部分。<br />
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==Overview 概览==<br />
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[[File:Barabasi Albert model.gif|thumb|left|Evolution graph of a social network: [[Barabási–Albert model|Barabási model]].]]<br />
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Barabási model.]]<br />
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Barabási model.]]<br />
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一个社会网络的演化图：巴拉巴西模型。<br />
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The social network is a [[Scientific theory|theoretical]] [[Construct (philosophy of science)|construct]] useful in the [[social sciences]] to study relationships between individuals, [[social groups|groups]], [[formal organizations|organizations]], or even entire [[society|societies]] ([[social unit]]s, see [[Differentiation (sociology)|differentiation]]). The term is used to describe a [[social structure]] determined by such [[social interactions|interactions]]. The ties through which any given social unit connects represent the convergence of the various social contacts of that unit. This theoretical approach is, necessarily, relational. An [[axiom]] of the social network approach to understanding [[social interaction]] is that social phenomena should be primarily conceived and investigated through the properties of relations between and within units, instead of the properties of these units themselves. Thus, one common criticism of social network theory is that [[Agency (sociology)|individual agency]] is often ignored<ref name="jscott">Scott, John P. (2000). ''Social Network Analysis: A Handbook'' (2nd edition). Thousand Oaks, CA: Sage Publications.</ref> although this may not be the case in practice (see [[agent-based model]]ing). Precisely because many different types of relations, singular or in combination, form these network configurations, [[Network science|network analytics]] are useful to a broad range of research enterprises. In social science, these fields of study include, but are not limited to [[anthropology]], [[biology]], [[communication studies]], [[economics]], [[geography]], [[information science]], [[organizational studies]], [[social psychology]], [[sociology]], and [[sociolinguistics]].<br />
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The social network is a theoretical construct useful in the social sciences to study relationships between individuals, groups, organizations, or even entire societies (social units, see differentiation). The term is used to describe a social structure determined by such interactions. The ties through which any given social unit connects represent the convergence of the various social contacts of that unit. This theoretical approach is, necessarily, relational. An axiom of the social network approach to understanding social interaction is that social phenomena should be primarily conceived and investigated through the properties of relations between and within units, instead of the properties of these units themselves. Thus, one common criticism of social network theory is that individual agency is often ignored although this may not be the case in practice (see agent-based modeling). Precisely because many different types of relations, singular or in combination, form these network configurations, network analytics are useful to a broad range of research enterprises. In social science, these fields of study include, but are not limited to anthropology, biology, communication studies, economics, geography, information science, organizational studies, social psychology, sociology, and sociolinguistics.<br />
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社会网络是社会科学中研究个人、团体、组织甚至整个社会（社会单位，见分化）之间关系的理论构造。这个术语用来描述由这种相互作用决定的社会结构。任何一个特定的社会单元之间的联系都代表着这个单元各种社会联系的聚合。这种理论方法必然是相关的。理解社会互动的社会网络方法的一个公理是，社会现象应该主要通过单元之间和单元内部关系的性质来构思和研究，而不是这些单元本身的性质。因此，社会网络理论总被诟病的一点是其常忽视个体代理，而实践中可能并非如此（见基于主体的建模）。正是因为许多不同类型的关系（单独或组合形式）形成这些网络配置，网络分析在广泛的研究中有用。在社会科学中，这些研究领域包括但不限于'''<font color="#FFD700">人类学 Anthropology</font>'''、生物学、'''<font color="#FFD700">传播学 Communication Studies</font>'''、经济学、地理学、信息科学、组织学、社会心理学、社会学和'''<font color="#FFD700">社会语言学 Sociolinguistics</font>''。<br />
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==History 历史==<br />
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In the late 1890s, both [[Émile Durkheim]] and [[Ferdinand Tönnies]] foreshadowed the idea of social networks in their theories and research of [[social group]]s. Tönnies argued that social groups can exist as personal and direct social ties that either link individuals who share values and belief (''[[Gemeinschaft]]'', German, commonly translated as "[[community]]") or impersonal, formal, and instrumental social links (''[[Gesellschaft]]'', German, commonly translated as "[[society]]").<ref>Tönnies, Ferdinand (1887). ''Gemeinschaft und Gesellschaft'', Leipzig: Fues's Verlag. (Translated, 1957 by Charles Price Loomis as ''Community and Society'', East Lansing: Michigan State University Press.)</ref> Durkheim gave a non-individualistic explanation of social facts, arguing that social phenomena arise when interacting individuals constitute a reality that can no longer be accounted for in terms of the properties of individual actors.<ref>Durkheim, Emile (1893). ''De la division du travail social: étude sur l'organisation des sociétés supérieures'', Paris: F. Alcan. (Translated, 1964, by Lewis A. Coser as ''The Division of Labor in Society'', New York: Free Press.)</ref> [[Georg Simmel]], writing at the turn of the twentieth century, pointed to the nature of networks and the effect of network size on interaction and examined the likelihood of interaction in loosely knit networks rather than groups.<ref>Simmel, Georg (1908). ''Soziologie'', Leipzig: Duncker & Humblot.</ref><br />
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In the late 1890s, both Émile Durkheim and Ferdinand Tönnies foreshadowed the idea of social networks in their theories and research of social groups. Tönnies argued that social groups can exist as personal and direct social ties that either link individuals who share values and belief (Gemeinschaft, German, commonly translated as "community") or impersonal, formal, and instrumental social links (Gesellschaft, German, commonly translated as "society"). Durkheim gave a non-individualistic explanation of social facts, arguing that social phenomena arise when interacting individuals constitute a reality that can no longer be accounted for in terms of the properties of individual actors. Georg Simmel, writing at the turn of the twentieth century, pointed to the nature of networks and the effect of network size on interaction and examined the likelihood of interaction in loosely knit networks rather than groups.<br />
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在19世纪90年代晚期，'''迪尔凯姆 Émile Durkheim'''和'''斐迪南·滕尼斯 Ferdinand Tönnies'''在他们关于社会群体的理论和研究中都预示了社会网络的概念。尼斯认为，社会群体可以作为个人和直接的社会关系存在，这种关系或者将具有共同价值观和信仰的个人联系在一起（德语 Gemeinschaft，通常翻译为“社区”），或者将非个人的、正式的和工具性的社会关系（德语 Gesellschaft，通常翻译为“社会”）。涂尔干对社会事实作了非个人主义的解释，认为当相互作用的个体构成一种现实时，社会现象就产生了，而这种现实再也不能用个体行为者的特性来解释。乔治 · 西梅尔在20世纪之交的著作中指出了网络的本质和网络规模对互动的影响，并研究了在松散的网络而不是群体中互动的可能性。<br />
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[[File:Moreno Sociogram 2nd Grade.png|thumb|Moreno's sociogram of a 2nd grade class]]<br />
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Moreno's sociogram of a 2nd grade class<br />
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莫雷诺二年级的社会记录<br />
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Major developments in the field can be seen in the 1930s by several groups in psychology, anthropology, and mathematics working independently.<ref name=jscott /><ref>For a historical overview of the development of social network analysis, see: {{cite book|last1=Carrington|first1=Peter J.|last2=Scott|first2=John|chapter=Introduction|title=The Sage Handbook of Social Network Analysis| publisher=Sage|year=2011|isbn=978-1-84787-395-8|page=1|chapter-url=https://books.google.com/books?id=2chSmLzClXgC&pg=PA1}}</ref><ref>See also the diagram in {{cite book|author=Scott, John|title=Social Network Analysis: A Handbook|publisher=Sage|year=2000|isbn=978-0-7619-6339-4|page=8|url=https://books.google.com/books?id=Ww3_bKcz6kgC&pg=PA8}}</ref> In [[psychology]], in the 1930s, [[Jacob L. Moreno]] began systematic recording and analysis of social interaction in small groups, especially classrooms and work groups (see [[sociometry]]). In [[anthropology]], the foundation for social network theory is the theoretical and [[ethnography|ethnographic]] work of [[Bronislaw Malinowski]],<ref>Malinowski, Bronislaw (1913). ''The Family Among the Australian Aborigines: A Sociological Study''. London: University of London Press.</ref> [[Radcliffe-Brown|Alfred Radcliffe-Brown]],<ref>Radcliffe-Brown, Alfred Reginald (1930) ''The social organization of Australian tribes''. Sydney, Australia: University of Sydney ''Oceania'' monographs, No.1.</ref><ref>{{cite journal | last1 = Radcliffe-Brown | first1 = A. R. | year = 1940 | title = On social structure | url = | journal = Journal of the Royal Anthropological Institute | volume = 70 | issue = 1| pages = 1–12 | doi=10.2307/2844197| jstor = 2844197 }}</ref> and [[Claude Lévi-Strauss]].<ref>Lévi-Strauss, Claude ([1947]1967). ''Les structures élémentaires de la parenté''. Paris: La Haye, Mouton et Co. (Translated, 1969 by J. H. Bell, J. R. von Sturmer, and R. Needham, 1969, as ''The Elementary Structures of Kinship'', Boston: Beacon Press.)</ref> A group of social anthropologists associated with [[Max Gluckman]] and the [[Manchester school (anthropology)|Manchester School]], including [[John Arundel Barnes|John A. Barnes]],<ref>Barnes, John (1954). "Class and Committees in a Norwegian Island Parish". ''Human Relations'', (7): 39–58.</ref> [[J. Clyde Mitchell]] and [[Elizabeth Bott Spillius]],<ref>{{cite journal | last1 = Freeman | first1 = Linton C. | last2 = Wellman | first2 = Barry | year = 1995 | title = A note on the ancestoral Toronto home of social network analysis | url = | journal = Connections | volume = 18 | issue = 2| pages = 15–19 }}</ref><ref>{{cite journal | last1 = Savage | first1 = Mike | year = 2008 | title = Elizabeth Bott and the formation of modern British sociology | url = | journal = The Sociological Review | volume = 56 | issue = 4| pages = 579–605 | doi=10.1111/j.1467-954x.2008.00806.x}}</ref> often are credited with performing some of the first fieldwork from which network analyses were performed, investigating community networks in southern Africa, India and the United Kingdom.<ref name=jscott /> Concomitantly, British anthropologist [[Siegfried Frederick Nadel|S. F. Nadel]] codified a theory of social structure that was influential in later network analysis.<ref>Nadel, S. F. 1957. ''The Theory of Social Structure''. London: Cohen and West.</ref> In [[sociology]], the early (1930s) work of [[Talcott Parsons]] set the stage for taking a relational approach to understanding social structure.<ref>Parsons, Talcott ([1937] 1949). ''The Structure of Social Action: A Study in Social Theory with Special Reference to a Group of European Writers''. New York: The Free Press.</ref><ref>Parsons, Talcott (1951). ''The Social System''. New York: The Free Press.</ref> Later, drawing upon Parsons' theory, the work of sociologist [[Peter Blau]] provides a strong impetus for analyzing the relational ties of social units with his work on [[social exchange theory]].<ref>Blau, Peter (1956). ''Bureaucracy in Modern Society''. New York: Random House, Inc.</ref><ref>Blau, Peter (1960). "A Theory of Social Integration". ''The American Journal of Sociology'', (65)6: 545–556, (May).</ref><ref>Blau, Peter (1964). ''Exchange and Power in Social Life''.</ref><br />
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Major developments in the field can be seen in the 1930s by several groups in psychology, anthropology, and mathematics working independently. In psychology, in the 1930s, Jacob L. Moreno began systematic recording and analysis of social interaction in small groups, especially classrooms and work groups (see sociometry). In anthropology, the foundation for social network theory is the theoretical and ethnographic work of Bronislaw Malinowski, Alfred Radcliffe-Brown, and Claude Lévi-Strauss. A group of social anthropologists associated with Max Gluckman and the Manchester School, including John A. Barnes, J. Clyde Mitchell and Elizabeth Bott Spillius, often are credited with performing some of the first fieldwork from which network analyses were performed, investigating community networks in southern Africa, India and the United Kingdom. In sociology, the early (1930s) work of Talcott Parsons set the stage for taking a relational approach to understanding social structure. Later, drawing upon Parsons' theory, the work of sociologist Peter Blau provides a strong impetus for analyzing the relational ties of social units with his work on social exchange theory.<br />
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20世纪30年代，心理学、人类学和数学领域的几个独立研究小组已经看到了这一领域的重大发展。在心理学方面，在20世纪30年代，雅各布 · l · 莫雷诺开始系统地记录和分析小团体中的社会互动，尤其是课堂和工作团体中的社会互动。在人类学中，社会网络理论的基础是 Bronislaw Malinowski，阿尔弗雷德·拉德克利夫-布朗和克洛德·列维-斯特劳斯的理论和人种学著作。包括约翰 · a · 巴恩斯(John a. Barnes)、 j · 克莱德 · 米切尔(j. Clyde Mitchell)和伊丽莎白 · 博特 · 斯皮利厄斯(Elizabeth Bott Spillius)在内的一群与马克斯 · 格拉克曼(Max Gluckman)和曼彻斯特学院(Manchester School)有关的社会人类学家，经常被认为是进行了一些最早的野。在社会学方面，塔尔科特 · 帕森斯早期(1930年代)的著作为采用关系方法理解社会结构奠定了基础。后来，社会学家彼得 · 布劳的社会交换理论为分析社会单位之间的关系提供了强大的动力。<br />
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By the 1970s, a growing number of scholars worked to combine the different tracks and traditions. One group consisted of sociologist [[Harrison White]] and his students at the [[Harvard Department of Social Relations|Harvard University Department of Social Relations]]. Also independently active in the Harvard Social Relations department at the time were [[Charles Tilly]], who focused on networks in political and community sociology and social movements, and [[Stanley Milgram]], who developed the "six degrees of separation" thesis.<ref>{{cite web |url=http://www.semioticon.com/semiotix/semiotix14/sem-14-05.html |title=The Networked Individual: A Profile of Barry Wellman |author=Bernie Hogan}}</ref> [[Mark Granovetter]]<ref name="Introduction for the French Reader">{{cite journal | last1 = Granovetter | first1 = Mark | year = 2007 | title = Introduction for the French Reader | url = | journal = Sociologica | volume = 2 | issue = | pages = 1–8 }}</ref> and [[Barry Wellman]]<ref>Wellman, Barry (1988). "Structural analysis: From method and metaphor to theory and substance". pp. 19–61 in B. Wellman and S. D. Berkowitz (eds.) ''Social Structures: A Network Approach'', Cambridge, UK: Cambridge University Press.</ref> are among the former students of White who elaborated and championed the analysis of social networks.<ref name="Introduction for the French Reader"/><ref>Mullins, Nicholas. ''Theories and Theory Groups in Contemporary American Sociology''. New York: Harper and Row, 1973.</ref><ref>Tilly, Charles, ed. ''An Urban World''. Boston: Little Brown, 1974.</ref><ref>Wellman, Barry. 1988. "Structural Analysis: From Method and Metaphor to Theory and Substance". pp. 19–61 in ''Social Structures: A Network Approach'', edited by Barry Wellman and S. D. Berkowitz. Cambridge: Cambridge University Press.</ref><br />
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By the 1970s, a growing number of scholars worked to combine the different tracks and traditions. One group consisted of sociologist Harrison White and his students at the Harvard University Department of Social Relations. Also independently active in the Harvard Social Relations department at the time were Charles Tilly, who focused on networks in political and community sociology and social movements, and Stanley Milgram, who developed the "six degrees of separation" thesis. Mark Granovetter and Barry Wellman are among the former students of White who elaborated and championed the analysis of social networks.<br />
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到了20世纪70年代，越来越多的学者致力于将不同的轨迹和传统结合起来。其中一组由社会学家哈里森 · 怀特和他在哈佛大学社会关系系的学生组成。当时在哈佛大学社会关系系独立活动的还有 Charles Tilly，他专注于政治和社区社会学和社会运动的网络，还有 Stanley Milgram，他发表了六度分隔理论的论文。马克·格兰诺维特和 Barry Wellman 是 White 以前的学生，他们阐述并支持对社交网络的分析。<br />
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Beginning in the late 1990s, social network analysis experienced work by sociologists, political scientists, and physicists such as [[Duncan J. Watts]], [[Albert-László Barabási]], [[Peter Bearman]], [[Nicholas A. Christakis]], [[James H. Fowler]], and others, developing and applying new models and methods to emerging data available about online social networks, as well as "digital traces" regarding face-to-face networks.<br />
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Beginning in the late 1990s, social network analysis experienced work by sociologists, political scientists, and physicists such as Duncan J. Watts, Albert-László Barabási, Peter Bearman, Nicholas A. Christakis, James H. Fowler, and others, developing and applying new models and methods to emerging data available about online social networks, as well as "digital traces" regarding face-to-face networks.<br />
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从20世纪90年代末开始，社会网络分析经历了社会学家、政治学家和物理学家的工作，如邓肯 · j · 瓦茨、阿尔伯特 · l · 斯兹尔 · 巴拉布 · 西、彼得 · 贝尔曼、尼古拉斯 · a · 克里斯塔基斯、詹姆斯 · h · 福勒等人，开发和应用新的模型和方法来获得有关在线社会网络的新兴数据，以及有关面对面网络的“数字痕迹”。<br />
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==Levels of analysis==<br />
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==Levels of analysis==<br />
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分析水平<br />
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[[File:Network self-organization stages.png|thumb|right|Self-organization of a network, based on Nagler, Levina, & Timme, (2011)<ref>{{cite journal|author1=Nagler, Jan|author2=Anna Levina|author3=Marc Timme|year=2011|title=Impact of single links in competitive percolation|journal=Nature Physics|volume=7|issue=3|pages=265–270|doi=10.1038/nphys1860|arxiv=1103.0922|bibcode=2011NatPh...7..265N}}</ref>]]<br />
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Self-organization of a network, based on Nagler, Levina, & Timme, (2011)<br />
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一个网络的自我组织，基于 Nagler，Levina，& Timme，(2011)<br />
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[[File:Social Network Diagram (large).svg|right|thumb|Centrality]]<br />
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Centrality<br />
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中心性<br />
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In general, social networks are [[self-organization|self-organizing]], [[emergence|emergent]], and [[social complexity|complex]], such that a globally coherent pattern appears from the local interaction of the elements that make up the system.<ref>Newman, Mark, Albert-László Barabási and Duncan J. Watts (2006). ''The Structure and Dynamics of Networks'' (Princeton Studies in Complexity). Oxford: Princeton University Press.</ref><ref>{{cite journal | last1 = Wellman | first1 = Barry | year = 2008 | title = Review: The development of social network analysis: A study in the sociology of science | url = | journal = Contemporary Sociology | volume = 37 | issue = 3| pages = 221–222 | doi=10.1177/009430610803700308}}</ref> These patterns become more apparent as network size increases. However, a global network analysis<ref>{{cite book|last=Faust|first=Stanley Wasserman; Katherine|title=Social network analysis : methods and applications|year=1998|publisher=Cambridge Univ. Press|location=Cambridge [u.a.]|isbn=978-0521382694|edition=Reprint.}}</ref> of, for example, all [[interpersonal relationships]] in the world is not feasible and is likely to contain so much [[Information theory|information]] as to be uninformative. Practical limitations of computing power, ethics and participant recruitment and payment also limit the scope of a social network analysis.<ref name="Kadu12">Kadushin, C. (2012). ''Understanding social networks: Theories, concepts, and findings''. Oxford: Oxford University Press.</ref><ref>{{Cite journal| author=Granovetter, M.|title= Network sampling: Some first steps| year=1976 |pages=1287–1303 |volume=81| issue=6 |journal= American Journal of Sociology | doi=10.1086/226224}}</ref> The nuances of a local system may be lost in a large network analysis, hence the quality of information may be more important than its scale for understanding network properties. Thus, social networks are analyzed at the scale relevant to the researcher's theoretical question. Although [[level of analysis|levels of analysis]] are not necessarily [[Mutually exclusive events|mutually exclusive]], there are three general levels into which networks may fall: [[Microsociology|micro-level]], [[wikt:meso-|meso-level]], and [[Macrosociology|macro-level]].<br />
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In general, social networks are self-organizing, emergent, and complex, such that a globally coherent pattern appears from the local interaction of the elements that make up the system. These patterns become more apparent as network size increases. However, a global network analysis of, for example, all interpersonal relationships in the world is not feasible and is likely to contain so much information as to be uninformative. Practical limitations of computing power, ethics and participant recruitment and payment also limit the scope of a social network analysis. The nuances of a local system may be lost in a large network analysis, hence the quality of information may be more important than its scale for understanding network properties. Thus, social networks are analyzed at the scale relevant to the researcher's theoretical question. Although levels of analysis are not necessarily mutually exclusive, there are three general levels into which networks may fall: micro-level, meso-level, and macro-level.<br />
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一般来说，社会网络是自组织的、涌现的和复杂的，以至于从组成系统的元素的局部相互作用中出现了一个全局连贯的模式。随着网络规模的增大，这些模式变得更加明显。然而，一个全球网络分析，例如，世界上所有的人际关系是不可行的，可能包含太多的信息，以至于没有提供信息。计算能力的实际限制、道德规范以及参与者的招募和支付也限制了社会网络分析的范围。本地系统的细微差别在大型网络分析中可能会丢失，因此对于理解网络属性来说，信息的质量可能比其规模更重要。因此，社会网络被分析在与研究者的理论问题相关的尺度上。虽然分析层次不一定相互排斥，但网络可以分为三个一般层次: 微观层次、中观层次和宏观层次。<br />
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===Micro level===<br />
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===Micro level===<br />
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微观层面<br />
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At the micro-level, social network research typically begins with an individual, [[Snowball sampling|snowballing]] as social relationships are traced, or may begin with a small group of individuals in a particular social context.<br />
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At the micro-level, social network research typically begins with an individual, snowballing as social relationships are traced, or may begin with a small group of individuals in a particular social context.<br />
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在微观层面上，社会网络研究通常从个人开始，随着社会关系的追踪而像滚雪球一样扩大，或者可能从特定社会背景下的一小群个人开始。<br />
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'''Dyadic level''': A [[Dyad (sociology)|dyad]] is a social relationship between two individuals. Network research on dyads may concentrate on [[Structural functionalism|structure]] of the relationship (e.g. multiplexity, strength), [[social equality]], and tendencies toward [[Reciprocity (social and political philosophy)|reciprocity/mutuality]].<br />
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Dyadic level: A dyad is a social relationship between two individuals. Network research on dyads may concentrate on structure of the relationship (e.g. multiplexity, strength), social equality, and tendencies toward reciprocity/mutuality.<br />
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二元层次: 二元是两个人之间的社会关系。网络对二元关系的研究可以集中在关系的结构上。多样性、力量)、社会平等以及互惠互利的倾向。<br />
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'''Triadic level''': Add one individual to a dyad, and you have a [[Triadic relation|triad]]. Research at this level may concentrate on factors such as [[Independence number|balance]] and [[Vertex-transitive graph|transitivity]], as well as [[social equality]] and tendencies toward [[Reciprocity (social and political philosophy)|reciprocity/mutuality]].<ref name="Kadu12"/> In the [[balance theory]] of [[Fritz Heider]] the triad is the key to social dynamics. The discord in a rivalrous [[love triangle]] is an example of an unbalanced triad, likely to change to a balanced triad by a change in one of the relations. The dynamics of social friendships in society has been modeled by balancing triads. The study is carried forward with the theory of [[signed graph]]s.<br />
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Triadic level: Add one individual to a dyad, and you have a triad. Research at this level may concentrate on factors such as balance and transitivity, as well as social equality and tendencies toward reciprocity/mutuality. In the balance theory of Fritz Heider the triad is the key to social dynamics. The discord in a rivalrous love triangle is an example of an unbalanced triad, likely to change to a balanced triad by a change in one of the relations. The dynamics of social friendships in society has been modeled by balancing triads. The study is carried forward with the theory of signed graphs.<br />
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三元等级: 把一个人加到一个二元组中，你就有了一个三元组。这一层次的研究可能集中在诸如平衡和传递性等因素，以及社会平等和互惠 / 互惠的倾向。在弗里茨 · 海德的平衡理论中，三元组是社会动力学的关键。争强好胜的三角恋中的不和谐是不平衡的三角关系的一个例子，很可能通过其中一种关系的改变而变成平衡的三角关系。社会中社会友谊的动态模型是通过平衡三合会建立起来的。本文利用符号图理论进行了研究。<br />
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'''Actor level''': The smallest unit of analysis in a social network is an individual in their social setting, i.e., an "actor" or "ego". Egonetwork analysis focuses on network characteristics such as size, relationship strength, density, [[centrality]], [[wikt:prestige|prestige]] and roles such as [[isolates|isolates, liaisons]], and [[Bridge (interpersonal)|bridges]].<ref name="Jone11"/> Such analyses, are most commonly used in the fields of [[psychology]] or [[Social psychology (sociology)|social psychology]], [[ethnographic]] [[kinship]] analysis or other [[genealogy|genealogical]] studies of relationships between individuals.<br />
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Actor level: The smallest unit of analysis in a social network is an individual in their social setting, i.e., an "actor" or "ego". Egonetwork analysis focuses on network characteristics such as size, relationship strength, density, centrality, prestige and roles such as isolates, liaisons, and bridges. Such analyses, are most commonly used in the fields of psychology or social psychology, ethnographic kinship analysis or other genealogical studies of relationships between individuals.<br />
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行为者层面: 社会网络中最小的分析单位是社会环境中的个体，即“行为者”或“自我”。Egonwork 分析主要关注网络特征，例如大小、关系强度、密度、中心性、声望和角色，例如隔离、联络和桥梁。这种分析最常用于心理学或社会心理学、人种学亲属关系分析或其他个人关系的系谱研究领域。<br />
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'''Subset level''': [[Subset]] levels of network research problems begin at the micro-level, but may cross over into the meso-level of analysis. Subset level research may focus on [[Distance (graph theory)|distance]] and reachability, [[cliques]], [[Cohesion (social policy)|cohesive]] subgroups, or other [[Group action (sociology)|group actions]] or [[Group behaviour|behavior]].<ref>{{cite book | title="Graph Theoretical Approaches to Social Network Analysis". in Computational Complexity: Theory, Techniques, and Applications (Robert A. Meyers, ed.) | publisher=Springer | author=de Nooy, Wouter |year=2012 | pages=2864–2877 | isbn=978-1-4614-1800-9|doi=10.1007/978-1-4614-1800-9_176| chapter=Social Network Analysis, Graph Theoretical Approaches to }}</ref><br />
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Subset level: Subset levels of network research problems begin at the micro-level, but may cross over into the meso-level of analysis. Subset level research may focus on distance and reachability, cliques, cohesive subgroups, or other group actions or behavior.<br />
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子集级别: 网络研究问题的子集级别开始于微观级别，但可能跨越到中观级别的分析。子集级别的研究可能集中在距离和可达性、派系、凝聚子群或其他群体行为或行为。<br />
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===Meso level===<br />
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中观水平<br />
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In general, meso-level theories begin with a [[Sample population|population]] size that falls between the micro- and macro-levels. However, meso-level may also refer to analyses that are specifically designed to reveal connections between micro- and macro-levels. Meso-level networks are low density and may exhibit causal processes distinct from interpersonal micro-level networks.<ref>{{cite journal | last1 = Hedström | first1 = Peter | last2 = Sandell | first2 = Rickard | last3 = Stern | first3 = Charlotta | year = 2000 | title = Mesolevel Networks and the Diffusion of Social Movements: The Case of the Swedish Social Democratic Party | url = http://www.nuffield.ox.ac.uk/users/hedstrom/ajs3.pdf | journal = American Journal of Sociology | volume = 106 | issue = 1| pages = 145–172 | doi = 10.1086/303109 }}</ref><br />
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In general, meso-level theories begin with a population size that falls between the micro- and macro-levels. However, meso-level may also refer to analyses that are specifically designed to reveal connections between micro- and macro-levels. Meso-level networks are low density and may exhibit causal processes distinct from interpersonal micro-level networks.<br />
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一般来说，中观层面的理论始于介于微观和宏观层面之间的人口规模。然而，中观层面也可以指专门为揭示微观和宏观层面之间的联系而设计的分析。中观层次的网络是低密度的，可能表现出不同于人际微观层次网络的因果过程。<br />
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[[File:Social Red.jpg|thumb|right|Social network diagram, meso-level]]<br />
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Social network diagram, meso-level<br />
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中层社会网络图<br />
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'''Organizations''': Formal [[organizations]] are [[social group]]s that distribute tasks for a collective [[goal]].<ref name="Riketta, M. 2007">{{cite journal | last1 = Riketta | first1 = M. | last2 = Nienber | first2 = S. | year = 2007 | title = Multiple identities and work motivation: The role of perceived compatibility between nested organizational units | url = | journal = British Journal of Management | volume = 18 | issue = | pages = S61–77 | doi=10.1111/j.1467-8551.2007.00526.x}}</ref> Network research on organizations may focus on either intra-organizational or inter-organizational ties in terms of [[Formal organization|formal]] or [[Informal organization|informal]] relationships. Intra-organizational networks themselves often contain multiple levels of analysis, especially in larger organizations with multiple branches, franchises or semi-autonomous departments. In these cases, research is often conducted at a work group level and organization level, focusing on the interplay between the two structures.<ref name="Riketta, M. 2007"/> Experiments with networked groups online have documented ways to optimize group-level coordination through diverse interventions, including the addition of autonomous agents to the groups.<ref>{{Cite journal|last=Shirado|first=Hirokazu|last2=Christakis|first2=Nicholas A|title=Locally noisy autonomous agents improve global human coordination in network experiments|journal=Nature|volume=545|issue=7654|pages=370–374|doi=10.1038/nature22332|pmid=28516927|pmc=5912653|bibcode=2017Natur.545..370S|year=2017}}</ref><br />
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Organizations: Formal organizations are social groups that distribute tasks for a collective goal. Network research on organizations may focus on either intra-organizational or inter-organizational ties in terms of formal or informal relationships. Intra-organizational networks themselves often contain multiple levels of analysis, especially in larger organizations with multiple branches, franchises or semi-autonomous departments. In these cases, research is often conducted at a work group level and organization level, focusing on the interplay between the two structures.<br />
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组织: 正式组织是为共同目标分配任务的社会团体。关于组织的网络研究可以侧重于正式或非正式关系方面的组织内或组织间联系。组织内网络本身往往包含多层次的分析，特别是在具有多个分支机构、特许经营权或半自治部门的较大组织中。在这些情况下，研究通常在工作组和组织层面进行，重点放在两个结构之间的相互作用。<br />
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'''Randomly distributed networks''': [[Exponential random graph models]] of social networks became state-of-the-art methods of social network analysis in the 1980s. This framework has the capacity to represent social-structural effects commonly observed in many human social networks, including general [[Degree (graph theory)|degree]]-based structural effects commonly observed in many human social networks as well as [[Reciprocity (social and political philosophy)|reciprocity]] and [[Transitive set|transitivity]], and at the node-level, [[homophily]] and [[Attribute-value system|attribute]]-based activity and popularity effects, as derived from explicit hypotheses about [[Dependency graph|dependencies]] among network ties. [[Parameter]]s are given in terms of the prevalence of small [[Induced subgraph|subgraph]] configurations in the network and can be interpreted as describing the combinations of local social processes from which a given network emerges. These probability models for networks on a given set of actors allow generalization beyond the restrictive dyadic independence assumption of micro-networks, allowing models to be built from theoretical structural foundations of social behavior.<ref>{{cite journal | last1 = Cranmer | first1 = Skyler J. | last2 = Desmarais | first2 = Bruce A. | year = 2011 | title = Inferential Network Analysis with Exponential Random Graph Models | url = | journal = Political Analysis | volume = 19 | issue = 1| pages = 66–86 | doi=10.1093/pan/mpq037| citeseerx = 10.1.1.623.751 }}</ref><br />
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Randomly distributed networks: Exponential random graph models of social networks became state-of-the-art methods of social network analysis in the 1980s. This framework has the capacity to represent social-structural effects commonly observed in many human social networks, including general degree-based structural effects commonly observed in many human social networks as well as reciprocity and transitivity, and at the node-level, homophily and attribute-based activity and popularity effects, as derived from explicit hypotheses about dependencies among network ties. Parameters are given in terms of the prevalence of small subgraph configurations in the network and can be interpreted as describing the combinations of local social processes from which a given network emerges. These probability models for networks on a given set of actors allow generalization beyond the restrictive dyadic independence assumption of micro-networks, allowing models to be built from theoretical structural foundations of social behavior.<br />
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随机分布的网络: 社会网络的指数随机图模型在20世纪80年代成为最先进的社会网络分析方法。这个框架有能力表示在许多人类社会网络中普遍观察到的社会结构效应，包括在许多人类社会网络中普遍观察到的基于程度的一般性结构效应以及互惠性和传递性，以及在节点一级、同相性和基于属性的活动和流行性效应，这些效应源于关于网络关系之间依赖性的明确假设。参数是根据网络中小型子图配置的流行程度给出的，可以解释为描述一个给定网络出现的局部社会过程的组合。这些网络的概率模型在给定的参与者集合上允许超越微型网络的限制性并元独立性假设的泛化，允许模型从社会行为的理论结构基础上建立。<br />
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[[File:Scale-free network sample.png|thumb|right|Examples of a random network and a scale-free network. Each graph has 32 nodes and 32 links. Note the "hubs" (shaded) in the scale-free diagram (on the right).]]<br />
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Examples of a random network and a scale-free network. Each graph has 32 nodes and 32 links. Note the "hubs" (shaded) in the scale-free diagram (on the right).<br />
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一个随机网络和一个无尺度网络的例子。每个图有32个节点和32个链接。注意无标度图中的“集线器”(阴影部分)(右侧)。<br />
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'''Scale-free networks''': A [[scale-free network]] is a [[complex network|network]] whose [[degree distribution]] follows a [[power law]], at least [[asymptotic]]ally. In [[network theory]] a scale-free ideal network is a [[random network]] with a [[degree distribution]] that unravels the size distribution of social groups.<ref>{{cite journal |author1=Moreira, André A.|author2=Demétrius R. Paula|author3=Raimundo N. Costa Filho|author4=José S. Andrade, Jr. |title=Competitive cluster growth in complex networks |year=2006 |doi=10.1103/PhysRevE.73.065101 |pmid=16906890 |journal=Physical Review E |volume=73 |issue=6 |pages=065101 |arxiv=cond-mat/0603272 |bibcode=2006PhRvE..73f5101M }}</ref> Specific characteristics of scale-free networks vary with the theories and analytical tools used to create them, however, in general, scale-free networks have some common characteristics. One notable characteristic in a scale-free network is the relative commonness of [[Vertex (graph theory)|vertices]] with a [[Maximum degree|degree]] that greatly exceeds the average. The highest-degree nodes are often called "hubs", and may serve specific purposes in their networks, although this depends greatly on the social context. Another general characteristic of scale-free networks is the [[clustering coefficient]] distribution, which decreases as the node degree increases. This distribution also follows a [[power law]].<ref>Barabási, Albert-László (2003). ''Linked: how everything is connected to everything else and what it means for business, science, and everyday life''. New York: Plum.</ref> The [[Barabási–Albert model|Barabási]] model of network evolution shown above is an example of a scale-free network.<br />
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Scale-free networks: A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. In network theory a scale-free ideal network is a random network with a degree distribution that unravels the size distribution of social groups. Specific characteristics of scale-free networks vary with the theories and analytical tools used to create them, however, in general, scale-free networks have some common characteristics. One notable characteristic in a scale-free network is the relative commonness of vertices with a degree that greatly exceeds the average. The highest-degree nodes are often called "hubs", and may serve specific purposes in their networks, although this depends greatly on the social context. Another general characteristic of scale-free networks is the clustering coefficient distribution, which decreases as the node degree increases. This distribution also follows a power law. The Barabási model of network evolution shown above is an example of a scale-free network.<br />
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无标度网络: 无尺度网络网络是一个度分布遵循幂律的网络，至少是渐近的。在网络理论中，无标度理想网络是一个具有度分布的随机网络，它揭示了社会群体的规模分布。无标度网络的具体特征随创建无标度网络的理论和分析工具的不同而不同，然而，一般来说，无标度网络具有一些共同的特征。无尺度网络的一个显著特征是，度大大超过平均值的顶点具有相对的共性。最高度的节点通常被称为“枢纽” ，并且可能在其网络中服务于特定的目的，尽管这在很大程度上取决于社会环境。无标度网络的另一个普遍特征是集聚系数分布，它随着节点度的增加而减少。这个分布也遵循一个幂定律。上面显示的网络演化的 barab si 模型就是无尺度网络的一个例子。<br />
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===Macro level===<br />
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===Macro level===<br />
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宏观层面<br />
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Rather than tracing interpersonal interactions, macro-level analyses generally trace the outcomes of interactions, such as [[economic]] or other [[resource]] [[Transfer function|transfer]] interactions over a large [[Sample population|population]].<br />
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Rather than tracing interpersonal interactions, macro-level analyses generally trace the outcomes of interactions, such as economic or other resource transfer interactions over a large population.<br />
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宏观层面的分析不是追踪人与人之间的相互作用，而是通常追踪相互作用的结果，例如经济或其他资源转移在一大群体中的相互作用。<br />
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[[File:Diagram of a social network.jpg|thumb|right|Diagram: section of a large-scale social network]]<br />
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Diagram: section of a large-scale social network<br />
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图: 一个大型社交网络的一部分<br />
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'''Large-scale networks''': [[Large-scale macroeconometric model|Large-scale network]] is a term somewhat synonymous with "macro-level" as used, primarily, in [[social science|social]] and [[Behavioural sciences|behavioral]] sciences, in [[economics]]. Originally, the term was used extensively in the [[computer sciences]] (see [[Network mapping#Large-scale mapping project|large-scale network mapping]]).<br />
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Large-scale networks: Large-scale network is a term somewhat synonymous with "macro-level" as used, primarily, in social and behavioral sciences, in economics. Originally, the term was used extensively in the computer sciences (see large-scale network mapping).<br />
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大规模网络: 大规模网络是一个与“宏观层面”同义的术语，主要用于社会和行为科学，经济学。最初，这个术语在计算机科学中被广泛使用(见大比例尺网络地图)。<br />
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'''Complex networks''': Most larger social networks display features of [[social complexity]], which involves substantial non-trivial features of [[network topology]], with patterns of complex connections between elements that are neither purely regular nor purely random (see, [[complexity science]], [[dynamical system]] and [[chaos theory]]), as do [[biological]], and [[Computer network|technological networks]]. Such [[complex network]] features include a heavy tail in the [[degree distribution]], a high [[clustering coefficient]], [[assortativity]] or disassortativity among vertices, [[community structure]] (see [[stochastic block model]]), and [[hierarchy|hierarchical structure]]. In the case of [[Agency (philosophy)|agency-directed]] networks these features also include [[Reciprocity in network|reciprocity]], triad significance profile (TSP, see [[network motif]]), and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as [[lattice graph|lattices]] and [[random graph]]s, do not show these features.<ref>{{cite journal|author=Strogatz, Steven H.|year=2001|title=Exploring complex networks|journal=Nature|volume=410|pages=268–276|doi=10.1038/35065725|pmid=11258382|issue=6825|bibcode=2001Natur.410..268S|doi-access=free}}</ref><br />
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Complex networks: Most larger social networks display features of social complexity, which involves substantial non-trivial features of network topology, with patterns of complex connections between elements that are neither purely regular nor purely random (see, complexity science, dynamical system and chaos theory), as do biological, and technological networks. Such complex network features include a heavy tail in the degree distribution, a high clustering coefficient, assortativity or disassortativity among vertices, community structure (see stochastic block model), and hierarchical structure. In the case of agency-directed networks these features also include reciprocity, triad significance profile (TSP, see network motif), and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as lattices and random graphs, do not show these features.<br />
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复杂网络: 大多数较大的社会网络呈现出社会复杂性的特征，包括网络拓扑的大量非琐碎特征，以及元素之间的复杂连接模式，这些元素既不是纯规则的也不是纯随机的(见复杂性科学、动力系统和混沌理论) ，生物学和技术网络也是如此。这些复杂的网络特征包括程度分布的重尾、高集聚系数、顶点之间的协调性或不协调性、社区结构(见随机块模型)和层次结构。在代理导向网络的情况下，这些特征还包括互惠性、三重显著性特征(TSP，见网络主题)和其他特征。相比之下，许多过去研究过的网络数学模型，如格和随机图，并没有表现出这些特征。<br />
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==Theoretical links==<br />
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==Theoretical links==<br />
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理论联系<br />
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===Imported theories===<br />
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===Imported theories===<br />
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输入的理论<br />
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Various theoretical frameworks have been imported for the use of social network analysis. The most prominent of these are [[Graph theory]], [[Balance theory]], Social comparison theory, and more recently, the [[Social identity approach]].<ref>{{cite book |last1=Kilduff|first1=M.|last2=Tsai|first2=W. |year=2003 |title= Social networks and organisations |publisher=Sage Publications}}</ref><br />
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Various theoretical frameworks have been imported for the use of social network analysis. The most prominent of these are Graph theory, Balance theory, Social comparison theory, and more recently, the Social identity approach.<br />
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为了使用社会网络分析，已经引入了各种理论框架。其中最突出的是图论、平衡论、社会比较论，以及最近的社会认同方法。<br />
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===Indigenous theories===<br />
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===Indigenous theories===<br />
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本土理论<br />
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Few complete theories have been produced from social network analysis. Two that have are [[role theory|structural role theory]] and [[heterophily|heterophily theory]].<br />
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Few complete theories have been produced from social network analysis. Two that have are structural role theory and heterophily theory.<br />
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很少有完整的理论产生于社会网络分析。结构角色理论和异质性理论。<br />
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The basis of Heterophily Theory was the finding in one study that more numerous weak ties can be important in seeking information and innovation, as cliques have a tendency to have more homogeneous opinions as well as share many common traits. This homophilic tendency was the reason for the members of the cliques to be attracted together in the first place. However, being similar, each member of the clique would also know more or less what the other members knew. To find new information or insights, members of the clique will have to look beyond the clique to its other friends and acquaintances. This is what Granovetter called "the strength of weak ties".<ref>{{Cite journal | author=Granovetter, M. |year=1973 |title= The strength of weak ties |journal= American Journal of Sociology |volume=78 |issue=6 |pages= 1360–1380 |doi=10.1086/225469}}</ref><br />
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The basis of Heterophily Theory was the finding in one study that more numerous weak ties can be important in seeking information and innovation, as cliques have a tendency to have more homogeneous opinions as well as share many common traits. This homophilic tendency was the reason for the members of the cliques to be attracted together in the first place. However, being similar, each member of the clique would also know more or less what the other members knew. To find new information or insights, members of the clique will have to look beyond the clique to its other friends and acquaintances. This is what Granovetter called "the strength of weak ties".<br />
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异质性理论的基础是在一项研究中发现，更多的弱关系可以在寻求信息和创新方面发挥重要作用，因为小团体倾向于有更同质化的观点，也有许多共同特征。这种亲同性倾向是小团体成员被吸引到一起的首要原因。然而，由于相似，小圈子里的每一个成员或多或少都知道其他成员所知道的事情。为了找到新的信息或见解，小圈子里的成员将不得不超越小圈子，关注其他的朋友和熟人。这就是格兰诺维特所说的“弱关系的力量”。<br />
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==Structural holes==<br />
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==Structural holes==<br />
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结构性漏洞<br />
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In the context of networks, [[social capital]] exists where people have an advantage because of their location in a network. Contacts in a network provide information, opportunities and perspectives that can be beneficial to the central player in the network. Most social structures tend to be characterized by dense clusters of strong connections.<ref name="Burt 2004">{{cite journal|last=Burt|first=Ronald|title=Structural Holes and Good Ideas|journal=American Journal of Sociology|year=2004|doi=10.1086/421787|volume=110|issue=2|pages=349–399|citeseerx=10.1.1.388.2251}}</ref> Information within these clusters tends to be rather homogeneous and redundant. Non-redundant information is most often obtained through contacts in different clusters.<ref name="Burt 1992">{{cite book|last=Burt|first=Ronald|title=Structural Holes: The Social Structure of Competition|year=1992|publisher=Harvard University Press|location=Cambridge, MA}}</ref> When two separate clusters possess non-redundant information, there is said to be a structural hole between them.<ref name="Burt 1992"/> Thus, a network that bridges [[structural holes]] will provide network benefits that are in some degree additive, rather than overlapping. An ideal network structure has a vine and cluster structure, providing access to many different clusters and structural holes.<ref name="Burt 1992"/><br />
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In the context of networks, social capital exists where people have an advantage because of their location in a network. Contacts in a network provide information, opportunities and perspectives that can be beneficial to the central player in the network. Most social structures tend to be characterized by dense clusters of strong connections. Information within these clusters tends to be rather homogeneous and redundant. Non-redundant information is most often obtained through contacts in different clusters. When two separate clusters possess non-redundant information, there is said to be a structural hole between them. Thus, a network that bridges structural holes will provide network benefits that are in some degree additive, rather than overlapping. An ideal network structure has a vine and cluster structure, providing access to many different clusters and structural holes.<br />
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在网络的背景下，社会资本存在于人们因为他们在网络中的位置而具有优势的地方。网络中的联系人提供的信息、机会和观点可能有利于网络中的核心参与者。大多数社会结构倾向于拥有属性密集的强连接。这些集群中的信息往往是相当均匀和冗余的。非冗余信息通常是通过不同集群中的联系人获得的。当两个独立的簇拥有非冗余信息时，我们称它们之间存在一个结构空洞。因此，一个桥接结构孔的网络将在某种程度上提供附加的网络效益，而不是重叠的。理想的网络结构具有蔓生结构和集群结构，提供对许多不同集群和结构洞的访问。<br />
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Networks rich in structural holes are a form of social capital in that they offer [[Information theory|information]] benefits. The main player in a network that bridges structural holes is able to access information from diverse sources and clusters.<ref name="Burt 1992"/> For example, in [[Business networking|business networks]], this is beneficial to an individual's career because he is more likely to hear of job openings and opportunities if his network spans a wide range of contacts in different industries/sectors. This concept is similar to Mark Granovetter's theory of [[Interpersonal ties|weak ties]], which rests on the basis that having a broad range of contacts is most effective for job attainment.<br />
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Networks rich in structural holes are a form of social capital in that they offer information benefits. The main player in a network that bridges structural holes is able to access information from diverse sources and clusters. For example, in business networks, this is beneficial to an individual's career because he is more likely to hear of job openings and opportunities if his network spans a wide range of contacts in different industries/sectors. This concept is similar to Mark Granovetter's theory of weak ties, which rests on the basis that having a broad range of contacts is most effective for job attainment.<br />
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富含结构性漏洞的网络是社会资本的一种形式，因为它们提供信息利益。桥接结构漏洞的网络中的主要参与者能够访问来自不同来源和集群的信息。例如，在商业网络中，这对个人的职业生涯是有益的，因为如果他的关系网涵盖不同行业 / 部门的广泛联系，他更有可能听到职位空缺和机会。这个概念类似于马克·格兰诺维特的弱关系理论，该理论的基础是拥有广泛的联系对于获得工作是最有效的。<br />
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==Research clusters==<br />
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==Research clusters==<br />
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研究集群<br />
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===Communication===<br />
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===Communication===<br />
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沟通<br />
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[[Communication Studies]] are often considered a part of both the social sciences and the humanities, drawing heavily on fields such as [[sociology]], [[psychology]], [[anthropology]], [[information science]], [[biology]], [[political science]], and [[economics]] as well as [[rhetoric]], [[literary studies]], and [[semiotics]]. Many communication concepts describe the transfer of information from one source to another, and can thus be conceived of in terms of a network.<br />
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Communication Studies are often considered a part of both the social sciences and the humanities, drawing heavily on fields such as sociology, psychology, anthropology, information science, biology, political science, and economics as well as rhetoric, literary studies, and semiotics. Many communication concepts describe the transfer of information from one source to another, and can thus be conceived of in terms of a network.<br />
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传播学通常被认为是社会科学和人文科学的一部分，主要集中在社会学、心理学、人类学、信息科学、生物学、政治学、经济学以及修辞学、文学研究和符号学等领域。许多通信概念描述了信息从一个来源到另一个来源的转移，因此可以从网络的角度来考虑。<br />
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===Community===<br />
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===Community===<br />
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社区<br />
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In J.A. Barnes' day, a "[[community]]" referred to a specific geographic location and studies of community ties had to do with who talked, associated, traded, and attended church with whom. Today, however, there are extended "online" communities developed through [[telecommunications]] devices and [[social network services]]. Such devices and services require extensive and ongoing maintenance and analysis, often using [[network science]] methods. [[Community development]] studies, today, also make extensive use of such methods.<br />
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In J.A. Barnes' day, a "community" referred to a specific geographic location and studies of community ties had to do with who talked, associated, traded, and attended church with whom. Today, however, there are extended "online" communities developed through telecommunications devices and social network services. Such devices and services require extensive and ongoing maintenance and analysis, often using network science methods. Community development studies, today, also make extensive use of such methods.<br />
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在耶路撒冷。巴恩斯日，一个“社区”指的是一个特定的地理位置和社区关系的研究与谁交谈，联系，贸易和参加教会与谁。然而，今天，通过电信设备和社交网络服务，有了扩展的“在线”社区。这样的设备和服务需要广泛和持续的维护和分析，通常使用网络科学的方法。社区发展研究，今天，也广泛使用这些方法。<br />
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===Complex networks===<br />
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===Complex networks===<br />
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复杂的网络<br />
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[[Complex networks]] require methods specific to modelling and interpreting [[social complexity]] and [[complex adaptive system]]s, including techniques of [[dynamic network analysis]].<br />
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Complex networks require methods specific to modelling and interpreting social complexity and complex adaptive systems, including techniques of dynamic network analysis.<br />
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复杂网络需要建模和解释社会复杂性和复杂适应系统的特定方法，包括动态网络分析技术。<br />
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Mechanisms such as [[Dual-phase evolution#social networks|Dual-phase evolution]] explain how temporal changes in connectivity contribute to the formation of structure in social networks.<br />
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Mechanisms such as Dual-phase evolution explain how temporal changes in connectivity contribute to the formation of structure in social networks.<br />
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双阶段进化等机制解释了连接性的时间变化如何促进社会网络结构的形成。<br />
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===Criminal networks===<br />
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===Criminal networks===<br />
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犯罪网络<br />
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In [[criminology]] and [[urban sociology]], much attention has been paid to the social networks among criminal actors. For example, Andrew Papachristos<ref>{{cite journal |last=Papachristos |first=Andrew |year=2009 |title=Murder by Structure: Dominance Relations and the Social Structure of Gang Homicide |journal=American Journal of Sociology |volume=115 |issue=1 |pages=74–128 |doi=10.2139/ssrn.855304 |url=http://www.papachristos.org/Publications_2_files/ajs_final_version.pdf |accessdate=29 March 2013 |url-status=dead |archiveurl=https://web.archive.org/web/20140407094725/http://www.papachristos.org/Publications_2_files/ajs_final_version.pdf |archivedate=7 April 2014 }}</ref> has studied gang murders as a series of exchanges between gangs. Murders can be seen to diffuse outwards from a single source, because weaker gangs cannot afford to kill members of stronger gangs in retaliation, but must commit other violent acts to maintain their reputation for strength.<br />
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In criminology and urban sociology, much attention has been paid to the social networks among criminal actors. For example, Andrew Papachristos has studied gang murders as a series of exchanges between gangs. Murders can be seen to diffuse outwards from a single source, because weaker gangs cannot afford to kill members of stronger gangs in retaliation, but must commit other violent acts to maintain their reputation for strength.<br />
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在犯罪学和城市社会学中，犯罪行为人之间的社会网络问题受到了广泛的关注。例如，安德鲁 · 帕帕克里斯托斯将帮派谋杀研究为帮派之间的一系列交流。谋杀可以被视为从单一来源向外扩散，因为较弱的帮派无法承担杀死较强帮派成员进行报复的代价，而必须采取其他暴力行动来维护他们强大的声誉。<br />
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===Diffusion of innovations===<br />
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===Diffusion of innovations===<br />
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创新产品渗透理论<br />
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[[Diffusion of innovations|Diffusion of ideas and innovations]] studies focus on the spread and use of ideas from one actor to another or one [[culture]] and another. This line of research seeks to explain why some become "early adopters" of ideas and innovations, and links social network structure with facilitating or impeding the spread of an innovation.<br />
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Diffusion of ideas and innovations studies focus on the spread and use of ideas from one actor to another or one culture and another. This line of research seeks to explain why some become "early adopters" of ideas and innovations, and links social network structure with facilitating or impeding the spread of an innovation.<br />
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思想和创新的传播研究的重点是思想从一个行动者传播到另一个行动者或一种文化和另一种文化。这一系列的研究试图解释为什么有些人成为创意和创新的“早期接受者” ，并将社交网络结构与促进或阻碍创新的传播联系起来。<br />
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===Demography===<br />
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===Demography===<br />
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人口统计<br />
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In [[demography]], the study of social networks has led to new sampling methods for estimating and reaching populations that are hard to enumerate (for example, homeless people or intravenous drug users.) For example, respondent driven sampling is a network-based sampling technique that relies on respondents to a survey recommending further respondents.<br />
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In demography, the study of social networks has led to new sampling methods for estimating and reaching populations that are hard to enumerate (for example, homeless people or intravenous drug users.) For example, respondent driven sampling is a network-based sampling technique that relies on respondents to a survey recommending further respondents.<br />
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在人口统计学方面，对社会网络的研究导致了新的抽样方法，用于估计和覆盖难以统计的人群(例如，无家可归者或静脉注射毒品者)例如，受访者驱动的抽样是一种基于网络的抽样技术，依赖于调查的受访者推荐更多的受访者。<br />
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===Economic sociology===<br />
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===Economic sociology===<br />
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经济社会学<br />
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The field of [[sociology]] focuses almost entirely on networks of outcomes of social interactions. More narrowly, [[economic sociology]] considers behavioral interactions of individuals and groups through [[social capital]] and social "markets". Sociologists, such as Mark Granovetter, have developed core principles about the interactions of social structure, information, ability to punish or reward, and trust that frequently recur in their analyses of political, economic and other institutions. Granovetter examines how social structures and social networks can affect economic outcomes like hiring, price, productivity and innovation and describes sociologists' contributions to analyzing the impact of social structure and networks on the economy.<ref>{{cite journal |last=Granovetter |first=Mark |year=2005 |title=The Impact of Social Structure on Economic Outcomes |journal=The Journal of Economic Perspectives |jstor=4134991 |volume=19 |issue=1 |pages=33–50|doi=10.1257/0895330053147958 }}</ref><br />
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The field of sociology focuses almost entirely on networks of outcomes of social interactions. More narrowly, economic sociology considers behavioral interactions of individuals and groups through social capital and social "markets". Sociologists, such as Mark Granovetter, have developed core principles about the interactions of social structure, information, ability to punish or reward, and trust that frequently recur in their analyses of political, economic and other institutions. Granovetter examines how social structures and social networks can affect economic outcomes like hiring, price, productivity and innovation and describes sociologists' contributions to analyzing the impact of social structure and networks on the economy.<br />
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社会学领域几乎完全关注社会互动的结果网络。更狭义地说，经济社会学通过社会资本和社会“市场”考虑个人和群体的行为互动。社会学家，如马克·格兰诺维特，已经发展出关于社会结构、信息、惩罚或奖励能力和信任相互作用的核心原则，这些原则在他们对政治、经济和其他制度的分析中经常出现。格兰诺维特研究了社会结构和社会网络如何影响经济结果，如雇佣、价格、生产力和创新，并描述了社会学家对分析社会结构和网络对经济的影响的贡献。<br />
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===Health care===<br />
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===Health care===<br />
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医疗保健<br />
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Analysis of social networks is increasingly incorporated into [[health care analytics]], not only in [[epidemiology|epidemiological]] studies but also in models of [[Health Communication|patient communication]] and education, disease prevention, mental health diagnosis and treatment, and in the study of health care organizations and [[health care systems|systems]].<ref>Levy, Judith and Bernice Pescosolido (2002). ''Social Networks and Health''. Boston, MA: JAI Press.</ref><br />
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Analysis of social networks is increasingly incorporated into health care analytics, not only in epidemiological studies but also in models of patient communication and education, disease prevention, mental health diagnosis and treatment, and in the study of health care organizations and systems.<br />
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社会网络分析不仅在流行病学研究中，而且在病人沟通和教育模型、疾病预防、心理健康诊断和治疗以及卫生保健组织和系统的研究中，越来越多地被纳入卫生保健分析中。<br />
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===Human ecology===<br />
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===Human ecology===<br />
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人类生态学<br />
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[[Human ecology]] is an [[interdisciplinary]] and [[transdisciplinary]] study of the relationship between [[human]]s and their [[natural environment|natural]], [[Social environment|social]], and [[built environment]]s. The scientific philosophy of human ecology has a diffuse history with connections to [[geography]], [[sociology]], [[psychology]], [[anthropology]], [[zoology]], and natural [[ecology]].<ref>Crona, Beatrice and Klaus Hubacek (eds.) (2010). [http://www.ecologyandsociety.org/issues/view.php?sf=48 "Special Issue: Social network analysis in natural resource governance"]. ''Ecology and Society'', 48.</ref><ref>Ernstson, Henrich (2010). "Reading list: Using social network analysis (SNA) in social-ecological studies". [http://rs.resalliance.org/2010/11/03/reading-list-using-social-network-analysis-sna-in-social-ecological-studies/ ''Resilience Science'']</ref><br />
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Human ecology is an interdisciplinary and transdisciplinary study of the relationship between humans and their natural, social, and built environments. The scientific philosophy of human ecology has a diffuse history with connections to geography, sociology, psychology, anthropology, zoology, and natural ecology.<br />
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人类生态学是一门跨学科、跨学科的学科，研究人类与自然、社会和人造环境之间的关系。人类生态学的科学哲学与地理学、社会学、心理学、人类学、动物学和自然生态学有着密切的联系。<br />
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===Language and linguistics===<br />
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===Language and linguistics===<br />
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语言与语言学<br />
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Studies of [[language]] and [[linguistics]], particularly [[evolutionary linguistics]], focus on the development of [[Morphology (linguistics)|linguistic forms]] and transfer of changes, [[Phonology|sounds]] or words, from one language system to another through networks of social interaction. Social networks are also important in [[language shift]], as groups of people add and/or abandon languages to their repertoire.<br />
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Studies of language and linguistics, particularly evolutionary linguistics, focus on the development of linguistic forms and transfer of changes, sounds or words, from one language system to another through networks of social interaction. Social networks are also important in language shift, as groups of people add and/or abandon languages to their repertoire.<br />
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语言学和语言学的研究，特别是演化语言学，集中在语言形式的发展和变化，声音或词汇的转移，从一个语言系统到另一个通过社会互动的网络。社交网络在语言转换中也很重要，因为一群人增加或者放弃了他们的语言。<br />
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===Literary networks===<br />
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===Literary networks===<br />
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文学网络<br />
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In the study of literary systems, network analysis has been applied by Anheier, Gerhards and Romo,<ref>{{cite journal | last1 = Anheier | first1 = H. K. | last2 = Romo | first2 = F. P. | year = 1995 | title = Forms of capital and social structure of fields: examining Bourdieu's social topography | url = | journal = American Journal of Sociology | volume = 100 | issue = 4| pages = 859–903 | doi=10.1086/230603}}</ref> De Nooy,<ref>{{cite journal | last1 = De Nooy | first1 = W | year = 2003| title = Fields and networks: Correspondence analysis and social network analysis in the framework of Field Theory | url = | journal = Poetics | volume = 31 | issue = 5–6| pages = 305–327 | doi = 10.1016/S0304-422X(03)00035-4 }}</ref> and Senekal,<ref>{{cite journal | last1 = Senekal | first1 = B. A. | year = 2012 | title = Die Afrikaanse literêre sisteem: ʼn Eksperimentele benadering met behulp van Sosiale-netwerk-analise (SNA) | url = | journal = LitNet Akademies | volume = 9 | issue = | page = 3 }}</ref> to study various aspects of how literature functions. The basic premise is that polysystem theory, which has been around since the writings of [[Even-Zohar]], can be integrated with network theory and the relationships between different actors in the literary network, e.g. writers, critics, publishers, literary histories, etc., can be mapped using [[Computer graphics (computer science)|visualization]] from SNA.<br />
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In the study of literary systems, network analysis has been applied by Anheier, Gerhards and Romo, De Nooy, and Senekal, to study various aspects of how literature functions. The basic premise is that polysystem theory, which has been around since the writings of Even-Zohar, can be integrated with network theory and the relationships between different actors in the literary network, e.g. writers, critics, publishers, literary histories, etc., can be mapped using visualization from SNA.<br />
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在文学体系的研究中，网络分析被阿海尔、格尔哈兹和罗姆、努伊和 Senekal 应用于研究文学功能的各个方面。其基本前提是多元系统理论可以与网络理论以及文学网络中不同行为者之间的关系相结合。作家、评论家、出版商、文学史等等，都可以通过使用 SNA 的可视化来绘制地图。<br />
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===Organizational studies===<br />
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===Organizational studies===<br />
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组织研究<br />
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Research studies of [[Formal organization|formal]] or [[informal organization]] [[Social relation|relationships]], [[organizational communication]], [[economics]], [[economic sociology]], and other [[resource]] [[Transfer function|transfers]]. Social networks have also been used to examine how organizations interact with each other, characterizing the many [[Interlocking directorate|informal connections]] that link executives together, as well as associations and connections between individual employees at different organizations.<ref>{{cite journal | last1 = Podolny | first1 = J. M. | last2 = Baron | first2 = J. N. | year = 1997 | title = Resources and relationships: Social networks and mobility in the workplace | url = | journal = American Sociological Review | volume = 62 | issue = 5| pages = 673–693 | doi=10.2307/2657354| jstor = 2657354 | citeseerx = 10.1.1.114.6822 }}</ref> Intra-organizational networks have been found to affect [[organizational commitment]],<ref>{{cite journal | last1 = Lee | first1 = J. | last2 = Kim | first2 = S. | year = 2011 | title = Exploring the role of social networks in affective organizational commitment: Network centrality, strength of ties, and structural holes | url = | journal = The American Review of Public Administration | volume = 41 | issue = 2| pages = 205–223 | doi=10.1177/0275074010373803}}</ref> [[organizational identification]],<ref name="Jone11">{{cite journal | last1 = Jones | first1 = C. | last2 = Volpe | first2 = E.H. | year = 2011 | title = Organizational identification: Extending our understanding of social identities through social networks | url = | journal = Journal of Organizational Behavior | volume = 32 | issue = 3| pages = 413–434 | doi=10.1002/job.694}}</ref> [[Organizational citizenship behavior|interpersonal citizenship behaviour]].<ref>{{cite journal | last1 = Bowler | first1 = W. M. | last2 = Brass | first2 = D. J. | year = 2011 | title = Relational correlates of interpersonal citizenship behaviour: A social network perspective | doi = 10.1037/0021-9010.91.1.70 | pmid = 16435939 | journal = Journal of Applied Psychology | volume = 91 | issue = 1| pages = 70–82 | citeseerx = 10.1.1.516.8746 }}</ref><br />
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Research studies of formal or informal organization relationships, organizational communication, economics, economic sociology, and other resource transfers. Social networks have also been used to examine how organizations interact with each other, characterizing the many informal connections that link executives together, as well as associations and connections between individual employees at different organizations. Intra-organizational networks have been found to affect organizational commitment, organizational identification, interpersonal citizenship behaviour.<br />
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研究正式或非正式的组织关系，组织沟通，经济，经济社会学和其他资源转移。社交网络也被用来研究组织之间如何相互作用，描述了许多非正式的联系，以及不同组织的个别雇员之间的联系和联系。研究发现，组织内网络对组织承诺、组织认同、人际公民行为有显著影响。<br />
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===Social capital===<br />
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===Social capital===<br />
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社会资本<br />
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[[Social capital]] is a form of [[Capital (economics)|economic]] and [[cultural capital]] in which social networks are central, [[Stock and flow|transactions]] are marked by [[Reciprocity (social psychology)|reciprocity]], [[Trust (social sciences)|trust]], and [[cooperation]], and [[Market (economics)|market]] [[Agent (economics)|agents]] produce [[goods and services]] not mainly for themselves, but for a [[common good]]. [[Social capital]] is split into three dimensions: the structural, the relational and the cognitive dimension. The structural dimension describes how partners interact with each other and which specific partners meet in a social network. Also The structural dimension of social capital indicates the level of ties among organizations.<ref>(Claridge, 2018).</ref>. This dimension is highly connected to the relational dimension which refers to trustworthiness, norms, expectations and idenfications of the bonds between partners.The relational dimension explains the nature of these ties which is mainly illustrated by the level of trust accorded to the network of organizations. <ref>(Claridge, 2018).</ref> The cognitive dimension analyses the extent to which organizations share common goals and objectives as a result of their ties and interactions. <ref>(Claridge, 2018).</ref><br />
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Social capital is a form of economic and cultural capital in which social networks are central, transactions are marked by reciprocity, trust, and cooperation, and market agents produce goods and services not mainly for themselves, but for a common good. Social capital is split into three dimensions: the structural, the relational and the cognitive dimension. The structural dimension describes how partners interact with each other and which specific partners meet in a social network. Also The structural dimension of social capital indicates the level of ties among organizations.. This dimension is highly connected to the relational dimension which refers to trustworthiness, norms, expectations and idenfications of the bonds between partners.The relational dimension explains the nature of these ties which is mainly illustrated by the level of trust accorded to the network of organizations. The cognitive dimension analyses the extent to which organizations share common goals and objectives as a result of their ties and interactions. <br />
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社会资本是一种经济和文化资本，其中社会网络是中心，交易以互惠、信任和合作为特征，市场代理人生产的商品和服务不主要是为他们自己，而是为了共同的利益。社会资本分为三个维度: 结构维度、关系维度和认知维度。结构维度描述了合作伙伴之间如何相互作用，以及哪些特定的合作伙伴在社交网络中相遇。社会资本的结构维度反映了组织之间的关系水平。这个维度与关系维度高度相关，关系维度指的是伴侣之间关系的可信度、规范、期望和相同性。 关系维度解释了这些联系的本质，主要表现在对组织网络的信任程度上。认知维度分析组织因其联系和相互作用而在多大程度上共享共同的目标和目的。<br />
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[[Social capital]] is a sociological concept about the value of [[social relation]]s and the role of cooperation and confidence to achieve positive outcomes. The term refers to the value one can get from their social ties. For example, newly arrived immigrants can make use of their social ties to established migrants to acquire jobs they may otherwise have trouble getting (e.g., because of unfamiliarity with the local language). A positive relationship exists between social capital and the intensity of social network use.<ref>{{cite journal|last=Sebastián|first=Valenzuela|author2=Namsu Park |author3=Kerk F. Kee |title=Is There Social Capital in a Social Network Site? Facebook Use and College Students' Life Satisfaction, Trust, and Participation|journal=Journal of Computer-Mediated Communication|year=2009|volume=14|issue=4|pages=875–901|doi=10.1111/j.1083-6101.2009.01474.x|doi-access=free}}</ref><ref>{{cite journal | title=Social Connectivity in America: Changes in Adult Friendship Network Size from 2002 to 2007 |author1=Wang, Hua |author2=Barry Wellman |lastauthoramp=yes | journal=American Behavioral Scientist | year=2010 | volume=53 | issue=8 | pages=1148–1169 | doi=10.1177/0002764209356247|url=https://semanticscholar.org/paper/402068f74efb35c093e7fd8c0f8268521219e242 }}</ref> In a dynamic framework, higher activity in a network feeds into higher social capital which itself encourages more activity.<ref name="GaudeulGiannetti2013">{{cite journal|last1=Gaudeul|first1=Alexia|last2=Giannetti|first2=Caterina|title=The role of reciprocation in social network formation, with an application to LiveJournal|journal=Social Networks|volume=35|issue=3|year=2013|pages=317–330|issn=03788733|doi=10.1016/j.socnet.2013.03.003}}</ref><br />
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Social capital is a sociological concept about the value of social relations and the role of cooperation and confidence to achieve positive outcomes. The term refers to the value one can get from their social ties. For example, newly arrived immigrants can make use of their social ties to established migrants to acquire jobs they may otherwise have trouble getting (e.g., because of unfamiliarity with the local language). A positive relationship exists between social capital and the intensity of social network use. In a dynamic framework, higher activity in a network feeds into higher social capital which itself encourages more activity.<br />
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社会资本是一个关于社会关系的价值和合作与信心的作用的社会学概念，以实现积极的成果。这个术语指的是一个人可以从他们的社会关系中获得的价值。例如，新来的移民可以利用他们与老移民的社会关系，获得他们在其他情况下可能难以获得的工作(例如，因为不熟悉当地语言)。社会资本与社会网络使用强度存在正相关关系。在一个动态的框架中，网络中更高的活动会产生更高的社会资本，而社会资本本身又会鼓励更多的活动。<br />
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=== Advertising ===<br />
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=== Advertising ===<br />
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广告<br />
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This particular cluster focuses on brand-image and promotional strategy effectiveness, taking into account the impact of customer participation on sales and brand-image. This is gauged through techniques such as sentiment analysis which rely on mathematical areas of study such as data mining and analytics. This area of research produces vast numbers of commercial applications as the main goal of any study is to understand consumer behaviour and drive sales.<br />
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This particular cluster focuses on brand-image and promotional strategy effectiveness, taking into account the impact of customer participation on sales and brand-image. This is gauged through techniques such as sentiment analysis which rely on mathematical areas of study such as data mining and analytics. This area of research produces vast numbers of commercial applications as the main goal of any study is to understand consumer behaviour and drive sales.<br />
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这个集群侧重于品牌形象和促销策略的有效性，同时考虑到顾客参与对销售和品牌形象的影响。这是通过诸如情感分析等技术来衡量的，这些技术依赖于数学领域的研究，如数据挖掘和分析。这一研究领域产生了大量的商业应用，因为任何研究的主要目标都是了解消费者行为并推动销售。<br />
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===Network position and benefits===<br />
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===Network position and benefits===<br />
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网络地位和利益<br />
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In many [[Formal organizations|organizations]], members tend to focus their activities inside their own groups, which stifles creativity and restricts opportunities. A player whose network bridges structural holes has an advantage in detecting and developing rewarding opportunities.<ref name="Burt 2004"/> Such a player can mobilize social capital by acting as a "broker" of information between two clusters that otherwise would not have been in contact, thus providing access to new ideas, opinions and opportunities. British philosopher and political economist [[John Stuart Mill]], writes, "it is hardly possible to overrate the value ... of placing human beings in contact with persons dissimilar to themselves.... Such communication [is] one of the primary sources of progress."<ref>{{cite book|last=Mill|first=John|title=Principles of Political Economy|year=1909|publisher=William J Ashley|location=Library of Economics and Liberty}}</ref> Thus, a player with a network rich in structural holes can add value to an organization through new ideas and opportunities. This in turn, helps an individual's career development and advancement.<br />
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In many organizations, members tend to focus their activities inside their own groups, which stifles creativity and restricts opportunities. A player whose network bridges structural holes has an advantage in detecting and developing rewarding opportunities. Thus, a player with a network rich in structural holes can add value to an organization through new ideas and opportunities. This in turn, helps an individual's career development and advancement.<br />
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在许多组织中，成员倾向于把他们的活动集中在他们自己的团队中，这抑制了创造力和限制了机会。一个玩家的网络桥梁结构洞有优势，在检测和发展奖励机会。因此，一个拥有丰富结构性漏洞的网络的玩家可以通过新的想法和机会为组织增加价值。这反过来又有助于个人的职业发展和晋升。<br />
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A social capital broker also reaps control benefits of being the facilitator of information flow between contacts. In the case of consulting firm Eden McCallum, the founders were able to advance their careers by bridging their connections with former big three consulting firm consultants and mid-size industry firms.<ref>{{cite journal|last=Gardner|first=Heidi|author2=Eccles, Robert |title=Eden McCallum: A Network Based Consulting Firm|journal=Harvard Business School Review|year=2011}}</ref> By bridging structural holes and mobilizing social capital, players can advance their careers by executing new opportunities between contacts.<br />
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A social capital broker also reaps control benefits of being the facilitator of information flow between contacts. In the case of consulting firm Eden McCallum, the founders were able to advance their careers by bridging their connections with former big three consulting firm consultants and mid-size industry firms. By bridging structural holes and mobilizing social capital, players can advance their careers by executing new opportunities between contacts.<br />
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社会资本经纪人还可以从促进联系人之间的信息流动中获得控制权利。在咨询公司 Eden McCallum 的案例中，创始人能够通过与前三大咨询公司顾问和中等规模的行业公司建立联系来提升自己的职业生涯。通过弥合结构漏洞和调动社会资本，玩家可以通过在联系人之间执行新的机会来提升自己的职业生涯。<br />
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There has been research that both substantiates and refutes the benefits of information brokerage. A study of high tech Chinese firms by Zhixing Xiao found that the control benefits of structural holes are "dissonant to the dominant firm-wide spirit of cooperation and the information benefits cannot materialize due to the communal sharing values" of such organizations.<ref>Xiao, Zhixing; Tsui, Anne (2007). "When Brokers May Not Work: The Cultural Contingency of [http://iniciarsesionentrar.com/facebook/ Social] Capital in Chinese High-tech Firms". ''Administrative Science Quarterly''.</ref> However, this study only analyzed Chinese firms, which tend to have strong communal sharing values. Information and control benefits of structural holes are still valuable in firms that are not quite as inclusive and cooperative on the firm-wide level. In 2004, Ronald Burt studied 673 managers who ran the supply chain for one of America's largest electronics companies. He found that managers who often discussed issues with other groups were better paid, received more positive job evaluations and were more likely to be promoted.<ref name="Burt 2004"/> Thus, bridging structural holes can be beneficial to an organization, and in turn, to an individual's career.<br />
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There has been research that both substantiates and refutes the benefits of information brokerage. A study of high tech Chinese firms by Zhixing Xiao found that the control benefits of structural holes are "dissonant to the dominant firm-wide spirit of cooperation and the information benefits cannot materialize due to the communal sharing values" of such organizations. However, this study only analyzed Chinese firms, which tend to have strong communal sharing values. Information and control benefits of structural holes are still valuable in firms that are not quite as inclusive and cooperative on the firm-wide level. In 2004, Ronald Burt studied 673 managers who ran the supply chain for one of America's largest electronics companies. He found that managers who often discussed issues with other groups were better paid, received more positive job evaluations and were more likely to be promoted. Thus, bridging structural holes can be beneficial to an organization, and in turn, to an individual's career.<br />
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有研究证实和反驳了信息经纪的好处。知行晓对中国高科技企业的一项研究发现，结构性漏洞的控制效益“与占主导地位的企业合作精神不协调，由于这类组织的共享价值观，信息效益不能实现”。然而，这项研究只分析了中国企业，这些企业往往有很强的公共共享价值观。结构性漏洞的信息和控制利益对于那些在整个公司层面上不那么具有包容性和合作性的公司来说仍然是有价值的。2004年，罗纳德 · 伯特研究了673名管理者，他们为美国最大的电子公司之一管理供应链。他发现，那些经常与其他团体讨论问题的管理人员收入更高，得到的工作评价更积极，也更有可能获得晋升。因此，架起结构性漏洞对组织有益，反过来对个人的职业生涯也有益。<br />
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===Social media===<br />
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===Social media===<br />
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社交媒体<br />
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{{main article|Social media}}<br />
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[[Computer network]]s combined with social networking software produces a new medium for social interaction.<ref>{{Cite book|title=The International Encyclopedia of Media Effects|last=Amichai-Hamburger|first=Yair|last2=Hayat|first2=Tsahi|date=2017|publisher=John Wiley & Sons, Inc.|isbn=9781118783764|language=en|doi=10.1002/9781118783764.wbieme0170}}</ref> A relationship over a computerized [[social networking service]] can be characterized by context, direction, and strength. The content of a relation refers to the resource that is exchanged. In a [[computer mediated communication]] context, social pairs exchange different kinds of information, including sending a data file or a computer program as well as providing emotional support or arranging a meeting. With the rise of [[electronic commerce]], information exchanged may also correspond to exchanges of money, goods or services in the "real" world.<ref>{{cite journal | title=Studying Online Social Networks | last1=Garton|first1=Laura|first2=Caroline |last2=Haythornthwaite|author2-link=Caroline Haythornthwaite|first3=Barry|last3=Wellman|author3-link=Barry Wellman | journal=Journal of Computer-Mediated Communication |date=23 June 2006 | volume=3 | issue=1 |pages = 0| doi=10.1111/j.1083-6101.1997.tb00062.x}}</ref> [[Social network analysis]] methods have become essential to examining these types of computer mediated communication.<br />
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Computer networks combined with social networking software produces a new medium for social interaction. A relationship over a computerized social networking service can be characterized by context, direction, and strength. The content of a relation refers to the resource that is exchanged. In a computer mediated communication context, social pairs exchange different kinds of information, including sending a data file or a computer program as well as providing emotional support or arranging a meeting. With the rise of electronic commerce, information exchanged may also correspond to exchanges of money, goods or services in the "real" world. Social network analysis methods have become essential to examining these types of computer mediated communication.<br />
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计算机网络与社交网络软件的结合为社交互动提供了一种新的媒介。通过计算机化的社交网络服务建立起来的关系可以是拥有属性、方向和力量。关系的内容指的是交换的资源。在计算机中介的交流环境中，社会对交换不同种类的信息，包括发送数据文件或计算机程序，以及提供情感支持或安排会议。随着电子商务的兴起，交换的信息也可能对应于“现实”世界中的货币、商品或服务的交换。社会网络分析方法已经成为检验这些类型的计算机中介交流的必要手段。<br />
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In addition, the sheer size and the volatile nature of [[social media]] has given rise to new network metrics. A key concern with networks extracted from social media is the lack of robustness of network metrics given missing data.<ref>{{cite journal | last1 = Wei | first1 = Wei | last2 = Joseph | first2 = Kenneth | last3 = Liu | first3 = Huan | last4 = Carley | first4 = Kathleen M. | year = 2016 | title = Exploring Characteristics of Suspended Users and Network Stability on Twitter | url = | journal = Social Network Analysis and Mining | volume = 6 | issue = | page = 51 | doi = 10.1007/s13278-016-0358-5 }}</ref><br />
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In addition, the sheer size and the volatile nature of social media has given rise to new network metrics. A key concern with networks extracted from social media is the lack of robustness of network metrics given missing data.<br />
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此外，社交媒体的庞大规模和不稳定性也催生了新的网络指标。从社交媒体中提取网络的一个关键问题是，在缺失数据的情况下，网络指标缺乏健壮性。<br />
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==See also==<br />
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==See also==<br />
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参见<br />
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* [[Bibliography of sociology]]<br />
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* [[Business networking]]<br />
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* [[Collective network]]<br />
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* [[International Network for Social Network Analysis]]<br />
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* [[Network society]]<br />
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* [[Network theory]]<br />
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*[[Network science]]<br />
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* [[Semiotics of social networking]]<br />
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* [[Scientific collaboration network]]<br />
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* [[Social network analysis]]<br />
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* [[Social network (sociolinguistics)]]<br />
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* [[Social networking service]]<br />
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* [[Social web]]<br />
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* [[Structural fold]]<br />
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==References==<br />
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==References==<br />
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参考资料<br />
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{{reflist}}<br />
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==Further reading==<br />
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==Further reading==<br />
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进一步阅读<br />
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* {{cite journal |last1=Aneja |first1=Nagender |last2=Gambhir |first2=Sapna |title=Ad-hoc Social Network: A Comprehensive Survey |url=http://www.ijser.org/researchpaper%5CAd-hoc-Social-Network-A-Comprehensive-Survey.pdf |journal=International Journal of Scientific & Engineering Research |volume=4 |issue=8 |date=August 2013 |pages=156–160|issn=2229-5518}}<br />
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* {{cite book |last=Barabási |first=Albert-László |year=2003 |title=Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life |publisher=Plum |isbn=978-0-452-28439-5 |url-access=registration |url=https://archive.org/details/linkedhoweveryth00bara }}<br />
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* {{cite book |last=Barnett |first=George A. |year=2011 |title=Encyclopedia of Social Networks |publisher=Sage |isbn=978-1-4129-7911-5}}<br />
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* {{cite book |last=Estrada |first=E |year=2011 |title=The Structure of Complex Networks: Theory and Applications |publisher=Oxford University Press |isbn=978-0-199-59175-6}}<br />
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* {{cite book |last=Ferguson |first=Niall |year=2018 |title=The Square and the Tower: Networks and Power, from the Freemasons to Facebook |isbn=978-0735222915 |publisher=Penguin Press}}<br />
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* {{cite book |last=Freeman |first=Linton C. |year=2004 |title=The Development of Social Network Analysis: A Study in the Sociology of Science |publisher=Empirical Press |isbn=978-1-59457-714-7}}<br />
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* {{cite book |last=Kadushin |first=Charles |year=2012 |title=Understanding Social Networks: Theories, Concepts, and Findings |location= |publisher=Oxford University Press |isbn=978-0-19-537946-4}}<br />
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* {{cite book |last1=Mauro |first1=Rios |last2=Petrella |first2=Carlos |year=2014 |title=La Quimera de las Redes Sociales |trans-title=The Chimera of Social Networks |language=Spanish |publisher=Bubok España |isbn=978-9974-99-637-3}}<br />
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* {{cite book |last1=Rainie |first1=Lee |last2=Wellman |first2=Barry |year=2012 |title=Networked: The New Social Operating System |url=https://archive.org/details/networkednewsoci0000rain |url-access=registration |location=Cambridge, Mass. |publisher=MIT Press |isbn=978-0262017190}}<br />
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* {{cite book |last=Scott |first=John |year=1991 |title=Social Network Analysis: A Handbook |publisher=Sage |isbn=978-0-7619-6338-7}}<br />
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* {{cite book |last1=Wasserman |first1=Stanley |last2=Faust |first2=Katherine |year=1994 |title=Social Network Analysis: Methods and Applications |publisher=Cambridge University Press |isbn=978-0-521-38269-4 |series=Structural Analysis in the Social Sciences }}<br />
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* {{cite book |last1=Wellman |first1=Barry |last2=Berkowitz |first2=S. D. |year=1988 |title=Social Structures: A Network Approach |series=Structural Analysis in the Social Sciences |publisher=Cambridge University Press |isbn=978-0-521-24441-1}}<br />
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==External links==<br />
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==External links==<br />
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外部链接<br />
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{{Commons category|Social networks}}<br />
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===Organizations===<br />
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* [http://www.insna.org/ International Network for Social Network Analysis]<br />
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===Peer-reviewed journals===<br />
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同行评审期刊<br />
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* ''[http://www.sciencedirect.com/science/journal/03788733 Social Networks]''<br />
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* ''[http://journals.cambridge.org/action/displaySpecialPage?pageId=3656 Network Science]''<br />
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* ''[http://www.cmu.edu/joss/content/articles/volindex.html Journal of Social Structure]''<br />
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* ''[http://www.tandfonline.com/toc/gmas20/current Journal of Mathematical Sociology]''<br />
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* ''[https://www.springer.com/computer/database+management+%26+information+retrieval/journal/13278 Social Network Analysis and Mining (SNAM)]''<br />
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* {{cite journal |title=INSNA - ''Connections'' Journal |journal=Connections Bulletin of the International Network for Social Network Analysis |location=Toronto |publisher=International Network for Social Network Analysis |url=http://www.insna.org/pubs/connections/ |issn=0226-1766 |url-status=dead |archiveurl=https://web.archive.org/web/20130718063641/http://www.insna.org/pubs/connections/ |archivedate=2013-07-18 }}<br />
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===Textbooks and educational resources===<br />
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课本及教育资源<br />
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* ''[http://www.cs.cornell.edu/home/kleinber/networks-book/ Networks, Crowds, and Markets]'' (2010) by D. Easley & J. Kleinberg<br />
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* ''[http://faculty.ucr.edu/~hanneman/nettext/ Introduction to Social Networks Methods]'' (2005) by R. Hanneman & M. Riddle<br />
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* [http://www.analytictech.com/networks/ Social Network Analysis Instructional Web Site] by S. Borgatti<br />
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* ''[https://web.archive.org/web/20160304063126/http://www.novagob.org/file/view/153259/ebook-gratuito-sobre-redes-sociales-virtuales Guide for virtual social networks for public administrations]'' (2015) by Mauro D. Ríos (in Spanish)<br />
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===Data sets===<br />
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===Data sets===<br />
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数据集<br />
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* [http://pajek.imfm.si/doku.php?id=data:urls:index Pajek's list of lists of datasets]<br />
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* [http://networkdata.ics.uci.edu/index.html UC Irvine Network Data Repository]<br />
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* [http://snap.stanford.edu/data/ Stanford Large Network Dataset Collection]<br />
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* [http://www-personal.umich.edu/~mejn/netdata/ M.E.J. Newman datasets]<br />
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* [http://vlado.fmf.uni-lj.si/pub/networks/data/ Pajek datasets]<br />
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* [http://wiki.gephi.org/index.php?title=Datasets#Social_networks Gephi datasets]<br />
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* [http://konect.uni-koblenz.de/networks KONECT – Koblenz network collection]<br />
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* [http://www.stats.ox.ac.uk/~snijders/siena/ RSiena datasets]<br />
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==编者推荐==<br />
集智俱乐部：[https://swarma.org/?p=18646 社会网络分析的四个维度：研究方法、应用和软件工具综述 | 网络科学论文速递25篇]<br />
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[http://arxiv.org/abs/2002.09485 The Four Dimensions of Social Network Analysis: An Overview of Research Methods, Applications, and Software Tools]：近年来，基于社交网络的应用呈指数级增长，其中一个方面的原因是，这个应用领域提供了一个特别肥沃的地方，可以测试和开发最先进的计算技术，从 Web 中提取有价值的信息。这项工作的主要贡献有三个方面：(1) 我们对社会网络分析 (SNA) 的最新发展水平进行了文献综述；(2) 我们提出了一套基于 SNA 四个基本特征（或维度）的新度量标准；(3) 最后，我们对一套流行的 SNA 工具和框架进行了定量分析。我们还进行了一项科学计量学研究，以检测该领域中最活跃的研究领域和应用领域。本工作提出了四个不同维度的定义，即(a)模式、(b)知识发现、(c)信息融合、(d)集成、可伸缩性和可视化，用于定义一组新的度量（称为度），以评估 SNA 的不同软件工具和框架（根据之前的度量对一组 20 个 SNA-软件工具进行分析和排序）。这些维度连同定义的程度，可以评价和衡量社会网络技术的成熟度，寻求对它们的定量评估，从而揭示这一活跃领域的挑战和未来发展趋势。<br />
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{{Social sciences}}<br />
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范畴: 传播理论<br />
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范畴: 网络理论<br />
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<small>This page was moved from [[wikipedia:en:Social network]]. Its edit history can be viewed at [[社会网络/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E5%9F%BA%E4%BA%8E%E4%B8%BB%E4%BD%93%E7%9A%84%E8%AE%A1%E7%AE%97%E7%BB%8F%E6%B5%8E%E5%AD%A6&diff=11552基于主体的计算经济学2020-08-02T14:34:29Z<p>Dorr：</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。<br />
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{{Use dmy dates|date=February 2020}}<br />
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'''Agent-based computational economics''' ('''ACE''') is the area of [[computational economics]] that studies economic processes, including whole [[economy|economies]], as [[dynamic system]]s of interacting [[Agent (economics)|agents]]. As such, it falls in the [[paradigm]] of [[complex adaptive system]]s.<ref>• [[W. Brian Arthur]], 1994. "[https://ocw.tudelft.nl/wp-content/uploads/ElFarolArtur1994.pdf Inductive Reasoning and Bounded Rationality]," ''American Economic Review'', 84(2), pp. [http://www-personal.umich.edu/~samoore/bit885f2011/arthur-inductive.pdf 406-411] {{Webarchive|url=https://web.archive.org/web/20130521145936/http://www-personal.umich.edu/~samoore/bit885f2011/arthur-inductive.pdf |date=21 May 2013 }}.<br/>&nbsp;&nbsp; • [[Leigh Tesfatsion]], 2003. "Agent-based Computational Economics: Modeling Economies as Complex Adaptive Systems," ''Information Sciences'', 149(4), pp. [http://copper.math.buffalo.edu/urgewiki/uploads/Literature/Tesfatsion2002.pdf 262-268] {{webarchive|url=https://web.archive.org/web/20120426000037/http://copper.math.buffalo.edu/urgewiki/uploads/Literature/Tesfatsion2002.pdf |date=26 April 2012 }}.</ref> In corresponding [[agent-based model]]s, the "[[agent (economics)|agents]]" are "computational objects modeled as interacting according to rules" over space and time, not real people. The rules are formulated to model behavior and social interactions based on incentives and information.<ref>Scott E. Page (2008). "agent-based models," ''[[The New Palgrave Dictionary of Economics]]'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_A000218&edition=current&q=agent-based%20computational%20modeling&topicid=&result_number=1 Abstract].</ref> Such rules could also be the result of optimization, realized through use of AI methods (such as [[Q-learning]] and other reinforcement learning techniques).<ref>Richard S. Sutton and Andrew G. Barto, Reinforcement Learning: An Introduction, The MIT Press, Cambridge, MA, 1998 [http://www.cs.ualberta.ca/~sutton/book/ebook/the-book.html] {{Webarchive|url=https://web.archive.org/web/20090904194934/http://www.cs.ualberta.ca/~sutton/book/ebook/the-book.html |date=4 September 2009 }}</ref><br />
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Agent-based computational economics (ACE) is the area of computational economics that studies economic processes, including whole economies, as dynamic systems of interacting agents. As such, it falls in the paradigm of complex adaptive systems. In corresponding agent-based models, the "agents" are "computational objects modeled as interacting according to rules" over space and time, not real people. The rules are formulated to model behavior and social interactions based on incentives and information. Such rules could also be the result of optimization, realized through use of AI methods (such as Q-learning and other reinforcement learning techniques).<br />
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'''基于主体的计算经济学 Agent-based Computational Economics (ACE)'''是[[计算经济学]]的一个领域，将整个经济视作不同主体相互作用的动态系统研究经济过程。因此，它属于复杂适应系统'''范式 Paradigm'''。在相应的'''智能体基模型 Agent-based Model(ABM)'''中，“主体”指时空中“根据规则建模的交互计算对象”，而非真实的人。这些规则是为了建立基于激励和信息的行为和社会互动模型而制定的。这些规则也可能是优化的结果，通过人工智能方法（如'''Q学习 Q-learning'''及其他强化学习技术）来实现。<br />
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The theoretical assumption of [[mathematical optimization]] by agents in [[equilibrium (economics)|equilibrium]] is replaced by the less restrictive postulate of agents with [[bounded rationality]] ''adapting'' to market forces.<ref>• [[John H. Holland]] and John H. Miller (1991). "Artificial Adaptive Agents in Economic Theory," ''American Economic Review'', 81(2), pp. [http://www.santafe.edu/media/workingpapers/91-05-025.pdf 365-370] {{Webarchive|url=https://web.archive.org/web/20110105015853/http://www.santafe.edu/media/workingpapers/91-05-025.pdf |date=5 January 2011 }} p. 366.<br/>&nbsp;&nbsp; • [[Thomas C. Schelling]] (1978 [2006]). ''Micromotives and Macrobehavior'', Norton. [http://books.wwnorton.com/books/978-0-393-32946-9/ Description] {{Webarchive|url=https://web.archive.org/web/20171102093240/http://books.wwnorton.com/books/978-0-393-32946-9/ |date=2 November 2017 }}, [https://books.google.com/books?id=DenWKRgqzWMC&printsec=find&pg=PA1=#v=onepage&q&f=false preview].<br/>&nbsp;&nbsp; •<br />
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The theoretical assumption of mathematical optimization by agents in equilibrium is replaced by the less restrictive postulate of agents with bounded rationality adapting to market forces.&nbsp;&nbsp; • Thomas C. Schelling (1978 [2006]). Micromotives and Macrobehavior, Norton. [http://books.wwnorton.com/books/978-0-393-32946-9/ Description] , [https://books.google.com/books?id=DenWKRgqzWMC&printsec=find&pg=PA1=#v=onepage&q&f=false preview].<br/>&nbsp;&nbsp; •<br />
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<br />
<br />
•托马斯 · c · 谢林(Thomas c. Schelling，1978[2006])。微动机和宏行为，诺顿。[ http://books.wwnorton.com/books/978-0-393-32946-9/ 描述] ，[ https://books.google.com/books?id=denwkrgqzwmc&printsec=find&pg=pa1=#v=onepage&q&f=false 预览] . br / •<br />
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[[Thomas J. Sargent]], 1994. ''Bounded Rationality in Macroeconomics'', Oxford. [http://www.oup.com/us/catalog/general/subject/Economics/MacroeconomicTheory/?view=usa&ci=9780198288695 Description] and chapter-preview 1st-page [https://www.questia.com/library/book/bounded-rationality-in-macroeconomics-thomas-j-sargent-by-thomas-j-sargent.jsp links.]</ref> ACE models apply [[numerical methods]] of analysis to [[Computer simulation|computer-based simulations]] of complex dynamic problems for which more conventional methods, such as theorem formulation, may not find ready use.<ref>• Kenneth L. Judd, 2006. "Computationally Intensive Analyses in Economics," ''Handbook of Computational Economics'', v. 2, ch. 17, Introduction, p. 883. [Pp. [https://books.google.com/books?id=6ITfRkNmKQcC&pg=PA881 881-] 893. Pre-pub [http://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF].<br />
<br />
Thomas J. Sargent, 1994. Bounded Rationality in Macroeconomics, Oxford. [http://www.oup.com/us/catalog/general/subject/Economics/MacroeconomicTheory/?view=usa&ci=9780198288695 Description] and chapter-preview 1st-page [https://www.questia.com/library/book/bounded-rationality-in-macroeconomics-thomas-j-sargent-by-thomas-j-sargent.jsp links.]</ref> ACE models apply numerical methods of analysis to computer-based simulations of complex dynamic problems for which more conventional methods, such as theorem formulation, may not find ready use.<ref>• Kenneth L. Judd, 2006. "Computationally Intensive Analyses in Economics," Handbook of Computational Economics, v. 2, ch. 17, Introduction, p. 883. [Pp. [https://books.google.com/books?id=6ITfRkNmKQcC&pg=PA881 881-] 893. Pre-pub [http://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF].<br />
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托马斯·萨金特，1994年。牛津大学宏观经济学有限理性。[ http://www.oup.com/us/catalog/general/subject/economics/macroeconomictheory/?view=usa&ci=9780198288695说明]和第一章预览页面[ https://www.questia.com/library/book/bounded-rationality-in-macroeconomics-thomas-j-sargent-by-thomas-j-sargent.jsp 链接]。] / ref <br />
。 文献• Kenneth l. Judd，2006。《经济学中的计算密集分析》 ，《计算经济学手册，第2卷，第5页。17，Introduction，p. 883.[ Pp.Https://books.google.com/books?id=6itfrknmkqcc&pg=pa881881-893.预订酒吧[ http://www2.econ.iastate.edu/tesfatsi/judd.finalrev.PDF PDF ]。<br />
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<br/>&nbsp;&nbsp; • _____, 1998. ''Numerical Methods in Economics'', MIT Press. Links to [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=3257 description] {{webarchive|url=https://web.archive.org/web/20120211061602/http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=3257 |date=11 February 2012 }} and [https://books.google.com/books?id=9Wxk_z9HskAC&pg=PR7 chapter previews].</ref> Starting from initial conditions specified by the modeler, the computational economy evolves over time as its constituent agents repeatedly interact with each other, including learning from interactions. In these respects, ACE has been characterized as a bottom-up culture-dish approach to the study of [[economic systems]].<ref>• Leigh Tesfatsion (2002). "Agent-Based Computational Economics: Growing Economies from the Bottom Up," ''Artificial Life'', 8(1), pp.55-82. [http://www.mitpressjournals.org/doi/abs/10.1162/106454602753694765 Abstract] and pre-pub [http://www.econ.brown.edu/fac/Peter_Howitt/SummerSchool/Agent.pdf PDF] {{webarchive|url=https://web.archive.org/web/20130514143904/http://www.econ.brown.edu/fac/Peter_Howitt/SummerSchool/Agent.pdf |date=14 May 2013 }}.<br/>&nbsp;&nbsp; • _____ (1997). "How Economists Can Get Alife," in W. B. Arthur, S. Durlauf, and D. Lane, eds., ''The Economy as an Evolving Complex System, II'', pp. 533-564. Addison-Wesley. Pre-pub [http://ageconsearch.umn.edu/bitstream/18196/1/er37.pdf PDF].</ref><br />
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<br/>&nbsp;&nbsp; • _____, 1998. Numerical Methods in Economics, MIT Press. Links to [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=3257 description] and [https://books.google.com/books?id=9Wxk_z9HskAC&pg=PR7 chapter previews].</ref> Starting from initial conditions specified by the modeler, the computational economy evolves over time as its constituent agents repeatedly interact with each other, including learning from interactions. In these respects, ACE has been characterized as a bottom-up culture-dish approach to the study of economic systems.<br />
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<br/>&nbsp;&nbsp; • _____, 1998.经济学数值方法，麻省理工学院出版社。链接到[ http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=3257说明]和[ https://books.google.com/books?id=9wxk_z9hskac&pg=pr7章预览]。<br />
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'''经济均衡 Economic Equilibrium'''中主体'''最优化 Mathematical Optimization'''的理论假设被限制较少的适应市场力量的'''有限理性 Bounded Rationality'''主体假设所取代。ACE 模型将数值分析方法应用于复杂动力学问题的计算机模拟，对其更传统的方法，如定理公式可能无法方便地使用。从建模者指定的初始条件开始，计算经济随着时间的推移而发展，其组成主体不断地相互作用，包括从交互中学习。在这些方面，ACE 被认为是研究'''经济体系 Economic System'''的一种自下而上的文化方法。<br />
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ACE has a similarity to, and overlap with, [[game theory]] as an agent-based method for modeling social interactions.<ref name="COMP&GT">• [[Joseph Y. Halpern]] (2008). "computer science and game theory," ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_C000566&edition=current&q=&topicid=&result_number=1 Abstract].<br/>&nbsp;&nbsp; • Yoav Shoham (2008). "Computer Science and Game Theory," ''Communications of the ACM'', 51(8), pp.<br />
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ACE has a similarity to, and overlap with, game theory as an agent-based method for modeling social interactions.&nbsp;&nbsp; • Yoav Shoham (2008). "Computer Science and Game Theory," Communications of the ACM, 51(8), pp.<br />
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• Yoav Shoham (2008)。《计算机科学与博弈论》 ，美国计算机学会通讯，51(8)页。<br />
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[http://www.robotics.stanford.edu/~shoham/www%20papers/CSGT-CACM-Shoham.pdf 75-79] {{Webarchive|url=https://web.archive.org/web/20120426005917/http://www.robotics.stanford.edu/~shoham/www%20papers/CSGT-CACM-Shoham.pdf |date=26 April 2012 }}.<br/>&nbsp;&nbsp; • [[Alvin E. Roth]] (2002). "The Economist as Engineer: Game Theory, Experimentation, and Computation as Tools for Design Economics," ''Econometrica'', 70(4), pp. [https://web.archive.org/web/20040414102216/http://kuznets.fas.harvard.edu/~aroth/papers/engineer.pdf 1341–1378].</ref> But practitioners have also noted differences from standard methods, for example in ACE events modeled being driven solely by initial conditions, whether or not equilibria exist or are computationally tractable, and in the modeling facilitation of agent autonomy and learning.<ref>Tesfatsion, Leigh (2006), "Agent-Based Computational Economics: A Constructive Approach to Economic Theory," ch. 16, ''Handbook of Computational Economics'', v. 2, part 2, ACE study of economic system. [http://www.sciencedirect.com/science/article/pii/S1574002105020162 Abstract] and pre-pub [http://econ2.econ.iastate.edu/tesfatsi/hbintlt.pdf PDF].</ref><br />
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[http://www.robotics.stanford.edu/~shoham/www%20papers/CSGT-CACM-Shoham.pdf 75-79] .<br/>&nbsp;&nbsp; • Alvin E. Roth (2002). "The Economist as Engineer: Game Theory, Experimentation, and Computation as Tools for Design Economics," Econometrica, 70(4), pp. [https://web.archive.org/web/20040414102216/http://kuznets.fas.harvard.edu/~aroth/papers/engineer.pdf 1341–1378].</ref> But practitioners have also noted differences from standard methods, for example in ACE events modeled being driven solely by initial conditions, whether or not equilibria exist or are computationally tractable, and in the modeling facilitation of agent autonomy and learning.<br />
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[http://www.robotics.stanford.edu/~shoham/www%20papers/CSGT-CACM-Shoham.pdf 75-79] .<br/>&nbsp;&nbsp; • Alvin E. Roth (2002).《作为工程师的经济学家: 博弈论、实验和计算作为设计经济学的工具》 ，《经济学》 ，70(4) ，页。[ https://web.archive.org/web/20040414102216/http://kuznets.fas.harvard.edu/~aroth/papers/engineer.pdf 1341-1378]. <br />
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ACE 与博弈论（作为基于主体的社会互动建模方法时）有相似之处和重叠部分。但实践者也注意到了与标准方法的区别，例如，在仅由初始条件驱动的ACE事件建模中，均衡是否存在或者在计算上是否易于处理，以及在建模过程中促进主体自主性和学习方面。<br />
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The method has benefited from continuing improvements in modeling techniques of [[computer science]] and increased computer capabilities. The ultimate scientific objective of the method is to "test theoretical findings against real-world data in ways that permit empirically supported theories to cumulate over time, with each researcher’s work building appropriately on the work that has gone before."<ref>• Leigh Tesfatsion (2006). "Agent-Based Computational Economics: A Constructive Approach to Economic Theory," ch. 16, ''Handbook of Computational Economics'', v. 2, [pp. 831-880] sect. 5. [http://www.sciencedirect.com/science/article/pii/S1574002105020162 Abstract] and pre-pub [http://econ2.econ.iastate.edu/tesfatsi/hbintlt.pdf PDF].<br/>&nbsp;&nbsp; • [[Kenneth L. Judd]] (2006). "Computationally Intensive Analyses in Economics," ''Handbook of Computational Economics'', v. 2, ch. 17, pp. [https://books.google.com/books?id=6ITfRkNmKQcC&pg=PA881 881-] 893. Pre-pub [http://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF].<br/>&nbsp;&nbsp; • Leigh Tesfatsion and Kenneth L. Judd, ed. (2006). ''Handbook of Computational Economics'', v. 2. [http://www.elsevier.com/wps/find/bookdescription.cws_home/660847/description#description Description] {{Webarchive|url=https://web.archive.org/web/20120306100156/http://www.elsevier.com/wps/find/bookdescription.cws_home/660847/description#description |date=6 March 2012 }} & and chapter-preview<br />
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The method has benefited from continuing improvements in modeling techniques of computer science and increased computer capabilities. The ultimate scientific objective of the method is to "test theoretical findings against real-world data in ways that permit empirically supported theories to cumulate over time, with each researcher’s work building appropriately on the work that has gone before."&nbsp;&nbsp; • Kenneth L. Judd (2006). "Computationally Intensive Analyses in Economics," Handbook of Computational Economics, v. 2, ch. 17, pp. [https://books.google.com/books?id=6ITfRkNmKQcC&pg=PA881 881-] 893. Pre-pub [http://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF].<br/>&nbsp;&nbsp; • Leigh Tesfatsion and Kenneth L. Judd, ed. (2006). Handbook of Computational Economics, v. 2. [http://www.elsevier.com/wps/find/bookdescription.cws_home/660847/description#description Description] & and chapter-preview<br />
<br />
• Kenneth l. Judd (2006)。《经济学中的计算密集分析》 ，计算经济学手册，第2卷，第5页。17页。Https://books.google.com/books?id=6itfrknmkqcc&pg=pa881881-893.Pre-pub [http://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF].<br/>&nbsp;&nbsp; • Leigh Tesfatsion and Kenneth L. Judd, ed.(2006).计算经济学手册第2节。[ http://www.elsevier.com/wps/find/bookdescription.cws_home/660847/Description#Description 说明]和章节预览<br />
<br />
[http://www.sciencedirect.com/science?_ob=PublicationURL&_hubEid=1-s2.0-S1574002105X02003&_cid=273377&_pubType=HS&_auth=y&_acct=C000228598&_version=1&_urlVersion=0&_userid=10&md5=e4757b4f65755ed6340a11fee9615200 links.]</ref> The subject has been applied to research areas like [[asset pricing]],<ref name=arthuretal>B. Arthur, J. Holland, B. LeBaron, R. Palmer, P. Taylor (1997), 'Asset pricing under endogenous expectations in an artificial stock market,' in ''The Economy as an Evolving Complex System II'', B. Arthur, S. Durlauf, and D. Lane, eds., Addison Wesley.</ref> [[competition]] and [[collaboration]],<ref>[[Robert Axelrod]] (1997). ''The Complexity of Cooperation: Agent-Based Models of Competition and Collaboration'', Princeton. [http://press.princeton.edu/titles/6144.html Description], [http://press.princeton.edu/titles/6144.html#TOC contents], and [https://books.google.com/books?id=J0dgRGMdjmQC&printsec=find&pg=PR11#v=onepage&q&f=false preview].</ref> [[transaction cost]]s,<ref>Tomas B. Klosa and [[Bart Nooteboom]], 2001. "Agent-based Computational Transaction Cost Economics," ''Journal of Economic Dynamics and Control'' 25(3–4), pp. 503–52. [http://www.sciencedirect.com/science/article/pii/S0165188900000348 Abstract.]</ref> [[market structure]] and [[industrial organization]] and dynamics,<ref>• Roberto Leombruni and Matteo Richiardi, ed. (2004), ''Industry and Labor Dynamics: The Agent-Based Computational Economics Approach.'' World Scientific Publishing {{ISBN|981-256-100-5}}. [http://www.worldscibooks.com/economics/5706.html Description] {{Webarchive|url=https://web.archive.org/web/20100727221149/http://www.worldscibooks.com/economics/5706.html |date=27 July 2010 }} and chapter-preview [https://books.google.com/books?id=P5O7A5D55nQC&printsec=fond&pg=PR5#v=onepage&q&f=false links].<br/>&nbsp;&nbsp; • [[Joshua M. Epstein]] (2006). "Growing Adaptive Organizations: An Agent-Based Computational Approach," in ''Generative Social Science: Studies in Agent-Based Computational Modeling'', pp. [https://books.google.com/books?id=543OS3qdxBYC&pg=PA309 309-] 344. [http://press.princeton.edu/titles/8277.html Description] {{Webarchive|url=https://web.archive.org/web/20120126180655/http://press.princeton.edu/titles/8277.html |date=26 January 2012 }} and [http://www.santafe.edu/research/working-papers/abstract/99895b6465e8b87656612f8e3570b34c/ abstract].</ref> [[welfare economics]],<ref>[[Robert Axtell]] (2005). "The Complexity of Exchange," ''Economic Journal'', 115(504, Features), pp. [http://econfaculty.gmu.edu/pboettke/workshop/archives/f05/Axtell.pdf F193-F210].</ref> and [[mechanism design]],<ref>• ''The New Palgrave Dictionary of Economics'' (2008), 2nd Edition: <br/>&nbsp;&nbsp;&nbsp;&nbsp; [[Roger B. Myerson]] "mechanism design." [http://www.dictionaryofeconomics.com/article?id=pde2008_M000132&edition=current&q=mechanism%20design&topicid=&result_number=3 Abstract.] <br/>&nbsp;&nbsp;&nbsp;&nbsp; _____. "revelation principle." [http://www.dictionaryofeconomics.com/article?id=pde2008_R000137&edition=current&q=moral&topicid=&result_number=1 Abstract.]<br/>&nbsp;&nbsp;&nbsp;&nbsp; Tuomas Sandholm. "computing in mechanism design." [http://www.dictionaryofeconomics.com/article?id=pde2008_C000563&edition=&field=keyword&q=algorithmic%20mechanism%20design&topicid=&result_number=1 Abstract.]<br/>&nbsp;&nbsp; • [[Noam Nisan]] and Amir Ronen (2001). "Algorithmic Mechanism Design," ''Games and Economic Behavior'', 35(1-2), pp. [https://www.cs.cmu.edu/~sandholm/cs15-892F09/Algorithmic%20mechanism%20design.pdf 166–196].<br/>&nbsp;&nbsp; • [[Noam Nisan]] ''et al''., ed. (2007). ''Algorithmic Game Theory'', Cambridge University Press. [http://www.cup.cam.ac.uk/asia/catalogue/catalogue.asp?isbn=9780521872829 Description] {{Webarchive|url=https://web.archive.org/web/20120505140924/http://www.cup.cam.ac.uk/asia/catalogue/catalogue.asp?isbn=9780521872829 |date=5 May 2012 }}.</ref> [[Information economics|information and uncertainty]],<ref>Tuomas W. Sandholm and Victor R. Lesser (2001). "Leveled Commitment Contracts and Strategic Breach," ''Games and Economic Behavior'', 35(1-2), pp. [https://www.cs.cmu.edu/afs/.cs.cmu.edu/Web/People/sandholm/leveled.geb.pdf 212-270].</ref> [[macroeconomics]],<ref>• [[David Colander]], [[Peter Howitt]], Alan Kirman, [[Axel Leijonhufvud]], and [[Perry Mehrling]], 2008. "Beyond DSGE Models: Toward an Empirically Based Macroeconomics," ''American Economic Review'', 98(2), pp. [https://www.jstor.org/pss/29730026 236]-240. Pre-pub [http://www.brown.edu/Departments/Economics/Faculty/Peter_Howitt/publication/complex%20macro6.pdf PDF].<br/>&nbsp;&nbsp; • [[Thomas J. Sargent]] (1994). ''Bounded Rationality in Macroeconomics'', Oxford. [http://www.oup.com/us/catalog/general/subject/Economics/MacroeconomicTheory/?view=usa&ci=9780198288695 Description] and chapter-preview 1st-page [https://www.questia.com/library/book/bounded-rationality-in-macroeconomics-thomas-j-sargent-by-thomas-j-sargent.jsp links].<br/>&nbsp;&nbsp; • M. Oeffner (2009). '[http://www.opus-bayern.de/uni-wuerzburg/volltexte/2009/3927/pdf/OeffnerDissohneAnhang.pdf Agent-based Keynesian Macroeconomics]'. PhD thesis, Faculty of Economics, University of Würzburg.</ref> and [[Marxist economics]].<ref>A. F. Cottrell, P. Cockshott, G. J. Michaelson, I. P. Wright, V. Yakovenko<br />
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[http://www.sciencedirect.com/science?_ob=PublicationURL&_hubEid=1-s2.0-S1574002105X02003&_cid=273377&_pubType=HS&_auth=y&_acct=C000228598&_version=1&_urlVersion=0&_userid=10&md5=e4757b4f65755ed6340a11fee9615200 links.]</ref> The subject has been applied to research areas like asset pricing, competition and collaboration, transaction costs, market structure and industrial organization and dynamics, welfare economics, and mechanism design, information and uncertainty, macroeconomics, and Marxist economics.<ref>A. F. Cottrell, P. Cockshott, G. J. Michaelson, I. P. Wright, V. Yakovenko<br />
<br />
[ http://www.sciencedirect.com/science?_ob=publicationurl&_hubeid=1-s2.0-s1574002105x02003&_cid=273377&_pubtype=hs&_auth=y&_acct=c000228598&_version=1&_urlversion=0&_userid=10&md5=e4757b4f65755ed6340a11fee9615200连结。参考文献<br />
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<br />
参考文献 a。科特雷尔，p. Cockshott，g. j. Michaelson，i. p. Wright，v. Yakovenko<br />
<br />
(2009), ''Classical Econophysics''. Routledge, {{ISBN|978-0-415-47848-9}}.</ref><ref>Leigh Tesfatsion (2006), "Agent-Based Computational Economics: A Constructive Approach to Economic Theory," ch. 16, ''Handbook of Computational Economics'', v. 2, part 2, ACE study of economic system. [http://www.sciencedirect.com/science/article/pii/S1574002105020162 Abstract] and pre-pub [http://econ2.econ.iastate.edu/tesfatsi/hbintlt.pdf PDF].</ref><br />
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(2009), Classical Econophysics. Routledge, .</ref><br />
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(2009) ，古典经济物理学。路特利奇，参考文献<br />
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该方法得益于计算机科学建模技术的不断改进和计算机能力的提高。其最终科学目标是“用实际数据来检验理论发现——以允许经验支持的理论随着时间的推移而累积的方式（每位研究者的工作恰当建立在之前工作的基础之上）。”其现已应用于如资产定价、竞争与合作、'''交易成本 Transaction Cost'''、市场形式与'''产业组织 Industrial Organization'''与动态、'''福利经济学 Welfare Economics'''、'''机制设计 Mechanism Design'''、信息与不确定性、'''宏观经济学 Macroeconomics'''和'''马克思主义政治经济学 Marxist economics'''等研究领域。<br />
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==Overview 概述==<br />
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The "[[Agent (economics)|agents]]" in ACE models can represent individuals (e.g. people), social groupings (e.g. firms), biological entities (e.g. growing crops), and/or physical systems (e.g. transport systems). The ACE modeler provides the initial configuration of a computational economic system comprising multiple interacting agents. The modeler then steps back to observe the development of the system over time without further intervention. In particular, system events should be driven by agent interactions without external imposition of equilibrium conditions.<ref>[http://www.socsci.aau.dk/ae2006/ Summary of methods] {{Webarchive|url=https://web.archive.org/web/20070526105550/http://www.socsci.aau.dk/ae2006/ |date=26 May 2007 }}: ''Department of Economics, Politics and Public Administration, Aalborg University, Denmark'' website.</ref> Issues include those common to [[experimental economics]] in general<ref>[[Vernon L. Smith]], 2008. "experimental economics," ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_E000277&q=experimental%20&topicid=&result_number=2 Abstract].</ref> and development of a common framework for empirical validation and resolving open questions in agent-based modeling.<ref>Giorgio Fagiolo, Alessio Moneta, and Paul Windrum, 2007. "A Critical Guide to Empirical Validation of Agent-Based Models in Economics: Methodologies, Procedures, and Open Problems," ''Computational Economics'', 30, pp. [https://doi.org/10.1007%2Fs10614-007-9104-4 195]–226.</ref><br />
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The "agents" in ACE models can represent individuals (e.g. people), social groupings (e.g. firms), biological entities (e.g. growing crops), and/or physical systems (e.g. transport systems). The ACE modeler provides the initial configuration of a computational economic system comprising multiple interacting agents. The modeler then steps back to observe the development of the system over time without further intervention. In particular, system events should be driven by agent interactions without external imposition of equilibrium conditions. Issues include those common to experimental economics in general and development of a common framework for empirical validation and resolving open questions in agent-based modeling.<br />
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ACE 模型中的“主体”可以代表个体（如人）、社会群体（如公司）、生物实体（如生长中的庄稼）和/或物理系统（如运输系统）。ACE 模型提供了由多个相互作用的主体组成的计算经济系统的初始配置。然后，建模人员返回来观察系统随着时间的推移而发展的情况，而不需要进一步的干预。注意，系统活动应由主体间的相互作用驱动，而不需要外部强加的平衡条件。研究问题包括'''实验经济学 Experimental economics'''通用的问题，以及开发用于经验验证的通用框架和解决基于主体的建模中尚未解决的问题。<br />
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ACE is an officially designated special interest group (SIG) of the Society for Computational Economics.<ref>[http://comp-econ.org/ Society for Computational Economics] website.</ref> Researchers at the [[Santa Fe Institute]] have contributed to the development of ACE.<br />
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ACE is an officially designated special interest group (SIG) of the Society for Computational Economics. Researchers at the Santa Fe Institute have contributed to the development of ACE.<br />
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ACE 是美国计算经济学学会的一个官方指定的特殊利益集团 (SIG) 。[[圣菲研究所 Santa Fe Institute]]的研究人员为 ACE 的发展做出了贡献。<br />
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==Example: finance 举例：金融==<br />
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One area where ACE methodology has frequently been applied is asset pricing. [[W. Brian Arthur]], Eric Baum, [[William A. Brock (economist)|William Brock]], Cars Hommes, and Blake LeBaron, among others, have developed computational models in which many agents choose from a set of possible forecasting strategies in order to predict stock prices, which affects their asset demands and thus affects stock prices. These models assume that agents are more likely to choose forecasting strategies which have recently been successful. The success of any strategy will depend on market conditions and also on the set of strategies that are currently being used. These models frequently find that large booms and busts in asset prices may occur as agents switch across forecasting strategies.<ref name=arthuretal/><ref>W. Brock and C. Hommes (1997), 'A rational route to randomness.' ''Econometrica'' 65 (5), pp. 1059-1095.</ref><ref>C. Hommes (2008), 'Interacting agents in finance,' in ''The New Palgrave Dictionary of Economics''.</ref> More recently, Brock, Hommes, and Wagener (2009) have used a model of this type to argue that the introduction of new hedging instruments may destabilize the market,<ref>{{cite journal |first=W. |last=Brock |first2=C. |last2=Hommes |first3=F. |last3=Wagener |year=2009 |title=More hedging instruments may destabilize markets |journal=Journal of Economic Dynamics and Control |volume=33 |issue=11 |pages=1912–1928 |doi=10.1016/j.jedc.2009.05.004 |url=http://cendef.uva.nl/binaries/content/assets/subsites/amsterdam-school-of-economics/amsterdam-school-of-economics-research-institute/cendef/working-papers-2006/brohomwag.pdf?1417181713282 }}</ref> and some papers have suggested that ACE might be a useful methodology for understanding the recent [[financial crisis]].<ref>M. Buchanan (2009), '[http://pagesperso-orange.fr/mark.buchanan/nature_economic_modelling.pdf Meltdown modelling. Could agent-based computer models prevent another financial crisis?].' Nature, Vol. 460, No. 7256. (5 August 2009), pp. 680-682.</ref><ref>J.D. Farmer, D. Foley (2009), 'The economy needs agent-based modelling.' Nature, Vol. 460, No. 7256. (5 August 2009), pp. 685-686.</ref><ref>M. Holcombe, S. Coakley, M.Kiran, S. Chin, C. Greenough, D.Worth, S.Cincotti, M.Raberto, A. Teglio, C. Deissenberg, S. van der Hoog, H. Dawid, S. Gemkow, P. Harting, M. Neugart. Large-scale Modeling of Economic Systems, Complex Systems, 22(2), 175-191, 2013</ref><br />
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One area where ACE methodology has frequently been applied is asset pricing. W. Brian Arthur, Eric Baum, William Brock, Cars Hommes, and Blake LeBaron, among others, have developed computational models in which many agents choose from a set of possible forecasting strategies in order to predict stock prices, which affects their asset demands and thus affects stock prices. These models assume that agents are more likely to choose forecasting strategies which have recently been successful. The success of any strategy will depend on market conditions and also on the set of strategies that are currently being used. These models frequently find that large booms and busts in asset prices may occur as agents switch across forecasting strategies. More recently, Brock, Hommes, and Wagener (2009) have used a model of this type to argue that the introduction of new hedging instruments may destabilize the market, and some papers have suggested that ACE might be a useful methodology for understanding the recent financial crisis.<br />
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ACE 方法已广泛应用的领域之一是资产定价。'''布莱恩·亚瑟 W. Brian Arthur'''，'''埃里克·鲍姆 Eric Baum'''，'''威廉·布洛克 William Brock'''，Cars Hommes，'''布莱克·勒巴朗 Blake LeBaron'''等人已开发出一个计算模型，其中许多主体从一组可能的预测策略中选择以预测股票价格，这影响了他们的资产需求，进而影响股票价格。这些模型假设主体更有可能选择最近成功的预测策略。任何策略的成功取决于市场条件，也取决于目前正在使用的一组策略。这些模型经常表明，随着主体改换预测策略，资产价格可能会出现大起大落。最近，布洛克、Hommes 和'''瓦格纳 Wagener''' （2009）使用了这类模型来论证引入新的对冲工具可能会破坏市场稳定，一些论文提出，ACE 可能是理解最近金融危机的有用方法。<br />
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==See also 参见==<br />
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* [[ACEGES]] 基于主体的全球能源系统的计算经济学<br />
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* [[Agent-based social simulation]] 基于主体的社会仿真<br />
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* [[Artificial economics]] 人工经济学<br />
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* [[Computational economics]] 计算经济学<br />
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* [[Econophysics]] 经济物理学<br />
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* [[Macroeconomic model]] 宏观经济模型<br />
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* [[Multi-agent system]] 多智能体系统<br />
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* [[Statistical finance]] 统计金融<br />
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==References 参考资料==<br />
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{{Reflist}}<br />
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==编者推荐==<br />
[https://campus.swarma.org/ 集智学园]课程：[https://campus.swarma.org/course/714 计算社会经济学的内容与方法]<br />
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本课程系统性地介绍计算社会经济学的主要研究内容和分析方法，阐述大规模社会经济数据（包括卫星遥感数据、手机通讯数据、社交媒体数据等）在解决全球贫困问题、刻画区域经济结构、推断个体财富和应急抢险救灾中的具体应用。<br />
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[[Category:Computational economics]]<br />
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Category:Computational economics<br />
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类别: 计算经济学<br />
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[[Category:Monte Carlo methods in finance]]<br />
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Category:Monte Carlo methods in finance<br />
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类别: 金融中的蒙特卡罗方法<br />
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[[Category:Computational fields of study]]<br />
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Category:Computational fields of study<br />
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类别: 研究的计算领域<br />
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<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Agent-based computational economics]]. Its edit history can be viewed at [[基于主体的计算经济学/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%B8%9F%E7%BE%A4%E7%AE%97%E6%B3%95&diff=11464鸟群算法2020-08-02T07:22:50Z<p>Dorr：</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。<br />
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{{short description|Artificial life program, developed by Craig Reynolds in 1986, which simulates the flocking behaviour of birds}}<br />
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{{Use mdy dates|date=July 2011}}<br />
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{{for|the reptilian family|Boidae}}<br />
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[[File:Boids OpenGL example.webm|thumb|A Boids example created in [[OpenGL]]]]<br />
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A Boids example created in [[OpenGL]]<br />
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在[[[ OpenGL ]]中创建的 Boids 示例<br />
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'''Boids''' is an [[artificial life]] program, developed by [[Craig Reynolds (computer graphics)|Craig Reynolds]] in 1986, which simulates the [[Flocking (behavior)|flocking]] behaviour of [[Bird|birds]]. His paper on this topic was published in 1987 in the proceedings of the [[Association for Computing Machinery|ACM]] [[SIGGRAPH]] conference. <ref>{{Cite book<br />
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Boids is an artificial life program, developed by Craig Reynolds in 1986, which simulates the flocking behaviour of birds. His paper on this topic was published in 1987 in the proceedings of the ACM SIGGRAPH conference. <ref>{{Cite book<br />
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最后1雷诺<br />
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第一名: 克雷格<br />
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| author1-link=Craig Reynolds (computer_graphics)<br />
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| author1-link=Craig Reynolds (computer_graphics)<br />
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| author1-link Craig Reynolds (计算机图形学)<br />
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| title=Flocks, herds and schools: A distributed behavioral model.<br />
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| title=Flocks, herds and schools: A distributed behavioral model.<br />
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羊群、畜群和学校: 一个分布式的行为模型。<br />
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87: 第14届计算机图形学与互动技术年度会议论文集<br />
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| publisher=[[Association for Computing Machinery]]<br />
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| publisher=Association for Computing Machinery<br />
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[国际标准图书编号978-0-89791-227-3]<br />
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| doi=10.1145/37401.37406<br />
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10.1145 / 37401.37406<br />
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The name "boid" corresponds to a shortened version of "bird-oid object", which refers to a bird-like object.<ref>{{Cite journal<br />
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The name "boid" corresponds to a shortened version of "bird-oid object", which refers to a bird-like object.<ref>{{Cite journal<br />
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第二名: 乔纳森<br />
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| title=A review of particle swarm optimization. Part I: background and development<br />
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| title=A review of particle swarm optimization. Part I: background and development<br />
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2012年10月12日 | 标题: 粒子群优化的回顾。第一部分: 背景与发展<br />
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自然计算杂志<br />
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}}</ref> Incidentally, "boid" is also a New York Metropolitan dialect pronunciation for "bird".<br />
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}}</ref> Incidentally, "boid" is also a New York Metropolitan dialect pronunciation for "bird".<br />
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'''鸟群算法 Boids'''是模拟鸟类群集行为的人工生命项目，由'''克雷格·雷诺兹 Craig Reynolds'''于1986年开发。该模型基于常用于计算机动画或计算机辅助设计的计算机三维几何。他关于这个主题的论文发表在1987年的 ACM SIGGRAPH（美国计算机协会计算机图形专业组组织的计算机图形学顶级年度会议）会议记录上。“ boid”是“ bird-oid object”的缩写，指一个类鸟对象。恰巧，“boid”也是纽约都市方言中“bird”的发音。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）补充自http://www.red3d.com/cwr/boids/：It was based on three dimensional computational geometry of the sort normally used in computer animation or computer aided design.<br />
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{{multiple image|direction=vertical|image1=Rule separation.gif|caption1=Separation|image2=Rule alignment.gif|caption2=Alignment|image3=Rule cohesion.gif|caption3=Cohesion|header=Rules applied in simple Boids}}<br />
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As with most artificial life simulations, Boids is an example of [[emergence|emergent]] behavior; that is, the complexity of Boids arises from the interaction of individual agents (the boids, in this case) adhering to a set of simple rules. The rules applied in the simplest Boids world are as follows:<br />
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As with most artificial life simulations, Boids is an example of emergent behavior; that is, the complexity of Boids arises from the interaction of individual agents (the boids, in this case) adhering to a set of simple rules. The rules applied in the simplest Boids world are as follows:<br />
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与大多数人工生命模拟一样，Boids 是'''涌现 Emergent'''行为的一个例子; 也就是说，Boids 的复杂性来自于遵循一系列简单规则的个体代理（这里是 Boids）的相互作用。在最简单的Bolds 世界中适用的规则如下，其描述了鸟群中的个体如何根据周边同伴的位置和速度移动:<br />
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* '''separation''': [[wikt:steer#Verb 2|steer]] to avoid crowding local flockmates<br />
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* '''alignment''': steer towards the average heading of local flockmates<br />
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* '''cohesion''': steer to move towards the average position (center of mass) of local flockmates<br />
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* '''分离 Separation''': 移动以避开群体拥挤处<br />
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* '''对齐 Alignment''': 转向群体的平均航向<br />
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* '''靠近 Cohesion''': 朝群体的平均位置（质心）移动<br />
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每个 boid 个体都可以得知整体的几何参数，但群体要求其只对其周围某个小邻近范围作出反应。该邻近范围由一个距离（从该个体的中心算起）和一个角度（从其飞行方向算起）决定。此范围外的同伴不予考虑。该临近范围可以认为是一个有限知觉的模型（就像浑浊的水中的鱼一样）；但更恰当的想法可能是，其定义了鸟群影响了个体转向的区域范围。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）补充自http://www.red3d.com/cwr/boids/：Each boid has direct access to the whole scene's geometric description, but flocking requires that it reacts only to flockmates within a certain small neighborhood around itself. The neighborhood is characterized by a distance (measured from the center of the boid) and an angle, measured from the boid's direction of flight. Flockmates outside this local neighborhood are ignored. The neighborhood could be considered a model of limited perception (as by fish in murky water) but it is probably more correct to think of it as defining the region in which flockmates influence a boids steering.<br />
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More complex rules can be added, such as obstacle avoidance and goal seeking.<br />
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More complex rules can be added, such as obstacle avoidance and goal seeking.<br />
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可以添加更复杂的规则，如避障和寻找目标。<br />
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The basic model has been extended in several different ways since Reynolds proposed it. For instance, Delgado-Mata et&nbsp;al.<ref><br />
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The basic model has been extended in several different ways since Reynolds proposed it. For instance, Delgado-Mata et&nbsp;al.<ref><br />
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|first1= Carlos | last1= Delgado-Mata<br />
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卡洛斯，德尔加多，马塔<br />
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|first2= Jesus Ibanez | last2= Martinez<br />
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|title= On the use of Virtual Animals with Artificial Fear in Virtual Environments<br />
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|title= On the use of Virtual Animals with Artificial Fear in Virtual Environments<br />
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论虚拟动物在虚拟环境中的人为恐惧<br />
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|journal= New Generation Computing<br />
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|journal= New Generation Computing<br />
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| 杂志新一代计算<br />
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|volume=25 |issue=2 |pages= 145–169<br />
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|doi= 10.1007/s00354-007-0009-5}}</ref><br />
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| doi 10.1007 / s00354-007-0009-5} / ref<br />
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extended the basic model to incorporate the effects of fear. Olfaction was used to transmit emotion between animals, through pheromones modelled as particles in a free expansion gas. Hartman and Benes<ref><br />
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extended the basic model to incorporate the effects of fear. Olfaction was used to transmit emotion between animals, through pheromones modelled as particles in a free expansion gas. Hartman and Benes<ref><br />
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{{cite journal<br />
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{{cite journal<br />
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{引用期刊<br />
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|first1= Christopher | last1= Hartman<br />
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1 Christopher | last 1 Hartman<br />
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|first2= Bedr̆ich |last2= Benes̆<br />
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|first2= Bedr̆ich |last2= Benes̆<br />
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|first2= Bedr̆ich |last2= Benes̆<br />
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|title= Autonomous boids<br />
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|title= Autonomous boids<br />
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标题自主 boids<br />
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|journal= Computer Animation and Virtual Worlds<br />
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|journal= Computer Animation and Virtual Worlds<br />
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计算机动画与虚拟世界<br />
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|volume=17 |issue=3–4 |pages= 199–206<br />
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|volume=17 |issue=3–4 |pages= 199–206<br />
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第17卷第3-4期，第199-206页<br />
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|date=July 2006<br />
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|doi= 10.1002/cav.123}}</ref><br />
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10.1002 / cav. 123} / ref<br />
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introduced a complementary force to the alignment that they call the change of leadership. This steer defines the chance of the boid to become a leader and try to escape.<br />
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introduced a complementary force to the alignment that they call the change of leadership. This steer defines the chance of the boid to become a leader and try to escape.<br />
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该基本模型自雷诺兹提出以来，已用多种方法扩展。例如，Delgado-Mata 等人扩展基本模型以加入恐惧的影响。利过一种可用自由膨胀气体中的粒子模拟的信息素，动物间用嗅觉来传递情感。'''哈特曼 Hartman'''和'''贝内斯 Benes'''为定位过程引入了一种补充力量，他们称之为领导权的更迭。这条决定了某只鸟成为领导者及试图逃跑的机会。<br />
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The movement of Boids can be characterized as either chaotic (splitting groups and wild behaviour) or orderly. Unexpected behaviours, such as splitting flocks and reuniting after avoiding obstacles, can be considered emergent.<br />
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The movement of Boids can be characterized as either chaotic (splitting groups and wild behaviour) or orderly. Unexpected behaviours, such as splitting flocks and reuniting after avoiding obstacles, can be considered emergent.<br />
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Boids 的运动可以表现为混乱（分裂的群体和狂野的行为）或有序。意想不到的行为，比如群体分散和避开障碍后的团聚可以被认为是应急的。<br />
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The boids framework is often used in computer graphics, providing realistic-looking representations of flocks of birds and other creatures, such as schools of fish or herds of animals. It was for instance used in the 1998 video game ''[[Half-Life (video game)|Half-Life]]'' for the flying bird-like creatures seen at the end of the game on [[Locations of Half-Life#Xen|Xen]], named "boid" in the game files.<br />
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The boids framework is often used in computer graphics, providing realistic-looking representations of flocks of birds and other creatures, such as schools of fish or herds of animals. It was for instance used in the 1998 video game Half-Life for the flying bird-like creatures seen at the end of the game on Xen, named "boid" in the game files.<br />
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Boids 框架通常用于计算机图形学，提供鸟群和其他生物（如鱼群）的逼真表现。例如，在1998年的视频游戏《'''半条命 Half-Life'''》中，Xen中游戏结束时出现的类似鸟类的飞行生物就使用了该框架（游戏文件中命名为“ boid”）。<br />
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The Boids model can be used for direct control and stabilization of teams of simple Unmanned Ground Vehicles (UGV)<ref>{{cite conference |title= Design and analysis of Group Escape Behavior for distributed autonomous mobile robots |first1= Hongkyu |last1= Min |first2= Zhidong |last2= Wang |year= 2011 |conference= IEEE International Conference on Robotics and Automation (ICRA)|doi= 10.1109/ICRA.2011.5980123 }}</ref> or [[Micro air vehicle|Micro Aerial Vehicles]] (MAV)<ref>{{cite conference |title= Swarms of micro aerial vehicles stabilized under a visual relative localization |first1= Martin |last1= Saska |first2= Vakula |last2= Jan |first3= Preucil |last3= Libor |year= 2014 |conference= IEEE International Conference on Robotics and Automation (ICRA)|doi= 10.1109/ICRA.2014.6907374 }}</ref> in [[swarm robotics]]. For stabilization of heterogeneous UAV-UGV teams, the model was adapted for using onboard relative localization by Saska et&nbsp;al.<ref>{{cite conference |title= Coordination and Navigation of Heterogeneous UAVs-UGVs Teams Localized by a Hawk-Eye Approach |first1= Martin |last1= Saska |first2= Vonasek |last2= Vojtech |first3= Krajnik |last3= Tomas |first4= Preucil |last4= Libor |year= 2012 |conference= IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)|url=http://labe.felk.cvut.cz/~tkrajnik/ardrone/articles/formace.pdf}}</ref><br />
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The Boids model can be used for direct control and stabilization of teams of simple Unmanned Ground Vehicles (UGV) or Micro Aerial Vehicles (MAV) in swarm robotics. For stabilization of heterogeneous UAV-UGV teams, the model was adapted for using onboard relative localization by Saska et&nbsp;al.<br />
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Boids 模型可用于'''[[集群机器人]] Swarm Robotics'''中简单的无人地面车辆（UGV）或'''微型飞行器 Micro Aerial Vehicles'''（MAV）群体的直接控制和稳定。为了异质 UAV-UGV 群体的稳定性，Saska 等人将该模型用于板载相对定位。<br />
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At the time of proposal, Reynolds' approach represented a giant step forward compared to the traditional techniques used in computer animation for motion pictures. The first animation created with the model was ''[[Stanley and Stella in: Breaking the Ice]]'' (1987), followed by a feature film debut in [[Tim Burton]]'s film ''[[Batman Returns]]'' (1992) with computer generated bat swarms and armies of penguins marching through the streets of Gotham City.<ref>{{cite journal | url=http://lrss.fri.uni-lj.si/people/ilbajec/papers/ilb_ab09.pdf | last1=Lebar Bajec |first1= Iztok |first2= Frank H. |last2= Heppner | date=2009 | title=Organized flight in birds | journal=Animal Behaviour | volume = 78 | issue = 4 | pages = 777–789 | doi = 10.1016/j.anbehav.2009.07.007 }}</ref><br />
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At the time of proposal, Reynolds' approach represented a giant step forward compared to the traditional techniques used in computer animation for motion pictures. The first animation created with the model was Stanley and Stella in: Breaking the Ice (1987), followed by a feature film debut in Tim Burton's film Batman Returns (1992) with computer generated bat swarms and armies of penguins marching through the streets of Gotham City.<br />
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在当时提出时，雷诺兹的方法相比于传统的计算机动画电影技术是一个巨大的进步。第一部利用了此模型的动画片是《'''史丹利和史黛拉: 破冰 Stanley and Stella in: Breaking The Ice'''》（1987），之后是'''蒂姆·伯顿 Tim Burton'''的故事片《'''蝙蝠侠归来 Batman Returns'''》（1992），电脑合成的蝙蝠群和成群的企鹅行进穿过'''哥谭市 Gotham City'''的街道。<br />
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The boids model has been used for other interesting applications. It has been applied to automatically program Internet multi-channel radio stations.<ref><br />
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The boids model has been used for other interesting applications. It has been applied to automatically program Internet multi-channel radio stations.<ref><br />
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{{cite conference<br />
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{{cite conference<br />
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{引用会议<br />
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| title = DJ-boids: emergent collective behavior as multichannel radio station programming<br />
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| title = DJ-boids: emergent collective behavior as multichannel radio station programming<br />
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多频道电台节目中的紧急集体行为<br />
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| first1 = Jesús | last1 = Ibáñez<br />
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| first1 = Jesús | last1 = Ibáñez<br />
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| first1 = Jesús | last1 = Ibáñez<br />
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| first2 = Antonio F. | last2 = Gómez-Skarmeta <br />
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| first2 = Antonio F. | last2 = Gómez-Skarmeta <br />
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| first2 = Antonio F. | last2 = Gómez-Skarmeta<br />
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| first3 = Josep | last3 = Blat<br />
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| first3 = Josep | last3 = Blat<br />
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| first3 = Josep | last3 = Blat<br />
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| date = 2003<br />
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| date = 2003<br />
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2003年<br />
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| booktitle = Proceedings of the 8th international conference on Intelligent User Interfaces<br />
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| booktitle = Proceedings of the 8th international conference on Intelligent User Interfaces<br />
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第八届智能用户界面国际会议论文集<br />
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| pages = 248–250<br />
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| pages = 248–250<br />
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第248-250页<br />
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| doi = 10.1145/604045.604089<br />
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| doi = 10.1145/604045.604089<br />
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10.1145 / 604045.604089<br />
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}}</ref><br />
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}}</ref><br />
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{} / ref<br />
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It has also been used for visualizing information<ref><br />
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It has also been used for visualizing information<ref><br />
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{{cite conference<br />
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{{cite conference<br />
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{引用会议<br />
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| title = Time-Varying Data Visualization Using Information Flocking Boids<br />
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| title = Time-Varying Data Visualization Using Information Flocking Boids<br />
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使用信息植群的时变数据可视化<br />
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| last = Moere | first = A V<br />
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| last = Moere | first = A V<br />
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最后一个 Moere | 第一个 a v<br />
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| date = 2004<br />
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| date = 2004<br />
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2004年<br />
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| booktitle = Proceedings of the IEEE Symposium on Information Visualization<br />
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| booktitle = Proceedings of the IEEE Symposium on Information Visualization<br />
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美国电气和电子工程师学会信息可视化研讨会论文集<br />
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| pages = 97–104<br />
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| pages = 97–104<br />
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第97-104页<br />
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| doi = 10.1109/INFVIS.2004.65<br />
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| doi = 10.1109/INFVIS.2004.65<br />
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10.1109 / INFVIS. 2004.65<br />
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}}</ref><br />
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}}</ref><br />
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{} / ref<br />
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and for optimization tasks.<ref><br />
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and for optimization tasks.<ref><br />
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{{cite journal<br />
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{{cite journal<br />
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{引用期刊<br />
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|first1 = Zhihua | last1 = Cui<br />
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|first1 = Zhihua | last1 = Cui<br />
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最新一期崔<br />
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|first2 = Zhongzhi | last2 = Shi<br />
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|first2 = Zhongzhi | last2 = Shi<br />
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|first2 = Zhongzhi | last2 = Shi<br />
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|title= Boid particle swarm optimisation<br />
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|title= Boid particle swarm optimisation<br />
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博伊德粒子群优化<br />
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|journal= International Journal of Innovative Computing and Applications<br />
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|journal= International Journal of Innovative Computing and Applications<br />
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国际创新计算与应用杂志<br />
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|volume=2 |issue=2 |pages= 77–85<br />
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|volume=2 |issue=2 |pages= 77–85<br />
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第二期，第77-85页<br />
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|date=2009<br />
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|date=2009<br />
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2009年<br />
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|doi= 10.1504/IJICA.2009.031778}}</ref><br />
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|doi= 10.1504/IJICA.2009.031778}}</ref><br />
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| doi 10.1504 / IJICA. 2009.031778} / ref<br />
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Boids 模型已经被用于其他有趣应用。该系统已应用于互联网多频道广播电台的自动节目编排，以及可视化信息和优化任务。<br />
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==See also 参见==<br />
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*[[Swarm intelligence]] 群体智能<br />
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*[[Collective cell migration]] 集体细胞迁移<br />
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==References 参考资料==<br />
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{{Reflist}}<br />
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==External links 外部链接==<br />
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* [http://www.red3d.com/cwr/boids/ Craig Reynolds' Boids page]<br />
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* [http://www.vergenet.net/~conrad/boids/pseudocode.html Explanation of algorithm in pseudocode]<br />
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* [https://gpolo.github.com/birdflocking JavaScript implementation]<br />
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*[https://lufemas.github.io/boid-ai-implementation/ JavaScript implementation with Phaser Framework]<br />
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* [https://web.archive.org/web/20080205014305/http://www.navgen.com/3d_boids/ 3D Boids Simulation using OpenGL, used by the BBC's Natural History Unit]<br />
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*[https://black-square.github.io/BirdFlock/ Live In-Browser 3D Simulation of Bird Flocking Behavior in Unity3D] – Open Source implementation for Windows, Linux and Mac<br />
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* [https://github.com/piskvorky/PredatorPrey UNIX+Windows open source implementation in C++, using OpenGL and simulation controls]<br />
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* [https://github.com/tofti/javafx-boids A java implementation using the javafx API]<br />
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*GLSchool - part of [[XScreenSaver]] software<br />
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{{collective animal behaviour}}<br />
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==编者推荐==<br />
===[https://campus.swarma.org/ 集智学园]课程：[https://campus.swarma.org/course/1104 透过人工鸟群Boid模型学习List的使用]===<br />
<br />
讲师：[[张江]](北师大系统科学学院教授、博士生导师，集智俱乐部、集智学园创始人。)<br />
<br />
本课程通过数个案例教会大家如何去动手搭建一个多主体仿真模型，以及如何利用NetLogo去实现。从生命游戏到人工鸟群，从模拟经济系统到病毒沿网络的传播，通过循序渐进的案例，该课程带你逐步走入NetLogo多主体建模的神奇世界。<br />
<br />
===[https://swarma.org/ 集智俱乐部]扩展阅读：[https://swarma.org/?p=16504 张江：乌合之众还是群智涌现？鸟群知道答案]===<br />
<br />
生命如何起源，智能如何涌现，因果如何反转，复杂谜题，引人深思。2019年7月13日，集智俱乐部创始人[[张江]]教授在混沌大学授课，详细解读了复杂系统中的混沌、涌现与进化，此文整理自课堂内容。<br />
<br />
===[https://v.qq.com/x/page/z13491sn7fy.html 动画片《Stanley and Stella in: Breaking the Ice (1987)》]===<br />
<br />
Craig Reynolds 与 Symbolics Graphics 和 Whitney / Demos Production 的同事合作制作的利用 boids 模型的小短片，讲述了飞鸟与鱼相爱，鸟儿冲破冰面与鱼相见的故事。影片最早于87年在 SIGGRAPH 电子剧院展出。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）补充自http://www.red3d.com/cwr/boids/：In cooperation with many coworkers at the Symbolics Graphics Division and Whitney / Demos Productions, we made an animated short featuring the boids model called Stanley and Stella in: Breaking the Ice. This film was first shown at the Electronic Theater at SIGGRAPH '87.<br />
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[[Category:Artificial life]]<br />
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Category:Artificial life<br />
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类别: 人工生命<br />
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[[Category:1986 software]]<br />
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Category:1986 software<br />
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类别: 1986软件<br />
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<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Boids]]. Its edit history can be viewed at [[鸟群算法/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E5%9F%BA%E4%BA%8E%E4%B8%BB%E4%BD%93%E7%9A%84%E8%AE%A1%E7%AE%97%E7%BB%8F%E6%B5%8E%E5%AD%A6&diff=11457基于主体的计算经济学2020-08-01T12:10:02Z<p>Dorr：</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。<br />
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{{Use dmy dates|date=February 2020}}<br />
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'''Agent-based computational economics''' ('''ACE''') is the area of [[computational economics]] that studies economic processes, including whole [[economy|economies]], as [[dynamic system]]s of interacting [[Agent (economics)|agents]]. As such, it falls in the [[paradigm]] of [[complex adaptive system]]s.<ref>• [[W. Brian Arthur]], 1994. "[https://ocw.tudelft.nl/wp-content/uploads/ElFarolArtur1994.pdf Inductive Reasoning and Bounded Rationality]," ''American Economic Review'', 84(2), pp. [http://www-personal.umich.edu/~samoore/bit885f2011/arthur-inductive.pdf 406-411] {{Webarchive|url=https://web.archive.org/web/20130521145936/http://www-personal.umich.edu/~samoore/bit885f2011/arthur-inductive.pdf |date=21 May 2013 }}.<br/>&nbsp;&nbsp; • [[Leigh Tesfatsion]], 2003. "Agent-based Computational Economics: Modeling Economies as Complex Adaptive Systems," ''Information Sciences'', 149(4), pp. [http://copper.math.buffalo.edu/urgewiki/uploads/Literature/Tesfatsion2002.pdf 262-268] {{webarchive|url=https://web.archive.org/web/20120426000037/http://copper.math.buffalo.edu/urgewiki/uploads/Literature/Tesfatsion2002.pdf |date=26 April 2012 }}.</ref> In corresponding [[agent-based model]]s, the "[[agent (economics)|agents]]" are "computational objects modeled as interacting according to rules" over space and time, not real people. The rules are formulated to model behavior and social interactions based on incentives and information.<ref>Scott E. Page (2008). "agent-based models," ''[[The New Palgrave Dictionary of Economics]]'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_A000218&edition=current&q=agent-based%20computational%20modeling&topicid=&result_number=1 Abstract].</ref> Such rules could also be the result of optimization, realized through use of AI methods (such as [[Q-learning]] and other reinforcement learning techniques).<ref>Richard S. Sutton and Andrew G. Barto, Reinforcement Learning: An Introduction, The MIT Press, Cambridge, MA, 1998 [http://www.cs.ualberta.ca/~sutton/book/ebook/the-book.html] {{Webarchive|url=https://web.archive.org/web/20090904194934/http://www.cs.ualberta.ca/~sutton/book/ebook/the-book.html |date=4 September 2009 }}</ref><br />
<br />
Agent-based computational economics (ACE) is the area of computational economics that studies economic processes, including whole economies, as dynamic systems of interacting agents. As such, it falls in the paradigm of complex adaptive systems. In corresponding agent-based models, the "agents" are "computational objects modeled as interacting according to rules" over space and time, not real people. The rules are formulated to model behavior and social interactions based on incentives and information. Such rules could also be the result of optimization, realized through use of AI methods (such as Q-learning and other reinforcement learning techniques).<br />
<br />
'''基于主体的计算经济学 Agent-based Computational Economics (ACE)'''是[[计算经济学]]的一个领域，将整个经济视作不同主体相互作用的动态系统研究经济过程。因此，它属于复杂适应系统'''范式 Paradigm'''。在相应的'''智能体基模型 Agent-based Model(ABM)'''中，“主体”指时空中“根据规则建模的交互计算对象”，而非真实的人。这些规则是为了建立基于激励和信息的行为和社会互动模型而制定的。这些规则也可能是优化的结果，通过人工智能方法（如'''Q学习 Q-learning'''及其他强化学习技术）来实现。<br />
<br />
<br />
<br />
The theoretical assumption of [[mathematical optimization]] by agents in [[equilibrium (economics)|equilibrium]] is replaced by the less restrictive postulate of agents with [[bounded rationality]] ''adapting'' to market forces.<ref>• [[John H. Holland]] and John H. Miller (1991). "Artificial Adaptive Agents in Economic Theory," ''American Economic Review'', 81(2), pp. [http://www.santafe.edu/media/workingpapers/91-05-025.pdf 365-370] {{Webarchive|url=https://web.archive.org/web/20110105015853/http://www.santafe.edu/media/workingpapers/91-05-025.pdf |date=5 January 2011 }} p. 366.<br/>&nbsp;&nbsp; • [[Thomas C. Schelling]] (1978 [2006]). ''Micromotives and Macrobehavior'', Norton. [http://books.wwnorton.com/books/978-0-393-32946-9/ Description] {{Webarchive|url=https://web.archive.org/web/20171102093240/http://books.wwnorton.com/books/978-0-393-32946-9/ |date=2 November 2017 }}, [https://books.google.com/books?id=DenWKRgqzWMC&printsec=find&pg=PA1=#v=onepage&q&f=false preview].<br/>&nbsp;&nbsp; •<br />
<br />
The theoretical assumption of mathematical optimization by agents in equilibrium is replaced by the less restrictive postulate of agents with bounded rationality adapting to market forces.&nbsp;&nbsp; • Thomas C. Schelling (1978 [2006]). Micromotives and Macrobehavior, Norton. [http://books.wwnorton.com/books/978-0-393-32946-9/ Description] , [https://books.google.com/books?id=DenWKRgqzWMC&printsec=find&pg=PA1=#v=onepage&q&f=false preview].<br/>&nbsp;&nbsp; •<br />
<br />
<br />
<br />
•托马斯 · c · 谢林(Thomas c. Schelling，1978[2006])。微动机和宏行为，诺顿。[ http://books.wwnorton.com/books/978-0-393-32946-9/ 描述] ，[ https://books.google.com/books?id=denwkrgqzwmc&printsec=find&pg=pa1=#v=onepage&q&f=false 预览] . br / •<br />
<br />
[[Thomas J. Sargent]], 1994. ''Bounded Rationality in Macroeconomics'', Oxford. [http://www.oup.com/us/catalog/general/subject/Economics/MacroeconomicTheory/?view=usa&ci=9780198288695 Description] and chapter-preview 1st-page [https://www.questia.com/library/book/bounded-rationality-in-macroeconomics-thomas-j-sargent-by-thomas-j-sargent.jsp links.]</ref> ACE models apply [[numerical methods]] of analysis to [[Computer simulation|computer-based simulations]] of complex dynamic problems for which more conventional methods, such as theorem formulation, may not find ready use.<ref>• Kenneth L. Judd, 2006. "Computationally Intensive Analyses in Economics," ''Handbook of Computational Economics'', v. 2, ch. 17, Introduction, p. 883. [Pp. [https://books.google.com/books?id=6ITfRkNmKQcC&pg=PA881 881-] 893. Pre-pub [http://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF].<br />
<br />
Thomas J. Sargent, 1994. Bounded Rationality in Macroeconomics, Oxford. [http://www.oup.com/us/catalog/general/subject/Economics/MacroeconomicTheory/?view=usa&ci=9780198288695 Description] and chapter-preview 1st-page [https://www.questia.com/library/book/bounded-rationality-in-macroeconomics-thomas-j-sargent-by-thomas-j-sargent.jsp links.]</ref> ACE models apply numerical methods of analysis to computer-based simulations of complex dynamic problems for which more conventional methods, such as theorem formulation, may not find ready use.<ref>• Kenneth L. Judd, 2006. "Computationally Intensive Analyses in Economics," Handbook of Computational Economics, v. 2, ch. 17, Introduction, p. 883. [Pp. [https://books.google.com/books?id=6ITfRkNmKQcC&pg=PA881 881-] 893. Pre-pub [http://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF].<br />
<br />
托马斯·萨金特，1994年。牛津大学宏观经济学有限理性。[ http://www.oup.com/us/catalog/general/subject/economics/macroeconomictheory/?view=usa&ci=9780198288695说明]和第一章预览页面[ https://www.questia.com/library/book/bounded-rationality-in-macroeconomics-thomas-j-sargent-by-thomas-j-sargent.jsp 链接]。] / ref <br />
。 文献• Kenneth l. Judd，2006。《经济学中的计算密集分析》 ，《计算经济学手册，第2卷，第5页。17，Introduction，p. 883.[ Pp.Https://books.google.com/books?id=6itfrknmkqcc&pg=pa881881-893.预订酒吧[ http://www2.econ.iastate.edu/tesfatsi/judd.finalrev.PDF PDF ]。<br />
<br />
<br/>&nbsp;&nbsp; • _____, 1998. ''Numerical Methods in Economics'', MIT Press. Links to [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=3257 description] {{webarchive|url=https://web.archive.org/web/20120211061602/http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=3257 |date=11 February 2012 }} and [https://books.google.com/books?id=9Wxk_z9HskAC&pg=PR7 chapter previews].</ref> Starting from initial conditions specified by the modeler, the computational economy evolves over time as its constituent agents repeatedly interact with each other, including learning from interactions. In these respects, ACE has been characterized as a bottom-up culture-dish approach to the study of [[economic systems]].<ref>• Leigh Tesfatsion (2002). "Agent-Based Computational Economics: Growing Economies from the Bottom Up," ''Artificial Life'', 8(1), pp.55-82. [http://www.mitpressjournals.org/doi/abs/10.1162/106454602753694765 Abstract] and pre-pub [http://www.econ.brown.edu/fac/Peter_Howitt/SummerSchool/Agent.pdf PDF] {{webarchive|url=https://web.archive.org/web/20130514143904/http://www.econ.brown.edu/fac/Peter_Howitt/SummerSchool/Agent.pdf |date=14 May 2013 }}.<br/>&nbsp;&nbsp; • _____ (1997). "How Economists Can Get Alife," in W. B. Arthur, S. Durlauf, and D. Lane, eds., ''The Economy as an Evolving Complex System, II'', pp. 533-564. Addison-Wesley. Pre-pub [http://ageconsearch.umn.edu/bitstream/18196/1/er37.pdf PDF].</ref><br />
<br />
<br/>&nbsp;&nbsp; • _____, 1998. Numerical Methods in Economics, MIT Press. Links to [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=3257 description] and [https://books.google.com/books?id=9Wxk_z9HskAC&pg=PR7 chapter previews].</ref> Starting from initial conditions specified by the modeler, the computational economy evolves over time as its constituent agents repeatedly interact with each other, including learning from interactions. In these respects, ACE has been characterized as a bottom-up culture-dish approach to the study of economic systems.<br />
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<br/>&nbsp;&nbsp; • _____, 1998.经济学数值方法，麻省理工学院出版社。链接到[ http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=3257说明]和[ https://books.google.com/books?id=9wxk_z9hskac&pg=pr7章预览]。<br />
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'''经济均衡 Economic Equilibrium'''中主体'''最优化 Mathematical Optimization'''的理论假设被限制较少的适应市场力量的'''有限理性 Bounded Rationality'''主体假设所取代。ACE 模型将数值分析方法应用于复杂动力学问题的计算机模拟，对其更传统的方法，如定理公式可能无法方便地使用。从建模者指定的初始条件开始，计算经济随着时间的推移而发展，其组成主体不断地相互作用，包括从交互中学习。在这些方面，ACE 被认为是研究'''经济体系 Economic System'''的一种自下而上的文化方法。<br />
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ACE has a similarity to, and overlap with, [[game theory]] as an agent-based method for modeling social interactions.<ref name="COMP&GT">• [[Joseph Y. Halpern]] (2008). "computer science and game theory," ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_C000566&edition=current&q=&topicid=&result_number=1 Abstract].<br/>&nbsp;&nbsp; • Yoav Shoham (2008). "Computer Science and Game Theory," ''Communications of the ACM'', 51(8), pp.<br />
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ACE has a similarity to, and overlap with, game theory as an agent-based method for modeling social interactions.&nbsp;&nbsp; • Yoav Shoham (2008). "Computer Science and Game Theory," Communications of the ACM, 51(8), pp.<br />
<br />
• Yoav Shoham (2008)。《计算机科学与博弈论》 ，美国计算机学会通讯，51(8)页。<br />
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[http://www.robotics.stanford.edu/~shoham/www%20papers/CSGT-CACM-Shoham.pdf 75-79] {{Webarchive|url=https://web.archive.org/web/20120426005917/http://www.robotics.stanford.edu/~shoham/www%20papers/CSGT-CACM-Shoham.pdf |date=26 April 2012 }}.<br/>&nbsp;&nbsp; • [[Alvin E. Roth]] (2002). "The Economist as Engineer: Game Theory, Experimentation, and Computation as Tools for Design Economics," ''Econometrica'', 70(4), pp. [https://web.archive.org/web/20040414102216/http://kuznets.fas.harvard.edu/~aroth/papers/engineer.pdf 1341–1378].</ref> But practitioners have also noted differences from standard methods, for example in ACE events modeled being driven solely by initial conditions, whether or not equilibria exist or are computationally tractable, and in the modeling facilitation of agent autonomy and learning.<ref>Tesfatsion, Leigh (2006), "Agent-Based Computational Economics: A Constructive Approach to Economic Theory," ch. 16, ''Handbook of Computational Economics'', v. 2, part 2, ACE study of economic system. [http://www.sciencedirect.com/science/article/pii/S1574002105020162 Abstract] and pre-pub [http://econ2.econ.iastate.edu/tesfatsi/hbintlt.pdf PDF].</ref><br />
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[http://www.robotics.stanford.edu/~shoham/www%20papers/CSGT-CACM-Shoham.pdf 75-79] .<br/>&nbsp;&nbsp; • Alvin E. Roth (2002). "The Economist as Engineer: Game Theory, Experimentation, and Computation as Tools for Design Economics," Econometrica, 70(4), pp. [https://web.archive.org/web/20040414102216/http://kuznets.fas.harvard.edu/~aroth/papers/engineer.pdf 1341–1378].</ref> But practitioners have also noted differences from standard methods, for example in ACE events modeled being driven solely by initial conditions, whether or not equilibria exist or are computationally tractable, and in the modeling facilitation of agent autonomy and learning.<br />
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[http://www.robotics.stanford.edu/~shoham/www%20papers/CSGT-CACM-Shoham.pdf 75-79] .<br/>&nbsp;&nbsp; • Alvin E. Roth (2002).《作为工程师的经济学家: 博弈论、实验和计算作为设计经济学的工具》 ，《经济学》 ，70(4) ，页。[ https://web.archive.org/web/20040414102216/http://kuznets.fas.harvard.edu/~aroth/papers/engineer.pdf 1341-1378]. <br />
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ACE 与博弈论（作为基于主体的社会互动建模方法时）有相似之处和重叠部分。但实践者也注意到了与标准方法的区别，例如，在仅由初始条件驱动的ACE事件建模中，均衡是否存在或者在计算上是否易于处理，以及在建模过程中促进主体自主性和学习方面。<br />
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The method has benefited from continuing improvements in modeling techniques of [[computer science]] and increased computer capabilities. The ultimate scientific objective of the method is to "test theoretical findings against real-world data in ways that permit empirically supported theories to cumulate over time, with each researcher’s work building appropriately on the work that has gone before."<ref>• Leigh Tesfatsion (2006). "Agent-Based Computational Economics: A Constructive Approach to Economic Theory," ch. 16, ''Handbook of Computational Economics'', v. 2, [pp. 831-880] sect. 5. [http://www.sciencedirect.com/science/article/pii/S1574002105020162 Abstract] and pre-pub [http://econ2.econ.iastate.edu/tesfatsi/hbintlt.pdf PDF].<br/>&nbsp;&nbsp; • [[Kenneth L. Judd]] (2006). "Computationally Intensive Analyses in Economics," ''Handbook of Computational Economics'', v. 2, ch. 17, pp. [https://books.google.com/books?id=6ITfRkNmKQcC&pg=PA881 881-] 893. Pre-pub [http://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF].<br/>&nbsp;&nbsp; • Leigh Tesfatsion and Kenneth L. Judd, ed. (2006). ''Handbook of Computational Economics'', v. 2. [http://www.elsevier.com/wps/find/bookdescription.cws_home/660847/description#description Description] {{Webarchive|url=https://web.archive.org/web/20120306100156/http://www.elsevier.com/wps/find/bookdescription.cws_home/660847/description#description |date=6 March 2012 }} & and chapter-preview<br />
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The method has benefited from continuing improvements in modeling techniques of computer science and increased computer capabilities. The ultimate scientific objective of the method is to "test theoretical findings against real-world data in ways that permit empirically supported theories to cumulate over time, with each researcher’s work building appropriately on the work that has gone before."&nbsp;&nbsp; • Kenneth L. Judd (2006). "Computationally Intensive Analyses in Economics," Handbook of Computational Economics, v. 2, ch. 17, pp. [https://books.google.com/books?id=6ITfRkNmKQcC&pg=PA881 881-] 893. Pre-pub [http://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF].<br/>&nbsp;&nbsp; • Leigh Tesfatsion and Kenneth L. Judd, ed. (2006). Handbook of Computational Economics, v. 2. [http://www.elsevier.com/wps/find/bookdescription.cws_home/660847/description#description Description] & and chapter-preview<br />
<br />
• Kenneth l. Judd (2006)。《经济学中的计算密集分析》 ，计算经济学手册，第2卷，第5页。17页。Https://books.google.com/books?id=6itfrknmkqcc&pg=pa881881-893.Pre-pub [http://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF].<br/>&nbsp;&nbsp; • Leigh Tesfatsion and Kenneth L. Judd, ed.(2006).计算经济学手册第2节。[ http://www.elsevier.com/wps/find/bookdescription.cws_home/660847/Description#Description 说明]和章节预览<br />
<br />
[http://www.sciencedirect.com/science?_ob=PublicationURL&_hubEid=1-s2.0-S1574002105X02003&_cid=273377&_pubType=HS&_auth=y&_acct=C000228598&_version=1&_urlVersion=0&_userid=10&md5=e4757b4f65755ed6340a11fee9615200 links.]</ref> The subject has been applied to research areas like [[asset pricing]],<ref name=arthuretal>B. Arthur, J. Holland, B. LeBaron, R. Palmer, P. Taylor (1997), 'Asset pricing under endogenous expectations in an artificial stock market,' in ''The Economy as an Evolving Complex System II'', B. Arthur, S. Durlauf, and D. Lane, eds., Addison Wesley.</ref> [[competition]] and [[collaboration]],<ref>[[Robert Axelrod]] (1997). ''The Complexity of Cooperation: Agent-Based Models of Competition and Collaboration'', Princeton. [http://press.princeton.edu/titles/6144.html Description], [http://press.princeton.edu/titles/6144.html#TOC contents], and [https://books.google.com/books?id=J0dgRGMdjmQC&printsec=find&pg=PR11#v=onepage&q&f=false preview].</ref> [[transaction cost]]s,<ref>Tomas B. Klosa and [[Bart Nooteboom]], 2001. "Agent-based Computational Transaction Cost Economics," ''Journal of Economic Dynamics and Control'' 25(3–4), pp. 503–52. [http://www.sciencedirect.com/science/article/pii/S0165188900000348 Abstract.]</ref> [[market structure]] and [[industrial organization]] and dynamics,<ref>• Roberto Leombruni and Matteo Richiardi, ed. (2004), ''Industry and Labor Dynamics: The Agent-Based Computational Economics Approach.'' World Scientific Publishing {{ISBN|981-256-100-5}}. [http://www.worldscibooks.com/economics/5706.html Description] {{Webarchive|url=https://web.archive.org/web/20100727221149/http://www.worldscibooks.com/economics/5706.html |date=27 July 2010 }} and chapter-preview [https://books.google.com/books?id=P5O7A5D55nQC&printsec=fond&pg=PR5#v=onepage&q&f=false links].<br/>&nbsp;&nbsp; • [[Joshua M. Epstein]] (2006). "Growing Adaptive Organizations: An Agent-Based Computational Approach," in ''Generative Social Science: Studies in Agent-Based Computational Modeling'', pp. [https://books.google.com/books?id=543OS3qdxBYC&pg=PA309 309-] 344. [http://press.princeton.edu/titles/8277.html Description] {{Webarchive|url=https://web.archive.org/web/20120126180655/http://press.princeton.edu/titles/8277.html |date=26 January 2012 }} and [http://www.santafe.edu/research/working-papers/abstract/99895b6465e8b87656612f8e3570b34c/ abstract].</ref> [[welfare economics]],<ref>[[Robert Axtell]] (2005). "The Complexity of Exchange," ''Economic Journal'', 115(504, Features), pp. [http://econfaculty.gmu.edu/pboettke/workshop/archives/f05/Axtell.pdf F193-F210].</ref> and [[mechanism design]],<ref>• ''The New Palgrave Dictionary of Economics'' (2008), 2nd Edition: <br/>&nbsp;&nbsp;&nbsp;&nbsp; [[Roger B. Myerson]] "mechanism design." [http://www.dictionaryofeconomics.com/article?id=pde2008_M000132&edition=current&q=mechanism%20design&topicid=&result_number=3 Abstract.] <br/>&nbsp;&nbsp;&nbsp;&nbsp; _____. "revelation principle." [http://www.dictionaryofeconomics.com/article?id=pde2008_R000137&edition=current&q=moral&topicid=&result_number=1 Abstract.]<br/>&nbsp;&nbsp;&nbsp;&nbsp; Tuomas Sandholm. "computing in mechanism design." [http://www.dictionaryofeconomics.com/article?id=pde2008_C000563&edition=&field=keyword&q=algorithmic%20mechanism%20design&topicid=&result_number=1 Abstract.]<br/>&nbsp;&nbsp; • [[Noam Nisan]] and Amir Ronen (2001). "Algorithmic Mechanism Design," ''Games and Economic Behavior'', 35(1-2), pp. [https://www.cs.cmu.edu/~sandholm/cs15-892F09/Algorithmic%20mechanism%20design.pdf 166–196].<br/>&nbsp;&nbsp; • [[Noam Nisan]] ''et al''., ed. (2007). ''Algorithmic Game Theory'', Cambridge University Press. [http://www.cup.cam.ac.uk/asia/catalogue/catalogue.asp?isbn=9780521872829 Description] {{Webarchive|url=https://web.archive.org/web/20120505140924/http://www.cup.cam.ac.uk/asia/catalogue/catalogue.asp?isbn=9780521872829 |date=5 May 2012 }}.</ref> [[Information economics|information and uncertainty]],<ref>Tuomas W. Sandholm and Victor R. Lesser (2001). "Leveled Commitment Contracts and Strategic Breach," ''Games and Economic Behavior'', 35(1-2), pp. [https://www.cs.cmu.edu/afs/.cs.cmu.edu/Web/People/sandholm/leveled.geb.pdf 212-270].</ref> [[macroeconomics]],<ref>• [[David Colander]], [[Peter Howitt]], Alan Kirman, [[Axel Leijonhufvud]], and [[Perry Mehrling]], 2008. "Beyond DSGE Models: Toward an Empirically Based Macroeconomics," ''American Economic Review'', 98(2), pp. [https://www.jstor.org/pss/29730026 236]-240. Pre-pub [http://www.brown.edu/Departments/Economics/Faculty/Peter_Howitt/publication/complex%20macro6.pdf PDF].<br/>&nbsp;&nbsp; • [[Thomas J. Sargent]] (1994). ''Bounded Rationality in Macroeconomics'', Oxford. [http://www.oup.com/us/catalog/general/subject/Economics/MacroeconomicTheory/?view=usa&ci=9780198288695 Description] and chapter-preview 1st-page [https://www.questia.com/library/book/bounded-rationality-in-macroeconomics-thomas-j-sargent-by-thomas-j-sargent.jsp links].<br/>&nbsp;&nbsp; • M. Oeffner (2009). '[http://www.opus-bayern.de/uni-wuerzburg/volltexte/2009/3927/pdf/OeffnerDissohneAnhang.pdf Agent-based Keynesian Macroeconomics]'. PhD thesis, Faculty of Economics, University of Würzburg.</ref> and [[Marxist economics]].<ref>A. F. Cottrell, P. Cockshott, G. J. Michaelson, I. P. Wright, V. Yakovenko<br />
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[http://www.sciencedirect.com/science?_ob=PublicationURL&_hubEid=1-s2.0-S1574002105X02003&_cid=273377&_pubType=HS&_auth=y&_acct=C000228598&_version=1&_urlVersion=0&_userid=10&md5=e4757b4f65755ed6340a11fee9615200 links.]</ref> The subject has been applied to research areas like asset pricing, competition and collaboration, transaction costs, market structure and industrial organization and dynamics, welfare economics, and mechanism design, information and uncertainty, macroeconomics, and Marxist economics.<ref>A. F. Cottrell, P. Cockshott, G. J. Michaelson, I. P. Wright, V. Yakovenko<br />
<br />
[ http://www.sciencedirect.com/science?_ob=publicationurl&_hubeid=1-s2.0-s1574002105x02003&_cid=273377&_pubtype=hs&_auth=y&_acct=c000228598&_version=1&_urlversion=0&_userid=10&md5=e4757b4f65755ed6340a11fee9615200连结。参考文献<br />
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参考文献 a。科特雷尔，p. Cockshott，g. j. Michaelson，i. p. Wright，v. Yakovenko<br />
<br />
(2009), ''Classical Econophysics''. Routledge, {{ISBN|978-0-415-47848-9}}.</ref><ref>Leigh Tesfatsion (2006), "Agent-Based Computational Economics: A Constructive Approach to Economic Theory," ch. 16, ''Handbook of Computational Economics'', v. 2, part 2, ACE study of economic system. [http://www.sciencedirect.com/science/article/pii/S1574002105020162 Abstract] and pre-pub [http://econ2.econ.iastate.edu/tesfatsi/hbintlt.pdf PDF].</ref><br />
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(2009), Classical Econophysics. Routledge, .</ref><br />
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(2009) ，古典经济物理学。路特利奇，参考文献<br />
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该方法得益于计算机科学建模技术的不断改进和计算机能力的提高。其最终科学目标是“用实际数据来检验理论发现——以允许经验支持的理论随着时间的推移而累积的方式（每位研究者的工作恰当建立在之前工作的基础之上）。”其现已应用于如资产定价、竞争与合作、'''交易成本 Transaction Cost'''、市场形式与'''产业组织 Industrial Organization'''与动态、'''福利经济学 Welfare Economics'''、'''机制设计 Mechanism Design'''、信息与不确定性、'''宏观经济学 Macroeconomics'''和'''马克思主义政治经济学 Marxist economics'''等研究领域。<br />
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==Overview 概述==<br />
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The "[[Agent (economics)|agents]]" in ACE models can represent individuals (e.g. people), social groupings (e.g. firms), biological entities (e.g. growing crops), and/or physical systems (e.g. transport systems). The ACE modeler provides the initial configuration of a computational economic system comprising multiple interacting agents. The modeler then steps back to observe the development of the system over time without further intervention. In particular, system events should be driven by agent interactions without external imposition of equilibrium conditions.<ref>[http://www.socsci.aau.dk/ae2006/ Summary of methods] {{Webarchive|url=https://web.archive.org/web/20070526105550/http://www.socsci.aau.dk/ae2006/ |date=26 May 2007 }}: ''Department of Economics, Politics and Public Administration, Aalborg University, Denmark'' website.</ref> Issues include those common to [[experimental economics]] in general<ref>[[Vernon L. Smith]], 2008. "experimental economics," ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_E000277&q=experimental%20&topicid=&result_number=2 Abstract].</ref> and development of a common framework for empirical validation and resolving open questions in agent-based modeling.<ref>Giorgio Fagiolo, Alessio Moneta, and Paul Windrum, 2007. "A Critical Guide to Empirical Validation of Agent-Based Models in Economics: Methodologies, Procedures, and Open Problems," ''Computational Economics'', 30, pp. [https://doi.org/10.1007%2Fs10614-007-9104-4 195]–226.</ref><br />
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The "agents" in ACE models can represent individuals (e.g. people), social groupings (e.g. firms), biological entities (e.g. growing crops), and/or physical systems (e.g. transport systems). The ACE modeler provides the initial configuration of a computational economic system comprising multiple interacting agents. The modeler then steps back to observe the development of the system over time without further intervention. In particular, system events should be driven by agent interactions without external imposition of equilibrium conditions. Issues include those common to experimental economics in general and development of a common framework for empirical validation and resolving open questions in agent-based modeling.<br />
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ACE 模型中的“主体”可以代表个体（如人）、社会群体（如公司）、生物实体（如生长中的庄稼）和/或物理系统（如运输系统）。ACE 模型提供了由多个相互作用的主体组成的计算经济系统的初始配置。然后，建模人员返回来观察系统随着时间的推移而发展的情况，而不需要进一步的干预。注意，系统活动应由主体间的相互作用驱动，而不需要外部强加的平衡条件。研究问题包括'''实验经济学 Experimental economics'''通用的问题，以及开发用于经验验证的通用框架和解决基于主体的建模中尚未解决的问题。<br />
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ACE is an officially designated special interest group (SIG) of the Society for Computational Economics.<ref>[http://comp-econ.org/ Society for Computational Economics] website.</ref> Researchers at the [[Santa Fe Institute]] have contributed to the development of ACE.<br />
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ACE is an officially designated special interest group (SIG) of the Society for Computational Economics. Researchers at the Santa Fe Institute have contributed to the development of ACE.<br />
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ACE 是美国计算经济学学会的一个官方指定的特殊利益集团 (SIG) 。[[圣菲研究所 Santa Fe Institute]]的研究人员为 ACE 的发展做出了贡献。<br />
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==Example: finance 举例：金融==<br />
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One area where ACE methodology has frequently been applied is asset pricing. [[W. Brian Arthur]], Eric Baum, [[William A. Brock (economist)|William Brock]], Cars Hommes, and Blake LeBaron, among others, have developed computational models in which many agents choose from a set of possible forecasting strategies in order to predict stock prices, which affects their asset demands and thus affects stock prices. These models assume that agents are more likely to choose forecasting strategies which have recently been successful. The success of any strategy will depend on market conditions and also on the set of strategies that are currently being used. These models frequently find that large booms and busts in asset prices may occur as agents switch across forecasting strategies.<ref name=arthuretal/><ref>W. Brock and C. Hommes (1997), 'A rational route to randomness.' ''Econometrica'' 65 (5), pp. 1059-1095.</ref><ref>C. Hommes (2008), 'Interacting agents in finance,' in ''The New Palgrave Dictionary of Economics''.</ref> More recently, Brock, Hommes, and Wagener (2009) have used a model of this type to argue that the introduction of new hedging instruments may destabilize the market,<ref>{{cite journal |first=W. |last=Brock |first2=C. |last2=Hommes |first3=F. |last3=Wagener |year=2009 |title=More hedging instruments may destabilize markets |journal=Journal of Economic Dynamics and Control |volume=33 |issue=11 |pages=1912–1928 |doi=10.1016/j.jedc.2009.05.004 |url=http://cendef.uva.nl/binaries/content/assets/subsites/amsterdam-school-of-economics/amsterdam-school-of-economics-research-institute/cendef/working-papers-2006/brohomwag.pdf?1417181713282 }}</ref> and some papers have suggested that ACE might be a useful methodology for understanding the recent [[financial crisis]].<ref>M. Buchanan (2009), '[http://pagesperso-orange.fr/mark.buchanan/nature_economic_modelling.pdf Meltdown modelling. Could agent-based computer models prevent another financial crisis?].' Nature, Vol. 460, No. 7256. (5 August 2009), pp. 680-682.</ref><ref>J.D. Farmer, D. Foley (2009), 'The economy needs agent-based modelling.' Nature, Vol. 460, No. 7256. (5 August 2009), pp. 685-686.</ref><ref>M. Holcombe, S. Coakley, M.Kiran, S. Chin, C. Greenough, D.Worth, S.Cincotti, M.Raberto, A. Teglio, C. Deissenberg, S. van der Hoog, H. Dawid, S. Gemkow, P. Harting, M. Neugart. Large-scale Modeling of Economic Systems, Complex Systems, 22(2), 175-191, 2013</ref><br />
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One area where ACE methodology has frequently been applied is asset pricing. W. Brian Arthur, Eric Baum, William Brock, Cars Hommes, and Blake LeBaron, among others, have developed computational models in which many agents choose from a set of possible forecasting strategies in order to predict stock prices, which affects their asset demands and thus affects stock prices. These models assume that agents are more likely to choose forecasting strategies which have recently been successful. The success of any strategy will depend on market conditions and also on the set of strategies that are currently being used. These models frequently find that large booms and busts in asset prices may occur as agents switch across forecasting strategies. More recently, Brock, Hommes, and Wagener (2009) have used a model of this type to argue that the introduction of new hedging instruments may destabilize the market, and some papers have suggested that ACE might be a useful methodology for understanding the recent financial crisis.<br />
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ACE 方法已广泛应用的领域之一是资产定价。'''布莱恩·亚瑟 W. Brian Arthur'''，'''埃里克·鲍姆 Eric Baum'''，'''威廉·布洛克 William Brock'''，Cars Hommes，'''布莱克·勒巴朗 Blake LeBaron'''等人已开发出一个计算模型，其中许多主体从一组可能的预测策略中选择以预测股票价格，这影响了他们的资产需求，进而影响股票价格。这些模型假设主体更有可能选择最近成功的预测策略。任何策略的成功取决于市场条件，也取决于目前正在使用的一组策略。这些模型经常表明，随着主体改换预测策略，资产价格可能会出现大起大落。最近，布洛克、Hommes 和'''瓦格纳 Wagener''' （2009）使用了这类模型来论证引入新的对冲工具可能会破坏市场稳定，一些论文提出，ACE 可能是理解最近金融危机的有用方法。<br />
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==See also 参见==<br />
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* [[ACEGES]]<br />
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* [[Agent-based social simulation]]<br />
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* [[Artificial economics]]<br />
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* [[Computational economics]]<br />
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* [[Econophysics]]<br />
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* [[Macroeconomic model]]<br />
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* [[Multi-agent system]]<br />
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* [[Statistical finance]]<br />
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==References 参考资料==<br />
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{{Reflist}}<br />
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==编者推荐==<br />
[https://campus.swarma.org/ 集智学园]课程：[https://campus.swarma.org/course/714 计算社会经济学的内容与方法]<br />
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本课程系统性地介绍计算社会经济学的主要研究内容和分析方法，阐述大规模社会经济数据（包括卫星遥感数据、手机通讯数据、社交媒体数据等）在解决全球贫困问题、刻画区域经济结构、推断个体财富和应急抢险救灾中的具体应用。<br />
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Category:Monte Carlo methods in finance<br />
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类别: 金融中的蒙特卡罗方法<br />
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[[Category:Computational fields of study]]<br />
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Category:Computational fields of study<br />
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类别: 研究的计算领域<br />
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<small>This page was moved from [[wikipedia:en:Agent-based computational economics]]. Its edit history can be viewed at [[基于主体的计算经济学/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E5%9F%BA%E4%BA%8E%E4%B8%BB%E4%BD%93%E7%9A%84%E8%AE%A1%E7%AE%97%E7%BB%8F%E6%B5%8E%E5%AD%A6&diff=11456基于主体的计算经济学2020-08-01T11:40:16Z<p>Dorr：</p>
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'''Agent-based computational economics''' ('''ACE''') is the area of [[computational economics]] that studies economic processes, including whole [[economy|economies]], as [[dynamic system]]s of interacting [[Agent (economics)|agents]]. As such, it falls in the [[paradigm]] of [[complex adaptive system]]s.<ref>• [[W. Brian Arthur]], 1994. "[https://ocw.tudelft.nl/wp-content/uploads/ElFarolArtur1994.pdf Inductive Reasoning and Bounded Rationality]," ''American Economic Review'', 84(2), pp. [http://www-personal.umich.edu/~samoore/bit885f2011/arthur-inductive.pdf 406-411] {{Webarchive|url=https://web.archive.org/web/20130521145936/http://www-personal.umich.edu/~samoore/bit885f2011/arthur-inductive.pdf |date=21 May 2013 }}.<br/>&nbsp;&nbsp; • [[Leigh Tesfatsion]], 2003. "Agent-based Computational Economics: Modeling Economies as Complex Adaptive Systems," ''Information Sciences'', 149(4), pp. [http://copper.math.buffalo.edu/urgewiki/uploads/Literature/Tesfatsion2002.pdf 262-268] {{webarchive|url=https://web.archive.org/web/20120426000037/http://copper.math.buffalo.edu/urgewiki/uploads/Literature/Tesfatsion2002.pdf |date=26 April 2012 }}.</ref> In corresponding [[agent-based model]]s, the "[[agent (economics)|agents]]" are "computational objects modeled as interacting according to rules" over space and time, not real people. The rules are formulated to model behavior and social interactions based on incentives and information.<ref>Scott E. Page (2008). "agent-based models," ''[[The New Palgrave Dictionary of Economics]]'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_A000218&edition=current&q=agent-based%20computational%20modeling&topicid=&result_number=1 Abstract].</ref> Such rules could also be the result of optimization, realized through use of AI methods (such as [[Q-learning]] and other reinforcement learning techniques).<ref>Richard S. Sutton and Andrew G. Barto, Reinforcement Learning: An Introduction, The MIT Press, Cambridge, MA, 1998 [http://www.cs.ualberta.ca/~sutton/book/ebook/the-book.html] {{Webarchive|url=https://web.archive.org/web/20090904194934/http://www.cs.ualberta.ca/~sutton/book/ebook/the-book.html |date=4 September 2009 }}</ref><br />
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Agent-based computational economics (ACE) is the area of computational economics that studies economic processes, including whole economies, as dynamic systems of interacting agents. As such, it falls in the paradigm of complex adaptive systems. In corresponding agent-based models, the "agents" are "computational objects modeled as interacting according to rules" over space and time, not real people. The rules are formulated to model behavior and social interactions based on incentives and information. Such rules could also be the result of optimization, realized through use of AI methods (such as Q-learning and other reinforcement learning techniques).<br />
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'''基于主体的计算经济学 Agent-based Computational Economics (ACE)'''是[[计算经济学]]的一个领域，将整个经济视作不同主体相互作用的动态系统研究经济过程。因此，它属于复杂适应系统'''范式 Paradigm'''。在相应的'''智能体基模型 Agent-based Model(ABM)'''中，“主体”指时空中“根据规则建模的交互计算对象”，而非真实的人。这些规则是为了建立基于激励和信息的行为和社会互动模型而制定的。这些规则也可能是优化的结果，通过人工智能方法(如'''Q学习 Q-learning'''及其他强化学习技术)来实现。<br />
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<br />
The theoretical assumption of [[mathematical optimization]] by agents in [[equilibrium (economics)|equilibrium]] is replaced by the less restrictive postulate of agents with [[bounded rationality]] ''adapting'' to market forces.<ref>• [[John H. Holland]] and John H. Miller (1991). "Artificial Adaptive Agents in Economic Theory," ''American Economic Review'', 81(2), pp. [http://www.santafe.edu/media/workingpapers/91-05-025.pdf 365-370] {{Webarchive|url=https://web.archive.org/web/20110105015853/http://www.santafe.edu/media/workingpapers/91-05-025.pdf |date=5 January 2011 }} p. 366.<br/>&nbsp;&nbsp; • [[Thomas C. Schelling]] (1978 [2006]). ''Micromotives and Macrobehavior'', Norton. [http://books.wwnorton.com/books/978-0-393-32946-9/ Description] {{Webarchive|url=https://web.archive.org/web/20171102093240/http://books.wwnorton.com/books/978-0-393-32946-9/ |date=2 November 2017 }}, [https://books.google.com/books?id=DenWKRgqzWMC&printsec=find&pg=PA1=#v=onepage&q&f=false preview].<br/>&nbsp;&nbsp; •<br />
<br />
The theoretical assumption of mathematical optimization by agents in equilibrium is replaced by the less restrictive postulate of agents with bounded rationality adapting to market forces.&nbsp;&nbsp; • Thomas C. Schelling (1978 [2006]). Micromotives and Macrobehavior, Norton. [http://books.wwnorton.com/books/978-0-393-32946-9/ Description] , [https://books.google.com/books?id=DenWKRgqzWMC&printsec=find&pg=PA1=#v=onepage&q&f=false preview].<br/>&nbsp;&nbsp; •<br />
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<br />
•托马斯 · c · 谢林(Thomas c. Schelling，1978[2006])。微动机和宏行为，诺顿。[ http://books.wwnorton.com/books/978-0-393-32946-9/ 描述] ，[ https://books.google.com/books?id=denwkrgqzwmc&printsec=find&pg=pa1=#v=onepage&q&f=false 预览] . br / •<br />
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[[Thomas J. Sargent]], 1994. ''Bounded Rationality in Macroeconomics'', Oxford. [http://www.oup.com/us/catalog/general/subject/Economics/MacroeconomicTheory/?view=usa&ci=9780198288695 Description] and chapter-preview 1st-page [https://www.questia.com/library/book/bounded-rationality-in-macroeconomics-thomas-j-sargent-by-thomas-j-sargent.jsp links.]</ref> ACE models apply [[numerical methods]] of analysis to [[Computer simulation|computer-based simulations]] of complex dynamic problems for which more conventional methods, such as theorem formulation, may not find ready use.<ref>• Kenneth L. Judd, 2006. "Computationally Intensive Analyses in Economics," ''Handbook of Computational Economics'', v. 2, ch. 17, Introduction, p. 883. [Pp. [https://books.google.com/books?id=6ITfRkNmKQcC&pg=PA881 881-] 893. Pre-pub [http://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF].<br />
<br />
Thomas J. Sargent, 1994. Bounded Rationality in Macroeconomics, Oxford. [http://www.oup.com/us/catalog/general/subject/Economics/MacroeconomicTheory/?view=usa&ci=9780198288695 Description] and chapter-preview 1st-page [https://www.questia.com/library/book/bounded-rationality-in-macroeconomics-thomas-j-sargent-by-thomas-j-sargent.jsp links.]</ref> ACE models apply numerical methods of analysis to computer-based simulations of complex dynamic problems for which more conventional methods, such as theorem formulation, may not find ready use.<ref>• Kenneth L. Judd, 2006. "Computationally Intensive Analyses in Economics," Handbook of Computational Economics, v. 2, ch. 17, Introduction, p. 883. [Pp. [https://books.google.com/books?id=6ITfRkNmKQcC&pg=PA881 881-] 893. Pre-pub [http://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF].<br />
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托马斯·萨金特，1994年。牛津大学宏观经济学有限理性。[ http://www.oup.com/us/catalog/general/subject/economics/macroeconomictheory/?view=usa&ci=9780198288695说明]和第一章预览页面[ https://www.questia.com/library/book/bounded-rationality-in-macroeconomics-thomas-j-sargent-by-thomas-j-sargent.jsp 链接]。] / ref <br />
。 文献• Kenneth l. Judd，2006。《经济学中的计算密集分析》 ，《计算经济学手册，第2卷，第5页。17，Introduction，p. 883.[ Pp.Https://books.google.com/books?id=6itfrknmkqcc&pg=pa881881-893.预订酒吧[ http://www2.econ.iastate.edu/tesfatsi/judd.finalrev.PDF PDF ]。<br />
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<br/>&nbsp;&nbsp; • _____, 1998. ''Numerical Methods in Economics'', MIT Press. Links to [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=3257 description] {{webarchive|url=https://web.archive.org/web/20120211061602/http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=3257 |date=11 February 2012 }} and [https://books.google.com/books?id=9Wxk_z9HskAC&pg=PR7 chapter previews].</ref> Starting from initial conditions specified by the modeler, the computational economy evolves over time as its constituent agents repeatedly interact with each other, including learning from interactions. In these respects, ACE has been characterized as a bottom-up culture-dish approach to the study of [[economic systems]].<ref>• Leigh Tesfatsion (2002). "Agent-Based Computational Economics: Growing Economies from the Bottom Up," ''Artificial Life'', 8(1), pp.55-82. [http://www.mitpressjournals.org/doi/abs/10.1162/106454602753694765 Abstract] and pre-pub [http://www.econ.brown.edu/fac/Peter_Howitt/SummerSchool/Agent.pdf PDF] {{webarchive|url=https://web.archive.org/web/20130514143904/http://www.econ.brown.edu/fac/Peter_Howitt/SummerSchool/Agent.pdf |date=14 May 2013 }}.<br/>&nbsp;&nbsp; • _____ (1997). "How Economists Can Get Alife," in W. B. Arthur, S. Durlauf, and D. Lane, eds., ''The Economy as an Evolving Complex System, II'', pp. 533-564. Addison-Wesley. Pre-pub [http://ageconsearch.umn.edu/bitstream/18196/1/er37.pdf PDF].</ref><br />
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<br/>&nbsp;&nbsp; • _____, 1998. Numerical Methods in Economics, MIT Press. Links to [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=3257 description] and [https://books.google.com/books?id=9Wxk_z9HskAC&pg=PR7 chapter previews].</ref> Starting from initial conditions specified by the modeler, the computational economy evolves over time as its constituent agents repeatedly interact with each other, including learning from interactions. In these respects, ACE has been characterized as a bottom-up culture-dish approach to the study of economic systems.<br />
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<br/>&nbsp;&nbsp; • _____, 1998.经济学数值方法，麻省理工学院出版社。链接到[ http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=3257说明]和[ https://books.google.com/books?id=9wxk_z9hskac&pg=pr7章预览]。 '''经济均衡 Economic Equilibrium'''中主体'''最优化 Mathematical Optimization'''的理论假设被限制较少的适应市场力量的'''有限理性 Bounded Rationality'''主体假设所取代。ACE 模型将数值分析方法应用于复杂动力学问题的计算机模拟，对其更传统的方法，如定理公式可能无法方便地使用。从建模者指定的初始条件开始，计算经济随着时间的推移而发展，其组成主体不断地相互作用，包括从交互中学习。在这些方面，ACE 被认为是研究'''经济体系 Economic System'''的一种自下而上的文化方法。<br />
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ACE has a similarity to, and overlap with, [[game theory]] as an agent-based method for modeling social interactions.<ref name="COMP&GT">• [[Joseph Y. Halpern]] (2008). "computer science and game theory," ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_C000566&edition=current&q=&topicid=&result_number=1 Abstract].<br/>&nbsp;&nbsp; • Yoav Shoham (2008). "Computer Science and Game Theory," ''Communications of the ACM'', 51(8), pp.<br />
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ACE has a similarity to, and overlap with, game theory as an agent-based method for modeling social interactions.&nbsp;&nbsp; • Yoav Shoham (2008). "Computer Science and Game Theory," Communications of the ACM, 51(8), pp.<br />
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• Yoav Shoham (2008)。《计算机科学与博弈论》 ，美国计算机学会通讯，51(8)页。<br />
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[http://www.robotics.stanford.edu/~shoham/www%20papers/CSGT-CACM-Shoham.pdf 75-79] {{Webarchive|url=https://web.archive.org/web/20120426005917/http://www.robotics.stanford.edu/~shoham/www%20papers/CSGT-CACM-Shoham.pdf |date=26 April 2012 }}.<br/>&nbsp;&nbsp; • [[Alvin E. Roth]] (2002). "The Economist as Engineer: Game Theory, Experimentation, and Computation as Tools for Design Economics," ''Econometrica'', 70(4), pp. [https://web.archive.org/web/20040414102216/http://kuznets.fas.harvard.edu/~aroth/papers/engineer.pdf 1341–1378].</ref> But practitioners have also noted differences from standard methods, for example in ACE events modeled being driven solely by initial conditions, whether or not equilibria exist or are computationally tractable, and in the modeling facilitation of agent autonomy and learning.<ref>Tesfatsion, Leigh (2006), "Agent-Based Computational Economics: A Constructive Approach to Economic Theory," ch. 16, ''Handbook of Computational Economics'', v. 2, part 2, ACE study of economic system. [http://www.sciencedirect.com/science/article/pii/S1574002105020162 Abstract] and pre-pub [http://econ2.econ.iastate.edu/tesfatsi/hbintlt.pdf PDF].</ref><br />
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[http://www.robotics.stanford.edu/~shoham/www%20papers/CSGT-CACM-Shoham.pdf 75-79] .<br/>&nbsp;&nbsp; • Alvin E. Roth (2002). "The Economist as Engineer: Game Theory, Experimentation, and Computation as Tools for Design Economics," Econometrica, 70(4), pp. [https://web.archive.org/web/20040414102216/http://kuznets.fas.harvard.edu/~aroth/papers/engineer.pdf 1341–1378].</ref> But practitioners have also noted differences from standard methods, for example in ACE events modeled being driven solely by initial conditions, whether or not equilibria exist or are computationally tractable, and in the modeling facilitation of agent autonomy and learning.<br />
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[http://www.robotics.stanford.edu/~shoham/www%20papers/CSGT-CACM-Shoham.pdf 75-79] .<br/>&nbsp;&nbsp; • Alvin E. Roth (2002).《作为工程师的经济学家: 博弈论、实验和计算作为设计经济学的工具》 ，《经济学》 ，70(4) ，页。[ https://web.archive.org/web/20040414102216/http://kuznets.fas.harvard.edu/~aroth/papers/engineer.pdf 1341-1378]. <br />
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ACE 与博弈论（作为基于主体的社会互动建模方法时）有相似之处和重叠部分。但实践者也注意到了与标准方法的区别，例如，在仅由初始条件驱动的ACE事件建模中，均衡是否存在或者在计算上是否易于处理，以及在建模过程中促进主体自主性和学习方面。<br />
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The method has benefited from continuing improvements in modeling techniques of [[computer science]] and increased computer capabilities. The ultimate scientific objective of the method is to "test theoretical findings against real-world data in ways that permit empirically supported theories to cumulate over time, with each researcher’s work building appropriately on the work that has gone before."<ref>• Leigh Tesfatsion (2006). "Agent-Based Computational Economics: A Constructive Approach to Economic Theory," ch. 16, ''Handbook of Computational Economics'', v. 2, [pp. 831-880] sect. 5. [http://www.sciencedirect.com/science/article/pii/S1574002105020162 Abstract] and pre-pub [http://econ2.econ.iastate.edu/tesfatsi/hbintlt.pdf PDF].<br/>&nbsp;&nbsp; • [[Kenneth L. Judd]] (2006). "Computationally Intensive Analyses in Economics," ''Handbook of Computational Economics'', v. 2, ch. 17, pp. [https://books.google.com/books?id=6ITfRkNmKQcC&pg=PA881 881-] 893. Pre-pub [http://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF].<br/>&nbsp;&nbsp; • Leigh Tesfatsion and Kenneth L. Judd, ed. (2006). ''Handbook of Computational Economics'', v. 2. [http://www.elsevier.com/wps/find/bookdescription.cws_home/660847/description#description Description] {{Webarchive|url=https://web.archive.org/web/20120306100156/http://www.elsevier.com/wps/find/bookdescription.cws_home/660847/description#description |date=6 March 2012 }} & and chapter-preview<br />
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The method has benefited from continuing improvements in modeling techniques of computer science and increased computer capabilities. The ultimate scientific objective of the method is to "test theoretical findings against real-world data in ways that permit empirically supported theories to cumulate over time, with each researcher’s work building appropriately on the work that has gone before."&nbsp;&nbsp; • Kenneth L. Judd (2006). "Computationally Intensive Analyses in Economics," Handbook of Computational Economics, v. 2, ch. 17, pp. [https://books.google.com/books?id=6ITfRkNmKQcC&pg=PA881 881-] 893. Pre-pub [http://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF].<br/>&nbsp;&nbsp; • Leigh Tesfatsion and Kenneth L. Judd, ed. (2006). Handbook of Computational Economics, v. 2. [http://www.elsevier.com/wps/find/bookdescription.cws_home/660847/description#description Description] & and chapter-preview<br />
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• Kenneth l. Judd (2006)。《经济学中的计算密集分析》 ，计算经济学手册，第2卷，第5页。17页。Https://books.google.com/books?id=6itfrknmkqcc&pg=pa881881-893.Pre-pub [http://www2.econ.iastate.edu/tesfatsi/Judd.finalrev.pdf PDF].<br/>&nbsp;&nbsp; • Leigh Tesfatsion and Kenneth L. Judd, ed.(2006).计算经济学手册第2节。[ http://www.elsevier.com/wps/find/bookdescription.cws_home/660847/Description#Description 说明]和章节预览<br />
<br />
[http://www.sciencedirect.com/science?_ob=PublicationURL&_hubEid=1-s2.0-S1574002105X02003&_cid=273377&_pubType=HS&_auth=y&_acct=C000228598&_version=1&_urlVersion=0&_userid=10&md5=e4757b4f65755ed6340a11fee9615200 links.]</ref> The subject has been applied to research areas like [[asset pricing]],<ref name=arthuretal>B. Arthur, J. Holland, B. LeBaron, R. Palmer, P. Taylor (1997), 'Asset pricing under endogenous expectations in an artificial stock market,' in ''The Economy as an Evolving Complex System II'', B. Arthur, S. Durlauf, and D. Lane, eds., Addison Wesley.</ref> [[competition]] and [[collaboration]],<ref>[[Robert Axelrod]] (1997). ''The Complexity of Cooperation: Agent-Based Models of Competition and Collaboration'', Princeton. [http://press.princeton.edu/titles/6144.html Description], [http://press.princeton.edu/titles/6144.html#TOC contents], and [https://books.google.com/books?id=J0dgRGMdjmQC&printsec=find&pg=PR11#v=onepage&q&f=false preview].</ref> [[transaction cost]]s,<ref>Tomas B. Klosa and [[Bart Nooteboom]], 2001. "Agent-based Computational Transaction Cost Economics," ''Journal of Economic Dynamics and Control'' 25(3–4), pp. 503–52. [http://www.sciencedirect.com/science/article/pii/S0165188900000348 Abstract.]</ref> [[market structure]] and [[industrial organization]] and dynamics,<ref>• Roberto Leombruni and Matteo Richiardi, ed. (2004), ''Industry and Labor Dynamics: The Agent-Based Computational Economics Approach.'' World Scientific Publishing {{ISBN|981-256-100-5}}. [http://www.worldscibooks.com/economics/5706.html Description] {{Webarchive|url=https://web.archive.org/web/20100727221149/http://www.worldscibooks.com/economics/5706.html |date=27 July 2010 }} and chapter-preview [https://books.google.com/books?id=P5O7A5D55nQC&printsec=fond&pg=PR5#v=onepage&q&f=false links].<br/>&nbsp;&nbsp; • [[Joshua M. Epstein]] (2006). "Growing Adaptive Organizations: An Agent-Based Computational Approach," in ''Generative Social Science: Studies in Agent-Based Computational Modeling'', pp. [https://books.google.com/books?id=543OS3qdxBYC&pg=PA309 309-] 344. [http://press.princeton.edu/titles/8277.html Description] {{Webarchive|url=https://web.archive.org/web/20120126180655/http://press.princeton.edu/titles/8277.html |date=26 January 2012 }} and [http://www.santafe.edu/research/working-papers/abstract/99895b6465e8b87656612f8e3570b34c/ abstract].</ref> [[welfare economics]],<ref>[[Robert Axtell]] (2005). "The Complexity of Exchange," ''Economic Journal'', 115(504, Features), pp. [http://econfaculty.gmu.edu/pboettke/workshop/archives/f05/Axtell.pdf F193-F210].</ref> and [[mechanism design]],<ref>• ''The New Palgrave Dictionary of Economics'' (2008), 2nd Edition: <br/>&nbsp;&nbsp;&nbsp;&nbsp; [[Roger B. Myerson]] "mechanism design." [http://www.dictionaryofeconomics.com/article?id=pde2008_M000132&edition=current&q=mechanism%20design&topicid=&result_number=3 Abstract.] <br/>&nbsp;&nbsp;&nbsp;&nbsp; _____. "revelation principle." [http://www.dictionaryofeconomics.com/article?id=pde2008_R000137&edition=current&q=moral&topicid=&result_number=1 Abstract.]<br/>&nbsp;&nbsp;&nbsp;&nbsp; Tuomas Sandholm. "computing in mechanism design." [http://www.dictionaryofeconomics.com/article?id=pde2008_C000563&edition=&field=keyword&q=algorithmic%20mechanism%20design&topicid=&result_number=1 Abstract.]<br/>&nbsp;&nbsp; • [[Noam Nisan]] and Amir Ronen (2001). "Algorithmic Mechanism Design," ''Games and Economic Behavior'', 35(1-2), pp. [https://www.cs.cmu.edu/~sandholm/cs15-892F09/Algorithmic%20mechanism%20design.pdf 166–196].<br/>&nbsp;&nbsp; • [[Noam Nisan]] ''et al''., ed. (2007). ''Algorithmic Game Theory'', Cambridge University Press. [http://www.cup.cam.ac.uk/asia/catalogue/catalogue.asp?isbn=9780521872829 Description] {{Webarchive|url=https://web.archive.org/web/20120505140924/http://www.cup.cam.ac.uk/asia/catalogue/catalogue.asp?isbn=9780521872829 |date=5 May 2012 }}.</ref> [[Information economics|information and uncertainty]],<ref>Tuomas W. Sandholm and Victor R. Lesser (2001). "Leveled Commitment Contracts and Strategic Breach," ''Games and Economic Behavior'', 35(1-2), pp. [https://www.cs.cmu.edu/afs/.cs.cmu.edu/Web/People/sandholm/leveled.geb.pdf 212-270].</ref> [[macroeconomics]],<ref>• [[David Colander]], [[Peter Howitt]], Alan Kirman, [[Axel Leijonhufvud]], and [[Perry Mehrling]], 2008. "Beyond DSGE Models: Toward an Empirically Based Macroeconomics," ''American Economic Review'', 98(2), pp. [https://www.jstor.org/pss/29730026 236]-240. Pre-pub [http://www.brown.edu/Departments/Economics/Faculty/Peter_Howitt/publication/complex%20macro6.pdf PDF].<br/>&nbsp;&nbsp; • [[Thomas J. Sargent]] (1994). ''Bounded Rationality in Macroeconomics'', Oxford. [http://www.oup.com/us/catalog/general/subject/Economics/MacroeconomicTheory/?view=usa&ci=9780198288695 Description] and chapter-preview 1st-page [https://www.questia.com/library/book/bounded-rationality-in-macroeconomics-thomas-j-sargent-by-thomas-j-sargent.jsp links].<br/>&nbsp;&nbsp; • M. Oeffner (2009). '[http://www.opus-bayern.de/uni-wuerzburg/volltexte/2009/3927/pdf/OeffnerDissohneAnhang.pdf Agent-based Keynesian Macroeconomics]'. PhD thesis, Faculty of Economics, University of Würzburg.</ref> and [[Marxist economics]].<ref>A. F. Cottrell, P. Cockshott, G. J. Michaelson, I. P. Wright, V. Yakovenko<br />
<br />
[http://www.sciencedirect.com/science?_ob=PublicationURL&_hubEid=1-s2.0-S1574002105X02003&_cid=273377&_pubType=HS&_auth=y&_acct=C000228598&_version=1&_urlVersion=0&_userid=10&md5=e4757b4f65755ed6340a11fee9615200 links.]</ref> The subject has been applied to research areas like asset pricing, competition and collaboration, transaction costs, market structure and industrial organization and dynamics, welfare economics, and mechanism design, information and uncertainty, macroeconomics, and Marxist economics.<ref>A. F. Cottrell, P. Cockshott, G. J. Michaelson, I. P. Wright, V. Yakovenko<br />
<br />
[ http://www.sciencedirect.com/science?_ob=publicationurl&_hubeid=1-s2.0-s1574002105x02003&_cid=273377&_pubtype=hs&_auth=y&_acct=c000228598&_version=1&_urlversion=0&_userid=10&md5=e4757b4f65755ed6340a11fee9615200连结。参考文献<br />
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该方法得益于计算机科学建模技术的不断改进和计算机能力的提高。其最终科学目标是“用实际数据来检验理论发现——以允许经验支持的理论随着时间的推移而累积的方式（每位研究者的工作恰当建立在之前工作的基础之上）。”其现已应用于如资产定价、竞争与合作、交易成本、市场结构与产业组织与动态、'''福利经济学 Welfare Economics'''、机制设计、信息与不确定性、宏观经济学和马克思主义政治经济学等研究领域。 <br />
<br />
参考文献 a。科特雷尔，p. Cockshott，g. j. Michaelson，i. p. Wright，v. Yakovenko<br />
<br />
(2009), ''Classical Econophysics''. Routledge, {{ISBN|978-0-415-47848-9}}.</ref><ref>Leigh Tesfatsion (2006), "Agent-Based Computational Economics: A Constructive Approach to Economic Theory," ch. 16, ''Handbook of Computational Economics'', v. 2, part 2, ACE study of economic system. [http://www.sciencedirect.com/science/article/pii/S1574002105020162 Abstract] and pre-pub [http://econ2.econ.iastate.edu/tesfatsi/hbintlt.pdf PDF].</ref><br />
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(2009), Classical Econophysics. Routledge, .</ref><br />
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(2009) ，古典经济物理学。路特利奇，参考文献<br />
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==Overview 概述==<br />
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The "[[Agent (economics)|agents]]" in ACE models can represent individuals (e.g. people), social groupings (e.g. firms), biological entities (e.g. growing crops), and/or physical systems (e.g. transport systems). The ACE modeler provides the initial configuration of a computational economic system comprising multiple interacting agents. The modeler then steps back to observe the development of the system over time without further intervention. In particular, system events should be driven by agent interactions without external imposition of equilibrium conditions.<ref>[http://www.socsci.aau.dk/ae2006/ Summary of methods] {{Webarchive|url=https://web.archive.org/web/20070526105550/http://www.socsci.aau.dk/ae2006/ |date=26 May 2007 }}: ''Department of Economics, Politics and Public Administration, Aalborg University, Denmark'' website.</ref> Issues include those common to [[experimental economics]] in general<ref>[[Vernon L. Smith]], 2008. "experimental economics," ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_E000277&q=experimental%20&topicid=&result_number=2 Abstract].</ref> and development of a common framework for empirical validation and resolving open questions in agent-based modeling.<ref>Giorgio Fagiolo, Alessio Moneta, and Paul Windrum, 2007. "A Critical Guide to Empirical Validation of Agent-Based Models in Economics: Methodologies, Procedures, and Open Problems," ''Computational Economics'', 30, pp. [https://doi.org/10.1007%2Fs10614-007-9104-4 195]–226.</ref><br />
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The "agents" in ACE models can represent individuals (e.g. people), social groupings (e.g. firms), biological entities (e.g. growing crops), and/or physical systems (e.g. transport systems). The ACE modeler provides the initial configuration of a computational economic system comprising multiple interacting agents. The modeler then steps back to observe the development of the system over time without further intervention. In particular, system events should be driven by agent interactions without external imposition of equilibrium conditions. Issues include those common to experimental economics in general and development of a common framework for empirical validation and resolving open questions in agent-based modeling.<br />
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ACE 模型中的“主体”可以代表个体（如人）、社会群体（如公司）、生物实体（如生长中的庄稼）和/或物理系统（如运输系统）。ACE 模型提供了由多个相互作用的主体组成的计算经济系统的初始配置。然后，建模人员返回来观察系统随着时间的推移而发展的情况，而不需要进一步的干预。注意，系统活动应由主体间的相互作用驱动，而不需要外部强加的平衡条件。这些问题包括实验经济学通用的问题，以及开发一个用于经验验证的通用框架和解决基于主体的建模中尚未解决的问题。<br />
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ACE is an officially designated special interest group (SIG) of the Society for Computational Economics.<ref>[http://comp-econ.org/ Society for Computational Economics] website.</ref> Researchers at the [[Santa Fe Institute]] have contributed to the development of ACE.<br />
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ACE is an officially designated special interest group (SIG) of the Society for Computational Economics. Researchers at the Santa Fe Institute have contributed to the development of ACE.<br />
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ACE 是美国计算经济学学会的一个官方指定的特殊利益集团。圣菲研究所的研究人员为 ACE 的发展做出了贡献。<br />
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==Example: finance 举例：金融==<br />
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One area where ACE methodology has frequently been applied is asset pricing. [[W. Brian Arthur]], Eric Baum, [[William A. Brock (economist)|William Brock]], Cars Hommes, and Blake LeBaron, among others, have developed computational models in which many agents choose from a set of possible forecasting strategies in order to predict stock prices, which affects their asset demands and thus affects stock prices. These models assume that agents are more likely to choose forecasting strategies which have recently been successful. The success of any strategy will depend on market conditions and also on the set of strategies that are currently being used. These models frequently find that large booms and busts in asset prices may occur as agents switch across forecasting strategies.<ref name=arthuretal/><ref>W. Brock and C. Hommes (1997), 'A rational route to randomness.' ''Econometrica'' 65 (5), pp. 1059-1095.</ref><ref>C. Hommes (2008), 'Interacting agents in finance,' in ''The New Palgrave Dictionary of Economics''.</ref> More recently, Brock, Hommes, and Wagener (2009) have used a model of this type to argue that the introduction of new hedging instruments may destabilize the market,<ref>{{cite journal |first=W. |last=Brock |first2=C. |last2=Hommes |first3=F. |last3=Wagener |year=2009 |title=More hedging instruments may destabilize markets |journal=Journal of Economic Dynamics and Control |volume=33 |issue=11 |pages=1912–1928 |doi=10.1016/j.jedc.2009.05.004 |url=http://cendef.uva.nl/binaries/content/assets/subsites/amsterdam-school-of-economics/amsterdam-school-of-economics-research-institute/cendef/working-papers-2006/brohomwag.pdf?1417181713282 }}</ref> and some papers have suggested that ACE might be a useful methodology for understanding the recent [[financial crisis]].<ref>M. Buchanan (2009), '[http://pagesperso-orange.fr/mark.buchanan/nature_economic_modelling.pdf Meltdown modelling. Could agent-based computer models prevent another financial crisis?].' Nature, Vol. 460, No. 7256. (5 August 2009), pp. 680-682.</ref><ref>J.D. Farmer, D. Foley (2009), 'The economy needs agent-based modelling.' Nature, Vol. 460, No. 7256. (5 August 2009), pp. 685-686.</ref><ref>M. Holcombe, S. Coakley, M.Kiran, S. Chin, C. Greenough, D.Worth, S.Cincotti, M.Raberto, A. Teglio, C. Deissenberg, S. van der Hoog, H. Dawid, S. Gemkow, P. Harting, M. Neugart. Large-scale Modeling of Economic Systems, Complex Systems, 22(2), 175-191, 2013</ref><br />
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One area where ACE methodology has frequently been applied is asset pricing. W. Brian Arthur, Eric Baum, William Brock, Cars Hommes, and Blake LeBaron, among others, have developed computational models in which many agents choose from a set of possible forecasting strategies in order to predict stock prices, which affects their asset demands and thus affects stock prices. These models assume that agents are more likely to choose forecasting strategies which have recently been successful. The success of any strategy will depend on market conditions and also on the set of strategies that are currently being used. These models frequently find that large booms and busts in asset prices may occur as agents switch across forecasting strategies. More recently, Brock, Hommes, and Wagener (2009) have used a model of this type to argue that the introduction of new hedging instruments may destabilize the market, and some papers have suggested that ACE might be a useful methodology for understanding the recent financial crisis.<br />
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Ace 方法经常应用的一个领域是资产定价。布莱恩 · 阿瑟，埃里克 · 鲍姆，威廉 · 布洛克，汽车人，布莱克 · 勒巴伦，等等，已经开发出计算模型，其中许多代理人选择从一组可能的预测策略，以预测股票价格，这影响了他们的资产需求，从而影响股票价格。这些模型假设代理人更有可能选择最近成功的预测策略。任何战略的成功将取决于市场条件，也取决于目前正在使用的一套战略。这些模型经常发现，随着经纪人转换预测策略，资产价格可能会出现大起大落。最近，Brock、 Hommes 和 Wagener (2009)使用了这类模型，认为引入新的对冲工具可能会破坏市场稳定，一些论文提出，ACE 可能是理解最近金融危机的有用方法。<br />
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==See also 参见==<br />
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* [[ACEGES]]<br />
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* [[Agent-based social simulation]]<br />
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* [[Artificial economics]]<br />
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* [[Computational economics]]<br />
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* [[Econophysics]]<br />
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* [[Macroeconomic model]]<br />
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* [[Multi-agent system]]<br />
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* [[Statistical finance]]<br />
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==References 参考资料==<br />
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{{Reflist}}<br />
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==编者推荐==<br />
[https://campus.swarma.org/ 集智学园]课程：[https://campus.swarma.org/course/714 计算社会经济学的内容与方法]<br />
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本课程系统性地介绍计算社会经济学的主要研究内容和分析方法，阐述大规模社会经济数据（包括卫星遥感数据、手机通讯数据、社交媒体数据等）在解决全球贫困问题、刻画区域经济结构、推断个体财富和应急抢险救灾中的具体应用。<br />
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[[Category:Monte Carlo methods in finance]]<br />
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Category:Monte Carlo methods in finance<br />
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类别: 金融中的蒙特卡罗方法<br />
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[[Category:Computational fields of study]]<br />
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Category:Computational fields of study<br />
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类别: 研究的计算领域<br />
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<small>This page was moved from [[wikipedia:en:Agent-based computational economics]]. Its edit history can be viewed at [[基于主体的计算经济学/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%B2%8D%E5%BE%B7%E6%B8%A9%E6%95%88%E5%BA%94&diff=11127鲍德温效应2020-07-24T16:10:33Z<p>Dorr：</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。<br />
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{{short description|Effect of learned behavior on evolution}}<br />
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{{Distinguish|Baldwin's rules}}<br />
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[[File:Lamarck Compared to Darwin, Baldwin, Waddington.svg|thumb|upright=2.5|The Baldwin effect compared to [[Lamarckism|Lamarck's theory of evolution]], [[Darwinian evolution]], and Waddington's [[genetic assimilation]]. All the theories offer explanations of how organisms respond to a changed environment with adaptive inherited change.]]<br />
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Lamarck's theory of evolution, Darwinian evolution, and Waddington's genetic assimilation. All the theories offer explanations of how organisms respond to a changed environment with adaptive inherited change.]]<br />
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拉马克的进化论、达尔文进化论和华丁顿的遗传同化论。[参考译文]所有的理论都解释了有机体是如何通过适应性遗传变化来应对环境变化的<br />
--~英文wiki无此段<br />
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In [[evolutionary biology]], the '''Baldwin effect''' describes the effect of learned behavior on evolution. In brief, [[James Mark Baldwin]] and others suggested during the [[eclipse of Darwinism]] in the late 19th century that an organism's ''ability to learn'' new behaviors (e.g. to acclimatise to a new stressor) will affect its reproductive success and will therefore have an effect on the genetic makeup of its species through [[natural selection]]. Though this process appears similar to [[Lamarckism|Lamarckian evolution]], Lamarck proposed that living things ''inherited'' their parents' acquired characteristics. The Baldwin effect has been independently proposed several times, and today it is generally recognized as part of the [[Modern synthesis (20th century)|modern synthesis]].<br />
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In evolutionary biology, the Baldwin effect describes the effect of learned behavior on evolution. In brief, James Mark Baldwin and others suggested during the eclipse of Darwinism in the late 19th century that an organism's ability to learn new behaviors (e.g. to acclimatise to a new stressor) will affect its reproductive success and will therefore have an effect on the genetic makeup of its species through natural selection. Though this process appears similar to Lamarckian evolution, Lamarck proposed that living things inherited their parents' acquired characteristics. The Baldwin effect has been independently proposed several times, and today it is generally recognized as part of the modern synthesis.<br />
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在'''演化生物学 Evolutionary Biology'''中，'''鲍德温效应 Baldwin Effect'''描述了习得行为对进化的影响。简而言之，在19世纪末'''达尔文主义 Darwinism'''的衰落期间，'''詹姆斯·马克·鲍德温 James Mark Baldwin'''及其他一些人提出，生物体学习新行为的能力（如为了适应新的压力）会影响其繁殖，进而会通过'''自然选择 Natural Selection'''影响其物种的基因组成。这个过程似乎与'''拉马克主义 Lamarckism'''中相似，但拉马克认为生物继承了其父母后天获得的特性。鲍德温效应已独立提出过多次，如今普遍认为其为'''现代进化综论 Modern Evolutionary Synthesis'''的一部分。<br />
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=="A New Factor in Evolution" “进化的新因素”==<br />
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The effect, then unnamed, was put forward in 1896 in a paper "A New Factor in Evolution" by [[United States|American]] [[psychology|psychologist]] [[James Mark Baldwin]]. The paper proposed a mechanism for specific selection for general learning ability. As [[Robert J. Richards|Robert Richards]] explains:<ref name=Richards1987>{{cite book | author=Richards, Robert J. | year=1987 | title=Darwin and the Emergence of Evolutionary Theories of Mind and Behavior| publisher=The University of Chicago Press |location=Chicago | isbn=978-0-226-71199-7 | page=399| author-link=Robert J. Richards }}</ref><br />
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The effect, then unnamed, was put forward in 1896 in a paper "A New Factor in Evolution" by American psychologist James Mark Baldwin. The paper proposed a mechanism for specific selection for general learning ability. As Robert Richards explains:<br />
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1896年，美国心理学家詹姆斯·马克·鲍德温在一篇名为《进化的新因素》的论文中提出了这一效应（1987年又发表了第二篇论文）。该文提出了一种针对一般学习能力的具体选择机制。正如'''罗伯特·理查兹 Robert Richards'''解释的那样:<br />
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块引号<br />
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If animals entered a new environment—or their old environment rapidly changed—those<br />
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If animals entered a new environment—or their old environment rapidly changed—those<br />
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that could flexibly respond by learning new behaviors or by ontogenetically adapting would<br />
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that could flexibly respond by learning new behaviors or by ontogenetically adapting would<br />
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be naturally preserved. This saved remnant would, over several generations, have the<br />
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be naturally preserved. This saved remnant would, over several generations, have the<br />
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opportunity to exhibit spontaneously congenital variations similar to their acquired traits and<br />
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opportunity to exhibit spontaneously congenital variations similar to their acquired traits and<br />
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have these variations naturally selected. It would look as though the acquired traits had sunk<br />
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have these variations naturally selected. It would look as though the acquired traits had sunk<br />
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into the hereditary substance in a Lamarckian fashion, but the process would really be [[Neo-Darwinism|neo-Darwinian]].<br />
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into the hereditary substance in a Lamarckian fashion, but the process would really be neo-Darwinian.<br />
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如果动物处于新环境或其原来所处的环境急剧变化，那些能够通过学习新行为或自身基因适应来灵活作出反应的动物会自然地留存下来。这些幸存者，经过几代之后，有机会自发地表现出与他们获得的特征相似的先天变异，并让其经自然选择而留存。看上去获得性特征以拉马克的方式留在了遗传物质上，但这个过程实际上是'''新达尔文主义 Neo-Darwinism'''的。<br />
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Selected offspring would tend to have an increased capacity for learning new skills rather than being confined to genetically coded, relatively fixed abilities. In effect, it places emphasis on the fact that the sustained behavior of a species or group can shape the evolution of that species. The "Baldwin effect" is better understood in [[evolutionary developmental biology]] literature as a scenario in which a character or [[biological trait|trait]] change occurring in an [[organism]] as a result of its interaction with its environment [[genetic assimilation|becomes gradually assimilated]] into its developmental genetic or epigenetic repertoire (Simpson, 1953; Newman, 2002). In the words of [[Daniel Dennett]],<ref name=dennett03/><br />
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Selected offspring would tend to have an increased capacity for learning new skills rather than being confined to genetically coded, relatively fixed abilities. In effect, it places emphasis on the fact that the sustained behavior of a species or group can shape the evolution of that species. The "Baldwin effect" is better understood in evolutionary developmental biology literature as a scenario in which a character or trait change occurring in an organism as a result of its interaction with its environment becomes gradually assimilated into its developmental genetic or epigenetic repertoire (Simpson, 1953; Newman, 2002). In the words of Daniel Dennett,<br />
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被选择留存的后代往往具有更强的学习新技能的能力，而不仅局限于基因编码的相对固定的能力。实际上，它强调以下事实：一个物种或群体的持续行为可以影响其进化。在'''演化发育生物学 Evolutionary developmental biology'''的文献中，“鲍德温效应”得到了更好的理解——生物体由于与其环境的相互作用而发生的的特征或性状变化逐渐被同化到其发育遗传或表观遗传库中。用'''丹尼尔·丹尼特 Daniel Dennett'''的话来说,<br />
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Thanks to the Baldwin effect, species can be said to pretest the efficacy of particular different designs by phenotypic (individual) exploration of the space of nearby possibilities. If a particularly winning setting is thereby discovered, this discovery will ''create'' a new selection pressure: organisms that are closer in the adaptive landscape to that discovery will have a clear advantage over those more distant.<br />
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Thanks to the Baldwin effect, species can be said to pretest the efficacy of particular different designs by phenotypic (individual) exploration of the space of nearby possibilities. If a particularly winning setting is thereby discovered, this discovery will create a new selection pressure: organisms that are closer in the adaptive landscape to that discovery will have a clear advantage over those more distant.<br />
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由于鲍德温效应，可以说物种通过表型（个体）探索附近可能性的空间，预先测试了特定不同设计的功效。若由此发现一个特别适宜的环境，该发现将导致新的选择压力: 在适应环境中，更接近此发现的生物体将较其他生物体有明显的优势。<br />
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An update to the Baldwin effect was developed by [[Jean Piaget]], [[Paul Alfred Weiss|Paul Weiss]], and [[Conrad Waddington]] in the 1960s–1970s. This new version included an explicit role for the social in shaping subsequent natural change in humans (both evolutionary and developmental), with reference to alterations of selection pressures.<ref>{{cite journal | author = Burman J. T. | year = 2013 | title = Updating the Baldwin Effect: The biological levels behind Piaget's new theory | url = | journal = New Ideas in Psychology | volume = 31 | issue = 3| pages = 363–373 | doi = 10.1016/j.newideapsych.2012.07.003 }}</ref><br />
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An update to the Baldwin effect was developed by Jean Piaget, Paul Weiss, and Conrad Waddington in the 1960s–1970s. This new version included an explicit role for the social in shaping subsequent natural change in humans (both evolutionary and developmental), with reference to alterations of selection pressures.<br />
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1960-1970年代，'''让·皮亚杰 Jean Piaget'''、'''保罗·维斯 Paul Weiss'''和'''康拉德·华丁顿 Conrad Waddington'''修正了鲍德温效应。该新版本明确了社会在塑造人类随后的自然变化（包括进化和发展）中的作用，并提到了选择压力的改变。<br />
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As is to be expected from [[Stigler's law]], subsequent research shows that Baldwin was not the first to identify the process; [[Douglas Spalding]] mentioned it in 1873.<ref>Noble, R.; [[Denis Noble|Noble, D.]] (2017) Was the Watchmaker Blind? Or Was She One-Eyed? ''Biology'' 2017, 6(4), 47; doi:10.3390/biology6040047, quoting Bateson, P. The adaptability driver: Links between behaviour and evolution. Biol. Theory 2006, 1, 342–345.</ref><br />
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As is to be expected from Stigler's law, subsequent research shows that Baldwin was not the first to identify the process; Douglas Spalding mentioned it in 1873.<br />
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正如'''斯蒂格勒定律 Stigler's Law'''所指，随后的研究表明，鲍德温并不是第一个发现这个过程的人; '''道格拉斯·斯波尔丁 Douglas Spalding'''在1873年提到过这个过程。<br />
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==Examples 举例==<br />
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Suppose a species is threatened by a new [[predator]] and there is a behavior that makes it more difficult for the predator to kill individuals of the species. Individuals who learn the behavior more quickly will obviously be at an advantage. As time goes on, the ability to learn the behavior will improve (by genetic selection), and at some point it will seem to be an [[instinct]]. Baldwin gives the following case involving cooperation: "Animals may be kept alive let us say in a given environment by social cooperation only; these transmit this social type of variation to posterity; thus social adaptation sets the direction of physical phylogeny and physical heredity is determined in part by this factor" (Baldwin, 1896, p.&nbsp;553).<br />
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Suppose a species is threatened by a new predator and there is a behavior that makes it more difficult for the predator to kill individuals of the species. Individuals who learn the behavior more quickly will obviously be at an advantage. As time goes on, the ability to learn the behavior will improve (by genetic selection), and at some point it will seem to be an instinct. Baldwin gives the following case involving cooperation: "Animals may be kept alive let us say in a given environment by social cooperation only; these transmit this social type of variation to posterity; thus social adaptation sets the direction of physical phylogeny and physical heredity is determined in part by this factor" (Baldwin, 1896, p.&nbsp;553).<br />
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假设一个物种受到一个新的捕食者的威胁，并且有一种行为使得捕食者更加难以杀死该物种的个体。更快学会这种行为的个体显然会处于有利地位。随着时间的推移，学习行为的能力将得到提高(通过基因选择) ，在某种程度上，这似乎是一种本能。鲍德温给出了以下涉及合作的案例: “动物可以保持活着，让我们说，在一个特定的环境中，只有通过社会合作; 这些传递这种社会类型的变异给子孙后代; 因此，社会适应确定了物理系统发育和物理遗传的方向，部分由这个因素决定”(鲍德温，1896年，第553页)。<br />
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The appearance of lactose tolerance (i.e., [[lactase persistence]]) in human populations with a long tradition of raising domesticated animals for milk production has been suggested as another example.<ref>[https://epdf.pub/evolution-and-learning-the-baldwin-effect-reconsidered.html Evolution and Learning: The Baldwin Effect Reconsidered]</ref> This argument holds that a feedback loop operates whereby a dairy culture increases the selective advantage from this genetic trait, while the average population genotype increases the collective rewards of a dairy culture.<br />
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The appearance of lactose tolerance (i.e., lactase persistence) in human populations with a long tradition of raising domesticated animals for milk production has been suggested as another example. This argument holds that a feedback loop operates whereby a dairy culture increases the selective advantage from this genetic trait, while the average population genotype increases the collective rewards of a dairy culture.<br />
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另一个例子是，在人类中出现了乳糖耐受性(即乳糖分解酶持续性) ，而人类在饲养家畜产奶方面有着悠久的传统。这种观点认为，乳制品文化通过一个反馈循环增加了这种遗传特性的选择优势，而普通群体的基因型增加了乳制品文化的集体回报。<br />
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==Controversy and acceptance 争议和接受==<br />
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Initially Baldwin's ideas were not incompatible with the prevailing, but uncertain, ideas about the mechanism of transmission of hereditary information and at least two other biologists put forward very similar ideas in 1896.<ref>{{Cite journal |last=Morgan |first=C. L. |authorlink=C. Lloyd Morgan |year=1896 |title=On modification and variation |journal=Science|volume=4|issue=99 |pages=733–740 |doi=10.1126/science.4.99.733|pmid=17735249 |url=https://zenodo.org/record/1448227 }}</ref><ref>{{Cite journal |last=Osborne |first=H. F. |authorlink=Henry Fairfield Osborne |year=1896 |title=Ontogenic and phylogenic variation |journal=Science |volume=4|issue=100 |pages=786–789 |doi=10.1126/science.4.100.786|pmid=17734840 |url=https://zenodo.org/record/1448151 }}</ref> In 1901, [[Maurice Maeterlinck]] referred to behavioural adaptations to prevailing climates in different species of bees as ‘what had merely been an idea, therefore, and opposed to instinct, has thus by slow degrees become an instinctive habit’.<ref>{{Cite book |title=The Life of the Bee|last=Materlinck |first=Maurice |authorlink=Maurice Maeterlinck |publisher=Dodd, Mead and Co. |year=1901 |location=New York |pages=Chapter VII section 102}}</ref> The Baldwin effect theory subsequently became more controversial, with scholars being split between "Baldwin boosters" and "Baldwin skeptics".<ref name="depew03">Depew, David J. (2003), "Baldwin Boosters, Baldwin Skeptics" in: {{cite book |title=Evolution and learning: The Baldwin effect reconsidered |last=Weber |first=Bruce H. |first2=David J. |last2=Depew |year=2003 |publisher=MIT Press |location=Cambridge, MA |isbn=978-0-262-23229-6 |pages=3–31 |url=https://books.google.com/books?id=yBtRzBilw1MC&pg=PR9&dq=baldwin+effect#v=onepage&q&f=false }}</ref> The theory was first called the "Baldwin effect" by [[George Gaylord Simpson]] in 1953.<ref name="depew03" /> Simpson "admitted that the idea was theoretically consistent, that is, not inconsistent with the [[Modern synthesis (20th century)|modern synthesis]]",<ref name="depew03" /> but he doubted that the phenomenon occurred very often, or if so, could be proven to occur. In his discussion of the reception of the Baldwin-effect theory Simpson points out that the theory appears to provide a reconciliation between a neo-Darwinian and a [[Lamarckism|neo-Lamarckian]] approach and that "Mendelism and later genetic theory so conclusively ruled out the extreme neo-Lamarckian position that reconciliation came to seem unnecessary".<ref>{{cite journal |title=The Baldwin effect |authorlink=George Gaylord Simpson |author=Simpson, George Gaylord |year=1953 |journal=Evolution |volume=7 | issue=2 |pages=110–117 |doi=10.2307/2405746|jstor=2405746 }}</ref> In 1942, the evolutionary biologist [[Julian Huxley]] promoted the Baldwin effect as part of the modern synthesis, saying the concept had been unduly neglected by evolutionists.<ref name=Julian1942>{{cite book | authorlink=Julian Huxley |author=Huxley, Julian |year=1942 |location=London |title=Evolution: The Modern Synthesis |publisher=George Allen & Unwin Ltd|title-link=Evolution: The Modern Synthesis }}</ref><br />
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Initially Baldwin's ideas were not incompatible with the prevailing, but uncertain, ideas about the mechanism of transmission of hereditary information and at least two other biologists put forward very similar ideas in 1896. In 1901, Maurice Maeterlinck referred to behavioural adaptations to prevailing climates in different species of bees as ‘what had merely been an idea, therefore, and opposed to instinct, has thus by slow degrees become an instinctive habit’. The Baldwin effect theory subsequently became more controversial, with scholars being split between "Baldwin boosters" and "Baldwin skeptics". The theory was first called the "Baldwin effect" by George Gaylord Simpson in 1953. In 1942, the evolutionary biologist Julian Huxley promoted the Baldwin effect as part of the modern synthesis, saying the concept had been unduly neglected by evolutionists.<br />
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最初，鲍德温的观点与广泛流传但未加验证的关于遗传信息传递机制的观点并不矛盾，且在1896年至少有两位生物学家提出了非常相似的观点。1901年，'''莫里斯·梅特林克 Maurice Maeterlinck'''提到不同种类的蜜蜂对当时气候的行为适应性，称其之前仅仅是一种想法，因此，与本能相反，慢慢地变成了一种本能的习惯。鲍德温效应理论随后变得更具争议性，学者们分为“鲍德温支持者”和“鲍德温怀疑论者”。1953年'''乔治·盖洛德·辛普森 George Gaylord Simpson'''首次命名该理论为“鲍德温效应”。1942年，进化生物学家'''朱利安·赫胥黎 Julian Huxley'''将鲍德温效应作为现代进化综论的一部分加以推广，称进化论者过度忽视了这一概念。<br />
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In the 1960s, the evolutionary biologist [[Ernst Mayr]] contended that the Baldwin effect theory was untenable because<br />
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In the 1960s, the evolutionary biologist Ernst Mayr contended that the Baldwin effect theory was untenable because<br />
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在20世纪60年代，进化生物学家'''恩斯特·迈尔'''称鲍德温效应理论是站不住脚的，因为<br />
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# the argument is stated in terms of the individual genotype, whereas what is really exposed to the selection pressure is a phenotypically and genetically variable population;<br />
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the argument is stated in terms of the individual genotype, whereas what is really exposed to the selection pressure is a phenotypically and genetically variable population;<br />
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这个论点是从个体基因型的角度来阐述的，而真正暴露在选择压力下的是表型和遗传可变的种群;<br />
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# it is not sufficiently emphasized that the degree of modification of the phenotype is in itself genetically controlled;<br />
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it is not sufficiently emphasized that the degree of modification of the phenotype is in itself genetically controlled;<br />
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没有充分强调表型的改变程度本身是受遗传控制的;<br />
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# it is assumed that phenotypic rigidity is selectively superior to phenotypic flexibility.<ref name=Mayr1963>{{cite book| authorlink=Ernst Mayr |author=Mayr, Ernst |year=1963 |location=Cambridge, MA |title=Animal Species and Evolution| url=https://archive.org/details/animalspeciesevo00mayr | url-access=registration |publisher=Harvard University Press|isbn=978-0-674-03750-2}}</ref><br />
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it is assumed that phenotypic rigidity is selectively superior to phenotypic flexibility.<br />
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假定表型刚性有选择性优于表型柔性。<br />
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In 1987 [[Geoffrey Hinton]] and Steven Nowlan demonstrated by computer simulation that learning can accelerate evolution, and they associated this with the Baldwin effect.<ref>{{cite journal |title=How learning can guide evolution |last=Hinton |first=Geoffrey E. |author-link=Geoffrey Hinton |last2=Nowlan |first2=Steven J. |year=1987 |journal=Complex Systems |volume=1 |pages=495–502}}</ref><ref>{{cite journal |title=When learning guides evolution|authorlink=John Maynard Smith |author=Maynard Smith, John |year=1987 |journal=Nature |volume=329 |issue=6142 |pages=761–762 |doi=10.1038/329761a0|pmid=3670381 }}</ref><ref name=Puentedura2003>{{cite book| author=Puentedura, Ruben R. |year=2003|location=Cambridge, MA |chapter=The Baldwin effect in the age of computation |title=Evolution and Learning: The Baldwin Effect Reconsidered |editor-last=Weber |editor-first=Bruce H. |editor-last2=Depew |editor-first2=David J.|publisher=MIT press |pages=219–234}}</ref><br />
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In 1987 Geoffrey Hinton and Steven Nowlan demonstrated by computer simulation that learning can accelerate evolution, and they associated this with the Baldwin effect.<br />
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1987年，'''杰弗里·辛顿 Geoffrey Hinton''' 和 '''史蒂芬·诺兰 Steven Nowlan''' 通过计算机模拟证明了学习可以加速进化，他们将此与鲍德温效应联系起来。<br />
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Paul Griffiths<ref name=Griffiths2003>{{cite book| author=Griffiths, Paul E. |year=2003|location=Cambridge, MA|chapter=Beyond the Baldwin effect: James Mark Baldwin’s ‘social heredity,’ epigenetic inheritance, and niche construction|title=Evolution and Learning: The Baldwin Effect Reconsidered|publisher=MIT press|pages=193–215|editor-last=Weber |editor-first=Bruce H. |editor-last2=Depew |editor-first2=David J.}}</ref> suggests two reasons for the continuing interest in the Baldwin effect. The first is the role ''mind'' is understood to play in the effect. The second is the connection between ''development'' and evolution in the effect. Baldwin’s account of how neurophysiological and conscious mental factors may contribute to the effect<ref>{{cite journal |title=Heredity and instinct|last=Baldwin |first=J. Mark |author-link=James Mark Baldwin|year=1896 |journal=Science |volume=3 |issue=64|pages=438–441, 558–561|doi=10.1126/science.3.64.438 |pmid=17780356|url=https://zenodo.org/record/1448082 }}</ref><ref>{{cite journal |title=Consciousness and evolution |last=Baldwin |first=J. Mark |author-link=James Mark Baldwin|year=1896 |journal=Psychological Review |volume=3 |issue=3|pages=300–309 |doi=10.1037/h0063996}}</ref><ref>{{cite journal |title=A new factor in evolution |last=Baldwin |first=J. Mark |author-link=James Mark Baldwin|year=1896 |journal=The American Naturalist |volume=30 |issue=354|pages=441–451, 536–553 |doi=10.1086/276408|url=https://semanticscholar.org/paper/0c1386d88cb54eb17c6a2745cd4bf15dbb3f2b09 }}</ref> brings into focus the question of the possible survival value of consciousness.<ref name=Lindahl2001>{{cite book| author=Lindahl, B. I. B. |year=2001 |location=Amsterdam |chapter=Consciousness, behavioural patterns and the direction of biological evolution: implications for the mind–brain problem |title=Dimensions of Conscious Experience |publisher=John Benjamins |pages=73–99 |editor-last=Pylkkänen |editor-first=Paavo |editor-last2=Vadén |editor-first2=Tere |isbn=978-90-272-5157-2}}</ref><br />
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Paul Griffiths suggests two reasons for the continuing interest in the Baldwin effect. The first is the role mind is understood to play in the effect. The second is the connection between development and evolution in the effect. Baldwin’s account of how neurophysiological and conscious mental factors may contribute to the effect brings into focus the question of the possible survival value of consciousness.<br />
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'''保罗·格里菲斯 Paul Griffiths'''指出了人们对鲍德温效应持续兴趣的两个原因。首先，我们可以理解心智在这种效果中所扮演的角色。二是其中发展与演变的关系。鲍德温关于神经生理学和意识心理因素如何促成这种效应的解释，使人们更加关注意识可能的存在价值问题。<br />
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[[File:Carpodacus mexicanus -Madison, Wisconsin, USA-8.jpg|thumb|The [[house finch]]'s colonisation of North America has provided empirical evidence of the Baldwin effect.<ref name=Badyaev/>]]<br />
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The [[house finch's colonisation of North America has provided empirical evidence of the Baldwin effect.]]<br />
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[图一：家朱雀在北美的定殖提供了鲍德温效应的经验证明。]<br />
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Still, observes David Depew, "it is striking that a rather diverse lot of contemporary evolutionary theorists, most of whom regard themselves as supporters of the Modern Synthesis, have of late become 'Baldwin boosters{{'"}}<ref name="depew03" /> These Baldwin boosters<br />
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Still, observes David Depew, "it is striking that a rather diverse lot of contemporary evolutionary theorists, most of whom regard themselves as supporters of the Modern Synthesis, have of late become 'Baldwin boosters These Baldwin boosters<br />
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are typically [[evolutionary psychologist]]s who are searching for scenarios in which a population can get itself by behavioral trial and error onto a "hard to find" part of the fitness landscape in which human brain, language, and mind can rapidly coevolve. They are searching for what [[Daniel Dennett]], himself a Baldwin booster, calls an "evolutionary crane," an instrument to do some heavy lifting fast.<ref name="depew03" /><br />
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are typically evolutionary psychologists who are searching for scenarios in which a population can get itself by behavioral trial and error onto a "hard to find" part of the fitness landscape in which human brain, language, and mind can rapidly coevolve. They are searching for what Daniel Dennett, himself a Baldwin booster, calls an "evolutionary crane," an instrument to do some heavy lifting fast.<br />
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不过，'''戴维•迪皮尤 David Depew'''评论道: “令人惊讶的是，很多当代进化论理论家（其中大多数人自认为是现代进化综论的支持者），最近都成为了‘鲍德温支持者’，这些人是典型的致力于寻找“很难找到的”，群体可以通过行为试错达到人脑、语言和思维快速共同进化的适应性图景之场景的'''进化心理学 Evolutionary Psychology'''家。他们正在寻找被丹尼尔·丹尼特（鲍德温的支持者）所称的“进化起重机”（快速升高重物的工具）。<br />
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According to [[Daniel Dennett]], recent work has rendered the Baldwin effect "no longer a controversial wrinkle in orthodox Darwinism".<ref name="dennett03">[[Daniel Dennett|Dennett, Daniel]] (2003), "The Baldwin Effect: a Crane, not a Skyhook" in: {{cite book |title=Evolution and learning: The Baldwin effect reconsidered |last=Weber |first=Bruce H. |first2=David J. |last2=Depew |year=2003 |publisher=MIT Press |location=Cambridge, MA |isbn=978-0-262-23229-6 |pages=69–106 |url=https://books.google.com/books?id=yBtRzBilw1MC&pg=PR9&dq=baldwin+effect#v=onepage&q&f=false }}</ref> Potential genetic mechanisms underlying the Baldwin effect have been proposed for the evolution of natural (genetically-determinant) antibodies.<ref>{{Cite journal|last=Anderson|first=Russell|date=1996|title=How the adaptive antibodies facilitate the evolution of natural antibodies|url=|journal=Immunology and Cell Biology|volume=74 | issue = 2 |pages=286–291|doi=10.1038/icb.1996.50|pmid=8799730}}</ref> In 2009, empirical evidence for the Baldwin effect was provided from the colonisation of North America by the [[house finch]].<ref name=Badyaev>{{cite journal |last1=Badyaev |first1=Alexander V. |title=Evolutionary significance of phenotypic accommodation in novel environments: an empirical test of the Baldwin effect |journal=Phil. Trans. R. Soc. Lond. B |date=March 2009 |volume=364 |issue=1520 |doi=10.1098/rstb.2008.0285 |pmid=19324617 |pages=1125–1141|pmc=2666683 }}</ref><br />
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According to Daniel Dennett, recent work has rendered the Baldwin effect "no longer a controversial wrinkle in orthodox Darwinism". Potential genetic mechanisms underlying the Baldwin effect have been proposed for the evolution of natural (genetically-determinant) antibodies. In 2009, empirical evidence for the Baldwin effect was provided from the colonisation of North America by the house finch.<br />
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丹尼尔•丹尼特表示，最近的研究工作已使鲍德温效应“在正统达尔文主义中不再是争议性问题”。科学家已提出鲍德温效应背后潜在的遗传机制用于自然（基因决定）抗体的进化。2009年，'''家朱雀 House Finch'''在北美的定殖为鲍德温效应提供了经验证明。<br />
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==Comparison with genetic assimilation 与遗传同化的区别==<br />
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{{further|Genetic assimilation}}<br />
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The Baldwin effect has been confused with, and sometimes conflated with, a different evolutionary theory also based on [[phenotypic plasticity]], [[C. H. Waddington]]'s [[genetic assimilation]]. The Baldwin effect includes genetic accommodation, of which one type is genetic assimilation.<ref>{{cite journal | last1=Crispo | first1=Erika | year=2007 | title=The Baldwin effect and genetic assimilation: revisiting two mechanisms of evolutionary change mediated by phenotypic plasticity | journal=Evolution | volume=61 | issue=11 | pages=2469–2479 | doi=10.1111/j.1558-5646.2007.00203.x| pmid=17714500 }}</ref><!--more needed--><br />
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The Baldwin effect has been confused with, and sometimes conflated with, a different evolutionary theory also based on phenotypic plasticity, C. H. Waddington's genetic assimilation. The Baldwin effect includes genetic accommodation, of which one type is genetic assimilation.<!--more needed--><br />
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鲍德温效应与另一种同样基于'''表型可塑性 Henotypic Plasticity'''的进化理论——C.H.华丁顿的'''遗传同化 Genetic Assimilation'''相混淆，有时甚至混为一谈。实际上，鲍德温效应包括遗传调节，其中一种类型是遗传同化。<br />
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==See also 参见==<br />
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* {{annotated link|Evolvability}}<br />
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* {{annotated link|Lamarckism}}<br />
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* {{annotated link|Maternal effect}}<br />
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* {{annotated link|Meme}}<br />
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* {{annotated link|Pangenesis}}<br />
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* {{annotated link|Weismann barrier}}<br />
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==Notes 注释==<br />
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{{reflist|30em}}<br />
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== References 参考资料==<br />
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* {{cite journal | last = Bateson |first=Patrick | year = 2004 | title = The Active Role of Behaviour in Evolution | url = | journal = Biology and Philosophy | volume = 19 | issue = 2| pages = 283–298 | doi=10.1023/b:biph.0000024468.12161.83}}<br />
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* {{cite journal | last1 = Baldwin | first1 = J. Mark | authorlink = James Mark Baldwin | year = 1896 | title = A New Factor in Evolution | url = http://www.brocku.ca/MeadProject/Baldwin/Baldwin_1896_h.html | journal = The American Naturalist | volume = 30 | issue = 354| pages = 441–451 | doi=10.1086/276408}}<br />
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* {{cite journal | last1 = Baldwin | first1 = J. Mark | authorlink = James Mark Baldwin | year = 1897 | title = Organic Selection | journal = Science | volume = 5 | issue = 121| pages = 634–636 | doi=10.1126/science.5.121.634| pmid = 17781159 | url = https://zenodo.org/record/1448253 }}<br />
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* {{cite book |last1=Cochran |first1=Gregory |last2=Harpending |first2=Henry |date=2009 |title=The 10,000 Year Explosion: How Civilization Accelerated Human Evolution |publisher=Basic Books |isbn=978-0-465-00221-4|title-link=The 10,000 Year Explosion }}<br />
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* {{cite journal | last = Hall |first=Brian K. | year = 2001 | title = Organic Selection: Proximate Environmental Effects on the Evolution of Morphology and Behaviour | doi = 10.1023/a:1006773408919 | journal = Biology and Philosophy | volume = 16 | issue = 2| pages = 215–237 }}<br />
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* {{cite journal|author=Newman, Stuart A |year=2002 |title=Putting Genes in their place |url=http://www.nymc.edu/sanewman/PDFs/Putting%20Genes%20in%20their%20Place.pdf |journal=Journal of Biosciences |volume=27 |issue= 2|pages=97–104 |doi=10.1007/bf02703765 |pmid=11937679 |url-status=dead |archiveurl=https://web.archive.org/web/20081217013308/http://www.nymc.edu/sanewman/PDFs/Putting%20Genes%20in%20their%20Place.pdf |archivedate=2008-12-17 }}<br />
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* {{cite journal | last1 = Osborn | first1 = Henry F. | authorlink = Henry Fairfield Osborn | year = 1896 | title = Ontogenic and Phylogenic Variation| journal = Science | volume = 4 | issue = 100| pages = 786–789 | doi=10.1126/science.4.100.786| pmid = 17734840 | url = https://zenodo.org/record/1448151 }}<br />
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* [[Robert J. Richards|Richards, Robert J.]]; ''Darwin and the Emergence of Evolutionary Theories of Mind and Behavior'', Chicago: The University of Chicago Press, 1987.<br />
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* {{cite journal | last = Simpson |first=G. Gaylord |authorlink=George Gaylord Simpson | year = 1953 | title = The Baldwin effect | url = | journal = Evolution | volume = 7 | issue = 2| pages = 110–117 | doi=10.2307/2405746|jstor=2405746 }}<br />
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* {{cite journal |last1=Sznajder |first1=B. |last2=Sabelis |first2=M. W. |last3=Egas |first3=M. |title=How Adaptive Learning Affects Evolution: Reviewing Theory on the Baldwin Effect |journal=Evolutionary Biology |date=September 2012|volume=39|issue=3 |pages=301–310 |doi=10.1007/s11692-011-9155-2 |pmid=22923852 |pmc=3423563}}<!--a review paper, a recent secondary source--><br />
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==External links 外部链接==<br />
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* [http://www.ecs.soton.ac.uk/~harnad/Hypermail/Cognition.Sociobiology.98/0101.html Baldwinian evolution]<br />
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* [https://web.archive.org/web/20060810005608/http://www2.create.human.nagoya-u.ac.jp/baldwin/ Bibliography]<br />
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[[Category:Evolutionary biology]]<br />
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Category:Evolutionary biology<br />
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分类: 进化生物学<br />
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[[Category:Selection]]<br />
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Category:Selection<br />
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类别: 选择<br />
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<small>This page was moved from [[wikipedia:en:Baldwin effect]]. Its edit history can be viewed at [[鲍德温效应/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%B8%9F%E7%BE%A4%E7%AE%97%E6%B3%95&diff=10832鸟群算法2020-07-19T13:26:55Z<p>Dorr：</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。<br />
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{{short description|Artificial life program, developed by Craig Reynolds in 1986, which simulates the flocking behaviour of birds}}<br />
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{{Use mdy dates|date=July 2011}}<br />
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{{for|the reptilian family|Boidae}}<br />
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[[File:Boids OpenGL example.webm|thumb|A Boids example created in [[OpenGL]]]]<br />
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A Boids example created in [[OpenGL]]<br />
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在[[[ OpenGL ]]中创建的 Boids 示例<br />
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'''Boids''' is an [[artificial life]] program, developed by [[Craig Reynolds (computer graphics)|Craig Reynolds]] in 1986, which simulates the [[Flocking (behavior)|flocking]] behaviour of [[Bird|birds]]. His paper on this topic was published in 1987 in the proceedings of the [[Association for Computing Machinery|ACM]] [[SIGGRAPH]] conference. <ref>{{Cite book<br />
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Boids is an artificial life program, developed by Craig Reynolds in 1986, which simulates the flocking behaviour of birds. His paper on this topic was published in 1987 in the proceedings of the ACM SIGGRAPH conference. <ref>{{Cite book<br />
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最后1雷诺<br />
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第一名: 克雷格<br />
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| author1-link=Craig Reynolds (computer_graphics)<br />
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| author1-link=Craig Reynolds (computer_graphics)<br />
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| author1-link Craig Reynolds (计算机图形学)<br />
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| title=Flocks, herds and schools: A distributed behavioral model.<br />
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| title=Flocks, herds and schools: A distributed behavioral model.<br />
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羊群、畜群和学校: 一个分布式的行为模型。<br />
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1987年<br />
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| journal=SIGGRAPH '87: Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques<br />
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87: 第14届计算机图形学与互动技术年度会议论文集<br />
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| publisher=[[Association for Computing Machinery]]<br />
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| publisher=Association for Computing Machinery<br />
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| 出版商计算机协会<br />
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| pages=25–34<br />
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| isbn=978-0-89791-227-3<br />
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| isbn=978-0-89791-227-3<br />
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[国际标准图书编号978-0-89791-227-3]<br />
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| doi=10.1145/37401.37406<br />
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| doi=10.1145/37401.37406<br />
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10.1145 / 37401.37406<br />
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| citeseerx=10.1.1.103.7187<br />
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The name "boid" corresponds to a shortened version of "bird-oid object", which refers to a bird-like object.<ref>{{Cite journal<br />
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The name "boid" corresponds to a shortened version of "bird-oid object", which refers to a bird-like object.<ref>{{Cite journal<br />
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第二名: 乔纳森<br />
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| title=A review of particle swarm optimization. Part I: background and development<br />
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| title=A review of particle swarm optimization. Part I: background and development<br />
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2012年10月12日 | 标题: 粒子群优化的回顾。第一部分: 背景与发展<br />
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|date=July 2007<br />
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2007年7月<br />
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| journal=Natural Computing<br />
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自然计算杂志<br />
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10.1007 / s11047-007-9049-5<br />
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}}</ref> Incidentally, "boid" is also a New York Metropolitan dialect pronunciation for "bird".<br />
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}}</ref> Incidentally, "boid" is also a New York Metropolitan dialect pronunciation for "bird".<br />
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'''鸟群算法 Boids'''是模拟鸟类群集行为的人工生命项目，由'''克雷格·雷诺兹 Craig Reynolds'''于1986年开发。该模型基于常用于计算机动画或计算机辅助设计的计算机三维几何。他关于这个主题的论文发表在1987年的 ACM SIGGRAPH（美国计算机协会计算机图形专业组组织的计算机图形学顶级年度会议）会议记录上。“ boid”是“ bird-oid object”的缩写，指一个类鸟对象。恰巧，“boid”也是纽约都市方言中“bird”的发音。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）补充自http://www.red3d.com/cwr/boids/：It was based on three dimensional computational geometry of the sort normally used in computer animation or computer aided design.<br />
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{{multiple image|direction=vertical|image1=Rule separation.gif|caption1=Separation|image2=Rule alignment.gif|caption2=Alignment|image3=Rule cohesion.gif|caption3=Cohesion|header=Rules applied in simple Boids}}<br />
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As with most artificial life simulations, Boids is an example of [[emergence|emergent]] behavior; that is, the complexity of Boids arises from the interaction of individual agents (the boids, in this case) adhering to a set of simple rules. The rules applied in the simplest Boids world are as follows:<br />
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As with most artificial life simulations, Boids is an example of emergent behavior; that is, the complexity of Boids arises from the interaction of individual agents (the boids, in this case) adhering to a set of simple rules. The rules applied in the simplest Boids world are as follows:<br />
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与大多数人工生命模拟一样，Boids 是'''涌现 Emergent'''行为的一个例子; 也就是说，Boids 的复杂性来自于遵循一系列简单规则的个体代理（这里是 Boids）的相互作用。在最简单的Bolds 世界中适用的规则如下，其描述了鸟群中的个体如何根据周边同伴的位置和速度移动:<br />
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* '''separation''': [[wikt:steer#Verb 2|steer]] to avoid crowding local flockmates<br />
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* '''alignment''': steer towards the average heading of local flockmates<br />
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* '''cohesion''': steer to move towards the average position (center of mass) of local flockmates<br />
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* '''分离 Separation''': 移动以避开群体拥挤处<br />
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* '''对齐 Alignment''': 转向群体的平均航向<br />
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* '''靠近 Cohesion''': 朝群体的平均位置（质心）移动<br />
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每个 boid 个体都可以得知整体的几何参数，但群体要求其只对其周围某个小邻近范围作出反应。该邻近范围由一个距离（从该个体的中心算起）和一个角度（从其飞行方向算起）决定。此范围外的同伴不予考虑。该临近范围可以认为是一个有限知觉的模型（就像浑浊的水中的鱼一样）；但更恰当的想法可能是，其定义了鸟群影响了个体转向的区域范围。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）补充自http://www.red3d.com/cwr/boids/：Each boid has direct access to the whole scene's geometric description, but flocking requires that it reacts only to flockmates within a certain small neighborhood around itself. The neighborhood is characterized by a distance (measured from the center of the boid) and an angle, measured from the boid's direction of flight. Flockmates outside this local neighborhood are ignored. The neighborhood could be considered a model of limited perception (as by fish in murky water) but it is probably more correct to think of it as defining the region in which flockmates influence a boids steering.<br />
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More complex rules can be added, such as obstacle avoidance and goal seeking.<br />
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More complex rules can be added, such as obstacle avoidance and goal seeking.<br />
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可以添加更复杂的规则，如避障和寻找目标。<br />
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The basic model has been extended in several different ways since Reynolds proposed it. For instance, Delgado-Mata et&nbsp;al.<ref><br />
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The basic model has been extended in several different ways since Reynolds proposed it. For instance, Delgado-Mata et&nbsp;al.<ref><br />
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|first1= Carlos | last1= Delgado-Mata<br />
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卡洛斯，德尔加多，马塔<br />
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|title= On the use of Virtual Animals with Artificial Fear in Virtual Environments<br />
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|title= On the use of Virtual Animals with Artificial Fear in Virtual Environments<br />
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论虚拟动物在虚拟环境中的人为恐惧<br />
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|journal= New Generation Computing<br />
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|journal= New Generation Computing<br />
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| 杂志新一代计算<br />
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|volume=25 |issue=2 |pages= 145–169<br />
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|volume=25 |issue=2 |pages= 145–169<br />
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|date=2007<br />
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|doi= 10.1007/s00354-007-0009-5}}</ref><br />
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|doi= 10.1007/s00354-007-0009-5}}</ref><br />
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| doi 10.1007 / s00354-007-0009-5} / ref<br />
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extended the basic model to incorporate the effects of fear. Olfaction was used to transmit emotion between animals, through pheromones modelled as particles in a free expansion gas. Hartman and Benes<ref><br />
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extended the basic model to incorporate the effects of fear. Olfaction was used to transmit emotion between animals, through pheromones modelled as particles in a free expansion gas. Hartman and Benes<ref><br />
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{{cite journal<br />
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{{cite journal<br />
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{引用期刊<br />
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|first1= Christopher | last1= Hartman<br />
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1 Christopher | last 1 Hartman<br />
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|first2= Bedr̆ich |last2= Benes̆<br />
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|first2= Bedr̆ich |last2= Benes̆<br />
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|first2= Bedr̆ich |last2= Benes̆<br />
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|title= Autonomous boids<br />
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|title= Autonomous boids<br />
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标题自主 boids<br />
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|journal= Computer Animation and Virtual Worlds<br />
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|journal= Computer Animation and Virtual Worlds<br />
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计算机动画与虚拟世界<br />
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|volume=17 |issue=3–4 |pages= 199–206<br />
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第17卷第3-4期，第199-206页<br />
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|doi= 10.1002/cav.123}}</ref><br />
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introduced a complementary force to the alignment that they call the change of leadership. This steer defines the chance of the boid to become a leader and try to escape.<br />
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introduced a complementary force to the alignment that they call the change of leadership. This steer defines the chance of the boid to become a leader and try to escape.<br />
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该基本模型自雷诺兹提出以来，已用多种方法扩展。例如，Delgado-Mata 等人扩展基本模型以加入恐惧的影响。利过一种可用自由膨胀气体中的粒子模拟的信息素，动物间用嗅觉来传递情感。'''哈特曼 Hartman'''和'''贝内斯 Benes'''为定位过程引入了一种补充力量，他们称之为领导权的更迭。这条决定了某只鸟成为领导者及试图逃跑的机会。<br />
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The movement of Boids can be characterized as either chaotic (splitting groups and wild behaviour) or orderly. Unexpected behaviours, such as splitting flocks and reuniting after avoiding obstacles, can be considered emergent.<br />
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The movement of Boids can be characterized as either chaotic (splitting groups and wild behaviour) or orderly. Unexpected behaviours, such as splitting flocks and reuniting after avoiding obstacles, can be considered emergent.<br />
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Boids 的运动可以表现为混乱（分裂的群体和狂野的行为）或有序。意想不到的行为，比如群体分散和避开障碍后的团聚可以被认为是应急的。<br />
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The boids framework is often used in computer graphics, providing realistic-looking representations of flocks of birds and other creatures, such as schools of fish or herds of animals. It was for instance used in the 1998 video game ''[[Half-Life (video game)|Half-Life]]'' for the flying bird-like creatures seen at the end of the game on [[Locations of Half-Life#Xen|Xen]], named "boid" in the game files.<br />
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The boids framework is often used in computer graphics, providing realistic-looking representations of flocks of birds and other creatures, such as schools of fish or herds of animals. It was for instance used in the 1998 video game Half-Life for the flying bird-like creatures seen at the end of the game on Xen, named "boid" in the game files.<br />
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Boids 框架通常用于计算机图形学，提供鸟群和其他生物（如鱼群）的逼真表现。例如，在1998年的视频游戏《'''半条命 Half-Life'''》中，Xen（游戏文件中命名为“ boid”）中游戏结束时出现的类似鸟类的飞行生物就使用了该框架。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）存疑<br />
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The Boids model can be used for direct control and stabilization of teams of simple Unmanned Ground Vehicles (UGV)<ref>{{cite conference |title= Design and analysis of Group Escape Behavior for distributed autonomous mobile robots |first1= Hongkyu |last1= Min |first2= Zhidong |last2= Wang |year= 2011 |conference= IEEE International Conference on Robotics and Automation (ICRA)|doi= 10.1109/ICRA.2011.5980123 }}</ref> or [[Micro air vehicle|Micro Aerial Vehicles]] (MAV)<ref>{{cite conference |title= Swarms of micro aerial vehicles stabilized under a visual relative localization |first1= Martin |last1= Saska |first2= Vakula |last2= Jan |first3= Preucil |last3= Libor |year= 2014 |conference= IEEE International Conference on Robotics and Automation (ICRA)|doi= 10.1109/ICRA.2014.6907374 }}</ref> in [[swarm robotics]]. For stabilization of heterogeneous UAV-UGV teams, the model was adapted for using onboard relative localization by Saska et&nbsp;al.<ref>{{cite conference |title= Coordination and Navigation of Heterogeneous UAVs-UGVs Teams Localized by a Hawk-Eye Approach |first1= Martin |last1= Saska |first2= Vonasek |last2= Vojtech |first3= Krajnik |last3= Tomas |first4= Preucil |last4= Libor |year= 2012 |conference= IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)|url=http://labe.felk.cvut.cz/~tkrajnik/ardrone/articles/formace.pdf}}</ref><br />
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The Boids model can be used for direct control and stabilization of teams of simple Unmanned Ground Vehicles (UGV) or Micro Aerial Vehicles (MAV) in swarm robotics. For stabilization of heterogeneous UAV-UGV teams, the model was adapted for using onboard relative localization by Saska et&nbsp;al.<br />
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Boids 模型可用于'''[[集群机器人]] Swarm Robotics'''中简单的无人地面车辆（UGV）或'''微型飞行器 Micro Aerial Vehicles'''（MAV）群体的直接控制和稳定。为了稳定异质 UAV-UGV 团队，Saska 等人将该模型用于板载相对定位。<br />
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At the time of proposal, Reynolds' approach represented a giant step forward compared to the traditional techniques used in computer animation for motion pictures. The first animation created with the model was ''[[Stanley and Stella in: Breaking the Ice]]'' (1987), followed by a feature film debut in [[Tim Burton]]'s film ''[[Batman Returns]]'' (1992) with computer generated bat swarms and armies of penguins marching through the streets of Gotham City.<ref>{{cite journal | url=http://lrss.fri.uni-lj.si/people/ilbajec/papers/ilb_ab09.pdf | last1=Lebar Bajec |first1= Iztok |first2= Frank H. |last2= Heppner | date=2009 | title=Organized flight in birds | journal=Animal Behaviour | volume = 78 | issue = 4 | pages = 777–789 | doi = 10.1016/j.anbehav.2009.07.007 }}</ref><br />
<br />
At the time of proposal, Reynolds' approach represented a giant step forward compared to the traditional techniques used in computer animation for motion pictures. The first animation created with the model was Stanley and Stella in: Breaking the Ice (1987), followed by a feature film debut in Tim Burton's film Batman Returns (1992) with computer generated bat swarms and armies of penguins marching through the streets of Gotham City.<br />
<br />
<br />
在当时提出时，雷诺兹的方法相比于传统的计算机动画电影技术是一个巨大的进步。第一部利用了此模型的动画片是《'''史丹利和史黛拉: 破冰 Stanley and Stella in: Breaking The Ice'''》（1987），之后是'''蒂姆·伯顿 Tim Burton'''的故事片《'''蝙蝠侠归来 Batman Returns'''》（1992），电脑合成的蝙蝠群和成群的企鹅行进穿过'''哥谭市 Gotham City'''的街道。<br />
<br />
<br />
<br />
The boids model has been used for other interesting applications. It has been applied to automatically program Internet multi-channel radio stations.<ref><br />
<br />
The boids model has been used for other interesting applications. It has been applied to automatically program Internet multi-channel radio stations.<ref><br />
<br />
<br />
<br />
{{cite conference<br />
<br />
{{cite conference<br />
<br />
{引用会议<br />
<br />
| title = DJ-boids: emergent collective behavior as multichannel radio station programming<br />
<br />
| title = DJ-boids: emergent collective behavior as multichannel radio station programming<br />
<br />
多频道电台节目中的紧急集体行为<br />
<br />
| first1 = Jesús | last1 = Ibáñez<br />
<br />
| first1 = Jesús | last1 = Ibáñez<br />
<br />
| first1 = Jesús | last1 = Ibáñez<br />
<br />
| first2 = Antonio F. | last2 = Gómez-Skarmeta <br />
<br />
| first2 = Antonio F. | last2 = Gómez-Skarmeta <br />
<br />
| first2 = Antonio F. | last2 = Gómez-Skarmeta<br />
<br />
| first3 = Josep | last3 = Blat<br />
<br />
| first3 = Josep | last3 = Blat<br />
<br />
| first3 = Josep | last3 = Blat<br />
<br />
| date = 2003<br />
<br />
| date = 2003<br />
<br />
2003年<br />
<br />
| booktitle = Proceedings of the 8th international conference on Intelligent User Interfaces<br />
<br />
| booktitle = Proceedings of the 8th international conference on Intelligent User Interfaces<br />
<br />
第八届智能用户界面国际会议论文集<br />
<br />
| pages = 248–250<br />
<br />
| pages = 248–250<br />
<br />
第248-250页<br />
<br />
| doi = 10.1145/604045.604089<br />
<br />
| doi = 10.1145/604045.604089<br />
<br />
10.1145 / 604045.604089<br />
<br />
}}</ref><br />
<br />
}}</ref><br />
<br />
{} / ref<br />
<br />
It has also been used for visualizing information<ref><br />
<br />
It has also been used for visualizing information<ref><br />
<br />
<br />
{{cite conference<br />
<br />
{{cite conference<br />
<br />
{引用会议<br />
<br />
| title = Time-Varying Data Visualization Using Information Flocking Boids<br />
<br />
| title = Time-Varying Data Visualization Using Information Flocking Boids<br />
<br />
使用信息植群的时变数据可视化<br />
<br />
| last = Moere | first = A V<br />
<br />
| last = Moere | first = A V<br />
<br />
最后一个 Moere | 第一个 a v<br />
<br />
| date = 2004<br />
<br />
| date = 2004<br />
<br />
2004年<br />
<br />
| booktitle = Proceedings of the IEEE Symposium on Information Visualization<br />
<br />
| booktitle = Proceedings of the IEEE Symposium on Information Visualization<br />
<br />
美国电气和电子工程师学会信息可视化研讨会论文集<br />
<br />
| pages = 97–104<br />
<br />
| pages = 97–104<br />
<br />
第97-104页<br />
<br />
| doi = 10.1109/INFVIS.2004.65<br />
<br />
| doi = 10.1109/INFVIS.2004.65<br />
<br />
10.1109 / INFVIS. 2004.65<br />
<br />
}}</ref><br />
<br />
}}</ref><br />
<br />
{} / ref<br />
<br />
and for optimization tasks.<ref><br />
<br />
and for optimization tasks.<ref><br />
<br />
<br />
<br />
{{cite journal<br />
<br />
{{cite journal<br />
<br />
{引用期刊<br />
<br />
|first1 = Zhihua | last1 = Cui<br />
<br />
|first1 = Zhihua | last1 = Cui<br />
<br />
最新一期崔<br />
<br />
|first2 = Zhongzhi | last2 = Shi<br />
<br />
|first2 = Zhongzhi | last2 = Shi<br />
<br />
|first2 = Zhongzhi | last2 = Shi<br />
<br />
|title= Boid particle swarm optimisation<br />
<br />
|title= Boid particle swarm optimisation<br />
<br />
博伊德粒子群优化<br />
<br />
|journal= International Journal of Innovative Computing and Applications<br />
<br />
|journal= International Journal of Innovative Computing and Applications<br />
<br />
国际创新计算与应用杂志<br />
<br />
|volume=2 |issue=2 |pages= 77–85<br />
<br />
|volume=2 |issue=2 |pages= 77–85<br />
<br />
第二期，第77-85页<br />
<br />
|date=2009<br />
<br />
|date=2009<br />
<br />
2009年<br />
<br />
|doi= 10.1504/IJICA.2009.031778}}</ref><br />
<br />
|doi= 10.1504/IJICA.2009.031778}}</ref><br />
<br />
| doi 10.1504 / IJICA. 2009.031778} / ref<br />
<br />
Boids 模型已经被用于其他有趣应用。该系统已应用于互联网多频道广播电台的自动节目编排，以及可视化信息和优化任务。<br />
<br />
==See also 参见==<br />
<br />
*[[Swarm intelligence]] 群体智能<br />
<br />
*[[Collective cell migration]] 集体细胞迁移<br />
<br />
<br />
<br />
==References 参考资料==<br />
<br />
{{Reflist}}<br />
<br />
<br />
<br />
==External links 外部链接==<br />
<br />
* [http://www.red3d.com/cwr/boids/ Craig Reynolds' Boids page]<br />
<br />
* [http://www.vergenet.net/~conrad/boids/pseudocode.html Explanation of algorithm in pseudocode]<br />
<br />
* [https://gpolo.github.com/birdflocking JavaScript implementation]<br />
<br />
*[https://lufemas.github.io/boid-ai-implementation/ JavaScript implementation with Phaser Framework]<br />
<br />
* [https://web.archive.org/web/20080205014305/http://www.navgen.com/3d_boids/ 3D Boids Simulation using OpenGL, used by the BBC's Natural History Unit]<br />
<br />
*[https://black-square.github.io/BirdFlock/ Live In-Browser 3D Simulation of Bird Flocking Behavior in Unity3D] – Open Source implementation for Windows, Linux and Mac<br />
<br />
* [https://github.com/piskvorky/PredatorPrey UNIX+Windows open source implementation in C++, using OpenGL and simulation controls]<br />
<br />
* [https://github.com/tofti/javafx-boids A java implementation using the javafx API]<br />
<br />
*GLSchool - part of [[XScreenSaver]] software<br />
<br />
<br />
<br />
{{collective animal behaviour}}<br />
<br />
<br />
<br />
==编者推荐==<br />
===[https://campus.swarma.org/ 集智学园]课程：[https://campus.swarma.org/course/1104 透过人工鸟群Boid模型学习List的使用]===<br />
<br />
讲师：[[张江]](北师大系统科学学院教授、博士生导师，集智俱乐部、集智学园创始人。)<br />
<br />
本课程通过数个案例教会大家如何去动手搭建一个多主体仿真模型，以及如何利用NetLogo去实现。从生命游戏到人工鸟群，从模拟经济系统到病毒沿网络的传播，通过循序渐进的案例，该课程带你逐步走入NetLogo多主体建模的神奇世界。<br />
<br />
===[https://swarma.org/ 集智俱乐部]扩展阅读：[https://swarma.org/?p=16504 张江：乌合之众还是群智涌现？鸟群知道答案]===<br />
<br />
生命如何起源，智能如何涌现，因果如何反转，复杂谜题，引人深思。2019年7月13日，集智俱乐部创始人[[张江]]教授在混沌大学授课，详细解读了复杂系统中的混沌、涌现与进化，此文整理自课堂内容。<br />
<br />
===[https://v.qq.com/x/page/z13491sn7fy.html 动画片《Stanley and Stella in: Breaking the Ice (1987)》]===<br />
<br />
Craig Reynolds 与 Symbolics Graphics 和 Whitney / Demos Production 的同事合作制作的利用 boids 模型的小短片，讲述了飞鸟与鱼相爱，鸟儿冲破冰面与鱼相见的故事。影片最早于87年在 SIGGRAPH 电子剧院展出。<br />
<br />
--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）补充自http://www.red3d.com/cwr/boids/：In cooperation with many coworkers at the Symbolics Graphics Division and Whitney / Demos Productions, we made an animated short featuring the boids model called Stanley and Stella in: Breaking the Ice. This film was first shown at the Electronic Theater at SIGGRAPH '87.<br />
<br />
[[Category:Artificial life]]<br />
<br />
Category:Artificial life<br />
<br />
类别: 人工生命<br />
<br />
[[Category:1986 software]]<br />
<br />
Category:1986 software<br />
<br />
类别: 1986软件<br />
<br />
<noinclude><br />
<br />
<small>This page was moved from [[wikipedia:en:Boids]]. Its edit history can be viewed at [[鸟群算法/edithistory]]</small></noinclude><br />
<br />
[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%B8%9F%E7%BE%A4%E7%AE%97%E6%B3%95&diff=10829鸟群算法2020-07-19T13:17:20Z<p>Dorr：</p>
<hr />
<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。<br />
<br />
{{short description|Artificial life program, developed by Craig Reynolds in 1986, which simulates the flocking behaviour of birds}}<br />
<br />
{{Use mdy dates|date=July 2011}}<br />
<br />
{{for|the reptilian family|Boidae}}<br />
<br />
[[File:Boids OpenGL example.webm|thumb|A Boids example created in [[OpenGL]]]]<br />
<br />
A Boids example created in [[OpenGL]]<br />
<br />
在[[[ OpenGL ]]中创建的 Boids 示例<br />
<br />
'''Boids''' is an [[artificial life]] program, developed by [[Craig Reynolds (computer graphics)|Craig Reynolds]] in 1986, which simulates the [[Flocking (behavior)|flocking]] behaviour of [[Bird|birds]]. His paper on this topic was published in 1987 in the proceedings of the [[Association for Computing Machinery|ACM]] [[SIGGRAPH]] conference. <ref>{{Cite book<br />
<br />
Boids is an artificial life program, developed by Craig Reynolds in 1986, which simulates the flocking behaviour of birds. His paper on this topic was published in 1987 in the proceedings of the ACM SIGGRAPH conference. <ref>{{Cite book<br />
<br />
<br />
<br />
| last1=Reynolds<br />
<br />
| last1=Reynolds<br />
<br />
最后1雷诺<br />
<br />
| first1=Craig<br />
<br />
| first1=Craig<br />
<br />
第一名: 克雷格<br />
<br />
| author1-link=Craig Reynolds (computer_graphics)<br />
<br />
| author1-link=Craig Reynolds (computer_graphics)<br />
<br />
| author1-link Craig Reynolds (计算机图形学)<br />
<br />
| title=Flocks, herds and schools: A distributed behavioral model.<br />
<br />
| title=Flocks, herds and schools: A distributed behavioral model.<br />
<br />
羊群、畜群和学校: 一个分布式的行为模型。<br />
<br />
| year=1987<br />
<br />
| year=1987<br />
<br />
1987年<br />
<br />
| number=<br />
<br />
| number=<br />
<br />
数字<br />
<br />
| journal=SIGGRAPH '87: Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques<br />
<br />
| journal=SIGGRAPH '87: Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques<br />
<br />
87: 第14届计算机图形学与互动技术年度会议论文集<br />
<br />
| publisher=[[Association for Computing Machinery]]<br />
<br />
| publisher=Association for Computing Machinery<br />
<br />
| 出版商计算机协会<br />
<br />
| pages=25–34<br />
<br />
| pages=25–34<br />
<br />
第25-34页<br />
<br />
| isbn=978-0-89791-227-3<br />
<br />
| isbn=978-0-89791-227-3<br />
<br />
[国际标准图书编号978-0-89791-227-3]<br />
<br />
| doi=10.1145/37401.37406<br />
<br />
| doi=10.1145/37401.37406<br />
<br />
10.1145 / 37401.37406<br />
<br />
| citeseerx=10.1.1.103.7187<br />
<br />
| citeseerx=10.1.1.103.7187<br />
<br />
10.1.1.103.7187<br />
<br />
}}</ref><br />
<br />
}}</ref><br />
<br />
{} / ref<br />
<br />
The name "boid" corresponds to a shortened version of "bird-oid object", which refers to a bird-like object.<ref>{{Cite journal<br />
<br />
The name "boid" corresponds to a shortened version of "bird-oid object", which refers to a bird-like object.<ref>{{Cite journal<br />
<br />
<br />
<br />
| last1=Banks<br />
<br />
| last1=Banks<br />
<br />
1 Banks<br />
<br />
| first1=Alec<br />
<br />
| first1=Alec<br />
<br />
首先 Alec<br />
<br />
| last2=Vincent<br />
<br />
| last2=Vincent<br />
<br />
最后2文森特<br />
<br />
| first2=Jonathan<br />
<br />
| first2=Jonathan<br />
<br />
第二名: 乔纳森<br />
<br />
| last3=Anyakoha<br />
<br />
| last3=Anyakoha<br />
<br />
3 / lastanyakoha<br />
<br />
| first3=Chukwudi<br />
<br />
| first3=Chukwudi<br />
<br />
| first3 Chukwudi<br />
<br />
| title=A review of particle swarm optimization. Part I: background and development<br />
<br />
| title=A review of particle swarm optimization. Part I: background and development<br />
<br />
2012年10月12日 | 标题: 粒子群优化的回顾。第一部分: 背景与发展<br />
<br />
|date=July 2007<br />
<br />
|date=July 2007<br />
<br />
2007年7月<br />
<br />
| journal=Natural Computing<br />
<br />
| journal=Natural Computing<br />
<br />
自然计算杂志<br />
<br />
| volume=6<br />
<br />
| volume=6<br />
<br />
第六卷<br />
<br />
| issue=4<br />
<br />
| issue=4<br />
<br />
第四期<br />
<br />
| pages=467–484<br />
<br />
| pages=467–484<br />
<br />
第467-484页<br />
<br />
| doi=10.1007/s11047-007-9049-5<br />
<br />
| doi=10.1007/s11047-007-9049-5<br />
<br />
10.1007 / s11047-007-9049-5<br />
<br />
| citeseerx=10.1.1.605.5879<br />
<br />
| citeseerx=10.1.1.605.5879<br />
<br />
10.1.1.605.5879<br />
<br />
}}</ref> Incidentally, "boid" is also a New York Metropolitan dialect pronunciation for "bird".<br />
<br />
}}</ref> Incidentally, "boid" is also a New York Metropolitan dialect pronunciation for "bird".<br />
<br />
} / ref <br />
'''鸟群算法 Boids'''是模拟鸟类群集行为的人工生命项目，由'''克雷格·雷诺兹 Craig Reynolds'''于1986年开发。该模型基于常用于计算机动画或计算机辅助设计的计算机三维几何。他关于这个主题的论文发表在1987年的 ACM SIGGRAPH（美国计算机协会计算机图形专业组组织的计算机图形学顶级年度会议）会议记录上。“ boid”是“ bird-oid object”的缩写，指一个类鸟对象。恰巧，“boid”也是纽约都市方言中“bird”的发音。<br />
<br />
--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）补充自http://www.red3d.com/cwr/boids/：It was based on three dimensional computational geometry of the sort normally used in computer animation or computer aided design.<br />
<br />
{{multiple image|direction=vertical|image1=Rule separation.gif|caption1=Separation|image2=Rule alignment.gif|caption2=Alignment|image3=Rule cohesion.gif|caption3=Cohesion|header=Rules applied in simple Boids}}<br />
<br />
As with most artificial life simulations, Boids is an example of [[emergence|emergent]] behavior; that is, the complexity of Boids arises from the interaction of individual agents (the boids, in this case) adhering to a set of simple rules. The rules applied in the simplest Boids world are as follows:<br />
<br />
As with most artificial life simulations, Boids is an example of emergent behavior; that is, the complexity of Boids arises from the interaction of individual agents (the boids, in this case) adhering to a set of simple rules. The rules applied in the simplest Boids world are as follows:<br />
<br />
与大多数人工生命模拟一样，Boids 是'''涌现 Emergent'''行为的一个例子; 也就是说，Boids 的复杂性来自于遵循一系列简单规则的个体代理（这里是 Boids）的相互作用。在最简单的Bolds 世界中适用的规则如下，其描述了鸟群中的个体如何根据周边同伴的位置和速度移动:<br />
<br />
<br />
<br />
* '''separation''': [[wikt:steer#Verb 2|steer]] to avoid crowding local flockmates<br />
<br />
* '''alignment''': steer towards the average heading of local flockmates<br />
<br />
* '''cohesion''': steer to move towards the average position (center of mass) of local flockmates<br />
<br />
<br />
* '''分离 Separation''': 移动以避开群体拥挤处<br />
<br />
* '''对齐 Alignment''': 转向群体的平均航向<br />
<br />
* '''靠近 Cohesion''': 朝群体的平均位置（质心）移动<br />
<br />
<br />
<br />
每个 boid 个体都可以得知整体的几何参数，但群体要求其只对其周围某个小邻近范围作出反应。该邻近范围由一个距离（从该个体的中心算起）和一个角度（从其飞行方向算起）决定。此范围外的同伴不予考虑。该临近范围可以认为是一个有限知觉的模型（就像浑浊的水中的鱼一样）；但更恰当的想法可能是，其定义了鸟群影响了个体转向的区域范围。<br />
<br />
--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）补充自http://www.red3d.com/cwr/boids/：Each boid has direct access to the whole scene's geometric description, but flocking requires that it reacts only to flockmates within a certain small neighborhood around itself. The neighborhood is characterized by a distance (measured from the center of the boid) and an angle, measured from the boid's direction of flight. Flockmates outside this local neighborhood are ignored. The neighborhood could be considered a model of limited perception (as by fish in murky water) but it is probably more correct to think of it as defining the region in which flockmates influence a boids steering.<br />
<br />
More complex rules can be added, such as obstacle avoidance and goal seeking.<br />
<br />
More complex rules can be added, such as obstacle avoidance and goal seeking.<br />
<br />
可以添加更复杂的规则，如避障和寻找目标。<br />
<br />
<br />
<br />
The basic model has been extended in several different ways since Reynolds proposed it. For instance, Delgado-Mata et&nbsp;al.<ref><br />
<br />
The basic model has been extended in several different ways since Reynolds proposed it. For instance, Delgado-Mata et&nbsp;al.<ref><br />
<br />
<br />
<br />
{{cite journal<br />
<br />
{{cite journal<br />
<br />
{引用期刊<br />
<br />
|first1= Carlos | last1= Delgado-Mata<br />
<br />
|first1= Carlos | last1= Delgado-Mata<br />
<br />
卡洛斯，德尔加多，马塔<br />
<br />
|first2= Jesus Ibanez | last2= Martinez<br />
<br />
|first2= Jesus Ibanez | last2= Martinez<br />
<br />
2 Jesus Ibanez | last 2 Martinez<br />
<br />
|first3= Simon | last3= Bee<br />
<br />
|first3= Simon | last3= Bee<br />
<br />
| first3 Simon | last3 Bee<br />
<br />
|first4= Rocio | last4= Ruiz-Rodarte<br />
<br />
|first4= Rocio | last4= Ruiz-Rodarte<br />
<br />
4 Rocio | last 4 Ruiz-Rodarte<br />
<br />
|first5= Ruth | last5= Aylett<br />
<br />
|first5= Ruth | last5= Aylett<br />
<br />
| first5 Ruth | last 5 Aylett<br />
<br />
|title= On the use of Virtual Animals with Artificial Fear in Virtual Environments<br />
<br />
|title= On the use of Virtual Animals with Artificial Fear in Virtual Environments<br />
<br />
论虚拟动物在虚拟环境中的人为恐惧<br />
<br />
|journal= New Generation Computing<br />
<br />
|journal= New Generation Computing<br />
<br />
| 杂志新一代计算<br />
<br />
|volume=25 |issue=2 |pages= 145–169<br />
<br />
|volume=25 |issue=2 |pages= 145–169<br />
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第25卷，第2期，第145-169页<br />
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|date=2007<br />
<br />
|date=2007<br />
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2007年<br />
<br />
|doi= 10.1007/s00354-007-0009-5}}</ref><br />
<br />
|doi= 10.1007/s00354-007-0009-5}}</ref><br />
<br />
| doi 10.1007 / s00354-007-0009-5} / ref<br />
<br />
extended the basic model to incorporate the effects of fear. Olfaction was used to transmit emotion between animals, through pheromones modelled as particles in a free expansion gas. Hartman and Benes<ref><br />
<br />
extended the basic model to incorporate the effects of fear. Olfaction was used to transmit emotion between animals, through pheromones modelled as particles in a free expansion gas. Hartman and Benes<ref><br />
<br />
<br />
<br />
{{cite journal<br />
<br />
{{cite journal<br />
<br />
{引用期刊<br />
<br />
|first1= Christopher | last1= Hartman<br />
<br />
|first1= Christopher | last1= Hartman<br />
<br />
1 Christopher | last 1 Hartman<br />
<br />
|first2= Bedr̆ich |last2= Benes̆<br />
<br />
|first2= Bedr̆ich |last2= Benes̆<br />
<br />
|first2= Bedr̆ich |last2= Benes̆<br />
<br />
|title= Autonomous boids<br />
<br />
|title= Autonomous boids<br />
<br />
标题自主 boids<br />
<br />
|journal= Computer Animation and Virtual Worlds<br />
<br />
|journal= Computer Animation and Virtual Worlds<br />
<br />
计算机动画与虚拟世界<br />
<br />
|volume=17 |issue=3–4 |pages= 199–206<br />
<br />
|volume=17 |issue=3–4 |pages= 199–206<br />
<br />
第17卷第3-4期，第199-206页<br />
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|date=July 2006<br />
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|date=July 2006<br />
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2006年7月<br />
<br />
|doi= 10.1002/cav.123}}</ref><br />
<br />
|doi= 10.1002/cav.123}}</ref><br />
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10.1002 / cav. 123} / ref<br />
<br />
introduced a complementary force to the alignment that they call the change of leadership. This steer defines the chance of the boid to become a leader and try to escape.<br />
<br />
introduced a complementary force to the alignment that they call the change of leadership. This steer defines the chance of the boid to become a leader and try to escape.<br />
<br />
该基本模型自雷诺兹提出以来，已用多种方法扩展。例如，Delgado-Mata 等人扩展基本模型以加入恐惧的影响。利过一种可用自由膨胀气体中的粒子模拟的信息素，动物间用嗅觉来传递情感。'''哈特曼 Hartman'''和'''贝内斯 Benes'''为定位过程引入了一种补充力量，他们称之为领导权的更迭。这条决定了某只鸟成为领导者及试图逃跑的机会。<br />
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<br />
<br />
The movement of Boids can be characterized as either chaotic (splitting groups and wild behaviour) or orderly. Unexpected behaviours, such as splitting flocks and reuniting after avoiding obstacles, can be considered emergent.<br />
<br />
The movement of Boids can be characterized as either chaotic (splitting groups and wild behaviour) or orderly. Unexpected behaviours, such as splitting flocks and reuniting after avoiding obstacles, can be considered emergent.<br />
<br />
Boids 的运动可以表现为混乱（分裂的群体和狂野的行为）或有序。意想不到的行为，比如群体分散和避开障碍后的团聚可以被认为是应急的。<br />
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The boids framework is often used in computer graphics, providing realistic-looking representations of flocks of birds and other creatures, such as schools of fish or herds of animals. It was for instance used in the 1998 video game ''[[Half-Life (video game)|Half-Life]]'' for the flying bird-like creatures seen at the end of the game on [[Locations of Half-Life#Xen|Xen]], named "boid" in the game files.<br />
<br />
The boids framework is often used in computer graphics, providing realistic-looking representations of flocks of birds and other creatures, such as schools of fish or herds of animals. It was for instance used in the 1998 video game Half-Life for the flying bird-like creatures seen at the end of the game on Xen, named "boid" in the game files.<br />
<br />
Boids 框架通常用于计算机图形学，提供鸟群和其他生物（如鱼群）的逼真表现。例如，在1998年的视频游戏《'''半条命 Half-Life'''》中，Xen（游戏文件中命名为“ boid”）中游戏结束时出现的类似鸟类的飞行生物就使用了该框架。<br />
<br />
--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）存疑<br />
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The Boids model can be used for direct control and stabilization of teams of simple Unmanned Ground Vehicles (UGV)<ref>{{cite conference |title= Design and analysis of Group Escape Behavior for distributed autonomous mobile robots |first1= Hongkyu |last1= Min |first2= Zhidong |last2= Wang |year= 2011 |conference= IEEE International Conference on Robotics and Automation (ICRA)|doi= 10.1109/ICRA.2011.5980123 }}</ref> or [[Micro air vehicle|Micro Aerial Vehicles]] (MAV)<ref>{{cite conference |title= Swarms of micro aerial vehicles stabilized under a visual relative localization |first1= Martin |last1= Saska |first2= Vakula |last2= Jan |first3= Preucil |last3= Libor |year= 2014 |conference= IEEE International Conference on Robotics and Automation (ICRA)|doi= 10.1109/ICRA.2014.6907374 }}</ref> in [[swarm robotics]]. For stabilization of heterogeneous UAV-UGV teams, the model was adapted for using onboard relative localization by Saska et&nbsp;al.<ref>{{cite conference |title= Coordination and Navigation of Heterogeneous UAVs-UGVs Teams Localized by a Hawk-Eye Approach |first1= Martin |last1= Saska |first2= Vonasek |last2= Vojtech |first3= Krajnik |last3= Tomas |first4= Preucil |last4= Libor |year= 2012 |conference= IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)|url=http://labe.felk.cvut.cz/~tkrajnik/ardrone/articles/formace.pdf}}</ref><br />
<br />
The Boids model can be used for direct control and stabilization of teams of simple Unmanned Ground Vehicles (UGV) or Micro Aerial Vehicles (MAV) in swarm robotics. For stabilization of heterogeneous UAV-UGV teams, the model was adapted for using onboard relative localization by Saska et&nbsp;al.<br />
<br />
Boids 模型可用于'''[[集群机器人]] Swarm Robotics'''中简单的无人地面车辆（UGV）或'''微型飞行器 Micro Aerial Vehicles'''（MAV）群体的直接控制和稳定。为了稳定异质 UAV-UGV 团队，Saska 等人将该模型用于板载相对定位。<br />
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<br />
At the time of proposal, Reynolds' approach represented a giant step forward compared to the traditional techniques used in computer animation for motion pictures. The first animation created with the model was ''[[Stanley and Stella in: Breaking the Ice]]'' (1987), followed by a feature film debut in [[Tim Burton]]'s film ''[[Batman Returns]]'' (1992) with computer generated bat swarms and armies of penguins marching through the streets of Gotham City.<ref>{{cite journal | url=http://lrss.fri.uni-lj.si/people/ilbajec/papers/ilb_ab09.pdf | last1=Lebar Bajec |first1= Iztok |first2= Frank H. |last2= Heppner | date=2009 | title=Organized flight in birds | journal=Animal Behaviour | volume = 78 | issue = 4 | pages = 777–789 | doi = 10.1016/j.anbehav.2009.07.007 }}</ref><br />
<br />
At the time of proposal, Reynolds' approach represented a giant step forward compared to the traditional techniques used in computer animation for motion pictures. The first animation created with the model was Stanley and Stella in: Breaking the Ice (1987), followed by a feature film debut in Tim Burton's film Batman Returns (1992) with computer generated bat swarms and armies of penguins marching through the streets of Gotham City.<br />
<br />
<br />
在提议的时候，雷诺兹的方法相比于传统的计算机动画电影技术是一个巨大的进步。第一部利用了此模型的动画片是《'''史丹利和史黛拉: 破冰 Stanley and Stella in: Breaking The Ice'''》（1987），之后是'''蒂姆·伯顿 Tim Burton'''的故事片《'''蝙蝠侠归来 Batman Returns'''》（1992），电脑合成的蝙蝠群和成群的企鹅行进穿过'''哥谭市 Gotham City'''的街道。<br />
<br />
<br />
<br />
The boids model has been used for other interesting applications. It has been applied to automatically program Internet multi-channel radio stations.<ref><br />
<br />
The boids model has been used for other interesting applications. It has been applied to automatically program Internet multi-channel radio stations.<ref><br />
<br />
<br />
<br />
{{cite conference<br />
<br />
{{cite conference<br />
<br />
{引用会议<br />
<br />
| title = DJ-boids: emergent collective behavior as multichannel radio station programming<br />
<br />
| title = DJ-boids: emergent collective behavior as multichannel radio station programming<br />
<br />
多频道电台节目中的紧急集体行为<br />
<br />
| first1 = Jesús | last1 = Ibáñez<br />
<br />
| first1 = Jesús | last1 = Ibáñez<br />
<br />
| first1 = Jesús | last1 = Ibáñez<br />
<br />
| first2 = Antonio F. | last2 = Gómez-Skarmeta <br />
<br />
| first2 = Antonio F. | last2 = Gómez-Skarmeta <br />
<br />
| first2 = Antonio F. | last2 = Gómez-Skarmeta<br />
<br />
| first3 = Josep | last3 = Blat<br />
<br />
| first3 = Josep | last3 = Blat<br />
<br />
| first3 = Josep | last3 = Blat<br />
<br />
| date = 2003<br />
<br />
| date = 2003<br />
<br />
2003年<br />
<br />
| booktitle = Proceedings of the 8th international conference on Intelligent User Interfaces<br />
<br />
| booktitle = Proceedings of the 8th international conference on Intelligent User Interfaces<br />
<br />
第八届智能用户界面国际会议论文集<br />
<br />
| pages = 248–250<br />
<br />
| pages = 248–250<br />
<br />
第248-250页<br />
<br />
| doi = 10.1145/604045.604089<br />
<br />
| doi = 10.1145/604045.604089<br />
<br />
10.1145 / 604045.604089<br />
<br />
}}</ref><br />
<br />
}}</ref><br />
<br />
{} / ref<br />
<br />
It has also been used for visualizing information<ref><br />
<br />
It has also been used for visualizing information<ref><br />
<br />
<br />
{{cite conference<br />
<br />
{{cite conference<br />
<br />
{引用会议<br />
<br />
| title = Time-Varying Data Visualization Using Information Flocking Boids<br />
<br />
| title = Time-Varying Data Visualization Using Information Flocking Boids<br />
<br />
使用信息植群的时变数据可视化<br />
<br />
| last = Moere | first = A V<br />
<br />
| last = Moere | first = A V<br />
<br />
最后一个 Moere | 第一个 a v<br />
<br />
| date = 2004<br />
<br />
| date = 2004<br />
<br />
2004年<br />
<br />
| booktitle = Proceedings of the IEEE Symposium on Information Visualization<br />
<br />
| booktitle = Proceedings of the IEEE Symposium on Information Visualization<br />
<br />
美国电气和电子工程师学会信息可视化研讨会论文集<br />
<br />
| pages = 97–104<br />
<br />
| pages = 97–104<br />
<br />
第97-104页<br />
<br />
| doi = 10.1109/INFVIS.2004.65<br />
<br />
| doi = 10.1109/INFVIS.2004.65<br />
<br />
10.1109 / INFVIS. 2004.65<br />
<br />
}}</ref><br />
<br />
}}</ref><br />
<br />
{} / ref<br />
<br />
and for optimization tasks.<ref><br />
<br />
and for optimization tasks.<ref><br />
<br />
<br />
<br />
{{cite journal<br />
<br />
{{cite journal<br />
<br />
{引用期刊<br />
<br />
|first1 = Zhihua | last1 = Cui<br />
<br />
|first1 = Zhihua | last1 = Cui<br />
<br />
最新一期崔<br />
<br />
|first2 = Zhongzhi | last2 = Shi<br />
<br />
|first2 = Zhongzhi | last2 = Shi<br />
<br />
|first2 = Zhongzhi | last2 = Shi<br />
<br />
|title= Boid particle swarm optimisation<br />
<br />
|title= Boid particle swarm optimisation<br />
<br />
博伊德粒子群优化<br />
<br />
|journal= International Journal of Innovative Computing and Applications<br />
<br />
|journal= International Journal of Innovative Computing and Applications<br />
<br />
国际创新计算与应用杂志<br />
<br />
|volume=2 |issue=2 |pages= 77–85<br />
<br />
|volume=2 |issue=2 |pages= 77–85<br />
<br />
第二期，第77-85页<br />
<br />
|date=2009<br />
<br />
|date=2009<br />
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2009年<br />
<br />
|doi= 10.1504/IJICA.2009.031778}}</ref><br />
<br />
|doi= 10.1504/IJICA.2009.031778}}</ref><br />
<br />
| doi 10.1504 / IJICA. 2009.031778} / ref<br />
<br />
Boids 模型已经被用于其他有趣应用。该系统已应用于互联网多频道广播电台的自动节目编排，以及可视化信息和优化任务。<br />
<br />
==See also 参见==<br />
<br />
*[[Swarm intelligence]] 群体智能<br />
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*[[Collective cell migration]] 集体细胞迁移<br />
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<br />
<br />
==References 参考资料==<br />
<br />
{{Reflist}}<br />
<br />
<br />
<br />
==External links 外部链接==<br />
<br />
* [http://www.red3d.com/cwr/boids/ Craig Reynolds' Boids page]<br />
<br />
* [http://www.vergenet.net/~conrad/boids/pseudocode.html Explanation of algorithm in pseudocode]<br />
<br />
* [https://gpolo.github.com/birdflocking JavaScript implementation]<br />
<br />
*[https://lufemas.github.io/boid-ai-implementation/ JavaScript implementation with Phaser Framework]<br />
<br />
* [https://web.archive.org/web/20080205014305/http://www.navgen.com/3d_boids/ 3D Boids Simulation using OpenGL, used by the BBC's Natural History Unit]<br />
<br />
*[https://black-square.github.io/BirdFlock/ Live In-Browser 3D Simulation of Bird Flocking Behavior in Unity3D] – Open Source implementation for Windows, Linux and Mac<br />
<br />
* [https://github.com/piskvorky/PredatorPrey UNIX+Windows open source implementation in C++, using OpenGL and simulation controls]<br />
<br />
* [https://github.com/tofti/javafx-boids A java implementation using the javafx API]<br />
<br />
*GLSchool - part of [[XScreenSaver]] software<br />
<br />
<br />
<br />
{{collective animal behaviour}}<br />
<br />
<br />
<br />
==编者推荐==<br />
===[https://campus.swarma.org/ 集智学园]课程：[https://campus.swarma.org/course/1104 透过人工鸟群Boid模型学习List的使用]===<br />
<br />
讲师：[[张江]](北师大系统科学学院教授、博士生导师，集智俱乐部、集智学园创始人。)<br />
<br />
本课程通过数个案例教会大家如何去动手搭建一个多主体仿真模型，以及如何利用NetLogo去实现。从生命游戏到人工鸟群，从模拟经济系统到病毒沿网络的传播，通过循序渐进的案例，该课程带你逐步走入NetLogo多主体建模的神奇世界。<br />
<br />
===[https://swarma.org/ 集智俱乐部]扩展阅读：[https://swarma.org/?p=16504 张江：乌合之众还是群智涌现？鸟群知道答案]===<br />
<br />
生命如何起源，智能如何涌现，因果如何反转，复杂谜题，引人深思。2019年7月13日，集智俱乐部创始人[[张江]]教授在混沌大学授课，详细解读了复杂系统中的混沌、涌现与进化，此文整理自课堂内容。<br />
<br />
==[https://v.qq.com/x/page/z13491sn7fy.html 动画片《Stanley and Stella in: Breaking the Ice (1987)》]<br />
<br />
[[Category:Artificial life]]<br />
<br />
Category:Artificial life<br />
<br />
类别: 人工生命<br />
<br />
[[Category:1986 software]]<br />
<br />
Category:1986 software<br />
<br />
类别: 1986软件<br />
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<noinclude><br />
<br />
<small>This page was moved from [[wikipedia:en:Boids]]. Its edit history can be viewed at [[鸟群算法/edithistory]]</small></noinclude><br />
<br />
[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%9D%9E%E7%BA%BF%E6%80%A7%E7%B3%BB%E7%BB%9F&diff=10824非线性系统2020-07-19T12:51:57Z<p>Dorr：</p>
<hr />
<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{Redirect|Nonlinear dynamics|the journal|Nonlinear Dynamics (journal)}}<br />
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{{Complex systems}}<br />
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{{About|"nonlinearity" in mathematics, physics and other sciences|video and film editing|Non-linear editing system|other uses|Nonlinearity (disambiguation)}}<br />
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In [[mathematics]] and [[science]], a '''nonlinear system''' is a [[system]] in which the change of the output is not [[proportionality (mathematics)|proportional]] to the change of the input.<ref>{{Cite news|url=https://news.mit.edu/2010/explained-linear-0226|title=Explained: Linear and nonlinear systems|work=MIT News|access-date=2018-06-30}}</ref><ref>{{Cite web|url=https://www.birmingham.ac.uk/research/activity/mathematics/applied-maths/nonlinear-systems.aspx|title=Nonlinear systems, Applied Mathematics - University of Birmingham|website=www.birmingham.ac.uk|language=en-gb|access-date=2018-06-30}}</ref> Nonlinear problems are of interest to [[engineer]]s, [[biologist]]s,<ref>{{Citation|date=2007|pages=181–276|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-540-34153-6_7|isbn=9783540341529|title = The Nonlinear Universe|series = The Frontiers Collection|chapter = Nonlinear Biology}}</ref><ref>{{cite journal|last=Korenberg|first=Michael J.|last2=Hunter|first2=Ian W.|date=March 1996|title=The identification of nonlinear biological systems: Volterra kernel approaches|journal=Annals of Biomedical Engineering|language=en|volume=24|issue=2|pages=250–268|doi=10.1007/bf02667354|issn=0090-6964}}</ref><ref>{{cite journal|last=Mosconi|first=Francesco|last2=Julou|first2=Thomas|last3=Desprat|first3=Nicolas|last4=Sinha|first4=Deepak Kumar|last5=Allemand|first5=Jean-François|last6=Vincent Croquette|last7=Bensimon|first7=David|date=2008|title=Some nonlinear challenges in biology|url=http://stacks.iop.org/0951-7715/21/i=8/a=T03|journal=Nonlinearity|language=en|volume=21|issue=8|pages=T131|doi=10.1088/0951-7715/21/8/T03|issn=0951-7715|bibcode=2008Nonli..21..131M}}</ref> [[physicist]]s,<ref>{{cite journal|last1=Gintautas|first1=V.|title=Resonant forcing of nonlinear systems of differential equations|journal=Chaos|date=2008|volume=18|issue=3|pages=033118|doi=10.1063/1.2964200|pmid=19045456|arxiv=0803.2252|bibcode=2008Chaos..18c3118G}}</ref><ref>{{cite journal|last1=Stephenson|first1=C.|last2=et.|first2=al.|title=Topological properties of a self-assembled electrical network via ab initio calculation|journal=Sci. Rep.|volume=7|pages=41621|date=2017|doi=10.1038/srep41621|pmid=28155863|pmc=5290745|bibcode=2017NatSR...741621S}}</ref> [[mathematician]]s, and many other [[scientist]]s because most systems are inherently nonlinear in nature.<ref>{{cite book|last1=de Canete|first1=Javier, Cipriano Galindo, and Inmaculada Garcia-Moral|title=System Engineering and Automation: An Interactive Educational Approach|date=2011|publisher=Springer|location=Berlin|isbn=978-3642202292|page=46|url=https://books.google.com/?id=h8rCQYXGGY8C&pg=PA46&lpg=PA46&dq=most+systems+are+inherently+nonlinear+in+nature#v=onepage&q=most%20systems%20are%20inherently%20nonlinear%20in%20nature&f=false|accessdate=20 January 2018}}</ref> Nonlinear [[dynamical system]]s, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler [[linear system]]s.<br />
<br />
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.<br />
<br />
在数学及科学中，'''非线性系统 Nonlinear System'''是一种输出的变化与输入的变化不成比例的系统。大多数系统在本质上是非线性的，因而非线性问题引起了工程师、生物学家、物理学家、数学家和许多其他科学家的兴趣。描述变量随时间变化的非线性动力系统与较之简单得多的线性系统相比，可能显得混沌、不可预测或违反直觉。<br />
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<br />
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Typically, the behavior of a nonlinear system is described in mathematics by a '''nonlinear system of equations''', which is a set of simultaneous [[equation]]s in which the unknowns (or the unknown functions in the case of [[differential equation]]s) appear as variables of a [[polynomial]] of degree higher than one or in the argument of a [[function (mathematics)|function]] which is not a polynomial of degree one.<br />
<br />
Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one.<br />
<br />
通常，非线性系统的行为在数学上被描述为一组非线性的联立方程组，其中未知数（或微分方程中的未知函数）作为一个高于一次的多项式变量出现，或者作为一个非一次多项式函数的参数出现。<br />
<br />
<br />
In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a [[linear combination]] of the unknown [[variable (mathematics)|variables]] or [[function (mathematics)|functions]] that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is ''linear'' if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
<br />
In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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换句话说，在非线性方程系统中，待解的方程不能被写成未知变量或函数的线性组合。无论方程中是否有已知的线性函数，系统都可以被定义为非线性。特别是当一个微分方程的未知函数及其导数是线性的，即使其他变量是非线性的，也称该方程是线性的。<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations ([[linearization]]). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as [[soliton]]s, [[chaos theory|chaos]],<ref>[http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm Nonlinear Dynamics I: Chaos] {{webarchive|url=https://web.archive.org/web/20080212045134/http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm |date=2008-02-12}} at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]</ref> and [[mathematical singularity|singularities]] are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble [[randomness|random]] behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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由于非线性动力学方程难以求解，通常用线性方程来近似非线性系统（'''线性化 Linearization'''）。这种方法对于一定范围的输入和某些精度要求下的效果不错，但一些有趣的现象如'''孤子 Soliton'''、'''混沌 Chaos'''和'''奇异性 Singularity'''在线性化后被隐藏。因此，非线性系统的动态行为在某些方面可能看起来违反直觉、不可预测、甚至混沌。尽管这种混沌行为可能感觉很像随机行为，但它实际上并不是随机的。例如，天气的某些方面被认为是混沌的，其系统某部分的微小扰动就会产生复杂的整体影响。这种非线性是目前技术无法进行精确长期预测的原因之一。<br />
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Some authors use the term '''nonlinear science''' for the study of nonlinear systems. This term is disputed by others:<br />
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Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:<br />
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有些作者用非线性科学这一术语来研究非线性系统。这一术语引起了其他人的争议:<br />
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{{quote|Using a term like nonlinear science is like referring to the bulk of zoology as the study of [[negation|non]]-elephant animals.|[[Stanislaw Ulam]]<ref>{{cite journal|last1=Campbell|first1=David K.|title=Nonlinear physics: Fresh breather|journal=Nature|date=25 November 2004|volume=432|issue=7016|pages=455–456|doi=10.1038/432455a|pmid=15565139|url=https://zenodo.org/record/1134179|language=en|issn=0028-0836|bibcode=2004Natur.432..455C}}</ref>}}<br />
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“使用‘非线性科学’这样的术语，就如同把动物学里大部分对象称作‘非大象动物’研究一样可笑。”<br />
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——'''斯塔尼斯拉夫·乌拉姆 Stanislaw Ulam'''<br />
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==Definition 定义==<br />
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In [[mathematics]], a [[linear map]] (or ''linear function'') <math>f(x)</math> is one which satisfies both of the following properties:<br />
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In mathematics, a linear map (or linear function) <math>f(x)</math> is one which satisfies both of the following properties:<br />
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在数学中，线性映射（或线性函数）<math>f (x)</math>满足以下两个性质:<br />
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*Additivity or [[superposition Principle]]: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*Homogeneity: <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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*'''可加性 Additivity'''（'''叠加性 Superposition principle'''）: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*'''齐次性 Homogeneity''': <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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Additivity implies homogeneity for any [[rational number|rational]] ''α'', and, for [[continuous function]]s, for any [[real number|real]] ''α''. For a [[complex number|complex]] ''α'', homogeneity does not follow from additivity. For example, an [[antilinear map]] is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle<br />
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Additivity implies homogeneity for any rational α, and, for continuous functions, for any real α. For a complex α, homogeneity does not follow from additivity. For example, an antilinear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle:<br />
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当α是有理数或实数，且<math>f(x)</math>是连续函数时，由可加性可以推出齐次性。但当α是复数时，可加性不能导出齐次性。例如，反线性映射是可加的，但不是齐次的。可加性和齐次性条件经常组合，称为叠加原理：<br />
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<math>f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)</math><br />
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An equation written as<br />
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对一个写成<br />
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<math>f(x) = C</math><br />
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is called '''linear''' if <math>f(x)</math> is a linear map (as defined above) and '''nonlinear''' otherwise. The equation is called ''homogeneous'' if <math>C = 0</math>.<br />
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is called linear if <math>f(x)</math> is a linear map (as defined above) and nonlinear otherwise. The equation is called homogeneous if <math>C = 0</math>.<br />
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的方程，若 <math> f (x) </math> 是线性映射（如上定义） ，则称其为'''线性的 Linear'''，否则称为'''非线性的 Nonlinear'''。若<math>C = 0</math>，该方程称为是齐次的。<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any [[map (mathematics)|mapping]], including integration or differentiation with associated constraints (such as [[boundary values]]). If <math>f(x)</math> contains [[derivative|differentiation]] with respect to <math>x</math>, the result will be a [[differential equation]].<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any mapping, including integration or differentiation with associated constraints (such as boundary values). If <math>f(x)</math> contains differentiation with respect to <math>x</math>, the result will be a differential equation.<br />
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定义 <math>f(x) = C</math> 是非常具有一般性的，因为 <math>x</math> 可以是任意合理的数学对象（数字、向量、函数等），函数 <math>f(x)</math> 实际上可以是任意映射，包括有相关约束（如给定边界值）的积分或微分。若 <math>f(x)</math> 包含对 <math>x</math> 的微分运算，则该方程为微分方程。<br />
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===相关概念辨析===<br />
'''线性Linear'''、'''非线性Nonlinear'''、'''亚线性 Sublinear'''、'''超线性 Superlinear'''：描述量与量之间的一种变化关系，例如<math>y=a+b*x^n</math>，其中<math>n>1</math>。当<math>n=1</math>时，表示为线性关系；<math>n\neq 1</math>时，表示为非线性关系；当<math>0<n<1</math>时，表示为亚线性关系；当<math>n>1</math>时，表示为超线性关系。<br />
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亚线性与超线性属于非线性变化关系范畴。当两个变量之间存在亚线性关系时，其典型特性是因变量的变化速率会随着自变量的增大而减小，即其一阶导数会随着自变量的增大而减小；两个变量之间存在超线性关系时，其典型特性是因变量的变化速率会随着自变量的增大而增大，即其一阶导数会随着自变量的增大而增大。<br />
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在线性系统中，整体等于部分和，描述线性系统的方程满足叠加原理，作用的总和正好等于每一部分作用相加的代数和，这意味着每一部分作用都是独立的、互不相关的；而在普遍存在的非线性系统中，作用的总和不等于每一部分作用相加的代数和，因为系统内部要素之间存在着复杂的非线性相互作用。<br />
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==Nonlinear algebraic equations 非线性代数方程==<br />
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{{Main|Algebraic equation|System of polynomial equations}}<br />
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Nonlinear [[algebraic equation]]s, which are also called ''[[polynomial equation]]s'', are defined by equating [[polynomial]]s (of degree greater than one) to zero. For example,<br />
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Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials (of degree greater than one) to zero. For example,<br />
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非线性'''代数方程 Algebraic Equation'''，又称'''多项式方程 Polynomial Equation'''，由某多项式（次数大于1）等于零定义。例如：<br />
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<math>x^2 + x - 1 = 0\,.</math><br />
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For a single polynomial equation, [[root-finding algorithm]]s can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of [[algebraic geometry]], a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see [[Hilbert's Nullstellensatz]]). Nevertheless, in the case of the systems with a finite number of complex solutions, these [[systems of polynomial equations]] are now well understood and efficient methods exist for solving them.<ref>{{cite journal |last1= Lazard |first1= D. |title= Thirty years of Polynomial System Solving, and now? |doi= 10.1016/j.jsc.2008.03.004 |journal= Journal of Symbolic Computation |volume= 44 |issue= 3 |pages= 222–231 |year= 2009 |pmid= |pmc=}}</ref><br />
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For a single polynomial equation, root-finding algorithms can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of algebraic geometry, a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see Hilbert's Nullstellensatz). Nevertheless, in the case of the systems with a finite number of complex solutions, these systems of polynomial equations are now well understood and efficient methods exist for solving them.<br />
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对于一个单一的多项式方程，'''求根算法 Root-finding Algorithms'''可用于其求解（即找到满足该方程的变量的值集）。而代数方程组则相对复杂，其研究是现代数学的较难分支——'''代数几何 Algebraic Geometry'''领域的动力之一。甚至很难判断一个给定的代数系统是否有复数解（见'''希尔伯特零点定律 Hilbert's Nullstellensatz'''）。不过，对于具有有限个复数解的系统的多项式方程组，我们现在已经有了充分的理解，并且找到了有效的求解方法。<br />
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==Nonlinear recurrence relations 非线性递推关系==<br />
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A nonlinear [[recurrence relation]] defines successive terms of a [[sequence]] as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the [[logistic map]] and the relations that define the various [[Hofstadter sequence]]s. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related [[nonlinear system identification]] and analysis procedures.<ref name="SAB1">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</ref> These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related nonlinear system identification and analysis procedures. These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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非线性递归关系中，序列的连续项被定义为其前项的非线性函数。非线性递归关系的例子有 [[logistic 映射]]和定义各种'''霍夫斯塔特序列 Hofstadter Sequences''' 的关系。非线性离散模型代表了一类广泛的非线性递归关系，包括 NARMAX（外部输入非线性自回归移动平均）模型和相关的非线性系统辨识和分析程序。这些方法可用于研究时域、频域和时空域的广泛复杂非线性行为。<br />
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==Nonlinear differential equations 非线性微分方程==<br />
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A [[simultaneous equations|system]] of [[differential equation]]s is said to be nonlinear if it is not a [[linear system]]. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the [[Navier–Stokes equations]] in fluid dynamics and the [[Lotka–Volterra equation]]s in biology.<br />
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A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology.<br />
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若一个微分方程组不是线性系统，则称其为非线性的。涉及非线性微分方程的问题非常多样，对不同问题的解决或分析方法也不相同。非线性微分方程的例子有流体力学中的 '''纳维-斯托克斯方程 Navier-Stokes Equations'''和生物学中的'''洛特卡－沃尔泰拉方程 Lotka-Volterra Equations'''。<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of [[linearly independent]] solutions can be used to construct general solutions through the [[superposition principle]]. A good example of this is one-dimensional heat transport with [[Dirichlet boundary conditions]], the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle. A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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非线性问题最大的困难之一是通常不可能将已知的解组合成新的解。例如，在线性问题中，可以根据叠加原理以一族线性独立的解构造通解。一个很好的例子是带有'''狄利克雷边界条件 Dirichlet Boundary Conditions'''的一维热传导问题，其解可以写成（随时间变化）不同频率的正弦波的线性组合，这使得解非常灵活。而对非线性方程，通常可以找到几个非常特殊的解，但是此时叠加原理不适用，故无法构造新的解。<br />
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===Ordinary differential equations 常微分方程===<br />
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First order [[ordinary differential equation]]s are often exactly solvable by [[separation of variables]], especially for autonomous equations. For example, the nonlinear equation<br />
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First order ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. For example, the nonlinear equation<br />
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一阶常微分方程，尤其是自治（自主）方程，通常可以用'''分离变量法 Separation of Variables'''来精确求解。例如，非线性方程<br />
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<math>\frac{d u}{d x} = -u^2</math><br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also ''u'' = 0 as a particular solution, corresponding to the limit of the general solution when ''C'' tends to infinity). The equation is nonlinear because it may be written as<br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also u = 0 as a particular solution, corresponding to the limit of the general solution when C tends to infinity). The equation is nonlinear because it may be written as<br />
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将 <math>u=\frac{1}{x+C}</math> 作为一般解（也有特解 ''u'' = 0，对应于 ''C'' 趋于无穷时的一般解的极限)。该方程是非线性的，因为它可以改写成<br />
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<math>\frac{du}{d x} + u^2=0</math><br />
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and the left-hand side of the equation is not a linear function of ''u'' and its derivatives. Note that if the ''u''<sup>2</sup> term were replaced with ''u'', the problem would be linear (the [[exponential decay]] problem).<br />
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and the left-hand side of the equation is not a linear function of u and its derivatives. Note that if the u<sup>2</sup> term were replaced with u, the problem would be linear (the exponential decay problem).<br />
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方程的左边不是 ''u'' 及其导数的线性函数。注意，若将 ''u''<sup>2</sup> 项替换为''u''，该问题将变为线性的（'''指数衰减 Exponential Decay'''问题）。<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield [[closed-form expression|closed-form]] solutions, though implicit solutions and solutions involving [[nonelementary integral]]s are encountered.<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered.<br />
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二阶和高阶常微分方程（更一般地说，非线性方程组）很少能产生封闭解，而隐式解和非初等函数积分形式的解较为常见。<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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非线性常微分方程定性分析的常用方法包括:<br />
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*Examination of any [[conserved quantities]], especially in [[Hamiltonian system]]s<br />
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*检查是否有任意'''守恒量 Conserved Quantities'''（特别是在'''哈密顿系统 Hamiltonian System'''中）<br />
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*Examination of dissipative quantities (see [[Lyapunov function]]) analogous to conserved quantities<br />
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*检查是否有类似守恒量的耗散量（见'''李亚普诺夫函数 Lyapunov Function'''）<br />
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*Linearization via [[Taylor expansion]]<br />
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*基于'''泰勒展开 Taylor Expansion'''的线性化<br />
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*Change of variables into something easier to study<br />
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*将变量进行代换以便更好的进行研究<br />
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*[[Bifurcation theory]]<br />
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*'''分岔理论 Bifurcation Theory'''<br />
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*[[Perturbation theory|Perturbation]] methods (can be applied to algebraic equations too)<br />
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*'''摄动理论 Perturbation Theory'''（也可应用于代数方程）<br />
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===Partial differential equations 偏微分方程===<br />
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{{main|Nonlinear partial differential equation}}<br />
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{{See also|List of nonlinear partial differential equations}}<br />
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参见：非线性偏微分列表<br />
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The most common basic approach to studying nonlinear [[partial differential equation]]s is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more [[ordinary differential equation]]s, as seen in [[separation of variables]], which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
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The most common basic approach to studying nonlinear partial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
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研究非线性偏微分方程最常用的基本方法是变量代换（或转换问题），使变换后的问题更简单（甚至可能变为线性的）。有时可以将此类方程转化成一或多个常微分方程（如同分离变量法所示），此时不论得到的常微分方程是否可解，对研究问题总是有用的。<br />
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Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use [[scale analysis (mathematics)|scale analysis]] to simplify a general, natural equation in a certain specific [[boundary value problem]]. For example, the (very) nonlinear [[Navier-Stokes equations]] can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
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Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem. For example, the (very) nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
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另一个流体力学和热力学中常见的策略（虽然不是数学上的）是利用'''尺度分析 Scale Analysis'''来在某一特定边界条件下简化一般自然方程。例如，在描述圆管内一维层流的瞬态时，非线性的纳维－斯托克斯方程可以简化为一个线性的偏微分方程; 尺度分析提供了层流和一维流动的条件，也产生了简化的方程。<br />
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Other methods include examining the [[method of characteristics|characteristics]] and using the methods outlined above for ordinary differential equations.<br />
<br />
Other methods include examining the characteristics and using the methods outlined above for ordinary differential equations.<br />
<br />
其他方法包括检查特征线法及前面所述研究常微分方程的方法。<br />
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===Pendula 摆===<br />
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{{Main|Pendulum (mathematics)}}<br />
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[[File:PendulumLayout.svg.png|thumb|图一：Illustration of a pendulum 摆图解|right|200px]]<br />
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[[File:PendulumLinearizations.png|thumb|图二：Linearizations of a pendulum 摆的线性化|right|200px]]<br />
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A classic, extensively studied nonlinear problem is the dynamics of a [[pendulum (mathematics)|pendulum]] under the influence of [[gravity]]. Using [[Lagrangian mechanics]], it may be shown<ref>[http://www.damtp.cam.ac.uk/user/tong/dynamics.html David Tong: Lectures on Classical Dynamics]</ref> that the motion of a pendulum can be described by the [[dimensionless]] nonlinear equation<br />
<br />
A classic, extensively studied nonlinear problem is the dynamics of a pendulum under the influence of gravity. Using Lagrangian mechanics, it may be shown that the motion of a pendulum can be described by the dimensionless nonlinear equation<br />
<br />
一个经典的被广泛研究的非线性问题是重力影响下的摆的动力学。利用'''拉格朗日力学 Lagrangian Mechanics'''，可以证明摆的运动可以用无量纲的非线性方程<br />
<br />
<math>\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0</math><br />
<br />
<br />
<br />
<br />
<br />
where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an [[integrating factor]], which would eventually yield<br />
<br />
where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an integrating factor, which would eventually yield<br />
<br />
描述，其中重力指向“下方”，<math>\theta</math> 是摆与其静止位置形成的角度，如右图所示。“解”这个方程的方法之一是用 <math>d\theta/dt</math> 作为'''积分因子 Integrating Factor'''，最终得<br />
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<math>\int{\frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)}}} = t + C_1</math><br />
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<br />
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<br />
which is an implicit solution involving an [[elliptic integral]]. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the [[nonelementary integral]] (nonelementary unless <math>C_0 = 2</math>).<br />
<br />
which is an implicit solution involving an elliptic integral. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral (nonelementary unless <math>C_0 = 2</math>).<br />
<br />
这是一个含'''椭圆积分 Elliptic Integral'''的隐式解。这个“解”通常没什么用，因为这个解的大部分性质都隐藏在非初等函数积分中（除非<math>C_0 = 2</math>，否则是非初等的）。<br />
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Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through [[Taylor expansion]]s. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
<br />
Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through Taylor expansions. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
<br />
另一种解决这个问题的方法是利用泰勒展开将任意非线性项（此时为正弦函数项)在某些感兴趣的点线性化。例如，在<math>\theta = 0</math> 的点附近线性化（称为小角度近似）为<br />
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<math>\frac{d^2 \theta}{d t^2} + \theta = 0</math><br />
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since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a [[simple harmonic oscillator]] corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
<br />
since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
<br />
因为 <math>\theta \approx 0</math> 时，有 <math>\sin(\theta) \approx \theta</math>。这是一个'''简谐振子 Simple Harmonic Oscillator''' ，对应于摆在其路径底部附近的摆动。另一种线性化方法是在 <math>\theta = \pi</math>附近线性化，对应于运动到最高点的摆:<br />
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<math>\frac{d^2 \theta}{d t^2} + \pi - \theta = 0</math><br />
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<br />
since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves [[hyperbolic sinusoid]]s, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
<br />
since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
<br />
因为 <math>\theta \approx \pi</math> 时，有 <math>\sin(\theta) \approx \pi - \theta</math>。这个问题的解含双曲正弦曲线；注意到不同于小角度近似，它是不稳定的，这意味着 <math>|\theta|</math> 通常会无限增长（但解也有可能是有界的）。这就解释了摆在最高点达到平衡的困难，此时实际上是一种不稳定的状态。<br />
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One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
<br />
One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
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一个更有趣的线性化可能是在 <math>\theta = \pi/2</math>附近，此时 <math>\sin(\theta) \approx 1</math>:<br />
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<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math><br />
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This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) [[phase portrait]]s and approximate periods.<br />
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This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) phase portraits and approximate periods.<br />
<br />
这相当于一个自由落体问题。把这样线性化的结果合在一起看，就能得到有关摆的运动的非常有用的图像，如右图所示。利用其他方法寻找（精确的）'''相图 Phase Portrait'''和估计周期。<br />
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==Types of nonlinear dynamic behaviors 非线性动力学行为的类型==<br />
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*[[Amplitude death]] – any oscillations present in the system cease due to some kind of interaction with other system or feedback by the same system<br />
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*'''振幅死亡 Amplitude Death'''——系统内的某振荡因系统的自回馈或与其他系统的某种相互作用而停止的现象<br />
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*[[Chaos theory|Chaos]] – values of a system cannot be predicted indefinitely far into the future, and fluctuations are [[aperiodic]]<br />
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*'''混沌 Chaos'''——系统内的值无法无限期地预测到遥远的未来；波动是非周期性的<br />
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*[[Multistability]] – the presence of two or more stable states<br />
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*'''多稳态 Multistability'''——两或多个稳态的存在<br />
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*[[Soliton]]s – self-reinforcing solitary waves<br />
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*'''孤波 Soliton'''s——自增强的孤立波<br />
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*[[Limit cycle|Limit cycles]] – asymptotic periodic orbits to which destabilized fixed points are attracted. <br />
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*'''极限环 Limit Cycles'''——吸引不稳定不动点的渐近周期轨道<br />
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*[[Self-oscillation|Self-oscillations]] - feedback oscillations taking place in open dissipative physical systems.<br />
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*'''自激振荡 Self-oscillations'''——开放耗散物理系统中的反馈振荡<br />
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==Examples of nonlinear equations 非线性方程的例子==<br />
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{{Div col|colwidth=25em}}<br />
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*[[Algebraic Riccati equation]] 代数黎卡提方程<br />
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*[[Ball and beam]] system 球杆系统<br />
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*[[Bellman equation]] for optimal policy 最佳策略的贝尔曼方程<br />
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*[[Boltzmann equation]] 玻尔兹曼方程<br />
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*[[Colebrook equation]] 科尔布鲁克方程<br />
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*[[General relativity]] 广义相对论<br />
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*[[Ginzburg–Landau theory]] 金兹堡-朗道方程<br />
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*[[Ishimori equation]] 石森方程<br />
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*[[Kadomtsev–Petviashvili equation]] 卡东穆塞夫-彼得韦亚斯维利方程<br />
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*[[Korteweg–de Vries equation]] kdV方程<br />
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*[[Landau–Lifshitz–Gilbert equation]] 朗道-利夫希兹-吉尔伯特方程<br />
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*[[Liénard equation]] 林纳德方程<br />
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*[[Navier–Stokes equations]] of [[fluid dynamics]] 流体力学的纳维－斯托克斯方程<br />
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*[[Nonlinear optics]] 非线性光学<br />
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*[[Nonlinear Schrödinger equation]] 非线性薛定谔方程<br />
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*[[Power-flow study]] 功率流研究<br />
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*[[Richards equation]] for unsaturated water flow 未饱和层水流的理查氏方程<br />
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<br />
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*[[Self-balancing unicycle]] 自平衡独轮车<br />
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*[[Sine-Gordon equation]] 正弦-戈尔登方程<br />
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*[[Van der Pol oscillator]] 范德波尔振荡器<br />
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*[[Vlasov equation]] 弗拉索夫方程<br />
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{{Div col end}}<br />
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==See also 参见==<br />
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<br />
*[[Aleksandr Mikhailovich Lyapunov]] 亚历山大·李亚普诺夫<br />
<br />
<br />
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*[[Dynamical system]] 动态系统<br />
<br />
<br />
<br />
*[[Feedback]] 反馈<br />
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<br />
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*[[Initial condition]] 初始条件<br />
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*[[Interaction]] 相互作用<br />
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*[[Linear system]] 线性系统<br />
<br />
<br />
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*[[Mode coupling]] 模式耦合<br />
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*[[Vector soliton]] 矢量孤子<br />
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<br />
*[[Volterra series]] 沃尔泰拉级数<br />
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<br />
==References 参考资料==<br />
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<br />
{{Reflist|35em}}<br />
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==Further reading 进一步阅读==<br />
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{{Refbegin|35em}}<br />
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*{{cite book<br />
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<br />
<br />
| author= [[Diederich Hinrichsen]] and Anthony J. Pritchard<br />
<br />
| author= Diederich Hinrichsen and Anthony J. Pritchard<br />
<br />
作者 Diederich Hinrichsen 和 Anthony j. Pritchard<br />
<br />
| year= 2005<br />
<br />
| year= 2005<br />
<br />
2005年<br />
<br />
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
<br />
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
<br />
数学系统理论 i-建模，状态空间分析，稳定性和鲁棒性<br />
<br />
| publisher= Springer Verlag<br />
<br />
| publisher= Springer Verlag<br />
<br />
出版商 Springer Verlag<br />
<br />
| isbn=9783540441250<br />
<br />
| isbn=9783540441250<br />
<br />
9783540441250<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Jordan<br />
<br />
| last= Jordan<br />
<br />
最后一个乔丹<br />
<br />
| first= D. W.<br />
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| first= D. W.<br />
<br />
首先是 D.w。<br />
<br />
| last2= Smith<br />
<br />
| last2= Smith<br />
<br />
最后2个史密斯<br />
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| first2= P.<br />
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| first2= P.<br />
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| first2 p.<br />
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| year= 2007<br />
<br />
| year= 2007<br />
<br />
2007年<br />
<br />
| edition= fourth<br />
<br />
| edition= fourth<br />
<br />
第四版<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
非线性常微分方程<br />
<br />
| publisher= Oxford University Press<br />
<br />
| publisher= Oxford University Press<br />
<br />
牛津大学出版社<br />
<br />
| isbn= 978-0-19-920824-1<br />
<br />
| isbn= 978-0-19-920824-1<br />
<br />
[国际标准图书编号978-0-19-920824-1]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Khalil<br />
<br />
| last= Khalil<br />
<br />
最后的哈利勒<br />
<br />
| first= Hassan K.<br />
<br />
| first= Hassan K.<br />
<br />
首先是哈桑 · k。<br />
<br />
| year= 2001<br />
<br />
| year= 2001<br />
<br />
2001年<br />
<br />
| title= Nonlinear Systems<br />
<br />
| title= Nonlinear Systems<br />
<br />
非线性系统<br />
<br />
| publisher= Prentice Hall<br />
<br />
| publisher= Prentice Hall<br />
<br />
出版商 Prentice Hall<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
[国际标准图书编号978-0-13-067389-3]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Kreyszig<br />
<br />
| last= Kreyszig<br />
<br />
| last= Kreyszig<br />
<br />
| first= Erwin<br />
<br />
| first= Erwin<br />
<br />
首先是欧文<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
作者 / 链接 Erwin Kreyszig<br />
<br />
| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
高等工程数学<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
Https://archive.org/details/advancedengineer0008krey<br />
<br />
| url-access= registration<br />
<br />
| url-access= registration<br />
<br />
访问注册<br />
<br />
| publisher= Wiley<br />
<br />
| publisher= Wiley<br />
<br />
出版商 Wiley<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
[国际标准图书编号978-0-471-15496-9]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Sontag<br />
<br />
| last= Sontag<br />
<br />
最后一个桑塔格<br />
<br />
| first= Eduardo<br />
<br />
| first= Eduardo<br />
<br />
首先是爱德华多<br />
<br />
| authorlink= Eduardo D. Sontag<br />
<br />
| authorlink= Eduardo D. Sontag<br />
<br />
作者: Eduardo d. Sontag<br />
<br />
| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
数学控制理论: 确定性有限维系统。第二版<br />
<br />
| publisher= Springer<br />
<br />
| publisher= Springer<br />
<br />
出版商斯普林格<br />
<br />
| isbn= 978-0-387-98489-6<br />
<br />
| isbn= 978-0-387-98489-6<br />
<br />
[国际标准图书编号978-0-387-98489-6]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
{{Refend}}<br />
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<br />
<br />
==External links 外部链接==<br />
<br />
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<br />
*[http://www.dodccrp.org/ Command and Control Research Program (CCRP)]<br />
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<br />
*[http://necsi.edu/guide/concepts/linearnonlinear.html New England Complex Systems Institute: Concepts in Complex Systems]<br />
<br />
<br />
<br />
*[http://ocw.mit.edu/courses/mathematics/18-353j-nonlinear-dynamics-i-chaos-fall-2012/ Nonlinear Dynamics I: Chaos] at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]<br />
<br />
<br />
<br />
*[http://www.hedengren.net/research/models.htm Nonlinear Model Library]{{snd}} (in [[MATLAB]]) a Database of Physical Systems<br />
<br />
<br />
<br />
*[http://cnls.lanl.gov/ The Center for Nonlinear Studies at Los Alamos National Laboratory]<br />
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{{Systems science}}<br />
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{{Differential equations topics}}<br />
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{{Authority control}}<br />
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<br />
==编者推荐==<br />
[https://campus.swarma.org/ 集智学园]课程：[https://campus.swarma.org/course/697 非线性动力学与混沌]<br />
<br />
非线性动力学和混沌理论是系统发展的，从一阶微分方程及其分岔开始，然后是相平面分析，极限环和它们的分岔，最终得到Lorenz方程，混沌，迭代映射，周期倍增，重整化，分形和奇怪吸引。<br />
<br />
<br />
此系列课程由[[斯蒂文·斯特罗加茨 Steven H. Strogatz]]主持，内容包括机械振动，激光，生物节律，超导电路，昆虫爆发，化学振荡器，遗传控制系统，混沌水轮，甚至是使用混乱发送秘密信息的技术。在每种情况下，科学背景都在初级阶段进行解释，并与数学理论紧密结合。<br />
<br />
[[Category:Nonlinear systems| ]]<br />
<br />
<br />
<br />
[[Category:Dynamical systems]]<br />
<br />
Category:Dynamical systems<br />
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类别: 动力系统<br />
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[[Category:Concepts in physics]]<br />
<br />
Category:Concepts in physics<br />
<br />
分类: 物理概念<br />
<br />
<noinclude><br />
<br />
<small>This page was moved from [[wikipedia:en:Nonlinear system]]. Its edit history can be viewed at [[非线性系统/edithistory]]</small></noinclude><br />
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{{short description|Artificial life program, developed by Craig Reynolds in 1986, which simulates the flocking behaviour of birds}}<br />
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{{Use mdy dates|date=July 2011}}<br />
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{{for|the reptilian family|Boidae}}<br />
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[[File:Boids OpenGL example.webm|thumb|A Boids example created in [[OpenGL]]]]<br />
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A Boids example created in [[OpenGL]]<br />
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在[[[ OpenGL ]]中创建的 Boids 示例<br />
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'''Boids''' is an [[artificial life]] program, developed by [[Craig Reynolds (computer graphics)|Craig Reynolds]] in 1986, which simulates the [[Flocking (behavior)|flocking]] behaviour of [[Bird|birds]]. His paper on this topic was published in 1987 in the proceedings of the [[Association for Computing Machinery|ACM]] [[SIGGRAPH]] conference. <ref>{{Cite book<br />
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Boids is an artificial life program, developed by Craig Reynolds in 1986, which simulates the flocking behaviour of birds. His paper on this topic was published in 1987 in the proceedings of the ACM SIGGRAPH conference. <ref>{{Cite book<br />
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'''鸟群算法 Boids'''是模拟鸟类群集行为的人工生命项目，由'''克雷格·雷诺兹 Craig Reynolds'''于1986年开发。他关于这个主题的论文发表在1987年的 ACM SIGGRAPH 会议记录上。引用{ Cite book<br />
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最后1雷诺<br />
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第一名: 克雷格<br />
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| author1-link=Craig Reynolds (computer_graphics)<br />
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| author1-link=Craig Reynolds (computer_graphics)<br />
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| author1-link Craig Reynolds (计算机图形学)<br />
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| title=Flocks, herds and schools: A distributed behavioral model.<br />
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| title=Flocks, herds and schools: A distributed behavioral model.<br />
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羊群、畜群和学校: 一个分布式的行为模型。<br />
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1987年<br />
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| journal=SIGGRAPH '87: Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques<br />
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| journal=SIGGRAPH '87: Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques<br />
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87: 第14届计算机图形学与互动技术年度会议论文集<br />
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| publisher=[[Association for Computing Machinery]]<br />
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| publisher=Association for Computing Machinery<br />
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| isbn=978-0-89791-227-3<br />
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[国际标准图书编号978-0-89791-227-3]<br />
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| doi=10.1145/37401.37406<br />
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10.1145 / 37401.37406<br />
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The name "boid" corresponds to a shortened version of "bird-oid object", which refers to a bird-like object.<ref>{{Cite journal<br />
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The name "boid" corresponds to a shortened version of "bird-oid object", which refers to a bird-like object.<ref>{{Cite journal<br />
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| title=A review of particle swarm optimization. Part I: background and development<br />
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2012年10月12日 | 标题: 粒子群优化的回顾。第一部分: 背景与发展<br />
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自然计算杂志<br />
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}}</ref> Incidentally, "boid" is also a New York Metropolitan dialect pronunciation for "bird".<br />
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}}</ref> Incidentally, "boid" is also a New York Metropolitan dialect pronunciation for "bird".<br />
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'''鸟群算法 Boids'''是模拟鸟类群集行为的人工生命项目，由'''克雷格·雷诺兹 Craig Reynolds'''于1986年开发。他关于这个主题的论文发表在1987年的 ACM SIGGRAPH（美国计算机协会计算机图形专业组组织的计算机图形学顶级年度会议）会议记录上。“ boid”是“ bird-oid object”的缩写，指一个类鸟对象。恰巧，“boid”也是纽约都市方言中“bird”的发音。<br />
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{{multiple image|direction=vertical|image1=Rule separation.gif|caption1=Separation|image2=Rule alignment.gif|caption2=Alignment|image3=Rule cohesion.gif|caption3=Cohesion|header=Rules applied in simple Boids}}<br />
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As with most artificial life simulations, Boids is an example of [[emergence|emergent]] behavior; that is, the complexity of Boids arises from the interaction of individual agents (the boids, in this case) adhering to a set of simple rules. The rules applied in the simplest Boids world are as follows:<br />
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As with most artificial life simulations, Boids is an example of emergent behavior; that is, the complexity of Boids arises from the interaction of individual agents (the boids, in this case) adhering to a set of simple rules. The rules applied in the simplest Boids world are as follows:<br />
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与大多数人工生命模拟一样，Boids 是'''涌现 Emergent'''行为的一个例子; 也就是说，Boids 的复杂性来自于遵循一系列简单规则的个体代理（这里是 Boids）的相互作用。在最简单的Bolds 世界中适用的规则如下:<br />
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* '''separation''': [[wikt:steer#Verb 2|steer]] to avoid crowding local flockmates<br />
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* '''alignment''': steer towards the average heading of local flockmates<br />
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* '''cohesion''': steer to move towards the average position (center of mass) of local flockmates<br />
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* '''分离 Separation''': 避开当地同类拥挤处<br />
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* '''定位 Alignment''': 转向当地同类的平均方向<br />
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* '''凝聚 Cohesion''': 朝当地同类的平均方向（质心）移动<br />
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More complex rules can be added, such as obstacle avoidance and goal seeking.<br />
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More complex rules can be added, such as obstacle avoidance and goal seeking.<br />
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可以添加更复杂的规则，如避障和寻找目标。<br />
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The basic model has been extended in several different ways since Reynolds proposed it. For instance, Delgado-Mata et&nbsp;al.<ref><br />
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The basic model has been extended in several different ways since Reynolds proposed it. For instance, Delgado-Mata et&nbsp;al.<ref><br />
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{{cite journal<br />
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|first1= Carlos | last1= Delgado-Mata<br />
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|first1= Carlos | last1= Delgado-Mata<br />
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卡洛斯，德尔加多，马塔<br />
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|title= On the use of Virtual Animals with Artificial Fear in Virtual Environments<br />
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|title= On the use of Virtual Animals with Artificial Fear in Virtual Environments<br />
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论虚拟动物在虚拟环境中的人为恐惧<br />
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|journal= New Generation Computing<br />
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|journal= New Generation Computing<br />
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| 杂志新一代计算<br />
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extended the basic model to incorporate the effects of fear. Olfaction was used to transmit emotion between animals, through pheromones modelled as particles in a free expansion gas. Hartman and Benes<ref><br />
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extended the basic model to incorporate the effects of fear. Olfaction was used to transmit emotion between animals, through pheromones modelled as particles in a free expansion gas. Hartman and Benes<ref><br />
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|title= Autonomous boids<br />
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|title= Autonomous boids<br />
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标题自主 boids<br />
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|journal= Computer Animation and Virtual Worlds<br />
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|journal= Computer Animation and Virtual Worlds<br />
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计算机动画与虚拟世界<br />
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|volume=17 |issue=3–4 |pages= 199–206<br />
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|volume=17 |issue=3–4 |pages= 199–206<br />
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第17卷第3-4期，第199-206页<br />
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|doi= 10.1002/cav.123}}</ref><br />
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introduced a complementary force to the alignment that they call the change of leadership. This steer defines the chance of the boid to become a leader and try to escape.<br />
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introduced a complementary force to the alignment that they call the change of leadership. This steer defines the chance of the boid to become a leader and try to escape.<br />
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该基本模型自雷诺兹提出以来，已用多种方法扩展。例如，Delgado-Mata 等人扩展基本模型以加入恐惧的影响。利过一种可用自由膨胀气体中的粒子模拟的信息素，动物间用嗅觉来传递情感。'''哈特曼 Hartman'''和'''贝内斯 Benes'''为定位过程引入了一种补充力量，他们称之为领导权的更迭。这条决定了某只鸟成为领导者及试图逃跑的机会。<br />
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The movement of Boids can be characterized as either chaotic (splitting groups and wild behaviour) or orderly. Unexpected behaviours, such as splitting flocks and reuniting after avoiding obstacles, can be considered emergent.<br />
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The movement of Boids can be characterized as either chaotic (splitting groups and wild behaviour) or orderly. Unexpected behaviours, such as splitting flocks and reuniting after avoiding obstacles, can be considered emergent.<br />
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Boids 的运动可以表现为混乱（分裂的群体和狂野的行为）或有序。意想不到的行为，比如群体分散和避开障碍后的团聚可以被认为是应急的。<br />
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The boids framework is often used in computer graphics, providing realistic-looking representations of flocks of birds and other creatures, such as schools of fish or herds of animals. It was for instance used in the 1998 video game ''[[Half-Life (video game)|Half-Life]]'' for the flying bird-like creatures seen at the end of the game on [[Locations of Half-Life#Xen|Xen]], named "boid" in the game files.<br />
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The boids framework is often used in computer graphics, providing realistic-looking representations of flocks of birds and other creatures, such as schools of fish or herds of animals. It was for instance used in the 1998 video game Half-Life for the flying bird-like creatures seen at the end of the game on Xen, named "boid" in the game files.<br />
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Boids 框架通常用于计算机图形学，提供鸟群和其他生物（如鱼群）的逼真表现。例如，在1998年的视频游戏《'''半条命 Half-Life'''》中，Xen（游戏文件中命名为“ boid”）中游戏结束时出现的类似鸟类的飞行生物就使用了该框架。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）存疑<br />
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The Boids model can be used for direct control and stabilization of teams of simple Unmanned Ground Vehicles (UGV)<ref>{{cite conference |title= Design and analysis of Group Escape Behavior for distributed autonomous mobile robots |first1= Hongkyu |last1= Min |first2= Zhidong |last2= Wang |year= 2011 |conference= IEEE International Conference on Robotics and Automation (ICRA)|doi= 10.1109/ICRA.2011.5980123 }}</ref> or [[Micro air vehicle|Micro Aerial Vehicles]] (MAV)<ref>{{cite conference |title= Swarms of micro aerial vehicles stabilized under a visual relative localization |first1= Martin |last1= Saska |first2= Vakula |last2= Jan |first3= Preucil |last3= Libor |year= 2014 |conference= IEEE International Conference on Robotics and Automation (ICRA)|doi= 10.1109/ICRA.2014.6907374 }}</ref> in [[swarm robotics]]. For stabilization of heterogeneous UAV-UGV teams, the model was adapted for using onboard relative localization by Saska et&nbsp;al.<ref>{{cite conference |title= Coordination and Navigation of Heterogeneous UAVs-UGVs Teams Localized by a Hawk-Eye Approach |first1= Martin |last1= Saska |first2= Vonasek |last2= Vojtech |first3= Krajnik |last3= Tomas |first4= Preucil |last4= Libor |year= 2012 |conference= IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)|url=http://labe.felk.cvut.cz/~tkrajnik/ardrone/articles/formace.pdf}}</ref><br />
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The Boids model can be used for direct control and stabilization of teams of simple Unmanned Ground Vehicles (UGV) or Micro Aerial Vehicles (MAV) in swarm robotics. For stabilization of heterogeneous UAV-UGV teams, the model was adapted for using onboard relative localization by Saska et&nbsp;al.<br />
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Boids 模型可用于'''[[集群机器人]] Swarm Robotics'''中简单的无人地面车辆（UGV）或'''微型飞行器 Micro Aerial Vehicles'''（MAV）群体的直接控制和稳定。为了稳定异质 UAV-UGV 团队，Saska 等人将该模型用于板载相对定位。<br />
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At the time of proposal, Reynolds' approach represented a giant step forward compared to the traditional techniques used in computer animation for motion pictures. The first animation created with the model was ''[[Stanley and Stella in: Breaking the Ice]]'' (1987), followed by a feature film debut in [[Tim Burton]]'s film ''[[Batman Returns]]'' (1992) with computer generated bat swarms and armies of penguins marching through the streets of Gotham City.<ref>{{cite journal | url=http://lrss.fri.uni-lj.si/people/ilbajec/papers/ilb_ab09.pdf | last1=Lebar Bajec |first1= Iztok |first2= Frank H. |last2= Heppner | date=2009 | title=Organized flight in birds | journal=Animal Behaviour | volume = 78 | issue = 4 | pages = 777–789 | doi = 10.1016/j.anbehav.2009.07.007 }}</ref><br />
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At the time of proposal, Reynolds' approach represented a giant step forward compared to the traditional techniques used in computer animation for motion pictures. The first animation created with the model was Stanley and Stella in: Breaking the Ice (1987), followed by a feature film debut in Tim Burton's film Batman Returns (1992) with computer generated bat swarms and armies of penguins marching through the streets of Gotham City.<br />
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在提议的时候，雷诺兹的方法相比于传统的计算机动画电影技术是一个巨大的进步。第一部利用了此模型的动画片是《'''史丹利和史黛拉: 破冰 Stanley and Stella in: Breaking The Ice'''》（1987），之后是'''蒂姆·伯顿 Tim Burton'''的故事片《'''蝙蝠侠归来 Batman Returns'''》（1992），电脑合成的蝙蝠群和成群的企鹅行进穿过'''哥谭市 Gotham City'''的街道。<br />
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The boids model has been used for other interesting applications. It has been applied to automatically program Internet multi-channel radio stations.<ref><br />
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The boids model has been used for other interesting applications. It has been applied to automatically program Internet multi-channel radio stations.<ref><br />
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{{cite conference<br />
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{引用会议<br />
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| title = DJ-boids: emergent collective behavior as multichannel radio station programming<br />
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| title = DJ-boids: emergent collective behavior as multichannel radio station programming<br />
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多频道电台节目中的紧急集体行为<br />
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| first1 = Jesús | last1 = Ibáñez<br />
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| date = 2003<br />
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| booktitle = Proceedings of the 8th international conference on Intelligent User Interfaces<br />
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| booktitle = Proceedings of the 8th international conference on Intelligent User Interfaces<br />
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第八届智能用户界面国际会议论文集<br />
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It has also been used for visualizing information<ref><br />
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It has also been used for visualizing information<ref><br />
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{{cite conference<br />
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| title = Time-Varying Data Visualization Using Information Flocking Boids<br />
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| title = Time-Varying Data Visualization Using Information Flocking Boids<br />
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使用信息植群的时变数据可视化<br />
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| booktitle = Proceedings of the IEEE Symposium on Information Visualization<br />
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| booktitle = Proceedings of the IEEE Symposium on Information Visualization<br />
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美国电气和电子工程师学会信息可视化研讨会论文集<br />
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| doi = 10.1109/INFVIS.2004.65<br />
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and for optimization tasks.<ref><br />
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and for optimization tasks.<ref><br />
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Boids 模型已经被用于其他有趣应用。该系统已应用于互联网多频道广播电台的自动节目编排，以及可视化信息和优化任务。<br />
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{{cite journal<br />
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{引用期刊<br />
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|first2 = Zhongzhi | last2 = Shi<br />
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|first2 = Zhongzhi | last2 = Shi<br />
<br />
|first2 = Zhongzhi | last2 = Shi<br />
<br />
|title= Boid particle swarm optimisation<br />
<br />
|title= Boid particle swarm optimisation<br />
<br />
博伊德粒子群优化<br />
<br />
|journal= International Journal of Innovative Computing and Applications<br />
<br />
|journal= International Journal of Innovative Computing and Applications<br />
<br />
国际创新计算与应用杂志<br />
<br />
|volume=2 |issue=2 |pages= 77–85<br />
<br />
|volume=2 |issue=2 |pages= 77–85<br />
<br />
第二期，第77-85页<br />
<br />
|date=2009<br />
<br />
|date=2009<br />
<br />
2009年<br />
<br />
|doi= 10.1504/IJICA.2009.031778}}</ref><br />
<br />
|doi= 10.1504/IJICA.2009.031778}}</ref><br />
<br />
| doi 10.1504 / IJICA. 2009.031778} / ref<br />
<br />
<br />
<br />
==See also==<br />
<br />
*[[Swarm intelligence]]<br />
<br />
*[[Collective cell migration]]<br />
<br />
<br />
<br />
==References==<br />
<br />
{{Reflist}}<br />
<br />
<br />
<br />
==External links==<br />
<br />
* [http://www.red3d.com/cwr/boids/ Craig Reynolds' Boids page]<br />
<br />
* [http://www.vergenet.net/~conrad/boids/pseudocode.html Explanation of algorithm in pseudocode]<br />
<br />
* [https://gpolo.github.com/birdflocking JavaScript implementation]<br />
<br />
*[https://lufemas.github.io/boid-ai-implementation/ JavaScript implementation with Phaser Framework]<br />
<br />
* [https://web.archive.org/web/20080205014305/http://www.navgen.com/3d_boids/ 3D Boids Simulation using OpenGL, used by the BBC's Natural History Unit]<br />
<br />
*[https://black-square.github.io/BirdFlock/ Live In-Browser 3D Simulation of Bird Flocking Behavior in Unity3D] – Open Source implementation for Windows, Linux and Mac<br />
<br />
* [https://github.com/piskvorky/PredatorPrey UNIX+Windows open source implementation in C++, using OpenGL and simulation controls]<br />
<br />
* [https://github.com/tofti/javafx-boids A java implementation using the javafx API]<br />
<br />
*GLSchool - part of [[XScreenSaver]] software<br />
<br />
<br />
<br />
{{collective animal behaviour}}<br />
<br />
<br />
<br />
[[Category:Artificial life]]<br />
<br />
Category:Artificial life<br />
<br />
类别: 人工生命<br />
<br />
[[Category:1986 software]]<br />
<br />
Category:1986 software<br />
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类别: 1986软件<br />
<br />
<noinclude><br />
<br />
<small>This page was moved from [[wikipedia:en:Boids]]. Its edit history can be viewed at [[鸟群算法/edithistory]]</small></noinclude><br />
<br />
[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%B8%9F%E7%BE%A4%E7%AE%97%E6%B3%95&diff=10682鸟群算法2020-07-18T15:22:30Z<p>Dorr：</p>
<hr />
<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。<br />
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{{short description|Artificial life program, developed by Craig Reynolds in 1986, which simulates the flocking behaviour of birds}}<br />
<br />
{{Use mdy dates|date=July 2011}}<br />
<br />
{{for|the reptilian family|Boidae}}<br />
<br />
[[File:Boids OpenGL example.webm|thumb|A Boids example created in [[OpenGL]]]]<br />
<br />
A Boids example created in [[OpenGL]]<br />
<br />
在[[[ OpenGL ]]中创建的 Boids 示例<br />
<br />
'''Boids''' is an [[artificial life]] program, developed by [[Craig Reynolds (computer graphics)|Craig Reynolds]] in 1986, which simulates the [[Flocking (behavior)|flocking]] behaviour of [[Bird|birds]]. His paper on this topic was published in 1987 in the proceedings of the [[Association for Computing Machinery|ACM]] [[SIGGRAPH]] conference. <ref>{{Cite book<br />
<br />
Boids is an artificial life program, developed by Craig Reynolds in 1986, which simulates the flocking behaviour of birds. His paper on this topic was published in 1987 in the proceedings of the ACM SIGGRAPH conference. <ref>{{Cite book<br />
<br />
'''鸟群算法 Boids'''是模拟鸟类群集行为的人工生命项目，由'''克雷格·雷诺兹 Craig Reynolds'''于1986年开发。他关于这个主题的论文发表在1987年的 ACM SIGGRAPH 会议记录上。引用{ Cite book<br />
<br />
| last1=Reynolds<br />
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| last1=Reynolds<br />
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最后1雷诺<br />
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| first1=Craig<br />
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| first1=Craig<br />
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第一名: 克雷格<br />
<br />
| author1-link=Craig Reynolds (computer_graphics)<br />
<br />
| author1-link=Craig Reynolds (computer_graphics)<br />
<br />
| author1-link Craig Reynolds (计算机图形学)<br />
<br />
| title=Flocks, herds and schools: A distributed behavioral model.<br />
<br />
| title=Flocks, herds and schools: A distributed behavioral model.<br />
<br />
羊群、畜群和学校: 一个分布式的行为模型。<br />
<br />
| year=1987<br />
<br />
| year=1987<br />
<br />
1987年<br />
<br />
| number=<br />
<br />
| number=<br />
<br />
数字<br />
<br />
| journal=SIGGRAPH '87: Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques<br />
<br />
| journal=SIGGRAPH '87: Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques<br />
<br />
87: 第14届计算机图形学与互动技术年度会议论文集<br />
<br />
| publisher=[[Association for Computing Machinery]]<br />
<br />
| publisher=Association for Computing Machinery<br />
<br />
| 出版商计算机协会<br />
<br />
| pages=25–34<br />
<br />
| pages=25–34<br />
<br />
第25-34页<br />
<br />
| isbn=978-0-89791-227-3<br />
<br />
| isbn=978-0-89791-227-3<br />
<br />
[国际标准图书编号978-0-89791-227-3]<br />
<br />
| doi=10.1145/37401.37406<br />
<br />
| doi=10.1145/37401.37406<br />
<br />
10.1145 / 37401.37406<br />
<br />
| citeseerx=10.1.1.103.7187<br />
<br />
| citeseerx=10.1.1.103.7187<br />
<br />
10.1.1.103.7187<br />
<br />
}}</ref><br />
<br />
}}</ref><br />
<br />
{} / ref<br />
<br />
The name "boid" corresponds to a shortened version of "bird-oid object", which refers to a bird-like object.<ref>{{Cite journal<br />
<br />
The name "boid" corresponds to a shortened version of "bird-oid object", which refers to a bird-like object.<ref>{{Cite journal<br />
<br />
<br />
<br />
| last1=Banks<br />
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| last1=Banks<br />
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1 Banks<br />
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| first1=Alec<br />
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| first1=Alec<br />
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首先 Alec<br />
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| last2=Vincent<br />
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| last2=Vincent<br />
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最后2文森特<br />
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| first2=Jonathan<br />
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| first2=Jonathan<br />
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第二名: 乔纳森<br />
<br />
| last3=Anyakoha<br />
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| last3=Anyakoha<br />
<br />
3 / lastanyakoha<br />
<br />
| first3=Chukwudi<br />
<br />
| first3=Chukwudi<br />
<br />
| first3 Chukwudi<br />
<br />
| title=A review of particle swarm optimization. Part I: background and development<br />
<br />
| title=A review of particle swarm optimization. Part I: background and development<br />
<br />
2012年10月12日 | 标题: 粒子群优化的回顾。第一部分: 背景与发展<br />
<br />
|date=July 2007<br />
<br />
|date=July 2007<br />
<br />
2007年7月<br />
<br />
| journal=Natural Computing<br />
<br />
| journal=Natural Computing<br />
<br />
自然计算杂志<br />
<br />
| volume=6<br />
<br />
| volume=6<br />
<br />
第六卷<br />
<br />
| issue=4<br />
<br />
| issue=4<br />
<br />
第四期<br />
<br />
| pages=467–484<br />
<br />
| pages=467–484<br />
<br />
第467-484页<br />
<br />
| doi=10.1007/s11047-007-9049-5<br />
<br />
| doi=10.1007/s11047-007-9049-5<br />
<br />
10.1007 / s11047-007-9049-5<br />
<br />
| citeseerx=10.1.1.605.5879<br />
<br />
| citeseerx=10.1.1.605.5879<br />
<br />
10.1.1.605.5879<br />
<br />
}}</ref> Incidentally, "boid" is also a New York Metropolitan dialect pronunciation for "bird".<br />
<br />
}}</ref> Incidentally, "boid" is also a New York Metropolitan dialect pronunciation for "bird".<br />
<br />
} / ref <br />
'''鸟群算法 Boids'''是模拟鸟类群集行为的人工生命项目，由'''克雷格·雷诺兹 Craig Reynolds'''于1986年开发。他关于这个主题的论文发表在1987年的 ACM SIGGRAPH（美国计算机协会计算机图形专业组组织的计算机图形学顶级年度会议）会议记录上。“ boid”是“ bird-oid object”的缩写，指一个类鸟对象。恰巧，“boid”也是纽约都市方言中“bird”的发音。<br />
<br />
<br />
<br />
{{multiple image|direction=vertical|image1=Rule separation.gif|caption1=Separation|image2=Rule alignment.gif|caption2=Alignment|image3=Rule cohesion.gif|caption3=Cohesion|header=Rules applied in simple Boids}}<br />
<br />
As with most artificial life simulations, Boids is an example of [[emergence|emergent]] behavior; that is, the complexity of Boids arises from the interaction of individual agents (the boids, in this case) adhering to a set of simple rules. The rules applied in the simplest Boids world are as follows:<br />
<br />
As with most artificial life simulations, Boids is an example of emergent behavior; that is, the complexity of Boids arises from the interaction of individual agents (the boids, in this case) adhering to a set of simple rules. The rules applied in the simplest Boids world are as follows:<br />
<br />
与大多数人工生命模拟一样，Boids 是'''涌现 Emergent'''行为的一个例子; 也就是说，Boids 的复杂性来自于遵循一系列简单规则的个体代理（这里是 Boids）的相互作用。在最简单的Bolds 世界中适用的规则如下:<br />
<br />
<br />
<br />
* '''separation''': [[wikt:steer#Verb 2|steer]] to avoid crowding local flockmates<br />
<br />
* '''alignment''': steer towards the average heading of local flockmates<br />
<br />
* '''cohesion''': steer to move towards the average position (center of mass) of local flockmates<br />
<br />
<br />
* '''分离 Separation''': 避开当地同类拥挤处<br />
<br />
* '''定位 Alignment''': 转向当地同类的平均方向<br />
<br />
* '''凝聚 Cohesion''': 朝当地同类的平均方向（质心）移动<br />
<br />
<br />
More complex rules can be added, such as obstacle avoidance and goal seeking.<br />
<br />
More complex rules can be added, such as obstacle avoidance and goal seeking.<br />
<br />
可以添加更复杂的规则，如避障和寻找目标。<br />
<br />
<br />
<br />
The basic model has been extended in several different ways since Reynolds proposed it. For instance, Delgado-Mata et&nbsp;al.<ref><br />
<br />
The basic model has been extended in several different ways since Reynolds proposed it. For instance, Delgado-Mata et&nbsp;al.<ref><br />
<br />
<br />
<br />
{{cite journal<br />
<br />
{{cite journal<br />
<br />
{引用期刊<br />
<br />
|first1= Carlos | last1= Delgado-Mata<br />
<br />
|first1= Carlos | last1= Delgado-Mata<br />
<br />
卡洛斯，德尔加多，马塔<br />
<br />
|first2= Jesus Ibanez | last2= Martinez<br />
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|first2= Jesus Ibanez | last2= Martinez<br />
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2 Jesus Ibanez | last 2 Martinez<br />
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|first3= Simon | last3= Bee<br />
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|first3= Simon | last3= Bee<br />
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| first3 Simon | last3 Bee<br />
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|first4= Rocio | last4= Ruiz-Rodarte<br />
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|first4= Rocio | last4= Ruiz-Rodarte<br />
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4 Rocio | last 4 Ruiz-Rodarte<br />
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|first5= Ruth | last5= Aylett<br />
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|first5= Ruth | last5= Aylett<br />
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| first5 Ruth | last 5 Aylett<br />
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|title= On the use of Virtual Animals with Artificial Fear in Virtual Environments<br />
<br />
|title= On the use of Virtual Animals with Artificial Fear in Virtual Environments<br />
<br />
论虚拟动物在虚拟环境中的人为恐惧<br />
<br />
|journal= New Generation Computing<br />
<br />
|journal= New Generation Computing<br />
<br />
| 杂志新一代计算<br />
<br />
|volume=25 |issue=2 |pages= 145–169<br />
<br />
|volume=25 |issue=2 |pages= 145–169<br />
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第25卷，第2期，第145-169页<br />
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|date=2007<br />
<br />
|date=2007<br />
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2007年<br />
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|doi= 10.1007/s00354-007-0009-5}}</ref><br />
<br />
|doi= 10.1007/s00354-007-0009-5}}</ref><br />
<br />
| doi 10.1007 / s00354-007-0009-5} / ref<br />
<br />
extended the basic model to incorporate the effects of fear. Olfaction was used to transmit emotion between animals, through pheromones modelled as particles in a free expansion gas. Hartman and Benes<ref><br />
<br />
extended the basic model to incorporate the effects of fear. Olfaction was used to transmit emotion between animals, through pheromones modelled as particles in a free expansion gas. Hartman and Benes<ref><br />
<br />
<br />
<br />
{{cite journal<br />
<br />
{{cite journal<br />
<br />
{引用期刊<br />
<br />
|first1= Christopher | last1= Hartman<br />
<br />
|first1= Christopher | last1= Hartman<br />
<br />
1 Christopher | last 1 Hartman<br />
<br />
|first2= Bedr̆ich |last2= Benes̆<br />
<br />
|first2= Bedr̆ich |last2= Benes̆<br />
<br />
|first2= Bedr̆ich |last2= Benes̆<br />
<br />
|title= Autonomous boids<br />
<br />
|title= Autonomous boids<br />
<br />
标题自主 boids<br />
<br />
|journal= Computer Animation and Virtual Worlds<br />
<br />
|journal= Computer Animation and Virtual Worlds<br />
<br />
计算机动画与虚拟世界<br />
<br />
|volume=17 |issue=3–4 |pages= 199–206<br />
<br />
|volume=17 |issue=3–4 |pages= 199–206<br />
<br />
第17卷第3-4期，第199-206页<br />
<br />
|date=July 2006<br />
<br />
|date=July 2006<br />
<br />
2006年7月<br />
<br />
|doi= 10.1002/cav.123}}</ref><br />
<br />
|doi= 10.1002/cav.123}}</ref><br />
<br />
10.1002 / cav. 123} / ref<br />
<br />
introduced a complementary force to the alignment that they call the change of leadership. This steer defines the chance of the boid to become a leader and try to escape.<br />
<br />
introduced a complementary force to the alignment that they call the change of leadership. This steer defines the chance of the boid to become a leader and try to escape.<br />
<br />
该基本模型自雷诺兹提出以来，已用多种方法扩展。例如，Delgado-Mata 等人扩展基本模型以加入恐惧的影响。利过一种可用自由膨胀气体中的粒子模拟的信息素，动物间用嗅觉来传递情感。'''哈特曼 Hartman'''和'''贝内斯 Benes'''为定位过程引入了一种补充力量，他们称之为领导权的更迭。这条决定了某只鸟成为领导者及试图逃跑的机会。<br />
<br />
<br />
<br />
The movement of Boids can be characterized as either chaotic (splitting groups and wild behaviour) or orderly. Unexpected behaviours, such as splitting flocks and reuniting after avoiding obstacles, can be considered emergent.<br />
<br />
The movement of Boids can be characterized as either chaotic (splitting groups and wild behaviour) or orderly. Unexpected behaviours, such as splitting flocks and reuniting after avoiding obstacles, can be considered emergent.<br />
<br />
Boids 的运动可以表现为混乱（分裂的群体和狂野的行为）或有序。意想不到的行为，比如群体分散和避开障碍后的团聚可以被认为是应急的。<br />
<br />
<br />
<br />
The boids framework is often used in computer graphics, providing realistic-looking representations of flocks of birds and other creatures, such as schools of fish or herds of animals. It was for instance used in the 1998 video game ''[[Half-Life (video game)|Half-Life]]'' for the flying bird-like creatures seen at the end of the game on [[Locations of Half-Life#Xen|Xen]], named "boid" in the game files.<br />
<br />
The boids framework is often used in computer graphics, providing realistic-looking representations of flocks of birds and other creatures, such as schools of fish or herds of animals. It was for instance used in the 1998 video game Half-Life for the flying bird-like creatures seen at the end of the game on Xen, named "boid" in the game files.<br />
<br />
Boids 框架通常用于计算机图形学，提供鸟群和其他生物（如鱼群）的逼真表现。例如，在1998年的视频游戏《'''半条命 Half-Life'''》中，Xen（游戏文件中命名为“ boid”）中游戏结束时出现的类似鸟类的飞行生物就使用了该框架。<br />
<br />
--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）存疑<br />
<br />
The Boids model can be used for direct control and stabilization of teams of simple Unmanned Ground Vehicles (UGV)<ref>{{cite conference |title= Design and analysis of Group Escape Behavior for distributed autonomous mobile robots |first1= Hongkyu |last1= Min |first2= Zhidong |last2= Wang |year= 2011 |conference= IEEE International Conference on Robotics and Automation (ICRA)|doi= 10.1109/ICRA.2011.5980123 }}</ref> or [[Micro air vehicle|Micro Aerial Vehicles]] (MAV)<ref>{{cite conference |title= Swarms of micro aerial vehicles stabilized under a visual relative localization |first1= Martin |last1= Saska |first2= Vakula |last2= Jan |first3= Preucil |last3= Libor |year= 2014 |conference= IEEE International Conference on Robotics and Automation (ICRA)|doi= 10.1109/ICRA.2014.6907374 }}</ref> in [[swarm robotics]]. For stabilization of heterogeneous UAV-UGV teams, the model was adapted for using onboard relative localization by Saska et&nbsp;al.<ref>{{cite conference |title= Coordination and Navigation of Heterogeneous UAVs-UGVs Teams Localized by a Hawk-Eye Approach |first1= Martin |last1= Saska |first2= Vonasek |last2= Vojtech |first3= Krajnik |last3= Tomas |first4= Preucil |last4= Libor |year= 2012 |conference= IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)|url=http://labe.felk.cvut.cz/~tkrajnik/ardrone/articles/formace.pdf}}</ref><br />
<br />
The Boids model can be used for direct control and stabilization of teams of simple Unmanned Ground Vehicles (UGV) or Micro Aerial Vehicles (MAV) in swarm robotics. For stabilization of heterogeneous UAV-UGV teams, the model was adapted for using onboard relative localization by Saska et&nbsp;al.<br />
<br />
Boids 模型可以用于群机器人中简单的无人地面车辆(UGV)或微型飞行器(MAV)团队的直接控制和稳定。为了稳定异质 UAV-UGV 团队，Saska 等人将该模型用于板载相对定位。<br />
<br />
<br />
<br />
At the time of proposal, Reynolds' approach represented a giant step forward compared to the traditional techniques used in computer animation for motion pictures. The first animation created with the model was ''[[Stanley and Stella in: Breaking the Ice]]'' (1987), followed by a feature film debut in [[Tim Burton]]'s film ''[[Batman Returns]]'' (1992) with computer generated bat swarms and armies of penguins marching through the streets of Gotham City.<ref>{{cite journal | url=http://lrss.fri.uni-lj.si/people/ilbajec/papers/ilb_ab09.pdf | last1=Lebar Bajec |first1= Iztok |first2= Frank H. |last2= Heppner | date=2009 | title=Organized flight in birds | journal=Animal Behaviour | volume = 78 | issue = 4 | pages = 777–789 | doi = 10.1016/j.anbehav.2009.07.007 }}</ref><br />
<br />
At the time of proposal, Reynolds' approach represented a giant step forward compared to the traditional techniques used in computer animation for motion pictures. The first animation created with the model was Stanley and Stella in: Breaking the Ice (1987), followed by a feature film debut in Tim Burton's film Batman Returns (1992) with computer generated bat swarms and armies of penguins marching through the streets of Gotham City.<br />
<br />
在提议的时候，雷诺兹的方法代表了一个巨大的进步相比，传统的技术用于电脑动画电影。第一部与模特合作的动画片是1987年的《史丹利和斯特拉: 破冰》(Stanley and Stella in: Breaking The Ice，1987) ，之后是1992年蒂姆 · 伯顿(Tim Burton)的电影《蝙蝠侠归来》(Batman Returns，1992) ，电脑合成的蝙蝠群和成群的企鹅穿过哥谭市。<br />
<br />
<br />
<br />
The boids model has been used for other interesting applications. It has been applied to automatically program Internet multi-channel radio stations.<ref><br />
<br />
The boids model has been used for other interesting applications. It has been applied to automatically program Internet multi-channel radio stations.<ref><br />
<br />
Boids 模型已经被用于其他有趣的应用程序。该系统已应用于互联网多频道广播电台的自动节目编排<br />
<br />
{{cite conference<br />
<br />
{{cite conference<br />
<br />
{引用会议<br />
<br />
| title = DJ-boids: emergent collective behavior as multichannel radio station programming<br />
<br />
| title = DJ-boids: emergent collective behavior as multichannel radio station programming<br />
<br />
多频道电台节目中的紧急集体行为<br />
<br />
| first1 = Jesús | last1 = Ibáñez<br />
<br />
| first1 = Jesús | last1 = Ibáñez<br />
<br />
| first1 = Jesús | last1 = Ibáñez<br />
<br />
| first2 = Antonio F. | last2 = Gómez-Skarmeta <br />
<br />
| first2 = Antonio F. | last2 = Gómez-Skarmeta <br />
<br />
| first2 = Antonio F. | last2 = Gómez-Skarmeta<br />
<br />
| first3 = Josep | last3 = Blat<br />
<br />
| first3 = Josep | last3 = Blat<br />
<br />
| first3 = Josep | last3 = Blat<br />
<br />
| date = 2003<br />
<br />
| date = 2003<br />
<br />
2003年<br />
<br />
| booktitle = Proceedings of the 8th international conference on Intelligent User Interfaces<br />
<br />
| booktitle = Proceedings of the 8th international conference on Intelligent User Interfaces<br />
<br />
第八届智能用户界面国际会议论文集<br />
<br />
| pages = 248–250<br />
<br />
| pages = 248–250<br />
<br />
第248-250页<br />
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| doi = 10.1145/604045.604089<br />
<br />
| doi = 10.1145/604045.604089<br />
<br />
10.1145 / 604045.604089<br />
<br />
}}</ref><br />
<br />
}}</ref><br />
<br />
{} / ref<br />
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It has also been used for visualizing information<ref><br />
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It has also been used for visualizing information<ref><br />
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它也被用于可视化信息参考<br />
<br />
{{cite conference<br />
<br />
{{cite conference<br />
<br />
{引用会议<br />
<br />
| title = Time-Varying Data Visualization Using Information Flocking Boids<br />
<br />
| title = Time-Varying Data Visualization Using Information Flocking Boids<br />
<br />
使用信息植群的时变数据可视化<br />
<br />
| last = Moere | first = A V<br />
<br />
| last = Moere | first = A V<br />
<br />
最后一个 Moere | 第一个 a v<br />
<br />
| date = 2004<br />
<br />
| date = 2004<br />
<br />
2004年<br />
<br />
| booktitle = Proceedings of the IEEE Symposium on Information Visualization<br />
<br />
| booktitle = Proceedings of the IEEE Symposium on Information Visualization<br />
<br />
美国电气和电子工程师学会信息可视化研讨会论文集<br />
<br />
| pages = 97–104<br />
<br />
| pages = 97–104<br />
<br />
第97-104页<br />
<br />
| doi = 10.1109/INFVIS.2004.65<br />
<br />
| doi = 10.1109/INFVIS.2004.65<br />
<br />
10.1109 / INFVIS. 2004.65<br />
<br />
}}</ref><br />
<br />
}}</ref><br />
<br />
{} / ref<br />
<br />
and for optimization tasks.<ref><br />
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and for optimization tasks.<ref><br />
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还有优化任务，裁判<br />
<br />
{{cite journal<br />
<br />
{{cite journal<br />
<br />
{引用期刊<br />
<br />
|first1 = Zhihua | last1 = Cui<br />
<br />
|first1 = Zhihua | last1 = Cui<br />
<br />
最新一期崔<br />
<br />
|first2 = Zhongzhi | last2 = Shi<br />
<br />
|first2 = Zhongzhi | last2 = Shi<br />
<br />
|first2 = Zhongzhi | last2 = Shi<br />
<br />
|title= Boid particle swarm optimisation<br />
<br />
|title= Boid particle swarm optimisation<br />
<br />
博伊德粒子群优化<br />
<br />
|journal= International Journal of Innovative Computing and Applications<br />
<br />
|journal= International Journal of Innovative Computing and Applications<br />
<br />
国际创新计算与应用杂志<br />
<br />
|volume=2 |issue=2 |pages= 77–85<br />
<br />
|volume=2 |issue=2 |pages= 77–85<br />
<br />
第二期，第77-85页<br />
<br />
|date=2009<br />
<br />
|date=2009<br />
<br />
2009年<br />
<br />
|doi= 10.1504/IJICA.2009.031778}}</ref><br />
<br />
|doi= 10.1504/IJICA.2009.031778}}</ref><br />
<br />
| doi 10.1504 / IJICA. 2009.031778} / ref<br />
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<br />
==See also==<br />
<br />
*[[Swarm intelligence]]<br />
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*[[Collective cell migration]]<br />
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==References==<br />
<br />
{{Reflist}}<br />
<br />
<br />
<br />
==External links==<br />
<br />
* [http://www.red3d.com/cwr/boids/ Craig Reynolds' Boids page]<br />
<br />
* [http://www.vergenet.net/~conrad/boids/pseudocode.html Explanation of algorithm in pseudocode]<br />
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* [https://gpolo.github.com/birdflocking JavaScript implementation]<br />
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*[https://lufemas.github.io/boid-ai-implementation/ JavaScript implementation with Phaser Framework]<br />
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* [https://web.archive.org/web/20080205014305/http://www.navgen.com/3d_boids/ 3D Boids Simulation using OpenGL, used by the BBC's Natural History Unit]<br />
<br />
*[https://black-square.github.io/BirdFlock/ Live In-Browser 3D Simulation of Bird Flocking Behavior in Unity3D] – Open Source implementation for Windows, Linux and Mac<br />
<br />
* [https://github.com/piskvorky/PredatorPrey UNIX+Windows open source implementation in C++, using OpenGL and simulation controls]<br />
<br />
* [https://github.com/tofti/javafx-boids A java implementation using the javafx API]<br />
<br />
*GLSchool - part of [[XScreenSaver]] software<br />
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<br />
{{collective animal behaviour}}<br />
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[[Category:Artificial life]]<br />
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Category:Artificial life<br />
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类别: 人工生命<br />
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[[Category:1986 software]]<br />
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Category:1986 software<br />
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类别: 1986软件<br />
<br />
<noinclude><br />
<br />
<small>This page was moved from [[wikipedia:en:Boids]]. Its edit history can be viewed at [[鸟群算法/edithistory]]</small></noinclude><br />
<br />
[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E7%83%AD%E5%8A%9B%E5%AD%A6%E7%AC%AC%E4%BA%8C%E5%AE%9A%E5%BE%8B&diff=10362热力学第二定律2020-07-16T09:44:10Z<p>Dorr：</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Law of physics}}<br />
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{{Thermodynamics|laws}}<br />
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The '''second law of thermodynamics''' states that the total [[entropy]] of an [[isolated system]] can never decrease over time, and is constant if and only if all processes are reversible.<ref>http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node38.html#SECTION05224000000000000000</ref> Isolated systems spontaneously evolve towards [[thermodynamic equilibrium]], the state with maximum entropy. <br />
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The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, and is constant if and only if all processes are reversible. Isolated systems spontaneously evolve towards thermodynamic equilibrium, the state with maximum entropy. <br />
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热力学第二定律指出，孤立系统的总熵永远不会随着时间而减少，而且只有当且仅当所有过程都是可逆的时，总熵才是恒定的。孤立的系统自发地进化到热力学平衡状态，即熵最大的状态。<br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in [[thermodynamic equilibrium]], or is undergoing a (fictive) [[Reversible process (thermodynamics)|reversible process]]. In all processes that occur, including [[spontaneous process]]es,<ref>Atkins and de Paula, p.78</ref> the total entropy of the system and its surroundings increases and the process is [[Irreversible process|irreversible in the thermodynamic sense]]. The increase in entropy accounts for the irreversibility of natural processes, and the [[Arrow of time|asymmetry between future and past]].<ref>{{cite book|last=Zohuri|first=Bahman|title=Dimensional Analysis Beyond the Pi Theorem|url=https://books.google.com/books?id=pRVuDQAAQBAJ|year=2016|publisher=Springer|isbn=978-3-319-45726-0|page=[https://books.google.com.ph/books?id=pRVuDQAAQBAJ&pg=PA111&dq=%22increase+in+entropy+accounts+for+the+irreversibility+of+natural+processes+and+the+asymmetry+between+future+and+past.%22&hl=en&sa=X&ved=0ahUKEwj6spb61tbaAhUFS7wKHfftDtIQ6AEIKDAA#v=onepage&q=%22increase%20in%20entropy%20accounts%20for%20the%20irreversibility%20of%20natural%20processes%20and%20the%20asymmetry%20between%20future%20and%20past.%22 111]}}</ref><br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in thermodynamic equilibrium, or is undergoing a (fictive) reversible process. In all processes that occur, including spontaneous processes, the total entropy of the system and its surroundings increases and the process is irreversible in the thermodynamic sense. The increase in entropy accounts for the irreversibility of natural processes, and the asymmetry between future and past.<br />
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系统及其周围环境的总熵在理想情况下可以保持不变，在这种情况下，系统处于热力学平衡期，或者正在经历一个(虚拟的)可逆过程期。在所有发生的过程中，包括自发过程，系统及其周围环境的总熵增加，这一过程在热力学意义上是不可逆的。熵的增加解释了自然过程的不可逆性，以及未来和过去之间的不对称性。<br />
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Historically, the second law was an [[Empirical evidence|empirical finding]] that was accepted as an axiom of [[thermodynamics|thermodynamic theory]]. [[Statistical mechanics]], classical or [[quantum thermodynamics|quantum]], explains the microscopic origin of the law.<br />
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Historically, the second law was an empirical finding that was accepted as an axiom of thermodynamic theory. Statistical mechanics, classical or quantum, explains the microscopic origin of the law.<br />
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从历史上看，第二定律是一个经验性的发现，被公认为热力学理论的公理。统计力学，无论是经典的还是量子的，都可以解释这个定律的微观起源。<br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist [[Nicolas Léonard Sadi Carnot|Sadi Carnot]], who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as [[Carnot's theorem (thermodynamics)|Carnot's rule]] or limit.<ref>Jaffe, R.L., Taylor, W. (2018). ''The Physics of Energy'', Cambridge University Press, Cambridge UK, pages 150, 151, 259, 772, 743.</ref><br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist Sadi Carnot, who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as Carnot's rule or limit.<br />
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第二定律已经用许多方式表达过了。法国科学家萨迪 · 卡诺在1824年证明了热能在热机中转化为功的效率是有上限的。这方面的第二定律也被称为卡诺的规则或限制。<br />
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==Introduction==<br />
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==Introduction==<br />
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引言<br />
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[[File:Heat flow hot to cold.png|thumb|upright|Heat flow from hot water to cold water.]]<br />
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Heat flow from hot water to cold water.<br />
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热量从热水流向冷水。<br />
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The [[first law of thermodynamics]] provides the definition of the [[internal energy]] of a [[thermodynamic system]], and expresses the law of [[conservation of energy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 40–41.</ref><ref>Munster A. (1970), pp. 8–9, 50–51.</ref> The second law is concerned with the direction of natural processes.<ref>{{harvnb|Mandl|1988}}</ref> It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such [[Phenomenon|phenomena]] are accounted for in terms of [[entropy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 79–107.</ref><ref>Bailyn, M. (1994), Section 71, pp. 113–154.</ref> If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, {{math|''S''}}, increases.<br />
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The first law of thermodynamics provides the definition of the internal energy of a thermodynamic system, and expresses the law of conservation of energy. The second law is concerned with the direction of natural processes. It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such phenomena are accounted for in terms of entropy. If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, , increases.<br />
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能量守恒定律提供了热力学系统内能的定义，并表达了能量守恒定律。第二定律与自然过程的方向有关。它断言自然过程只在一种意义上运行，而且是不可逆的。例如，当有了传导和辐射的路径时，热总是自发地从一个较热的物体流向一个较冷的物体。这种现象可以用熵来解释。如果一个孤立的系统最初是通过内部分隔不可渗透的墙来保持在内部热力学平衡，然后一些操作使墙更具渗透性，那么系统自发地进化到最终达到一个新的内部热力学平衡，其总熵增加。<br />
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In a fictive reversible process, an infinitesimal increment in the entropy ({{math|d''S''}}) of a system is defined to result from an infinitesimal transfer of heat ({{math|δ''Q''}}) to a [[closed system]] (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature ({{math|''T''}}) of the system in equilibrium and the surroundings which supply the heat:<ref>Bailyn, M. (1994), p. 120.</ref><br />
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In a fictive reversible process, an infinitesimal increment in the entropy () of a system is defined to result from an infinitesimal transfer of heat () to a closed system (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature () of the system in equilibrium and the surroundings which supply the heat:<br />
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在虚构的可逆过程中，系统熵的无穷小增量被定义为由无穷小的热量传递()到一个封闭系统(允许能量进入或出去，但不允许物质传递)除以平衡系统和提供热量的环境的共同温度() :<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, idealized fictive reversible process)}.</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, idealized fictive reversible process)}.</math><br />
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(闭合系统，理想化的可逆过程) / math<br />
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Different notations are used for infinitesimal amounts of heat ({{mvar|δ}}) and infinitesimal amounts of entropy ({{mvar|d}}) because entropy is a [[function of state]], while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the [[Clausius Theorem|inequality]] <ref name="MortimerBook"> Mortimer, R. G. ''Physical Chemistry''. 3rd ed., p. 120, Academic Press, 2008. </ref><ref name="FermiBook"> Fermi, E. ''Thermodynamics'', footnote on p. 48, Dover Publications,1956 (still in print). </ref> <br />
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Different notations are used for infinitesimal amounts of heat () and infinitesimal amounts of entropy () because entropy is a function of state, while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the inequality <br />
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用不同的符号表示无穷小量的热量()和无穷小量的熵() ，因为熵是状态的函数，而热量和功一样不是。第二定律要求系统熵的增量满足不等式，对于实际上可能存在的不与环境发生质量交换的无穷小过程，系统熵增量满足不等式<br />
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: <math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process).}</math><br />
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<math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process).}</math><br />
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(实际上可能是封闭系统，不可逆性)。数学<br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature ({{math|''T''}}) and the temperature of the surroundings ({{math|''T''}}<sub>''surr''</sub>).<ref name=":0">Adkins, C.J. (1968/1983), p. 75.</ref><ref name="Munster 45"/> Note that the equality still applies for pure heat flow,<ref name="Schmidt-Rohr 14"> Schmidt-Rohr, K. (2014). "Expansion Work without the External Pressure, and Thermodynamics in Terms of Quasistatic Irreversible Processes" ''J. Chem. Educ.'' '''91''': 402-409. https://dx.doi.org/10.1021/ed3008704 </ref><br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature () and the temperature of the surroundings (<sub>surr</sub>).<br />
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这是因为这种情况下的一般过程可能包括周围环境对系统所做的功，这可能在系统内部产生摩擦或粘滞效应，因为一个化学反应可能正在进行，或者因为热传递实际上只是不可逆地发生，由系统温度()和周围环境温度(sub surr / sub)之间的差分驱动。<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change).}</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change).}</math><br />
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(实际上可能是不改变成分的准静态不可逆性)。数学<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry.<ref name="Oxtoby8th">Oxtoby, D. W; Gillis, H.P., [[Laurie Butler|Butler, L. J.]] (2015).''Principles of Modern Chemistry'', Brooks Cole. p. 617. {{ISBN|978-1305079113}}</ref> <ref name="MortimerBook"></ref> Introducing a set of internal variables <math>\xi</math> to describe the deviation of a thermodynamic system in physical equilibrium (with the required well-defined uniform pressure ''P'' and temperature ''T'')<ref name="Schmidt-Rohr 14"></ref> from the chemical equilibrium state, one can record the equality<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry. from the chemical equilibrium state, one can record the equality<br />
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这是根据测得的热容曲线和相变时的熵变精确确定纯物质的绝对熵的基础。用量热法。从化学平衡来看，人们可以记录平等<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process).}</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process).}</math><br />
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- frac { t }-frac {1}{ j } ， Xi { j } ， delta Xi j， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ,.数学<br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of ''the second principle of thermodynamics'' – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<ref> Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769–781.</ref><ref>{{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}</ref><br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of the second principle of thermodynamics – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<br />
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第二项代表内部变量的工作，这些内部变量可以受到外部影响的干扰，但是系统不能通过内部变量执行任何正功。这种说法介绍了热力学系统在时间上不可能逆转演化的事实，并且可以被认为是热力学第二原理的表述---- 这个表述，当然，相当于熵原理的表述。<br />
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The [[zeroth law of thermodynamics]] in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body.<ref name=dugdale>{{cite book|author=J. S. Dugdale|title=Entropy and its Physical Meaning|publisher=Taylor & Francis|year=1996|isbn=978-0-7484-0569-5|page=13|quote=This law is the basis of temperature.}}</ref> For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows{{how?|date=August 2018}} a distinguished temperature scale, which defines an absolute, [[thermodynamic temperature]], independent of the properties of any particular reference thermometric body.<ref>[[Mark Zemansky|Zemansky, M.W.]] (1968), pp. 207–209.</ref><ref>Quinn, T.J. (1983), p. 8.</ref><br />
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The zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body. For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows a distinguished temperature scale, which defines an absolute, thermodynamic temperature, independent of the properties of any particular reference thermometric body.<br />
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美国热力学第零定律协会在其通常的简短声明中允许承认，相对于热平衡的两个物体具有相同的温度，特别是测试物体具有与参考温度测量物体相同的温度。对于一个物体和另一个物体之间的热平衡，存在着无限多的经验温度标度，通常分别取决于特定参考温度标度的性质。第二定律允许区分温度标度，它定义了一个绝对的热力学温度，与任何特定的参考温度体的性质无关。<br />
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==Various statements of the law==<br />
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==Various statements of the law==<br />
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各种各样的法律陈述<br />
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The second law of thermodynamics may be expressed in many specific ways,<ref name=MIT>{{cite web|title=Concept and Statements of the Second Law|url=http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node37.html|accessdate=2010-10-07 |publisher=web.mit.edu}}</ref> the most prominent classical statements{{sfnp|Lieb|Yngvason|1999}} being the statement by [[Rudolf Clausius]] (1854), the statement by [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] (1851), and the statement in axiomatic thermodynamics by [[Constantin Carathéodory]] (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.{{sfnp|Rao|2004|p=213}}<br />
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The second law of thermodynamics may be expressed in many specific ways, the most prominent classical statements being the statement by Rudolf Clausius (1854), the statement by Lord Kelvin (1851), and the statement in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.<br />
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热力学第二定律可以用许多特定的方式来表达，最突出的经典陈述是 Rudolf Clausius (1854)的陈述，Kelvin 勋爵(1851)的陈述，以及康斯坦丁·卡拉西奥多里(1909)在公理化热力学中的陈述。这些陈述用一般的物理术语描述法律，引用某些过程的不可能性。克劳修斯和开尔文陈述被证明是等价的。<br />
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===Carnot's principle===<br />
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===Carnot's principle===<br />
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卡诺原理<br />
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The historical origin<ref>[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986).</ref> of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a [[Carnot heat engine]], fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the [[caloric theory]] of heat was seriously considered, before the recognition of the [[first law of thermodynamics]], and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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The historical origin of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a Carnot heat engine, fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the caloric theory of heat was seriously considered, before the recognition of the first law of thermodynamics, and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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热力学第二定律的历史起源源于卡诺的原则。它指的是一个卡诺热机周期，虚拟地以被称为准静态的极慢极限模式运行，因此热量和功量在子系统之间传递，这些子系统始终处于它们自己的内部热力学平衡状态。卡诺发动机是热机效率工程师特别感兴趣的理想设备。卡诺认识到卡诺原理的时候，正是热量热理论被认真考虑的时候，正是能量守恒定律理论被认识之前，正是熵概念的数学表达之前。根据第一定律的解释，它在物理上等同于热力学第二定律，并且在今天仍然有效。卡诺最初的论点是从卡路里理论的观点提出的，直到能量守恒定律的发现。以下是他书中的一些例子:<br />
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::...''wherever there exists a difference of temperature, motive power can be produced.''<ref>Carnot, S. (1824/1986), p. 51.</ref><br />
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...wherever there exists a difference of temperature, motive power can be produced.<br />
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只要有温差，就能产生动力。<br />
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::The production of motive power is then due in steam engines not to an actual consumption of caloric, but ''to its transportation from a warm body to a cold body ...''<ref>Carnot, S. (1824/1986), p. 46.</ref><br />
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The production of motive power is then due in steam engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body ...<br />
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动力的产生不是由于蒸汽机实际消耗的热量，而是由于它从一个温暖的物体运输到一个寒冷的物体..。<br />
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::''The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.''<ref>Carnot, S. (1824/1986), p. 68.</ref><br />
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The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.<br />
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热的动力与实现热的媒介无关，热的数量完全取决于两个物体之间的温度，最后是热量的传递。<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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用现代术语来说，卡诺的原则可能更为准确:<br />
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::The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<ref>[[Clifford Truesdell|Truesdell, C.]] (1980), Chapter 5.</ref><ref>Adkins, C.J. (1968/1983), pp. 56–58.</ref><ref>Münster, A. (1970), p. 11.</ref><ref>Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998), pp.67–75.</ref><ref>Lebon, G., Jou, D., Casas-Vázquez, J. (2008), p. 10.</ref><ref>Eu, B.C. (2002), pp. 32–35.</ref><br />
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The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<br />
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准静态或可逆卡诺循环的效率只取决于两个热源的温度，而且无论工作物质是什么，效率是相同的。以这种方式运行的卡诺发动机是使用这两种温度的最有效的热机。<br />
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===Clausius statement===<br />
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===Clausius statement===<br />
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克劳修斯声明<br />
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The German scientist [[Rudolf Clausius]] laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work.{{sfnp|Clausius|1850}} His formulation of the second law, which was published in German in 1854, is known as the ''Clausius statement'':<br />
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The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work. His formulation of the second law, which was published in German in 1854, is known as the Clausius statement:<br />
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1850年，德国科学家 Rudolf Clausius 通过研究热传递和功之间的关系，为热力学第二定律实验室奠定了基础。他在1854年用德语发表的第二定律的提法被称为克劳修斯声明:<br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.{{sfnp|Clausius|1854|p=86}}</blockquote><br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.</blockquote><br />
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热量不可能从一个较冷的物体传递到一个较热的物体，除非同时发生一些其他的变化。 / blockquote<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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克劳修斯的陈述使用了“热量通道”的概念。正如通常在热力学的讨论，这意味着“能量作为热量的净转移” ，并没有提到促成转移的一种方式和其他。<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of [[refrigeration]], for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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如果不对系统进行外部功，热就不能自发地从冷区流向热区，这一点从制冷的普通经验中可以看出。在冰箱中，热量从冷到热，但只有在外部介质——制冷系统的强制作用下才会发生变化。<br />
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===Kelvin statements===<br />
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===Kelvin statements===<br />
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开尔文陈述<br />
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[[William Thomson, 1st Baron Kelvin|Lord Kelvin]] expressed the second law in several wordings.<br />
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Lord Kelvin expressed the second law in several wordings.<br />
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开尔文勋爵用几个字表达了第二定律。<br />
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::It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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没有任何外部机构的帮助，自动机器是不可能把热量从一个物体传递到另一个物体的较高温度。<br />
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::It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.{{sfnp|Thomson|1851}}<br />
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It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.<br />
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通过无生命物质的作用，把物质冷却到周围最冷的物体的温度以下，从物质的任何部分获得机械效应是不可能的。<br />
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===Equivalence of the Clausius and the Kelvin statements===<br />
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===Equivalence of the Clausius and the Kelvin statements===<br />
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克劳修斯和开尔文陈述的等价性<br />
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[[Image:Deriving Kelvin Statement from Clausius Statement.svg|thumb|Derive Kelvin Statement from Clausius Statement]]<br />
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Derive Kelvin Statement from Clausius Statement<br />
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从克劳修斯陈述推导出开尔文陈述<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed [[Carnot engine]] as shown by the figure. The [[Heat engine#Efficiency|efficiency]] of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the [[first law of thermodynamics]], as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the figure. The efficiency of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the first law of thermodynamics, as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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假设有一个引擎违反了开尔文定理: 也就是说，这个引擎以循环的方式吸收热量并将其转化为功，没有任何其他结果。现在配对与反向卡诺发动机如图所示。普通热机的效率为，反向热机的效率为1 / 。这对组合发动机的净效应和唯一效应是将热量从较冷的贮存器转移到较热的贮存器，这违反了克劳修斯的说法。(这是能量守恒定律的结果，因为系统的总能量保持不变，数学文本{输入} + 文本{输出}0暗示 q-frac { eta }-qc / math，所以数学 qc 向左(frac { eta-1 right) / math)。因此，违反开尔文陈述意味着违反克劳修斯陈述，即。克劳修斯的说法暗示了凯尔文的说法。我们可以用类似的方式证明开尔文陈述暗示了克劳修斯陈述，因此两者是等价的。<br />
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===Planck's proposition===<br />
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===Planck's proposition===<br />
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普朗克的命题<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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普朗克提出了直接来自经验的如下命题。这有时被认为是他对第二定律的陈述，但他认为这是第二定律推导的起点。<br />
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::It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 86.</ref><ref>Roberts, J.K., Miller, A.R. (1928/1960), p. 319.</ref><br />
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It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<br />
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建造一台发动机是不可能的，这台发动机将在一个完整的循环中工作，并且除了提高重量和冷却热储以外不会产生任何效果。<br />
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===Relation between Kelvin's statement and Planck's proposition 开尔文表述与普朗克命题的关系===<br />
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It is almost customary in textbooks to speak of the "[[Kelvin-Planck statement]]" of the law, as for example in the text by [[Dirk ter Haar|ter Haar]] and [[Harald Wergeland|Wergeland]].<ref>[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966), p. 17.</ref><br />
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It is almost customary in textbooks to speak of the "Kelvin-Planck statement" of the law, as for example in the text by ter Haar and Wergeland.<br />
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教科书中几乎惯常提及该定律的“'''开尔文-普朗克表述 Kelvin-Planck Statement'''” ，例如'''德克·特哈尔 Diek ter Haar''' 和'''哈拉尔德·沃格兰 Harald Wergeland''' 在文中就是这样表述的。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）Diek ter Haar 中间是小写的<br />
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The '''Kelvin–Planck statement''' (or the ''heat engine statement'') of the second law of thermodynamics states that <br />
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The Kelvin–Planck statement (or the heat engine statement) of the second law of thermodynamics states that <br />
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热力学第二定律的开尔文-普朗克表述（或称“'''热机表述 Heat Engine Statement'''”）指出<br />
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::It is impossible to devise a [[thermodynamic cycle|cyclically]] operating device, the sole effect of which is to absorb energy in the form of heat from a single [[heat reservoir|thermal reservoir]] and to deliver an equivalent amount of [[Work (physics)|work]].<ref name="Rao">{{cite book|last=Rao|first=Y. V. C.|title=Chemical Engineering Thermodynamics|publisher=Universities Press|isbn=978-81-7371-048-3|page=158|year=1997}}</ref><br />
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It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.<br />
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设计一种唯一效果是从单一热源吸收热量并提供等量的功的循环运行装置是不可能的。<br />
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===Planck's statement 普朗克表述===<br />
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Planck stated the second law as follows.<br />
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Planck stated the second law as follows.<br />
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普朗克表述第二定律如下。<br />
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::Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<ref name="Planck 100">[[Max Planck|Planck, M.]] (1897/1903), p. 100.</ref><ref name="Planck 463">[[Max Planck|Planck, M.]] (1926), p. 463, translation by Uffink, J. (2003), p. 131.</ref><ref name="Roberts & Miller 382">Roberts, J.K., Miller, A.R. (1928/1960), p. 382. This source is partly verbatim from Planck's statement, but does not cite Planck. This source calls the statement the principle of the increase of entropy.</ref><br />
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Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<br />
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自然界中发生的任一过程都是沿参与其中的所有物体的熵之和增加的方向进行的。在某些限制下——即对于可逆过程，熵的总和保持不变。<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for ''irreversible phenomena''.<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for irreversible phenomena.<br />
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与普朗克表述非常相似的是'''乌伦贝克 Uhlenbeck'''和'''福特 Ford'''关于不可逆现象的表述。<br />
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::... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<ref>[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963), p. 16.</ref><br />
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... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<br />
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在从一个平衡态到另一个平衡态的不可逆或自发的变化中（例如，当两个物体 A 和 B 接触时的温度平衡过程），熵总是增加。<br />
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===Principle of Carathéodory 卡拉西奥多里原理===<br />
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<!-- [[Caratheodory's principle]] redirects here --><br />
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—— caratheodory 的原理在这里重新定向——<br />
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[[Constantin Carathéodory]] formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<ref>[[Constantin Carathéodory|Carathéodory, C.]] (1909).</ref><br />
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Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<br />
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'''康斯坦丁·卡拉西奥多里 Constantin Carathéodory'''在纯数学公理的基础上进行了热力学公理化。他对第二定律的陈述被称为'''卡拉西奥多里原理 Principle of Carathéodory'''，可以这样表述:<br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.<ref>Buchdahl, H.A. (1966), p. 68.</ref></blockquote><br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.</blockquote><br />
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在绝热封闭系统的任意状态 S 附近，总有从 S 出发不可达的状态。<br />
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With this formulation, he described the concept of [[adiabatic accessibility]] for the first time and provided the foundation for a new subfield of classical thermodynamics, often called [[Ruppeiner geometry|geometrical thermodynamics]]. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic [[process function]], in other words, <math>\delta Q=TdS</math>.<ref name="Sychev1991">{{cite book |last=Sychev |first=V. V. |title=The Differential Equations of Thermodynamics |year=1991 |publisher=Taylor & Francis |isbn=978-1-56032-121-7}}</ref> {{clarify|date=February 2014}}<br />
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With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, <math>\delta Q=TdS</math>. <br />
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公理化处理后，他第一次描述了'''绝热可达性 Adiabatic Accessibility'''的概念，并为经典热力学的一个新的子领域，即通常所说的'''几何热力学 Geometrical Thermodynamics'''奠定了基础。由卡拉西奥多里原理可以推出，准静态转移的热量值是一个可积过程函数，即<math>\delta Q=TdS</math>。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）准静态转移的热量值是一个可积过程函数 存疑<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.<ref name="Munster 45">Münster, A. (1970), p. 45.</ref>{{sfnp|Lieb|Yngvason|1999|p=49}}<ref name="Planck 1926">[[Max Planck|Planck, M.]] (1926).</ref><ref>Buchdahl, H.A. (1966), p. 69.</ref> {{clarify|date=February 2014}}<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium. <br />
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尽管在教科书几乎惯称卡拉西奥多里原理表述了第二定律，并认为其与克劳修斯表述或开尔文-普朗克表述等价，但事实并非如此。为了得到第二定律的所有内容，需要对卡拉西奥多里原理补充普朗克表述，即等量功总是增加一个最初处于自身内部热力学平衡的封闭系统的内部能量。<br />
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===Planck's principle 普朗克原理===<br />
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In 1926, [[Max Planck]] wrote an important paper on the basics of thermodynamics.<ref name="Planck 1926"/><ref>Uffink, J. (2003), pp. 129–132.</ref> He indicated the principle<br />
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In 1926, Max Planck wrote an important paper on the basics of thermodynamics. He indicated the principle<br />
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1926年，'''马克斯·普朗克 Max Planck'''写了一篇关于热力学基础的重要论文。他指出了以下原理<br />
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::The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<ref name="Munster 45"/>{{sfnp |Lieb|Yngvason|1999|p=49}}<br />
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The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<br />
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一个封闭系统的内部能量因绝热过程增加，在整个过程中，系统的体积保持不变。<br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work."<ref>[[Clifford Truesdell|Truesdell, C.]], Muncaster, R.G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 15.</ref> Planck wrote: "The production of heat by friction is irreversible."<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 81.</ref><ref>[[Max Planck|Planck, M.]] (1926), p. 457, Wikipedia editor's translation.</ref><br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work." Planck wrote: "The production of heat by friction is irreversible."<br />
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这个公式没有提到热，没有提到温度，甚至没有提到熵，也不一定隐含地依赖于这些概念，但它暗示了第二定律的内容。一个密切相关的表述为，“摩擦力从来不做正功。”普朗克写道: “摩擦生热是不可逆的。”<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above.<ref>Lieb, E.H., Yngvason, J. (2003), p. 149.</ref> It is relevant that for a system at constant volume and [[Mole (unit)|mole numbers]], the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above. It is relevant that for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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这个普朗克定理没有提到熵，是用物理术语来表述的。它与上面给出的开尔文表述密切相关。相关的是，对于恒定体积和摩尔数的系统，熵是内能的单调函数。然而，普朗克的这个原理实际上并不是普朗克对第二定律的首选表述（见前面小节），而是依赖于熵的概念。<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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Borgnakke 和 Sonntag 提出的以下表述在某种意义上是对普朗克原理的补充。他们没有将其作为第二定律的完整表述:<br />
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::... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<ref>Borgnakke, C., Sonntag., R.E. (2009), p. 304.</ref><br />
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... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<br />
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只有一种方法可以减少[闭合]系统的熵——从系统中转移热量。<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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与普朗克之前提出的原理不同，这一原理明确地用熵的变化来表示。从系统中去除物质也可以减少系统的熵。<br />
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===Statement for a system that has a known expression of its internal energy as a function of its extensive state variables 一个其内能有已知表达式（其扩展状态变量的函数）的系统的表述===<br />
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The second law has been shown to be equivalent to the [[internal energy]] ''U'' being a weakly [[convex function]], when written as a function of extensive properties (mass, volume, entropy, ...).<ref>{{cite book |last1=van Gool |first1=W. |last2=Bruggink |first2=J.J.C. (Eds) |url= |title=Energy and time in the economic and physical sciences |publisher=North-Holland |year=1985 |pages=41–56 |quote= |isbn=978-0-444-87748-2}}</ref><ref>{{Cite journal | last1 = Grubbström | first1 = Robert W. | doi = 10.1016/j.apenergy.2007.01.003 | title = An Attempt to Introduce Dynamics Into Generalised Exergy Considerations| journal = Applied Energy| volume = 84| issue = 7–8 | pages = 701–718 | year = 2007}}</ref> {{clarify|date=February 2014}}<br />
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The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...). <br />
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第二定律被证明等价于弱凸函数内能 U （写成广泛性质（质量，体积，熵，...）的函数时）。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）存疑<br />
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==Corollaries 推论==<br />
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===Perpetual motion of the second kind 第二类永动机===<br />
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{{main article|Perpetual motion}}<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of [[first law of thermodynamics]] by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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在第二定律建立之前，许多志在发明永动机的人试图通过提取环境的巨大内能作为机器动力，以突破'''热力学第一定律 First Law of Thermodynamics''' 的限制。这种机器被称为“'''第二类永动机 Perpetual Motion Machine of the Second Kind'''”。第二定律表明这种机器是不可能的。<br />
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===Carnot theorem 卡诺定理===<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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'''卡诺定理 Carnot's Theorem'''（1824）是一条限制任何可能的发动机的最大效率的原理。效率完全取决于热库和冷库之间的温差。卡诺定理指出:<br />
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*All irreversible heat engines between two heat reservoirs are less efficient than a [[Carnot engine]] operating between the same reservoirs.<br />
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*All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs.<br />
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* 所有不可逆热机的效率低于在两相同热源之间工作的'''卡诺机 Carnot Engine'''。<br />
* 所有可逆热机的效率与在两相同热源之间工作的卡诺机相等。<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as [[thermodynamic reversibility]]. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the [[Carnot cycle]]'s reversibility and was condemned to be less efficient.<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the Carnot cycle's reversibility and was condemned to be less efficient.<br />
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在他的理想模型中，热转换成功可以通过逆转循环的运动而恢复，这个概念后来被称为'''热力学可逆性 Thermodynamic Reversibility'''。然而，卡诺进一步假定，一些热量损失了，并没有转化为机械功。因此，没有一个真实的热机能够实现'''卡诺循环 Carnot Cycle'''的可逆性，并且被认为效率较低。<br />
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Though formulated in terms of caloric (see the obsolete [[caloric theory]]), rather than [[entropy]], this was an early insight into the second law.<br />
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Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law.<br />
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该理论尽管是用热量表述的（见被取代的'''热质说 Caloric Theory'''），而不是熵，但是它是对第二定律的早期认识。<br />
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===Clausius inequality 克劳修斯不等式===<br />
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The [[Clausius theorem]] (1854) states that in a cyclic process<br />
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The Clausius theorem (1854) states that in a cyclic process<br />
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'''克劳修斯定理 Clausius Theorem'''（1854）指出，在一个循环的过程中<br />
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: <math>\oint \frac{\delta Q}{T} \leq 0.</math><br />
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The equality holds in the reversible case<ref>[http://scienceworld.wolfram.com/physics/ClausiusTheorem.html ''Clausius theorem''] at [[Wolfram Research]]</ref> and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function [[entropy]]. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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The equality holds in the reversible case and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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等号在可逆情况下成立，严格不等号在不可逆情况下成立。可逆情况下引入状态函数熵。这是因为在循环过程中，状态函数的变化为零。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）末句存疑<br />
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===Thermodynamic temperature 热力学温度===<br />
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{{main article|Thermodynamic temperature}}<br />
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For an arbitrary heat engine, the efficiency is:<br />
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For an arbitrary heat engine, the efficiency is:<br />
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对于任意热机，效率为:<br />
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: <math>\eta = \frac {W_n}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H} \qquad (1)</math><br />
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where ''W''<sub>n</sub> is for the net work done per cycle. Thus the efficiency depends only on q<sub>C</sub>/q<sub>H</sub>.<br />
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其中 ''W''<sub>n</sub> 表示每个循环所做的净功。因此效率只取决于 q<sub>C</sub>/q<sub>H</sub>。<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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卡诺定理指出，所有在相同热源间运行的可逆发动机的效率是相同的。<br />
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Thus, any reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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Thus, any reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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因此，任何在温度''T''<sub>1</sub>和''T''<sub>2</sub>之间运行的可逆热机必须具有相同的效率，也就是说，效率只是温度的函数:<br />
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<math>\frac{q_C}{q_H} = f(T_H,T_C)\qquad (2).</math><br />
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In addition, a reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between ''T''<sub>1</sub> and another (intermediate) temperature ''T''<sub>2</sub>, and the second between ''T''<sub>2</sub> and''T''<sub>3</sub>. This can only be the case if<br />
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In addition, a reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between T<sub>1</sub> and another (intermediate) temperature T<sub>2</sub>, and the second between T<sub>2</sub> andT<sub>3</sub>. This can only be the case if<br />
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另外，在温度 ''T''<sub>1</sub> 和 ''T''<sub>3</sub> 之间工作的可逆热机必须具有与由分别在温度''T''<sub>1</sub> 和（中间）温度 ''T''<sub>2</sub>之间、在 ''T''<sub>2</sub> 和 ''T''<sub>3</sub> 之间的两个循环组成的系统效率相同。只有下式成立才会出现这种情况:<br />
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: <math>f(T_1,T_3) = \frac{q_3}{q_1} = \frac{q_2 q_3} {q_1 q_2} = f(T_1,T_2)f(T_2,T_3).</math><br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the [[triple point]] of water. Then for any ''T''<sub>2</sub> and ''T''<sub>3</sub>,<br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the triple point of water. Then for any T<sub>2</sub> and T<sub>3</sub>,<br />
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现在考虑如下情形，<math>T_1</math> 是一个固定的参考温度: 水的'''三相点 Triple Point'''的温度。则对于任意 T<sub>2</sub> 和 T<sub>3</sub>,<br />
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: <math>f(T_2,T_3) = \frac{f(T_1,T_3)}{f(T_1,T_2)} = \frac{273.16 \cdot f(T_1,T_3)}{273.16 \cdot f(T_1,T_2)}.</math><br />
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Therefore, if thermodynamic temperature is defined by<br />
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Therefore, if thermodynamic temperature is defined by<br />
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因此，如果热力学温度被定义为<br />
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: <math>T = 273.16 \cdot f(T_1,T) \,</math><br />
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then the function ''f'', viewed as a function of thermodynamic temperature, is simply<br />
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then the function f, viewed as a function of thermodynamic temperature, is simply<br />
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那么函数 ''f'' 作为热力学温度的函数，为<br />
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: <math>f(T_2,T_3) = \frac{T_3}{T_2},</math><br />
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and the reference temperature ''T''<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be used{{snd}}the choice here corresponds to the [[Kelvin]] scale.)<br />
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and the reference temperature T<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be usedthe choice here corresponds to the Kelvin scale.)<br />
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参考温度 ''T''<sub>1</sub> 的值为273.16。（任何参考温度和任何正值均可用——此处的选择对应开尔文标度。）<br />
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===Entropy 熵===<br />
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{{main article|Entropy (classical thermodynamics)}}<br />
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According to the [[Clausius theorem|Clausius equality]], for a reversible process<br />
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According to the Clausius equality, for a reversible process<br />
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根据克劳修斯定理，对可逆过程有<br />
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: <math>\oint \frac{\delta Q}{T}=0</math><br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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这意味着线积分 <math>\int_L \frac{\delta Q}{T}</math> 对于可逆过程是路径无关的。<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer<ref name="Schmidt-Rohr 14" /> satisfies<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer satisfies<br />
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所以我们可以定义一个叫做熵的状态函数 S，对于可逆过程或者纯热传递满足<br />
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: <math>dS = \frac{\delta Q}{T} \!</math><br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the [[third law of thermodynamics]], which states that ''S'' = 0 at [[absolute zero]] for perfect crystals.<br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the third law of thermodynamics, which states that S = 0 at absolute zero for perfect crystals.<br />
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据此，只有对上述公式进行积分，才能得到熵的差值。为了获得绝对值，我们需要'''热力学第三定律 Third Law of Thermodynamics'''，它指出'''绝对零度 Absolute Zero'''下完美晶体的 ''S'' = 0。<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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对于任意不可逆过程，由于熵是一个状态函数，我们总是可以将初始状态和最终状态与一个虚拟的可逆过程联系起来，并在这条路径上积分以计算熵的差值。<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the [[Clausius inequality]] on this loop,<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the Clausius inequality on this loop,<br />
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现在把可逆过程逆过来，将其与上述不可逆过程结合。把克劳修斯不等式应用到这个循环,<br />
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: <math>-\Delta S+\int\frac{\delta Q}{T}=\oint\frac{\delta Q}{T}< 0</math><br />
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Thus,<br />
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Thus,<br />
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故,<br />
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: <math>\Delta S \ge \int \frac{\delta Q}{T} \,\!</math><br />
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where the equality holds if the transformation is reversible.<br />
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where the equality holds if the transformation is reversible.<br />
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如果变换可逆，等号成立。<br />
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Notice that if the process is an [[adiabatic process]], then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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Notice that if the process is an adiabatic process, then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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注意，若该过程是一个'''绝热过程 Adiabatic Process'''，则<math>\delta Q=0</math>，故<math>\Delta S\ge 0</math>。<br />
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===Energy, available useful work===<br />
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===Energy, available useful work===<br />
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能量，有用的工作<br />
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{{See also|Exergy}}<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an ''unlimited'' heat reservoir at temperature ''T<sub>R</sub>'' and pressure ''P<sub>R</sub>'' {{snd}}so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain ''T<sub>R</sub>''; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain ''P<sub>R</sub>''.<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an unlimited heat reservoir at temperature T<sub>R</sub> and pressure P<sub>R</sub> so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain T<sub>R</sub>; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain P<sub>R</sub>.<br />
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考虑将第二定律应用于孤立系统(称为整体系统或宇宙)的情形，是一个重要的、具有启发性的理想情形。孤立系统由两部分组成: 感兴趣的子系统和子系统的环境。这些环境被想象为如此之大，以至于它们可以被认为是一个温度为 t 子 r / sub 和压力为 p 子 r / sub 的无限热源，因此无论有多少热量被转移到(或来自)子系统，周围环境的温度将保持 t 子 r / sub; 无论子系统的体积扩大(或收缩)多少，周围环境的压力将保持 p 子 r / sub。<br />
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Whatever changes to ''dS'' and ''dS<sub>R</sub>'' occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy ''S<sub>tot</sub>'' of the isolated total system must not decrease:<br />
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Whatever changes to dS and dS<sub>R</sub> occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy S<sub>tot</sub> of the isolated total system must not decrease:<br />
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根据第二定律，孤立总体系统的熵 s 子系统 / 子系统的熵不能减小，无论子系统和周围环境的熵对 dS 和 dS 子系统 r / 子系统发生什么变化:<br />
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: <math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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<math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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数学 dS + dS r ge 0 / math<br />
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According to the [[first law of thermodynamics]], the change ''dU'' in the internal energy of the sub-system is the sum of the heat ''δq'' added to the sub-system, ''less'' any work ''δw'' done ''by'' the sub-system, ''plus'' any net chemical energy entering the sub-system ''d ∑μ<sub>iR</sub>N<sub>i</sub>'', so that:<br />
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According to the first law of thermodynamics, the change dU in the internal energy of the sub-system is the sum of the heat δq added to the sub-system, less any work δw done by the sub-system, plus any net chemical energy entering the sub-system d ∑μ<sub>iR</sub>N<sub>i</sub>, so that:<br />
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根据能量守恒定律，子系统内部能量的变化 dU 是子系统内部能量 q 的总和，减去子系统所做的任何功，再加上进入子系统 d ∑ sub iR / sub n sub i / sub 的任何净化学能，因此:<br />
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: <math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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<math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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数学 dU delta q- delta w + d ( sum mu iR } n i) ，/ math<br />
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where μ<sub>iR</sub> are the [[chemical potential]]s of chemical species in the external surroundings.<br />
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where μ<sub>iR</sub> are the chemical potentials of chemical species in the external surroundings.<br />
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其中亚 iR / 亚是外部环境中化学物种的化学势。<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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现在热量离开储存器进入子系统是<br />
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: <math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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<math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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数学△ qtr (- dS r) le tr dS / math<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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其中我们首先使用了经典热力学中熵的定义(在统计热力学中，熵变、温度和吸收热量之间的关系可以推导出来) ，然后从上面推导出第二定律的不等式。<br />
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It therefore follows that any net work ''δw'' done by the sub-system must obey<br />
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It therefore follows that any net work δw done by the sub-system must obey<br />
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因此，子系统所做的任何净功都必须服从<br />
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: <math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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<math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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数学增量 w le-dU + tds + sum mu { iR } dN i，/ math<br />
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It is useful to separate the work ''δw'' done by the subsystem into the ''useful'' work ''δw<sub>u</sub>'' that can be done ''by'' the sub-system, over and beyond the work ''p<sub>R</sub> dV'' done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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It is useful to separate the work δw done by the subsystem into the useful work δw<sub>u</sub> that can be done by the sub-system, over and beyond the work p<sub>R</sub> dV done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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将子系统所做的功与子系统所能做的功分离为子系统所能做的有用功，超过了子系统对外界压力所做的功，给出了子系统所能做的有用功(火用)的下列关系式:<br />
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: <math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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<math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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D (u-t r s + p r v-sum { iR } n i) ，/ math<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the ''availability'' or ''[[exergy]]'' ''E'' of the subsystem,<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the availability or exergy E of the subsystem,<br />
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可以很方便地将子系统的右侧定义为热动力位能的精确衍生物，称为子系统的可用性或火用 e,<br />
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: <math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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<math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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数学 e u-t r s + p r v- sum { iR } n i / math<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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因此，第二定律意味着，对于任何可以简单地被视为分为一个子系统和一个与之接触的无限温度和压力储存器的过程,<br />
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: <math> dE + \delta w_u \le 0 \, </math><br />
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<math> dE + \delta w_u \le 0 \, </math><br />
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数学德 delta u le 0，/ math<br />
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i.e. the change in the subsystem's exergy plus the useful work done ''by'' the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done ''on'' the system) must be less than or equal to zero.<br />
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i.e. the change in the subsystem's exergy plus the useful work done by the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done on the system) must be less than or equal to zero.<br />
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也就是。子系统(火用)的变化加上子系统所做的有用功(或者，子系统(火用)的变化不包括任何功，除了压力贮存器所做的功之外，在系统中所做的功)必须小于或等于零。<br />
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In sum, if a proper ''infinite-reservoir-like'' reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in ''E'' for an irreversible process and no change for a reversible process.<br />
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In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in E for an irreversible process and no change for a reversible process.<br />
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总之，如果选择一个合适的类似于无限库的参考状态作为现实世界中的系统环境，那么第二定律预测一个不可逆性的 e 值会减少，而一个可逆过程的 e 值不会变化。<br />
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: <math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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<math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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数学 d = 0 = 0 = 0 = 0 = 0<br />
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This expression together with the associated reference state permits a [[design engineer]] working at the macroscopic scale (above the [[thermodynamic limit]]) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (''Also, see [[process engineer]]''). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (''See [[Exergy efficiency|second law efficiency]]''.)<br />
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This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (Also, see process engineer). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (See second law efficiency.)<br />
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这个表达式连同相关的参考状态允许在宏观的设计工程师利用第二定律，而不需要直接测量或考虑整个孤立系统的熵变。(另外，请参阅流程工程师)。这些变化已经被考虑，假设被考虑的系统可以达到平衡的参考状态而不改变参考状态。还可以找到一个过程或过程集合的效率，将其与可逆的理想状态相比较<br />
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This approach to the Second Law is widely utilized in [[engineering]] practice, [[environmental accounting]], [[systems ecology]], and other disciplines.<br />
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This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines.<br />
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第二定律的这种方法被广泛应用于工程实践、环境会计、系统生态学和其他学科。<br />
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===The second law in chemical thermodynamics===<br />
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===The second law in chemical thermodynamics===<br />
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化学热力学的第二定律<br />
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For a [[spontaneous process|spontaneous chemical process]] in a closed system at constant temperature and pressure without non-''PV'' work, the Clausius inequality Δ''S > Q/T<sub>surr</sub>'' transforms into a condition for the change in [[Gibbs free energy]] <br />
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For a spontaneous chemical process in a closed system at constant temperature and pressure without non-PV work, the Clausius inequality ΔS > Q/T<sub>surr</sub> transforms into a condition for the change in Gibbs free energy <br />
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对于一个封闭系统中的自发化学过程，在没有非 pv 功的情况下，在恒定温度和压力下，将克劳修斯不等式 s q / t sub surr / sub 转化为吉布斯自由能变化的条件<br />
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: <math>\Delta G < 0 </math><br />
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数学 Delta g 0 / 数学<br />
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or d''G'' < 0. For a similar process at constant temperature and volume, the change in [[Helmholtz free energy]] must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products.<ref name="Oxtoby8th"/> <ref name="MortimerBook"></ref> The chemical equilibrium condition at constant ''T'' and ''p'' without electrical work is d''G'' = 0.<br />
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or dG < 0. For a similar process at constant temperature and volume, the change in Helmholtz free energy must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products. The chemical equilibrium condition at constant T and p without electrical work is dG = 0.<br />
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或者 dG 0。对于一个相似的过程，在恒定的温度和体积下，亥姆霍兹自由能的变化一定是负的，这是 Delta a 0 / math。因此，自由能(g 或 a)变化的负值是过程自发的必要条件。这是热力学第二定律在化学中最有用的形式，其中自由能的变化可以通过列表的生成焓和反应物和产物的标准摩尔熵来计算。在 t 和 p 不变的情况下，化学平衡条件是 dG 0。<br />
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==History==<br />
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==History==<br />
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历史<br />
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{{See also|History of entropy}}<br />
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[[File:Sadi Carnot.jpeg|thumb|upright|Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique]].]]<br />
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Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique.]]<br />
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尼古拉·莱昂纳尔·萨迪·卡诺穿着传统的巴黎综合理工学院制服<br />
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The first theory of the conversion of heat into mechanical work is due to [[Nicolas Léonard Sadi Carnot]] in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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The first theory of the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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第一个把热转化为机械功的理论是在1824年由尼古拉·莱昂纳尔·萨迪·卡诺提出的。他是第一个正确认识到这种转换的效率取决于发动机和环境之间的温差。<br />
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Recognizing the significance of [[James Prescott Joule]]'s work on the conservation of energy, [[Rudolf Clausius]] was the first to formulate the second law during 1850, in this form: heat does not flow ''spontaneously'' from cold to hot bodies. While common knowledge now, this was contrary to the [[caloric theory]] of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law during 1850, in this form: heat does not flow spontaneously from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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认识到詹姆斯·普雷斯科特·焦耳在能量守恒方面工作的重要性，Rudolf Clausius 在1850年第一个提出了第二定律: 热不会自发地从冷物体流向热物体。虽然现在的常识是，这与当时流行的热量理论相反，热量理论认为热量是一种流体。从那里他能够推断出萨迪卡诺定律和熵的定义(1865年)。<br />
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Established during the 19th century, the [[Kelvin-Planck statement|Kelvin-Planck statement of the Second Law]] says, "It is impossible for any device that operates on a [[cyclic process|cycle]] to receive heat from a single [[heat reservoir|reservoir]] and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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Established during the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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建立于19世纪的开尔文-普朗克第二定律(Kelvin-Planck)说，“任何循环运行的设备都不可能从单个蓄热体接收热量，并产生净功。”这被证明相当于克劳修斯的陈述。<br />
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The [[ergodic hypothesis]] is also important for the [[Boltzmann]] approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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遍历假设对玻尔兹曼方法也很重要。它说，在很长一段时间内，在具有相同能量的微观态的相空间的某些区域所花费的时间与这个区域的体积成正比，即。在很长一段时间内，所有可访问的微观状态的可能性都是一样的。同样的，它表明时间平均值和系综平均值是相同的。<br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a [[Macroscopic bodies|macroscopic system]]. This doctrine is obsolescent.<ref>Denbigh, K.G., Denbigh, J.S. (1985). ''Entropy in Relation to Incomplete Knowledge'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-25677-1}}, pp. 43–44.</ref><ref>Grandy, W.T., Jr (2008). ''Entropy and the Time Evolution of Macroscopic Systems'', Oxford University Press, Oxford, {{ISBN|978-0-19-954617-6}}, pp. 55–58.</ref><ref name=Lambert>[http://entropysite.oxy.edu Entropy Sites — A Guide] Content selected by [[Frank L. Lambert]]</ref><br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a macroscopic system. This doctrine is obsolescent.<br />
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有一种传统的学说，从克劳修斯开始，认为熵可以被理解为宏观系统中的分子“无序”。这种学说已经过时了。<br />
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===Account given by Clausius===<br />
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===Account given by Clausius===<br />
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作者: 克劳修斯<br />
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[[File:Clausius-1.jpg|thumb|upright|Rudolf Clausius]]<br />
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Rudolf Clausius<br />
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鲁道夫 · 克劳修斯<br />
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In 1856, the German physicist [[Rudolf Clausius]] stated what he called the "second fundamental theorem in the [[mechanical theory of heat]]" in the following form:{{sfnp|Clausius|1867}}<br />
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In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:<br />
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1856年，德国物理学家鲁道夫 · 克劳修斯阐述了他所谓的“热力学理论中的第二个基本定理” ，其形式如下:<br />
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: <math>\int \frac{\delta Q}{T} = -N</math><br />
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<math>\int \frac{\delta Q}{T} = -N</math><br />
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- n / math<br />
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where ''Q'' is heat, ''T'' is temperature and ''N'' is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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其中 q 是热，t 是温度，n 是一个循环过程中所有未补偿的相变的“等价值”。后来，在1865年，克劳修斯将“等价值”定义为熵。紧接着这个定义，也就是在同一年，第二定律最著名的版本在4月24日苏黎世哲学学会的一次演讲中被宣读，在演讲的最后，克劳修斯总结道:<br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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宇宙的熵趋于最大<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. [[universe]], as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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这句话是第二定律中最广为人知的措辞。由于其语言的松散性，例如:。这个问题的答案是肯定的，因为这个问题的答案是肯定的。开放，封闭，或孤立，许多人认为这个简单的陈述意味着热力学第二定律适用于几乎所有可以想象的主题。这是不正确的; 这个陈述只是一个更加扩展和精确描述的简化版本。<br />
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In terms of time variation, the mathematical statement of the second law for an [[isolated system]] undergoing an arbitrary transformation is:<br />
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In terms of time variation, the mathematical statement of the second law for an isolated system undergoing an arbitrary transformation is:<br />
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就时间变化而言，经历任意变换的孤立系统第二定律的数学表述是:<br />
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: <math>\frac{dS}{dt} \ge 0</math><br />
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<math>\frac{dS}{dt} \ge 0</math><br />
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0 / math<br />
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where<br />
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where<br />
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: ''S'' is the entropy of the system and<br />
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S is the entropy of the system and<br />
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S 是系统的熵<br />
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: ''t'' is [[time]].<br />
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t is time.<br />
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时间到了。<br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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!-- 可逆过程需要与周围环境保持平衡。这对于一个孤立的系统是不可能的。因此，关于可逆过程的讨论已经转移到对封闭系统的分析----<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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平衡后应用等式符号。建立孤立系统第二定律的另一种方法是:<br />
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: <math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math<br />
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with <math> \dot S_{i}</math> the sum of the rate of [[entropy production]] by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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with <math> \dot S_{i}</math> the sum of the rate of entropy production by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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用数学 s { i } / 数学表示系统内所有进程的产生熵之和。这种配方的优点是它显示了产生熵的效果。产生熵是一个非常重要的概念，因为它决定(限制)热机器的效率。与环境温度数学 t { a } / math 相乘，得到所谓的耗散能量数学 p { diss } t { a } dot s { i } / math。<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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封闭系统的第二定律(允许热量交换和移动边界，但不允许物质交换)的表达式是:<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math 来计算数学点 q } + 点 s { i } / math<br />
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给你<br />
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: <math>\dot Q</math> is the heat flow into the system<br />
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<math>\dot Q</math> is the heat flow into the system<br />
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数学 q / math 是进入系统的热流<br />
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: <math>T</math> is the temperature at the point where the heat enters the system.<br />
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<math>T</math> is the temperature at the point where the heat enters the system.<br />
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Math 是热量进入系统时的温度。<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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等式符号在只有可逆过程在系统内发生的情况下成立。如果发生不可逆过程(在实际操作系统中就是这种情况) ,-符号保持不变。如果系统有多处供热，我们必须求相应项的代数和。<br />
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For open systems (also allowing exchange of matter):<br />
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For open systems (also allowing exchange of matter):<br />
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对于开放系统(也允许物质交换) :<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math 表示数学点 q } + 点 s + 点 s { i } / math<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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这里的数学 s / math 是熵流进入系统，与物质流进入系统有关。它不应该与熵的时间导数混淆。如果物质在几个地方提供，我们必须取这些贡献的代数和。<br />
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==Statistical mechanics==<br />
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==Statistical mechanics==<br />
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统计力学<br />
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[[Statistical mechanics]] gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a [[microstate (statistical mechanics)|microstate]] of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/{{radic|''N''}} where ''N'' is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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Statistical mechanics gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a microstate of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/ where N is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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通过假设物质是由不断运动的原子和分子组成的，统计力学对第二定律给出了解释。系统中每个粒子的一组特定的位置和速度称为系统的微观状态，由于系统的不断运动，系统不断地改变其微观状态。统计力学假设，在平衡状态下，系统可能处于的每个微观状态发生的可能性是相等的，当这个假设被提出时，它直接导致了第二定律必须在统计意义上成立的结论。也就是说，第二定律平均成立，其中 n 是系统中粒子数的1 / 数量级的统计变化。在日常(宏观)情况下，违反第二定律的概率几乎为零。然而，对于粒子数量很少的系统，热力学参数，包括熵，可能显示出与第二定律预测的显著的统计偏差。经典热力学理论没有处理这些统计变量。<br />
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（by Jie XIANG) 在通过假定一种物质由不断运动的原子和分子组成的前提下，统计力学对热力学第二定律进行了解释。系统中每个粒子其特定的位置和速度组成为系统的微观状态，由于（原子和分子的）不断运动，系统也持续改变其微观状态。统计力学假设，在系统力平衡的情况下，其微观状态其实是等可能发生的，进而能够得出结论：热力学第二定律必须在统计学意义上成立。也就是说，第二定律将保持均值不变， 其统计变化为1 /√N，N为系统中的粒子数。在日常(宏观)情况下，违反热力学第二定律的概率几乎为零，但是，对于粒子数量非常少的系统，其热力学参数（包括熵）可能会显示出与第二定律所预测的明显统计偏差。然而经典热力学理论并不处理这些统计变量。<br />
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==Derivation from statistical mechanics==<br />
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==Derivation from statistical mechanics==<br />
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从统计力学导出<br />
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{{Further information|H-theorem}}<br />
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The first mechanical argument of the [[Kinetic theory of gases]] that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to [[James Clerk Maxwell]] in 1860;<ref>{{Cite journal | last1 = Gyenis | first1 = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G }}</ref> [[Ludwig Boltzmann]] with his [[H-theorem]] of 1872 also argued that due to collisions gases should over time tend toward the [[Maxwell–Boltzmann distribution]].<br />
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The first mechanical argument of the Kinetic theory of gases that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to James Clerk Maxwell in 1860; Ludwig Boltzmann with his H-theorem of 1872 also argued that due to collisions gases should over time tend toward the Maxwell–Boltzmann distribution.<br />
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气体动力学理论的第一个力学论证由詹姆斯·克拉克·麦克斯韦在1860年给出，指出分子碰撞引起温度均衡，因此整体趋向于平衡； 路德维希·玻尔兹曼在1872年提出的 H 定理也认为，气体由于碰撞应该随着时间的推移趋向于麦克斯韦-波兹曼分布。<br />
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Due to [[Loschmidt's paradox]], derivations of the Second Law have to make an assumption regarding the past, namely that the system is [[Correlation and dependence|uncorrelated]] at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a [[boundary condition]], and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the [[Big Bang]]), though [[Boltzmann brain|other scenarios]] have also been suggested.<ref name="Hawking AOT">{{cite journal|last=Hawking|first=SW|title=Arrow of time in cosmology|journal=Phys. Rev. D|year=1985|volume=32|issue=10|pages=2489–2495|doi=10.1103/PhysRevD.32.2489|pmid=9956019|bibcode = 1985PhRvD..32.2489H }}</ref><ref>{{cite book | last = Greene | first = Brian | authorlink = Brian Greene | title = The Fabric of the Cosmos | url = https://archive.org/details/fabricofcosmossp00gree | url-access = registration | publisher = Alfred A. Knopf | year = 2004 | page = [https://archive.org/details/fabricofcosmossp00gree/page/171 171] | isbn = 978-0-375-41288-2}}</ref><ref name=Lebowitz>{{cite journal|last=Lebowitz|first=Joel L.|title= Boltzmann's Entropy and Time's Arrow|journal=Physics Today|date=September 1993|volume=46|issue=9|pages=32–38|url=http://users.df.uba.ar/ariel/materias/FT3_2008_1C/papers_pdf/lebowitz_370.pdf|accessdate=2013-02-22|doi=10.1063/1.881363|bibcode = 1993PhT....46i..32L }}</ref><br />
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Due to Loschmidt's paradox, derivations of the Second Law have to make an assumption regarding the past, namely that the system is uncorrelated at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a boundary condition, and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the Big Bang), though other scenarios have also been suggested.<br />
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由于洛施密特悖论，第二定律的导出必须对过去做出一个假设，即系统在过去的某个时刻是不相关的；这样的假设允许进行简单的概率处理。这个假设通常被认为是一个边界条件，因此热力学第二定律最终是过去某个地方的初始条件的结果，可能是在宇宙的开始(大爆炸) 。也有人提出了其他假设。<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the [[Statistical mechanics#Fundamental postulate|fundamental postulate]], also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy.{{citation needed|date=August 2012}} The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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考虑到这些假设，在统计力学中，第二定律不是一个假设，而是统计力学基本假设的一个结果，也被称为等先验概率假设。这个基本假设表明，只要一个人清楚地知道，简单的概率论证只适用于未来，而对于过去，有辅助的信息来源告诉我们，它是低熵的。热力学第二定律的第一部分指出，热孤立系统的熵只能增加。如果我们把熵的概念限制在热平衡系统中，那么热力学第二定律的第一部分是等先验概率假设的一个显然结果。处于热平衡状态的孤立系统包含能量<math>E</math>的熵表示为：<br />
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: <math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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<math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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其中<math>\Omega\left(E\right)</math> 是处于 <math>E</math>和<math>E +\delta E</math这个小区间内的量子态数目。这里的 <math>\delta E</math> 是一个宏观上很小的固定能量区间。严格地说，这意味着熵取决于对<math>\delta E</math>的选择。然而在热力学极限下（例如无穷大系统的极限），狭义的熵(单位体积或单位质量的熵)不依赖于 <math>\delta E</math>。<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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假设我们有一个孤立系统，其宏观状态由许多变量描述。这些宏观变量可以是总体积、活塞在系统中的位置等。从而<math>\Omega</math>将取决于这些变量的值。如果某个变量不是固定的（我们不会在某个位置夹住活塞) ，那么因为在平衡状态下所有可到达状态的等几率到达的，平衡状态下的自由变量会使 <math>\Omega</math> 最大，因为这是平衡状态下最可能的情况。<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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如果该变量最初固定到某个值然后释放，当达到新的平衡时，变量将自我调整使得 <math>\Omega</math> 最大，这意味着熵将增加或保持不变(如果变量初始固定的值恰好是平衡值)。<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's [[H-theorem]], however, proves that the quantity {{math|''H''}} increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the quantity increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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假设我们初始位于一个平衡状态，突然移除了对一个变量的约束。我们做完这件事的时候，可达到的微观状态的数为<math>\Omega</math>，但是系统还没有达到平衡，所以系统处于某些可达到的状态的实际概率还不等于先验概率 <math>1/\Omega</math>。我们已经知道，最终的平衡状态相对于之前的平衡状态，熵会增加或者保持不变。然而，玻耳兹曼的H定理证明系统在不处于平衡态的期间，那个量作为时间的函数单调增加。<br />
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===Derivation of the entropy change for reversible processes===<br />
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===Derivation of the entropy change for reversible processes===<br />
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可逆过程熵变的推导<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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热力学第二定律的第二部分指出，经历可逆过程的系统的熵变为:<br />
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: <math>dS =\frac{\delta Q}{T}</math><br />
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<math>dS =\frac{\delta Q}{T}</math><br />
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where the temperature is defined as:<br />
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where the temperature is defined as:<br />
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其中温度定义为:<br />
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:<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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See [[Microcanonical ensemble|here]] for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the [[adiabatic theorem]] of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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请参阅此处查看该定义的正当性。假设系统有一些可以改变的外部参数 x。一般来说，系统的能量本征态将依赖于 x。根据量子力学的绝热定理，在系统哈密顿量无限缓慢变化的极限下，系统将保持在相同的能量本征态，因此系统的能量会随着其所在能量本征态的能量变化而变化。<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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对于外部变量 x 可以定义广义力 X ，因此如果 x 增加 dx，那么<math>X dx</math> 就是系统所做的功。例如，如果 x 是体积，那么 X就 是压强。一个已知处于能量本征态<math>E_{r}</math>的系统的广义力为：<br />
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: <math>X = -\frac{dE_{r}}{dx}</math><br />
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<math>X = -\frac{dE_{r}}{dx}</math><br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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由于系统可以在<math>\delta E</math> 区间内的任意能量本征态内，我们将系统的广义力定义为上述表达式的期望值:<br />
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: <math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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<math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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为了计算平均值，我们按照能量本征态来划分<math>\Omega\left(E\right)</math>，通过计算有多少能量本征态的<math>\frac{dE_{r}}{dx}</math>值在区间在<math>Y</math> 到 <math>Y + \delta Y</math>之间。把这个数字叫做为 <math>\Omega_{Y}\left(E\right)</math>，我们有:<br />
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: <math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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<math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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The average defining the generalized force can now be written:<br />
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The average defining the generalized force can now be written:<br />
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根据平均值定义的广义力现在可以写成:<br />
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: <math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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<math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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我们可以把它和熵对 x 在恒定能量 e 下的导数联系起来，如下所示。假设我们把 x 改成 x + dx。然后 math Omega left (e right) / math 将会改变，因为能量本征态依赖于 x，导致能量本征态进入或超出数学 e / math 和数学 e + delta e / math 之间的范围。让我们再次关注数学 y / math 和数学 y + delta y / math 之间的能量本征态。由于这些能量本征态的能量增加了 y dx，所有这些能量本征态在 e-y dx 到 e 之间的区间内从 e 以下移动到 e 以上。 有<br />
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: <math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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<math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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数学 n { y }左(e 右) frac { y }左(e 右)}{ delta e } y dx ，/ math<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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这样的能量本征态。如果数学 y dx leq delta e / math，所有这些能量本征态将移动到数学 e / math 和数学 e + delta e / math 之间的范围内，有助于数学 ω / math 的增加。从数学 e + △ e / math 下移到数学 e + △ e / math 上的能量本征态的个数是由数学 n { y }左(e + △ e 右) / math 给出的。区别在于<br />
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: <math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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<math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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数学 n { y }左(e 右)-n { y }左(e + delta e 右) ，/ math<br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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因此是数学增长的净贡献欧米茄 / 数学。请注意，如果 y dx 大于 math delta e / math，那么将会有能量本征态从 e 下面移动到 e + delta e / math 上面。它们在数学 n { y } left (e right) / math 和数学 n { y } left (e + delta e right) / math 中都有计算，因此上述表达式在这种情况下也是有效的。<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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将上面的表达式表示为对 e 的导数，对 y 的求和得到表达式:<br />
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: <math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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数学左(部分欧米茄左(部分欧米茄右){ e }-和 y 左(部分欧米茄右){ x 左(部分欧米茄左(部分欧米茄右)}{ x 右) ，/ 数学<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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关于 x 的欧米茄 / 数学对数导数由以下方式给出:<br />
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: <math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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Math 左( frac { partial ln 左( Omega 右)}{部分 x 右){ e } beta x + 左( frac { partial x 右){ x } ，/ math<br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
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第一个术语是密集型的，即。它不能根据系统大小进行缩放。相比之下，最后一项的规模与逆系统的规模一样，因此将在热力学极限中消失。因此，我们发现:<br />
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: <math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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<math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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数学左(部分 s }右){ e } frac { x }{ t } ，/ math<br />
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Combining this with<br />
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Combining this with<br />
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结合这一点<br />
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: <math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
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<math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
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数学左(部分 s }右){ x } frac {1}{ t } ，/ math<br />
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Gives:<br />
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给予:<br />
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: <math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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<math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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数学 dS 左(部分 s }部分 e }右){ x } dE + 左(部分 s }右){ e } dx frac { t } + frac { t }{ delta q }{ t } ，/ math<br />
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===Derivation for systems described by the canonical ensemble===<br />
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===Derivation for systems described by the canonical ensemble===<br />
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正则系系统的推导<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the [[canonical ensemble]]:<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:<br />
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如果一个系统与某个温度为T的热浴热接触，那么在平衡状态下，关于能量本征值的概率分布值由正则系综给出:<br />
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: <math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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<math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the [[Partition function (statistical mechanics)|partition function]]. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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这里Z是一个使所有概率之和归一化的因子，这个函数被称为配分函数。现在我们考虑对温度和能级所依赖的外部参数的无限小的可逆改变。它遵循熵的一般公式:<br />
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: <math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
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<math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
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that<br />
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that<br />
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以及<br />
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: <math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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<math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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把正则系综的数学形式 <math>P_{j}</math> 插入，可以得到:<br />
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: <math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
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<math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
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==Living organisms==<br />
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==Living organisms==<br />
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生命体<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as [[What is Life?|Erwin Schrödinger]], [[Léon Brillouin]]<ref>Léon Brillouin [https://books.google.com/books?hl=en&lr=&id=tPXVbiw_1P0C ''Science and Information Theory''] (Academic Press, 1962) (Dover, 2004)</ref> and [[Life and Energy|Isaac Asimov]]. It is also the topic of current{{When|date=July 2019}} research.{{Citation needed|date=July 2019}}<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as Erwin Schrödinger, Léon Brillouin and Isaac Asimov. It is also the topic of current research.<br />
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表述热力学主要有两种方式：(a)描述从一种热力学平衡状态到另一种状态的路径，(b)通过使系统保持不变，同时增加周围环境的总熵的循环过程。这两种方法有助于理解生命过程。这个话题大大超出了本文的范围，但是已经有一些研究者考虑过这个问题，比如埃尔温·薛定谔、莱昂·布里渊和艾萨克.阿西莫夫。这也是当前研究的主题。<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of [[metabolism]], give out breakdown products and heat. Plants [[Photosynthesis|take in radiative energy]] from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of metabolism, give out breakdown products and heat. Plants take in radiative energy from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
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从近似的角度来看，生命体可以被认为是(b)的一个例子。近似地，一只动物的身体状态每天循环，使得它几乎没有什么变化。动物吸收食物、水和氧气，经过新陈代谢，输出分解的产物和热量。植物吸收来自太阳的辐射能量，这可以认为是热量，以及二氧化碳和水，然后它们释放氧气。它们就是这样生长的，最终会死亡，尸体腐烂，大部分重新变成二氧化碳和水。这可以看作是一个循环过程。总的来说，阳光来自一个高温的源——太阳，它的能量被传递到一个较低的温度汇，例如向太空辐射。这个过程使得植物周围环境的熵增加。因此从循环过程的角度来看，动物和植物服从热力学第二定律。简单的热机效率概念很难适用于这个问题，因为它们假定系统是封闭的。<br />
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From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to [[Non-equilibrium thermodynamics|unsettled current debate or research]]. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
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From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to unsettled current debate or research. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
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从(a)的热力学观点考虑 ，即考虑从一个平衡状态到另一个平衡状态的路径，只会产生一个粗略近似的图像，因为生命体从来不会处于热力学平衡状态。生物体经常必须被认为是开放的系统，因为它们吸收营养物质并排出废弃物。开放系统热力学目前通常从一个热力学平衡状态到另一个状态的路径的角度来考虑，或者是考虑局部热力学平衡近似下的流。生命体的问题可以通过假定一个流不变的稳态来进一步简化。对于这样的近似，熵产生的一般原理在目前有争论和研究的不确定性。尽管如此，从热力学第二定律的这个角度出发衍生出来的想法对生物是有启发意义的。<br />
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==Gravitational systems==<br />
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==Gravitational systems==<br />
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重力系统<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive [[heat capacity]], meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it.<ref name="Grandy 151">Grandy, W.T. (Jr) (2008), p. 151.</ref> Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive heat capacity, meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it. Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
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在不需要描述广义相对论的系统中，物体总是具有正的热容，这意味着温度随着能量的增加而升高。因此，当能量从高温物体流向低温物体时，源温度降低，而汇温度升高; 因此温差随时间而减小。这种情况并不总是发生在那些需要引力和广义相对论的系统中。这样的系统可以自发地朝着质量和能量不均匀分布的方向变化。这在大尺度上适用于宇宙，因此可能很难或不可能将第二定律应用于宇宙。除此之外，广义相对论所描述的系统热力学超出了本文的范围。<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
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{{main article|Non-equilibrium thermodynamics}}<br />
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The theory of classical or [[thermodynamic equilibrium|equilibrium thermodynamics]] is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<ref name="Grandy 151"/><ref>Callen, H.B. (1960/1985), p. 15.</ref><br />
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The theory of classical or equilibrium thermodynamics is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<br />
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经典热力学或平衡热力学的理想化。一个主要的假设或假设，往往甚至没有明确说明，是系统的存在，在他们自己的内部状态的热力学平衡。一般来说，在给定时间内包含一个物理系统的空间区域---- 可能在自然界中找到---- 不是在热力学平衡中，用最严格的术语来解释。从更宽松的角度来说，整个宇宙中没有任何东西是或者曾经是完全精确的热力学平衡。<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of [[thermodynamic equilibrium]]. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable [[thermal fluctuations|fluctuations]]. There is an exception, the case of [[critical point (thermodynamics)|critical states]], which exhibit to the naked eye the phenomenon of [[critical opalescence]]. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of thermodynamic equilibrium. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable fluctuations. There is an exception, the case of critical states, which exhibit to the naked eye the phenomenon of critical opalescence. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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为了物理分析的目的，假设热力学平衡是非常方便的。这种假设可能依赖于试错来证明其正当性。如果这个假设是正确的，它通常是非常有价值和有用的，因为它提供了热力学理论。平衡假设的要素是，一个系统被观察到在无限长的时间内是不变的，而且一个系统中有如此多的粒子，以至于它的微粒性质可以被完全忽略。在这样的平衡假设下，一般来说，没有宏观上可以探测到的波动。有一个例外，即临界状态的例子，它们用肉眼就可以看到临界乳光现象。对于临界状态的实验室研究，需要非常长的观察时间。<br />
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In all cases, the assumption of [[thermodynamic equilibrium]], once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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In all cases, the assumption of thermodynamic equilibrium, once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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在所有情况下，热力学平衡的假设，一旦做出，就意味着没有假定的候选者“涨落”改变系统的熵。<br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<ref>Lieb, E.H., Yngvason, J. (2003), p. 190.</ref><br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<br />
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一个物理系统很容易发生内部的宏观变化，其速度之快足以使熵的恒定性假设失效。或者一个物理系统的粒子太少，以至于微粒的本质在可观测的波动中是显而易见的。那么热力学平衡的假设就会被放弃。对于非平衡态，熵没有绝对的一般定义。<br />
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There are intermediate cases, in which the assumption of local [[thermodynamic equilibrium]] is a very good approximation,<ref>Gyarmati, I. (1967/1970), pp. 4-14.</ref><ref>Glansdorff, P., Prigogine, I. (1971).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003).</ref> but strictly speaking it is still an approximation, not theoretically ideal.<br />
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There are intermediate cases, in which the assumption of local thermodynamic equilibrium is a very good approximation, but strictly speaking it is still an approximation, not theoretically ideal.<br />
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还有一些中间情况，其中局部热力学平衡的假设是一个非常好的近似值，但严格地说它仍然是一个近似值，不是理论上的理想值。<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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对于一般的非平衡状态，考虑其他量的统计力学定义可能是有用的，这些量可以方便地称为‘熵’ ，但是它们不应该与第二定律正确定义的熵混淆或混淆。这些其他的量确实属于统计力学，而不是热力学---- 第二定律的主要领域。<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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宏观观测波动的物理学超出了本文的范围。<br />
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==Arrow of time==<br />
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==Arrow of time==<br />
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时间之箭<br />
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{{Further information|Entropy (arrow of time)}}Thermodynamic Asymmetry in Time<br />
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Thermodynamic Asymmetry in Time<br />
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时间中的热力学不对称性<br />
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{{See also|Arrow of time}}<br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely [[CPT symmetry]], since the second law applies statistically, it is hypothesized, on time-asymmetric ''boundary conditions''.<ref>{{cite web|first=Craig|last=Callender|url=https://plato.stanford.edu/archives/fall2011/entries/time-thermo/|title=Thermodynamic Asymmetry in Time|encyclopedia=Stanford Encyclopedia of Philosophy|date=29 July 2011}}</ref><br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely CPT symmetry, since the second law applies statistically, it is hypothesized, on time-asymmetric boundary conditions.<br />
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热力学第二定律是一个物理定律，它不对称于时间方向的反转。这与已经观察到的物理学基本定律，即 CPT 对称性的概念并不冲突，因为第二定律在统计上适用，它是假设的，是关于时间非对称的边界条件。<br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect ([[Arrow of time#The causal arrow of time|the causal arrow of time]]).<ref>{{cite book | first = J.J.| last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1|display-authors=etal}} chapter 6</ref><br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect (the causal arrow of time).<br />
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第二定律被提出来部分解释时间向前和向后移动的区别，例如为什么原因先于结果(因果关系的时间箭头)。<br />
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==Irreversibility==<br />
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==Irreversibility==<br />
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不可逆性<br />
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Irreversibility in [[thermodynamic process]]es is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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Irreversibility in thermodynamic processes is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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热力学过程的不可逆性是热力学作用不对称性的结果，而不是物体内部不可逆的微观性质的结果。热力学操作是施加在参与机构上的宏观外部干预，而不是从它们的内部性质中得到的。由于没有认识到这一点，产生了被称为“悖论”的说法。<br />
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===Loschmidt's paradox===<br />
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===Loschmidt's paradox===<br />
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洛斯密特悖论<br />
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{{main article|Loschmidt's paradox}}<br />
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[[Loschmidt's paradox]], also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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洛斯密特悖论，也被称为可逆性悖论，其目的在于不可能从描述宏观系统微观演化的时间对称动力学推导出不可逆性。<br />
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In the opinion of [[Erwin Schrödinger|Schrödinger]], "It is now quite obvious in what manner you have to reformulate the law of entropy{{snd}}or for that matter, all other irreversible statements{{snd}}so that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other."<ref>[[Erwin Schrödinger|Schrödinger, E.]] (1950), p.&nbsp;192.</ref> The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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In the opinion of Schrödinger, "It is now quite obvious in what manner you have to reformulate the law of entropyor for that matter, all other irreversible statementsso that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other." The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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根据薛定谔的观点，“现在很明显，你必须用什么方式来重新定义该物质的熵定律，所有其他不可逆的状态，这样它们才能从可逆模型中导出。你不能只谈论一个孤立的系统，而应该至少谈论两个，你可以暂时认为它们与世界其它地方是孤立的，但并不总是相互孤立的。”这两个系统被隔离墙隔离开来，直到按照法律的设想被热力学操作拆除。热力学操作是外部强加的，不受支配系统组成部分的可逆微观动力学定律的制约。它是不可逆转性的原因。本文中的法律陈述符合薛定谔的建议。因果关系在逻辑上优先于第二定律，而不是从第二定律派生出来的。<br />
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===Poincaré recurrence theorem===<br />
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===Poincaré recurrence theorem===<br />
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庞加莱始态复现定理<br />
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{{Main article|Poincaré recurrence theorem}}<br />
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The [[Poincaré recurrence theorem]] considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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The Poincaré recurrence theorem considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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庞加莱始态复现定理认为这是一个孤立的物理系统的理论微观描述。这可以被认为是一个热力学系统的模型后，一个热力学操作已经移除了一个内墙。在足够长的时间之后，系统将返回到一个非常接近初始状态的显微定义状态。庞加莱回归时间是指回归前经过的时间长度。它极其漫长，可能比宇宙的寿命还要长，并且敏感地依赖于被热力学操作拆除的墙体的几何形状。递归定理可能被认为是明显与热力学第二定律相矛盾的。然而，更明显的是，它只不过是一个微观模型，热力学平衡在一个孤立的系统中，通过移除两个系统之间的一道墙而形成。对于一个典型的热力学系统，重现时间是如此之大(比宇宙的生命周期长许多倍) ，以至于在所有的实际目的中，人们都不能观察到重现。然而，人们可能希望想象一下，可以等待庞加莱复发，然后重新插入被热力学操作移除的壁。然后很明显，不可逆性的出现是由于庞加莱循环的完全不可预测性，只要初始状态是热力学平衡，就像宏观热力学的情况一样。即使可以等待，也没有实际可能性选择合适的时间重新插入墙壁。庞加莱始态复现定理为 Loschmidt 悖论提供了一个解决方案。如果一个孤立的热力学系统可以被监测到在越来越多的平均 poincar 递归时间的倍数上，系统的热力学行为将在时间反转下变得不变。<br />
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[[File:James-clerk-maxwell3.jpg|thumb|upright|James Clerk Maxwell]]<br />
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James Clerk Maxwell<br />
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詹姆斯·克拉克·麦克斯韦<br />
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===Maxwell's demon===<br />
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===Maxwell's demon===<br />
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麦克斯韦的魔鬼<br />
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{{unreferenced|section|date=August 2018}}<br />
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{{main article|Maxwell's demon}}<br />
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[[James Clerk Maxwell]] imagined one container divided into two parts, ''A'' and ''B''. Both parts are filled with the same [[gas]] at equal temperatures and placed next to each other, separated by a wall. Observing the [[molecule]]s on both sides, an imaginary [[demon]] guards a microscopic trapdoor in the wall. When a faster-than-average molecule from ''A'' flies towards the trapdoor, the demon opens it, and the molecule will fly from ''A'' to ''B''. The average [[speed]] of the molecules in ''B'' will have increased while in ''A'' they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in ''A'' and increases in ''B'', contrary to the second law of thermodynamics.<br />
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James Clerk Maxwell imagined one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other, separated by a wall. Observing the molecules on both sides, an imaginary demon guards a microscopic trapdoor in the wall. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics.<br />
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詹姆斯·克拉克·麦克斯韦想象一个容器分成两部分，a 和 b。两个部分在相同的温度下充入相同的气体，并被一堵墙隔开，挨着放置。通过观察两边的分子，一个想象中的恶魔守卫着墙上的一个微型暗门。当一个来自 a 的比平均速度快的分子飞向活板门时，魔鬼打开了它，分子就会从 a 飞到 b。B 中分子的平均速度会增加，而 a 中分子的平均速度会减慢。由于平均分子速度与温度相对应，因此 a 中的温度降低，b 中的温度升高，这与热力学第二定律相反。<br />
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One response to this question was suggested in 1929 by [[Leó Szilárd]] and later by [[Léon Brillouin]]. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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One response to this question was suggested in 1929 by Leó Szilárd and later by Léon Brillouin. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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1929年，Leó Szilárd 和后来的 l on brilouin 对这个问题提出了一个答案。Szil rd 指出，现实生活中的麦克斯韦恶魔需要一些测量分子速度的方法，而获取信息需要消耗大量能量。<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between ''A'' and ''B''. It is therefore performing [[thermodynamic operation]]s on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between A and B. It is therefore performing thermodynamic operations on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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麦克斯韦的“魔鬼”不断地改变 a 和 b 之间墙体的渗透性。因此，它是在微观尺度上执行热力学操作，而不仅仅是观察普通的自发或自然的宏观热力学过程。<br />
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==Quotations==<br />
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==Quotations==<br />
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语录<br />
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{{Wikiquote}}<br />
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{{quote|The law that entropy always increases holds, I think, the supreme position among the [[Laws of science|laws of Nature]]. If someone points out to you that your pet theory of the [[universe]] is in disagreement with [[Maxwell's equations]] – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.|Sir [[Arthur Stanley Eddington]], ''The Nature of the Physical World'' (1927)}}<br />
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{{quote|There have been nearly as many formulations of the second law as there have been discussions of it.|Philosopher / Physicist [[Percy Williams Bridgman|P.W. Bridgman]], (1941)}}<br />
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{{quote|Clausius is the author of the sibyllic utterance, "The energy of the universe is constant; the entropy of the universe tends to a maximum." The objectives of continuum thermomechanics stop far short of explaining the "universe", but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size.|[[Clifford Truesdell|Truesdell, C.]], Muncaster, R. G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 17.}}<br />
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==See also==<br />
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==See also==<br />
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参见<br />
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*[[Clausius–Duhem inequality]]<br />
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*[[Fluctuation theorem]]<br />
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*[[Heat death of the universe]]<br />
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*[[History of thermodynamics]]<br />
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*[[Jarzynski equality]]<br />
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*[[Laws of thermodynamics]]<br />
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*[[Maximum entropy thermodynamics]]<br />
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*[[Quantum thermodynamics]]<br />
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*[[Reflections on the Motive Power of Fire]]<br />
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*[[Relativistic heat conduction]]<br />
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*[[Thermal diode]]<br />
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*[[Thermodynamic equilibrium]]<br />
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==References==<br />
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==References==<br />
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参考资料<br />
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===Sources===<br />
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===Sources===<br />
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资料来源<br />
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{{refbegin|50em}}<br />
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*Adkins, C.J. (1968/1983). ''Equilibrium Thermodynamics'', (1st edition 1968), third edition 1983, Cambridge University Press, Cambridge UK, {{ISBN|0-521-25445-0}}.<br />
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*[[Peter Atkins|Atkins, P.W.]], de Paula, J. (2006). ''Atkins' Physical Chemistry'', eighth edition, W.H. Freeman, New York, {{ISBN|978-0-7167-8759-4}}.<br />
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*Attard, P. (2012). ''Non-equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications'', Oxford University Press, Oxford UK, {{ISBN|978-0-19-966276-0}}.<br />
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*Baierlein, R. (1999). ''Thermal Physics'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-59082-5}}.<br />
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*Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics, New York, {{ISBN|0-88318-797-3}}.<br />
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*{{Cite book|title=Concepts in thermal physics|last=Blundell|first=Stephen J.|last2=Blundell|authorlink1=Stephen Blundell|first2=Katherine M.|authorlink2=Katherine Blundell|publisher=[[Oxford University Press]]|year=2010|isbn=9780199562107|edition=2nd|location=Oxford|pages=|oclc=607907330|doi=10.1093/acprof:oso/9780199562091.001.0001 |url=http://cds.cern.ch/record/1235139}}<br />
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*[[Ludwig Boltzmann|Boltzmann, L.]] (1896/1964). ''Lectures on Gas Theory'', translated by S.G. Brush, University of California Press, Berkeley.<br />
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*Borgnakke, C., Sonntag., R.E. (2009). ''Fundamentals of Thermodynamics'', seventh edition, Wiley, {{ISBN|978-0-470-04192-5}}.<br />
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*Buchdahl, H.A. (1966). ''The Concepts of Classical Thermodynamics'', Cambridge University Press, Cambridge UK.<br />
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*[[Percy Williams Bridgman|Bridgman, P.W.]] (1943). ''The Nature of Thermodynamics'', Harvard University Press, Cambridge MA.<br />
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*[[Herbert Callen|Callen, H.B.]] (1960/1985). ''Thermodynamics and an Introduction to Thermostatistics'', (1st edition 1960) 2nd edition 1985, Wiley, New York, {{ISBN|0-471-86256-8}}.<br />
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*{{cite journal|author=C. Carathéodory|author1-link=Constantin Carathéodory<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
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|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
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2010年3月15日 | 第355-386页 | http://gdz.sub.uni-goettingen.de/index.php?id=11&ppn=ppn235181684_0067&dmdid=dmdlog_0033&l=1<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). ''The Second Law of Thermodynamics'', Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können.(p. 363) | doi 10.1007 / bf01450409}.译文可以在这里找到[ http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf ]。同时，在 Kestin，j。(1976).热力学第二定律，Dowden，Hutchinson & Ross，Stroudsburg PA. 。<br />
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*[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986). [http://www.worldcat.org/title/reflections-on-the-motive-power-of-fire-a-critical-edition-with-the-surviving-scientific-manuscripts-translated-and-edited-by-fox-robert/oclc/812944517&referer=brief_results ''Reflections on the motive power of fire''], Manchester University Press, Manchester UK, {{ISBN|0-7190-1741-6}}. [https://archive.org/stream/reflectionsonmot00carnrich#page/n7/mode/2up Also here.]<br />
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*[[Sydney Chapman (mathematician)|Chapman, S.]], [[Thomas George Cowling|Cowling, T.G.]] (1939/1970). ''The Mathematical Theory of Non-uniform gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases'', third edition 1970, Cambridge University Press, London.<br />
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*{{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Ueber Die Bewegende Kraft Der Wärme Und Die Gesetze, Welche Sich Daraus Für Die Wärmelehre Selbst Ableiten Lassen|journal=Annalen der Physik|year=1850|volume=79|issue=4|pages=368–397, 500–524|url=http://gallica.bnf.fr/ark:/12148/bpt6k15164w/f518.image|accessdate=26 June 2012|doi=10.1002/andp.18501550403|bibcode = 1850AnP...155..500C |hdl=2027/uc1.$b242250|hdl-access=free}} Translated into English: {{cite journal|last=Clausius|first=R.|title=On the Moving Force of Heat, and the Laws regarding the Nature of Heat itself which are deducible therefrom|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1851| volume=2|series=4th|issue=VIII|pages=1–21; 102–119 |url=https://archive.org/stream/londonedinburghd02lond#page/1/mode/1up|accessdate=26 June 2012<br />
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10.1080 / 14786445108646819}<br />
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* {{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie|year=1854|journal=Annalen der Physik|volume=xciii|issue=12|pages=481–506|url=http://zfbb.thulb.uni-jena.de/servlets/MCRFileNodeServlet/jportal_derivate_00140956/18541691202_ftp.pdf|accessdate=24 March 2014|doi=10.1002/andp.18541691202<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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| bibcode 1854AnP... 169. 1856年7月 | 第2卷 | 第4页 | 第86页 | https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=242014年3月<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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热力学理论及其在蒸汽机和物体物理性质中的应用 | 1867年 | 出版商 John van Voorst | 位置: 伦敦 | https://archive.org/details/mechanicaltheor04claugoog|quote=editions:pwr_sbkwa8ic。 2012年6月19日<br />
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*Denbigh, K. (1954/1981). ''The Principles of Chemical Equilibrium. With Applications in Chemistry and Chemical Engineering'', fourth edition, Cambridge University Press, Cambridge UK, {{ISBN|0-521-23682-7}}.<br />
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*[[Josiah Willard Gibbs|Gibbs, J.W.]] (1876/1878). On the equilibrium of heterogeneous substances, ''Trans. Conn. Acad.'', '''3''': 108-248, 343-524, reprinted in ''The Collected Works of J. Willard Gibbs, Ph.D, LL. D.'', edited by W.R. Longley, R.G. Van Name, Longmans, Green & Co., New York, 1928, volume 1, pp.&nbsp;55–353.<br />
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*Glansdorff, P., Prigogine, I. (1971). ''Thermodynamic Theory of Structure, Stability, and Fluctuations'', Wiley-Interscience, London, 1971, {{ISBN|0-471-30280-5}}.<br />
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*Gyarmati, I. (1967/1970) ''Non-equilibrium Thermodynamics. Field Theory and Variational Principles'', translated by E. Gyarmati and W.F. Heinz, Springer, New York.<br />
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*[[Charles Kittel|Kittel, C.]], [[Herbert Kroemer|Kroemer, H.]] (1969/1980). ''Thermal Physics'', second edition, Freeman, San Francisco CA, {{ISBN|0-7167-1088-9}}.<br />
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*Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998). ''Modern Thermodynamics: From Heat Engines to Dissipative Structures'', John Wiley & Sons, Chichester, {{ISBN|0-471-97393-9}}.<br />
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*Lebon, G., Jou, D., Casas-Vázquez, J. (2008). ''Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers'', Springer-Verlag, Berlin, {{ISBN|978-3-540-74252-4}}.<br />
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*{{cite journal|ref=harv|title=The Physics and Mathematics of the Second Law of Thermodynamics|last1=Lieb |first1=E. H. |last2=Yngvason |first2=J.|journal=Physics Reports|volume=310|issue=1|pages=1–96 |year=1999|doi=10.1016/S0370-1573(98)00082-9|arxiv = cond-mat/9708200 |bibcode = 1999PhR...310....1L}}<br />
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*Lieb, E.H., Yngvason, J. (2003). The Entropy of Classical Thermodynamics, pp.&nbsp;147–195, Chapter 8 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*{{cite book|ref=harv|first=F.|last=Mandl|title=Statistical physics|edition=second|year=1988|publisher=[[Wiley & Sons]]|isbn=978-0-471-91533-1}}<br />
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*{{cite journal | last1 = Maxwell | first1 = J.C. | authorlink = James Clerk Maxwell | year = 1867 | title = On the dynamical theory of gases | url = | journal = Phil. Trans. R. Soc. Lond. | volume = 157 | issue = | pages = 49–88 | doi = 10.1098/rstl.1867.0004 }}<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985). ''Thermodynamics'', Pitman, London, {{ISBN|0-273-08577-8}}.<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003). Entropy in Nonequilibrium, pp.&nbsp;79–109, Chapter 5 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*Münster, A. (1970), ''Classical Thermodynamics'', translated by E.S. Halberstadt, Wiley–Interscience, London, {{ISBN|0-471-62430-6}}.<br />
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*[[Brian Pippard|Pippard, A.B.]] (1957/1966). ''Elements of Classical Thermodynamics for Advanced Students of Physics'', original publication 1957, reprint 1966, Cambridge University Press, Cambridge UK.<br />
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*[[Max Planck|Planck, M.]] (1897/1903). [https://archive.org/stream/treatiseonthermo00planrich#page/100/mode/2up ''Treatise on Thermodynamics'', translated by A. Ogg, Longmans Green, London, p. 100.]<br />
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*[[Max Planck|Planck. M.]] (1914). [https://archive.org/details/theoryofheatradi00planrich ''The Theory of Heat Radiation''], a translation by Masius, M. of the second German edition, P. Blakiston's Son & Co., Philadelphia.<br />
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*[[Max Planck|Planck, M.]] (1926). Über die Begründung des zweiten Hauptsatzes der Thermodynamik, ''Sitzungsberichte der Preussischen Akademie der Wissenschaften: Physikalisch-mathematische Klasse'': 453–463.<br />
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* Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769-781.<br />
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* {{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}<br />
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*Quinn, T.J. (1983). ''Temperature'', Academic Press, London, {{ISBN|0-12-569680-9}}.<br />
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*{{cite book|ref=harv|first=Y.V.C.|last=Rao|title=An Introduction to thermodynamics|url=https://books.google.com/books?id=iYWiCXziWsEC&pg=PA213|date=2004|publisher=Universities Press|isbn=978-81-7371-461-0|page=213}}<br />
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*Roberts, J.K., Miller, A.R. (1928/1960). ''Heat and Thermodynamics'', (first edition 1928), fifth edition, Blackie & Son Limited, Glasgow.<br />
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*[[Erwin Schrödinger|Schrödinger, E.]] (1950). Irreversibility, ''Proc. R. Ir. Acad.'', '''A53''': 189–195.<br />
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*[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966). ''Elements of Thermodynamics'', Addison-Wesley Publishing, Reading MA.<br />
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*{{cite journal|ref=harv|last=Thomson|first=W.|author-link=William Thomson, 1st Baron Kelvin|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Transactions of the Royal Society of Edinburgh|year=1851|volume=XX|issue=part II|pages=261–268; 289–298|url=https://www.biodiversitylibrary.org/item/126047#page/295/mode/1up}} Also published in {{cite journal|last=Thomson|first=W.|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Philos. Mag. |date=December 1852 |volume=IV |series=4 |issue=22 |page=13|url=https://archive.org/stream/londonedinburghp04maga#page/12/mode/2up |accessdate=25 June 2012}}<br />
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*[[William Thomson, 1st Baron Kelvin|Thomson, W.]] (1852). ''On the universal tendency in nature to the dissipation of mechanical energy'' Philosophical Magazine, Ser. 4, p.&nbsp;304.<br />
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*[[László Tisza|Tisza, L.]] (1966). ''Generalized Thermodynamics'', M.I.T Press, Cambridge MA.<br />
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*[[Clifford Truesdell|Truesdell, C.]] (1980). ''The Tragicomical History of Thermodynamics 1822–1854'', Springer, New York, {{ISBN|0-387-90403-4}}.<br />
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*Uffink, J. (2001). Bluff your way in the second law of thermodynamics, ''Stud. Hist. Phil. Mod. Phys.'', '''32'''(3): 305–394.<br />
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*Uffink, J. (2003). Irreversibility and the Second Law of Thermodynamics, Chapter 7 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003), Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963). ''Lectures in Statistical Mechanics'', American Mathematical Society, Providence RI.<br />
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*[[Mark Zemansky|Zemansky, M.W.]] (1968). ''Heat and Thermodynamics. An Intermediate Textbook'', fifth edition, McGraw-Hill Book Company, New York.<br />
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==Further reading==<br />
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==Further reading==<br />
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进一步阅读<br />
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*Goldstein, Martin, and Inge F., 1993. ''The Refrigerator and the Universe''. Harvard Univ. Press. Chpts. 4–9 contain an introduction to the Second Law, one a bit less technical than this entry. {{ISBN|978-0-674-75324-2}}<br />
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*Leff, Harvey S., and Rex, Andrew F. (eds.) 2003. ''Maxwell's Demon 2 : Entropy, classical and quantum information, computing''. Bristol UK; Philadelphia PA: [[Institute of Physics]]. {{ISBN|978-0-585-49237-7}}<br />
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*{{Cite book | first = J.J. | last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1}}(technical).<br />
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*{{cite book |title=Reflections on the Motive Power of Heat and on Machines Fitted to Develop That Power |last=Carnot |first=Sadi |editor=[[Robert Henry Thurston|Thurston, Robert Henry]] |year=1890 |publisher=J. Wiley & Sons |location=New York |isbn= |pages= }} ([https://books.google.com/books?id=tgdJAAAAIAAJ full text of 1897 ed.]) ([http://www.history.rochester.edu/steam/carnot/1943/ html])<br />
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*Stephen Jay Kline (1999). ''The Low-Down on Entropy and Interpretive Thermodynamics'', La Cañada, CA: DCW Industries. {{ISBN|1-928729-01-0}}.<br />
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*{{cite book | last1 = Kostic | first1 = M | year = 2011 | title = Revisiting The Second Law of Energy Degradation and Entropy Generation: From Sadi Carnot's Ingenious Reasoning to Holistic Generalization | journal = AIP Conf. Proc. | volume = 1411 | issue = 1| pages = 327–350 | doi = 10.1063/1.3665247 | isbn = 978-0-7354-0985-9 | bibcode = 2011AIPC.1411..327K | series = AIP Conference Proceedings | citeseerx = 10.1.1.405.1945 }} also at [https://web.archive.org/web/20130420222450/http://www.kostic.niu.edu/2ndLaw/Revisiting%20The%20Second%20Law%20of%20Energy%20Degradation%20and%20Entropy%20Generation%20-%20From%20Carnot%20to%20Holistic%20Generalization-4.pdf].<br />
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==External links==<br />
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==External links==<br />
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外部链接<br />
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*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics]" – by Lawrence Sklar.<br />
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*[http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node30.html ''Second law of thermodynamics''] in the MIT Course [http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/notes.html ''Unified Thermodynamics and Propulsion''] from Prof. Z. S. Spakovszky<br />
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*[[E.T. Jaynes]], 1988, "[http://bayes.wustl.edu/etj/articles/ccarnot.pdf The evolution of Carnot's principle]," in G. J. Erickson and C. R. Smith (eds.)''Maximum-Entropy and Bayesian Methods in Science and Engineering, Vol'' 1: p.&nbsp;267.<br />
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*[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf Caratheodory, C., "Examination of the foundations of thermodynamics," trans. by D. H. Delphenich]<br />
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* [https://www.bbc.co.uk/programmes/p004y2bm The Second Law of Thermodynamics], BBC Radio 4 discussion with John Gribbin, Peter Atkins & Monica Grady (''In Our Time'', Dec. 16, 2004)<br />
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* [http://mdpi.org/entropy/papers/e6010001.pdf [[Entropy (journal)]], 2004]<br />
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* [https://www.journals.uchicago.edu/doi/abs/10.1086/663835 The Journal of the International Society for the History of Philosophy of Science, 2012]<br />
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{{DEFAULTSORT:Second Law Of Thermodynamics}}<br />
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[[Category:Concepts in physics]]<br />
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Category:Concepts in physics<br />
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分类: 物理概念<br />
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[[Category:Laws of thermodynamics|2]]<br />
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2<br />
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[[Category:Non-equilibrium thermodynamics]]<br />
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Category:Non-equilibrium thermodynamics<br />
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类别: 非平衡态热力学<br />
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[[Category:Philosophy of thermal and statistical physics]]<br />
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Category:Philosophy of thermal and statistical physics<br />
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类别: 热力学和统计物理学哲学<br />
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<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Second law of thermodynamics]]. Its edit history can be viewed at [[热力学第二定律/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E7%83%AD%E5%8A%9B%E5%AD%A6%E7%AC%AC%E4%BA%8C%E5%AE%9A%E5%BE%8B&diff=10360热力学第二定律2020-07-16T09:38:31Z<p>Dorr：/* Thermodynamic temperature 热力学温度 */</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Law of physics}}<br />
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{{Thermodynamics|laws}}<br />
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The '''second law of thermodynamics''' states that the total [[entropy]] of an [[isolated system]] can never decrease over time, and is constant if and only if all processes are reversible.<ref>http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node38.html#SECTION05224000000000000000</ref> Isolated systems spontaneously evolve towards [[thermodynamic equilibrium]], the state with maximum entropy. <br />
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The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, and is constant if and only if all processes are reversible. Isolated systems spontaneously evolve towards thermodynamic equilibrium, the state with maximum entropy. <br />
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热力学第二定律指出，孤立系统的总熵永远不会随着时间而减少，而且只有当且仅当所有过程都是可逆的时，总熵才是恒定的。孤立的系统自发地进化到热力学平衡状态，即熵最大的状态。<br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in [[thermodynamic equilibrium]], or is undergoing a (fictive) [[Reversible process (thermodynamics)|reversible process]]. In all processes that occur, including [[spontaneous process]]es,<ref>Atkins and de Paula, p.78</ref> the total entropy of the system and its surroundings increases and the process is [[Irreversible process|irreversible in the thermodynamic sense]]. The increase in entropy accounts for the irreversibility of natural processes, and the [[Arrow of time|asymmetry between future and past]].<ref>{{cite book|last=Zohuri|first=Bahman|title=Dimensional Analysis Beyond the Pi Theorem|url=https://books.google.com/books?id=pRVuDQAAQBAJ|year=2016|publisher=Springer|isbn=978-3-319-45726-0|page=[https://books.google.com.ph/books?id=pRVuDQAAQBAJ&pg=PA111&dq=%22increase+in+entropy+accounts+for+the+irreversibility+of+natural+processes+and+the+asymmetry+between+future+and+past.%22&hl=en&sa=X&ved=0ahUKEwj6spb61tbaAhUFS7wKHfftDtIQ6AEIKDAA#v=onepage&q=%22increase%20in%20entropy%20accounts%20for%20the%20irreversibility%20of%20natural%20processes%20and%20the%20asymmetry%20between%20future%20and%20past.%22 111]}}</ref><br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in thermodynamic equilibrium, or is undergoing a (fictive) reversible process. In all processes that occur, including spontaneous processes, the total entropy of the system and its surroundings increases and the process is irreversible in the thermodynamic sense. The increase in entropy accounts for the irreversibility of natural processes, and the asymmetry between future and past.<br />
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系统及其周围环境的总熵在理想情况下可以保持不变，在这种情况下，系统处于热力学平衡期，或者正在经历一个(虚拟的)可逆过程期。在所有发生的过程中，包括自发过程，系统及其周围环境的总熵增加，这一过程在热力学意义上是不可逆的。熵的增加解释了自然过程的不可逆性，以及未来和过去之间的不对称性。<br />
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Historically, the second law was an [[Empirical evidence|empirical finding]] that was accepted as an axiom of [[thermodynamics|thermodynamic theory]]. [[Statistical mechanics]], classical or [[quantum thermodynamics|quantum]], explains the microscopic origin of the law.<br />
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Historically, the second law was an empirical finding that was accepted as an axiom of thermodynamic theory. Statistical mechanics, classical or quantum, explains the microscopic origin of the law.<br />
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从历史上看，第二定律是一个经验性的发现，被公认为热力学理论的公理。统计力学，无论是经典的还是量子的，都可以解释这个定律的微观起源。<br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist [[Nicolas Léonard Sadi Carnot|Sadi Carnot]], who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as [[Carnot's theorem (thermodynamics)|Carnot's rule]] or limit.<ref>Jaffe, R.L., Taylor, W. (2018). ''The Physics of Energy'', Cambridge University Press, Cambridge UK, pages 150, 151, 259, 772, 743.</ref><br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist Sadi Carnot, who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as Carnot's rule or limit.<br />
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第二定律已经用许多方式表达过了。法国科学家萨迪 · 卡诺在1824年证明了热能在热机中转化为功的效率是有上限的。这方面的第二定律也被称为卡诺的规则或限制。<br />
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==Introduction==<br />
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==Introduction==<br />
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引言<br />
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[[File:Heat flow hot to cold.png|thumb|upright|Heat flow from hot water to cold water.]]<br />
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Heat flow from hot water to cold water.<br />
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热量从热水流向冷水。<br />
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The [[first law of thermodynamics]] provides the definition of the [[internal energy]] of a [[thermodynamic system]], and expresses the law of [[conservation of energy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 40–41.</ref><ref>Munster A. (1970), pp. 8–9, 50–51.</ref> The second law is concerned with the direction of natural processes.<ref>{{harvnb|Mandl|1988}}</ref> It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such [[Phenomenon|phenomena]] are accounted for in terms of [[entropy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 79–107.</ref><ref>Bailyn, M. (1994), Section 71, pp. 113–154.</ref> If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, {{math|''S''}}, increases.<br />
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The first law of thermodynamics provides the definition of the internal energy of a thermodynamic system, and expresses the law of conservation of energy. The second law is concerned with the direction of natural processes. It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such phenomena are accounted for in terms of entropy. If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, , increases.<br />
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能量守恒定律提供了热力学系统内能的定义，并表达了能量守恒定律。第二定律与自然过程的方向有关。它断言自然过程只在一种意义上运行，而且是不可逆的。例如，当有了传导和辐射的路径时，热总是自发地从一个较热的物体流向一个较冷的物体。这种现象可以用熵来解释。如果一个孤立的系统最初是通过内部分隔不可渗透的墙来保持在内部热力学平衡，然后一些操作使墙更具渗透性，那么系统自发地进化到最终达到一个新的内部热力学平衡，其总熵增加。<br />
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In a fictive reversible process, an infinitesimal increment in the entropy ({{math|d''S''}}) of a system is defined to result from an infinitesimal transfer of heat ({{math|δ''Q''}}) to a [[closed system]] (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature ({{math|''T''}}) of the system in equilibrium and the surroundings which supply the heat:<ref>Bailyn, M. (1994), p. 120.</ref><br />
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In a fictive reversible process, an infinitesimal increment in the entropy () of a system is defined to result from an infinitesimal transfer of heat () to a closed system (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature () of the system in equilibrium and the surroundings which supply the heat:<br />
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在虚构的可逆过程中，系统熵的无穷小增量被定义为由无穷小的热量传递()到一个封闭系统(允许能量进入或出去，但不允许物质传递)除以平衡系统和提供热量的环境的共同温度() :<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, idealized fictive reversible process)}.</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, idealized fictive reversible process)}.</math><br />
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(闭合系统，理想化的可逆过程) / math<br />
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Different notations are used for infinitesimal amounts of heat ({{mvar|δ}}) and infinitesimal amounts of entropy ({{mvar|d}}) because entropy is a [[function of state]], while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the [[Clausius Theorem|inequality]] <ref name="MortimerBook"> Mortimer, R. G. ''Physical Chemistry''. 3rd ed., p. 120, Academic Press, 2008. </ref><ref name="FermiBook"> Fermi, E. ''Thermodynamics'', footnote on p. 48, Dover Publications,1956 (still in print). </ref> <br />
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Different notations are used for infinitesimal amounts of heat () and infinitesimal amounts of entropy () because entropy is a function of state, while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the inequality <br />
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用不同的符号表示无穷小量的热量()和无穷小量的熵() ，因为熵是状态的函数，而热量和功一样不是。第二定律要求系统熵的增量满足不等式，对于实际上可能存在的不与环境发生质量交换的无穷小过程，系统熵增量满足不等式<br />
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: <math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process).}</math><br />
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<math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process).}</math><br />
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(实际上可能是封闭系统，不可逆性)。数学<br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature ({{math|''T''}}) and the temperature of the surroundings ({{math|''T''}}<sub>''surr''</sub>).<ref name=":0">Adkins, C.J. (1968/1983), p. 75.</ref><ref name="Munster 45"/> Note that the equality still applies for pure heat flow,<ref name="Schmidt-Rohr 14"> Schmidt-Rohr, K. (2014). "Expansion Work without the External Pressure, and Thermodynamics in Terms of Quasistatic Irreversible Processes" ''J. Chem. Educ.'' '''91''': 402-409. https://dx.doi.org/10.1021/ed3008704 </ref><br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature () and the temperature of the surroundings (<sub>surr</sub>).<br />
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这是因为这种情况下的一般过程可能包括周围环境对系统所做的功，这可能在系统内部产生摩擦或粘滞效应，因为一个化学反应可能正在进行，或者因为热传递实际上只是不可逆地发生，由系统温度()和周围环境温度(sub surr / sub)之间的差分驱动。<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change).}</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change).}</math><br />
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(实际上可能是不改变成分的准静态不可逆性)。数学<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry.<ref name="Oxtoby8th">Oxtoby, D. W; Gillis, H.P., [[Laurie Butler|Butler, L. J.]] (2015).''Principles of Modern Chemistry'', Brooks Cole. p. 617. {{ISBN|978-1305079113}}</ref> <ref name="MortimerBook"></ref> Introducing a set of internal variables <math>\xi</math> to describe the deviation of a thermodynamic system in physical equilibrium (with the required well-defined uniform pressure ''P'' and temperature ''T'')<ref name="Schmidt-Rohr 14"></ref> from the chemical equilibrium state, one can record the equality<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry. from the chemical equilibrium state, one can record the equality<br />
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这是根据测得的热容曲线和相变时的熵变精确确定纯物质的绝对熵的基础。用量热法。从化学平衡来看，人们可以记录平等<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process).}</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process).}</math><br />
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- frac { t }-frac {1}{ j } ， Xi { j } ， delta Xi j， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ,.数学<br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of ''the second principle of thermodynamics'' – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<ref> Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769–781.</ref><ref>{{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}</ref><br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of the second principle of thermodynamics – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<br />
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第二项代表内部变量的工作，这些内部变量可以受到外部影响的干扰，但是系统不能通过内部变量执行任何正功。这种说法介绍了热力学系统在时间上不可能逆转演化的事实，并且可以被认为是热力学第二原理的表述---- 这个表述，当然，相当于熵原理的表述。<br />
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The [[zeroth law of thermodynamics]] in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body.<ref name=dugdale>{{cite book|author=J. S. Dugdale|title=Entropy and its Physical Meaning|publisher=Taylor & Francis|year=1996|isbn=978-0-7484-0569-5|page=13|quote=This law is the basis of temperature.}}</ref> For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows{{how?|date=August 2018}} a distinguished temperature scale, which defines an absolute, [[thermodynamic temperature]], independent of the properties of any particular reference thermometric body.<ref>[[Mark Zemansky|Zemansky, M.W.]] (1968), pp. 207–209.</ref><ref>Quinn, T.J. (1983), p. 8.</ref><br />
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The zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body. For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows a distinguished temperature scale, which defines an absolute, thermodynamic temperature, independent of the properties of any particular reference thermometric body.<br />
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美国热力学第零定律协会在其通常的简短声明中允许承认，相对于热平衡的两个物体具有相同的温度，特别是测试物体具有与参考温度测量物体相同的温度。对于一个物体和另一个物体之间的热平衡，存在着无限多的经验温度标度，通常分别取决于特定参考温度标度的性质。第二定律允许区分温度标度，它定义了一个绝对的热力学温度，与任何特定的参考温度体的性质无关。<br />
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==Various statements of the law==<br />
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==Various statements of the law==<br />
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各种各样的法律陈述<br />
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The second law of thermodynamics may be expressed in many specific ways,<ref name=MIT>{{cite web|title=Concept and Statements of the Second Law|url=http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node37.html|accessdate=2010-10-07 |publisher=web.mit.edu}}</ref> the most prominent classical statements{{sfnp|Lieb|Yngvason|1999}} being the statement by [[Rudolf Clausius]] (1854), the statement by [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] (1851), and the statement in axiomatic thermodynamics by [[Constantin Carathéodory]] (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.{{sfnp|Rao|2004|p=213}}<br />
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The second law of thermodynamics may be expressed in many specific ways, the most prominent classical statements being the statement by Rudolf Clausius (1854), the statement by Lord Kelvin (1851), and the statement in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.<br />
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热力学第二定律可以用许多特定的方式来表达，最突出的经典陈述是 Rudolf Clausius (1854)的陈述，Kelvin 勋爵(1851)的陈述，以及康斯坦丁·卡拉西奥多里(1909)在公理化热力学中的陈述。这些陈述用一般的物理术语描述法律，引用某些过程的不可能性。克劳修斯和开尔文陈述被证明是等价的。<br />
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===Carnot's principle===<br />
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===Carnot's principle===<br />
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卡诺原理<br />
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The historical origin<ref>[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986).</ref> of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a [[Carnot heat engine]], fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the [[caloric theory]] of heat was seriously considered, before the recognition of the [[first law of thermodynamics]], and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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The historical origin of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a Carnot heat engine, fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the caloric theory of heat was seriously considered, before the recognition of the first law of thermodynamics, and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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热力学第二定律的历史起源源于卡诺的原则。它指的是一个卡诺热机周期，虚拟地以被称为准静态的极慢极限模式运行，因此热量和功量在子系统之间传递，这些子系统始终处于它们自己的内部热力学平衡状态。卡诺发动机是热机效率工程师特别感兴趣的理想设备。卡诺认识到卡诺原理的时候，正是热量热理论被认真考虑的时候，正是能量守恒定律理论被认识之前，正是熵概念的数学表达之前。根据第一定律的解释，它在物理上等同于热力学第二定律，并且在今天仍然有效。卡诺最初的论点是从卡路里理论的观点提出的，直到能量守恒定律的发现。以下是他书中的一些例子:<br />
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::...''wherever there exists a difference of temperature, motive power can be produced.''<ref>Carnot, S. (1824/1986), p. 51.</ref><br />
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...wherever there exists a difference of temperature, motive power can be produced.<br />
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只要有温差，就能产生动力。<br />
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::The production of motive power is then due in steam engines not to an actual consumption of caloric, but ''to its transportation from a warm body to a cold body ...''<ref>Carnot, S. (1824/1986), p. 46.</ref><br />
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The production of motive power is then due in steam engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body ...<br />
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动力的产生不是由于蒸汽机实际消耗的热量，而是由于它从一个温暖的物体运输到一个寒冷的物体..。<br />
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::''The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.''<ref>Carnot, S. (1824/1986), p. 68.</ref><br />
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The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.<br />
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热的动力与实现热的媒介无关，热的数量完全取决于两个物体之间的温度，最后是热量的传递。<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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用现代术语来说，卡诺的原则可能更为准确:<br />
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::The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<ref>[[Clifford Truesdell|Truesdell, C.]] (1980), Chapter 5.</ref><ref>Adkins, C.J. (1968/1983), pp. 56–58.</ref><ref>Münster, A. (1970), p. 11.</ref><ref>Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998), pp.67–75.</ref><ref>Lebon, G., Jou, D., Casas-Vázquez, J. (2008), p. 10.</ref><ref>Eu, B.C. (2002), pp. 32–35.</ref><br />
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The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<br />
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准静态或可逆卡诺循环的效率只取决于两个热源的温度，而且无论工作物质是什么，效率是相同的。以这种方式运行的卡诺发动机是使用这两种温度的最有效的热机。<br />
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===Clausius statement===<br />
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===Clausius statement===<br />
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克劳修斯声明<br />
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The German scientist [[Rudolf Clausius]] laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work.{{sfnp|Clausius|1850}} His formulation of the second law, which was published in German in 1854, is known as the ''Clausius statement'':<br />
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The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work. His formulation of the second law, which was published in German in 1854, is known as the Clausius statement:<br />
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1850年，德国科学家 Rudolf Clausius 通过研究热传递和功之间的关系，为热力学第二定律实验室奠定了基础。他在1854年用德语发表的第二定律的提法被称为克劳修斯声明:<br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.{{sfnp|Clausius|1854|p=86}}</blockquote><br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.</blockquote><br />
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热量不可能从一个较冷的物体传递到一个较热的物体，除非同时发生一些其他的变化。 / blockquote<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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克劳修斯的陈述使用了“热量通道”的概念。正如通常在热力学的讨论，这意味着“能量作为热量的净转移” ，并没有提到促成转移的一种方式和其他。<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of [[refrigeration]], for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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如果不对系统进行外部功，热就不能自发地从冷区流向热区，这一点从制冷的普通经验中可以看出。在冰箱中，热量从冷到热，但只有在外部介质——制冷系统的强制作用下才会发生变化。<br />
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===Kelvin statements===<br />
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===Kelvin statements===<br />
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开尔文陈述<br />
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[[William Thomson, 1st Baron Kelvin|Lord Kelvin]] expressed the second law in several wordings.<br />
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Lord Kelvin expressed the second law in several wordings.<br />
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开尔文勋爵用几个字表达了第二定律。<br />
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::It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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没有任何外部机构的帮助，自动机器是不可能把热量从一个物体传递到另一个物体的较高温度。<br />
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::It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.{{sfnp|Thomson|1851}}<br />
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It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.<br />
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通过无生命物质的作用，把物质冷却到周围最冷的物体的温度以下，从物质的任何部分获得机械效应是不可能的。<br />
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===Equivalence of the Clausius and the Kelvin statements===<br />
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===Equivalence of the Clausius and the Kelvin statements===<br />
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克劳修斯和开尔文陈述的等价性<br />
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[[Image:Deriving Kelvin Statement from Clausius Statement.svg|thumb|Derive Kelvin Statement from Clausius Statement]]<br />
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Derive Kelvin Statement from Clausius Statement<br />
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从克劳修斯陈述推导出开尔文陈述<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed [[Carnot engine]] as shown by the figure. The [[Heat engine#Efficiency|efficiency]] of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the [[first law of thermodynamics]], as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the figure. The efficiency of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the first law of thermodynamics, as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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假设有一个引擎违反了开尔文定理: 也就是说，这个引擎以循环的方式吸收热量并将其转化为功，没有任何其他结果。现在配对与反向卡诺发动机如图所示。普通热机的效率为，反向热机的效率为1 / 。这对组合发动机的净效应和唯一效应是将热量从较冷的贮存器转移到较热的贮存器，这违反了克劳修斯的说法。(这是能量守恒定律的结果，因为系统的总能量保持不变，数学文本{输入} + 文本{输出}0暗示 q-frac { eta }-qc / math，所以数学 qc 向左(frac { eta-1 right) / math)。因此，违反开尔文陈述意味着违反克劳修斯陈述，即。克劳修斯的说法暗示了凯尔文的说法。我们可以用类似的方式证明开尔文陈述暗示了克劳修斯陈述，因此两者是等价的。<br />
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===Planck's proposition===<br />
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===Planck's proposition===<br />
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普朗克的命题<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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普朗克提出了直接来自经验的如下命题。这有时被认为是他对第二定律的陈述，但他认为这是第二定律推导的起点。<br />
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::It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 86.</ref><ref>Roberts, J.K., Miller, A.R. (1928/1960), p. 319.</ref><br />
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It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<br />
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建造一台发动机是不可能的，这台发动机将在一个完整的循环中工作，并且除了提高重量和冷却热储以外不会产生任何效果。<br />
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===Relation between Kelvin's statement and Planck's proposition 开尔文表述与普朗克命题的关系===<br />
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It is almost customary in textbooks to speak of the "[[Kelvin-Planck statement]]" of the law, as for example in the text by [[Dirk ter Haar|ter Haar]] and [[Harald Wergeland|Wergeland]].<ref>[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966), p. 17.</ref><br />
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It is almost customary in textbooks to speak of the "Kelvin-Planck statement" of the law, as for example in the text by ter Haar and Wergeland.<br />
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教科书中几乎惯常提及该定律的“'''开尔文-普朗克表述 Kelvin-Planck Statement'''” ，例如'''德克·特哈尔 Diek ter Haar''' 和'''哈拉尔德·沃格兰 Harald Wergeland''' 在文中就是这样表述的。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）Diek ter Haar 中间是小写的<br />
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The '''Kelvin–Planck statement''' (or the ''heat engine statement'') of the second law of thermodynamics states that <br />
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The Kelvin–Planck statement (or the heat engine statement) of the second law of thermodynamics states that <br />
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热力学第二定律的开尔文-普朗克表述（或称“'''热机表述 Heat Engine Statement'''”）指出<br />
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::It is impossible to devise a [[thermodynamic cycle|cyclically]] operating device, the sole effect of which is to absorb energy in the form of heat from a single [[heat reservoir|thermal reservoir]] and to deliver an equivalent amount of [[Work (physics)|work]].<ref name="Rao">{{cite book|last=Rao|first=Y. V. C.|title=Chemical Engineering Thermodynamics|publisher=Universities Press|isbn=978-81-7371-048-3|page=158|year=1997}}</ref><br />
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It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.<br />
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设计一种唯一效果是从单一热源吸收热量并提供等量的功的循环运行装置是不可能的。<br />
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===Planck's statement 普朗克表述===<br />
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Planck stated the second law as follows.<br />
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Planck stated the second law as follows.<br />
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普朗克表述第二定律如下。<br />
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::Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<ref name="Planck 100">[[Max Planck|Planck, M.]] (1897/1903), p. 100.</ref><ref name="Planck 463">[[Max Planck|Planck, M.]] (1926), p. 463, translation by Uffink, J. (2003), p. 131.</ref><ref name="Roberts & Miller 382">Roberts, J.K., Miller, A.R. (1928/1960), p. 382. This source is partly verbatim from Planck's statement, but does not cite Planck. This source calls the statement the principle of the increase of entropy.</ref><br />
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Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<br />
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自然界中发生的任一过程都是沿参与其中的所有物体的熵之和增加的方向进行的。在某些限制下——即对于可逆过程，熵的总和保持不变。<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for ''irreversible phenomena''.<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for irreversible phenomena.<br />
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与普朗克表述非常相似的是'''乌伦贝克 Uhlenbeck'''和'''福特 Ford'''关于不可逆现象的表述。<br />
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::... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<ref>[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963), p. 16.</ref><br />
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... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<br />
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在从一个平衡态到另一个平衡态的不可逆或自发的变化中（例如，当两个物体 A 和 B 接触时的温度平衡过程），熵总是增加。<br />
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===Principle of Carathéodory 卡拉西奥多里原理===<br />
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<!-- [[Caratheodory's principle]] redirects here --><br />
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—— caratheodory 的原理在这里重新定向——<br />
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[[Constantin Carathéodory]] formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<ref>[[Constantin Carathéodory|Carathéodory, C.]] (1909).</ref><br />
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Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<br />
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'''康斯坦丁·卡拉西奥多里 Constantin Carathéodory'''在纯数学公理的基础上进行了热力学公理化。他对第二定律的陈述被称为'''卡拉西奥多里原理 Principle of Carathéodory'''，可以这样表述:<br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.<ref>Buchdahl, H.A. (1966), p. 68.</ref></blockquote><br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.</blockquote><br />
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在绝热封闭系统的任意状态 S 附近，总有从 S 出发不可达的状态。<br />
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With this formulation, he described the concept of [[adiabatic accessibility]] for the first time and provided the foundation for a new subfield of classical thermodynamics, often called [[Ruppeiner geometry|geometrical thermodynamics]]. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic [[process function]], in other words, <math>\delta Q=TdS</math>.<ref name="Sychev1991">{{cite book |last=Sychev |first=V. V. |title=The Differential Equations of Thermodynamics |year=1991 |publisher=Taylor & Francis |isbn=978-1-56032-121-7}}</ref> {{clarify|date=February 2014}}<br />
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With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, <math>\delta Q=TdS</math>. <br />
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公理化处理后，他第一次描述了'''绝热可达性 Adiabatic Accessibility'''的概念，并为经典热力学的一个新的子领域，即通常所说的'''几何热力学 Geometrical Thermodynamics'''奠定了基础。由卡拉西奥多里原理可以推出，准静态转移的热量值是一个可积过程函数，即<math>\delta Q=TdS</math>。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）准静态转移的热量值是一个可积过程函数 存疑<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.<ref name="Munster 45">Münster, A. (1970), p. 45.</ref>{{sfnp|Lieb|Yngvason|1999|p=49}}<ref name="Planck 1926">[[Max Planck|Planck, M.]] (1926).</ref><ref>Buchdahl, H.A. (1966), p. 69.</ref> {{clarify|date=February 2014}}<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium. <br />
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尽管在教科书几乎惯称卡拉西奥多里原理表述了第二定律，并认为其与克劳修斯表述或开尔文-普朗克表述等价，但事实并非如此。为了得到第二定律的所有内容，需要对卡拉西奥多里原理补充普朗克表述，即等量功总是增加一个最初处于自身内部热力学平衡的封闭系统的内部能量。<br />
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===Planck's principle 普朗克原理===<br />
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In 1926, [[Max Planck]] wrote an important paper on the basics of thermodynamics.<ref name="Planck 1926"/><ref>Uffink, J. (2003), pp. 129–132.</ref> He indicated the principle<br />
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In 1926, Max Planck wrote an important paper on the basics of thermodynamics. He indicated the principle<br />
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1926年，'''马克斯·普朗克 Max Planck'''写了一篇关于热力学基础的重要论文。他指出了以下原理<br />
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::The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<ref name="Munster 45"/>{{sfnp |Lieb|Yngvason|1999|p=49}}<br />
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The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<br />
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一个封闭系统的内部能量因绝热过程增加，在整个过程中，系统的体积保持不变。<br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work."<ref>[[Clifford Truesdell|Truesdell, C.]], Muncaster, R.G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 15.</ref> Planck wrote: "The production of heat by friction is irreversible."<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 81.</ref><ref>[[Max Planck|Planck, M.]] (1926), p. 457, Wikipedia editor's translation.</ref><br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work." Planck wrote: "The production of heat by friction is irreversible."<br />
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这个公式没有提到热，没有提到温度，甚至没有提到熵，也不一定隐含地依赖于这些概念，但它暗示了第二定律的内容。一个密切相关的表述为，“摩擦力从来不做正功。”普朗克写道: “摩擦生热是不可逆的。”<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above.<ref>Lieb, E.H., Yngvason, J. (2003), p. 149.</ref> It is relevant that for a system at constant volume and [[Mole (unit)|mole numbers]], the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above. It is relevant that for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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这个普朗克定理没有提到熵，是用物理术语来表述的。它与上面给出的开尔文表述密切相关。相关的是，对于恒定体积和摩尔数的系统，熵是内能的单调函数。然而，普朗克的这个原理实际上并不是普朗克对第二定律的首选表述（见前面小节），而是依赖于熵的概念。<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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Borgnakke 和 Sonntag 提出的以下表述在某种意义上是对普朗克原理的补充。他们没有将其作为第二定律的完整表述:<br />
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::... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<ref>Borgnakke, C., Sonntag., R.E. (2009), p. 304.</ref><br />
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... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<br />
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只有一种方法可以减少[闭合]系统的熵——从系统中转移热量。<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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与普朗克之前提出的原理不同，这一原理明确地用熵的变化来表示。从系统中去除物质也可以减少系统的熵。<br />
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===Statement for a system that has a known expression of its internal energy as a function of its extensive state variables 一个其内能有已知表达式（其扩展状态变量的函数）的系统的表述===<br />
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The second law has been shown to be equivalent to the [[internal energy]] ''U'' being a weakly [[convex function]], when written as a function of extensive properties (mass, volume, entropy, ...).<ref>{{cite book |last1=van Gool |first1=W. |last2=Bruggink |first2=J.J.C. (Eds) |url= |title=Energy and time in the economic and physical sciences |publisher=North-Holland |year=1985 |pages=41–56 |quote= |isbn=978-0-444-87748-2}}</ref><ref>{{Cite journal | last1 = Grubbström | first1 = Robert W. | doi = 10.1016/j.apenergy.2007.01.003 | title = An Attempt to Introduce Dynamics Into Generalised Exergy Considerations| journal = Applied Energy| volume = 84| issue = 7–8 | pages = 701–718 | year = 2007}}</ref> {{clarify|date=February 2014}}<br />
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The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...). <br />
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第二定律被证明等价于弱凸函数内能 U （写成广泛性质（质量，体积，熵，...）的函数时）。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）存疑<br />
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==Corollaries 推论==<br />
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===Perpetual motion of the second kind 第二类永动机===<br />
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{{main article|Perpetual motion}}<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of [[first law of thermodynamics]] by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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在第二定律建立之前，许多志在发明永动机的人试图通过提取环境的巨大内能作为机器动力，以突破'''热力学第一定律 First Law of Thermodynamics''' 的限制。这种机器被称为“'''第二类永动机 Perpetual Motion Machine of the Second Kind'''”。第二定律表明这种机器是不可能的。<br />
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===Carnot theorem 卡诺定理===<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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'''卡诺定理 Carnot's theorem'''（1824）是一条限制任何可能的发动机的最大效率的原理。效率完全取决于热库和冷库之间的温差。卡诺定理指出:<br />
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*All irreversible heat engines between two heat reservoirs are less efficient than a [[Carnot engine]] operating between the same reservoirs.<br />
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*All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs.<br />
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* 所有不可逆热机的效率低于在两相同热源之间工作的'''卡诺机 Carnot engine'''。<br />
* 所有可逆热机的效率与在两相同热源之间工作的卡诺机相等。<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as [[thermodynamic reversibility]]. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the [[Carnot cycle]]'s reversibility and was condemned to be less efficient.<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the Carnot cycle's reversibility and was condemned to be less efficient.<br />
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在他的理想模型中，热转换成功可以通过逆转循环的运动而恢复，这个概念后来被称为'''热力学可逆性 Thermodynamic Reversibility'''。然而，卡诺进一步假定，一些热量损失了，并没有转化为机械功。因此，没有一个真实的热机能够实现'''卡诺循环 Carnot cycle'''的可逆性，并且被认为效率较低。<br />
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Though formulated in terms of caloric (see the obsolete [[caloric theory]]), rather than [[entropy]], this was an early insight into the second law.<br />
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Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law.<br />
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该理论尽管是用热量表述的（见被取代的'''热质说 Caloric Theory'''），而不是熵，但是它是对第二定律的早期认识。<br />
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===Clausius inequality 克劳修斯不等式===<br />
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The [[Clausius theorem]] (1854) states that in a cyclic process<br />
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The Clausius theorem (1854) states that in a cyclic process<br />
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'''克劳修斯定理 Clausius theorem'''（1854）指出，在一个循环的过程中<br />
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: <math>\oint \frac{\delta Q}{T} \leq 0.</math><br />
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The equality holds in the reversible case<ref>[http://scienceworld.wolfram.com/physics/ClausiusTheorem.html ''Clausius theorem''] at [[Wolfram Research]]</ref> and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function [[entropy]]. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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The equality holds in the reversible case and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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等号在可逆情况下成立，严格不等号在不可逆情况下成立。可逆情况下引入状态函数熵。这是因为在循环过程中，状态函数的变化为零。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）末句存疑<br />
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===Thermodynamic temperature 热力学温度===<br />
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{{main article|Thermodynamic temperature}}<br />
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For an arbitrary heat engine, the efficiency is:<br />
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For an arbitrary heat engine, the efficiency is:<br />
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对于任意热机，效率为:<br />
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: <math>\eta = \frac {W_n}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H} \qquad (1)</math><br />
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where ''W''<sub>n</sub> is for the net work done per cycle. Thus the efficiency depends only on q<sub>C</sub>/q<sub>H</sub>.<br />
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其中 ''W''<sub>n</sub> 表示每个循环所做的净功。因此效率只取决于 q<sub>C</sub>/q<sub>H</sub>。<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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卡诺定理指出，所有在相同热源间运行的可逆发动机的效率是相同的。<br />
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Thus, any reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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Thus, any reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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因此，任何在温度''T''<sub>1</sub>和''T''<sub>2</sub>之间运行的可逆热机必须具有相同的效率，也就是说，效率只是温度的函数:<br />
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<math>\frac{q_C}{q_H} = f(T_H,T_C)\qquad (2).</math><br />
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In addition, a reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between ''T''<sub>1</sub> and another (intermediate) temperature ''T''<sub>2</sub>, and the second between ''T''<sub>2</sub> and''T''<sub>3</sub>. This can only be the case if<br />
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In addition, a reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between T<sub>1</sub> and another (intermediate) temperature T<sub>2</sub>, and the second between T<sub>2</sub> andT<sub>3</sub>. This can only be the case if<br />
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另外，在温度 ''T''<sub>1</sub> 和 ''T''<sub>3</sub> 之间工作的可逆热机必须具有与由分别在温度''T''<sub>1</sub> 和（中间）温度 ''T''<sub>2</sub>之间、在 ''T''<sub>2</sub> 和 ''T''<sub>3</sub> 之间的两个循环组成的系统效率相同。只有下式成立才会出现这种情况:<br />
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: <math>f(T_1,T_3) = \frac{q_3}{q_1} = \frac{q_2 q_3} {q_1 q_2} = f(T_1,T_2)f(T_2,T_3).</math><br />
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<br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the [[triple point]] of water. Then for any ''T''<sub>2</sub> and ''T''<sub>3</sub>,<br />
<br />
Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the triple point of water. Then for any T<sub>2</sub> and T<sub>3</sub>,<br />
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现在考虑如下情形，<math>T_1</math> 是一个固定的参考温度: 水的'''三相点 Triple Point'''的温度。则对于任意 T<sub>2</sub> 和 T<sub>3</sub>,<br />
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: <math>f(T_2,T_3) = \frac{f(T_1,T_3)}{f(T_1,T_2)} = \frac{273.16 \cdot f(T_1,T_3)}{273.16 \cdot f(T_1,T_2)}.</math><br />
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Therefore, if thermodynamic temperature is defined by<br />
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Therefore, if thermodynamic temperature is defined by<br />
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因此，如果热力学温度被定义为<br />
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: <math>T = 273.16 \cdot f(T_1,T) \,</math><br />
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then the function ''f'', viewed as a function of thermodynamic temperature, is simply<br />
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then the function f, viewed as a function of thermodynamic temperature, is simply<br />
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那么函数 ''f'' 作为热力学温度的函数，为<br />
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: <math>f(T_2,T_3) = \frac{T_3}{T_2},</math><br />
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and the reference temperature ''T''<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be used{{snd}}the choice here corresponds to the [[Kelvin]] scale.)<br />
<br />
and the reference temperature T<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be usedthe choice here corresponds to the Kelvin scale.)<br />
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参考温度 ''T''<sub>1</sub> 的值为273.16。（任何参考温度和任何正值均可用——此处的选择对应开尔文标度。）<br />
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===Entropy 熵===<br />
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{{main article|Entropy (classical thermodynamics)}}<br />
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According to the [[Clausius theorem|Clausius equality]], for a reversible process<br />
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According to the Clausius equality, for a reversible process<br />
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根据克劳修斯定理，对可逆过程有<br />
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: <math>\oint \frac{\delta Q}{T}=0</math><br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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这意味着线积分 <math>\int_L \frac{\delta Q}{T}</math> 对于可逆过程是路径无关的。<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer<ref name="Schmidt-Rohr 14" /> satisfies<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer satisfies<br />
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所以我们可以定义一个叫做熵的状态函数 S，对于可逆过程或者纯热传递满足<br />
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: <math>dS = \frac{\delta Q}{T} \!</math><br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the [[third law of thermodynamics]], which states that ''S'' = 0 at [[absolute zero]] for perfect crystals.<br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the third law of thermodynamics, which states that S = 0 at absolute zero for perfect crystals.<br />
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据此，只有对上述公式进行积分，才能得到熵的差值。为了获得绝对值，我们需要'''热力学第三定律 Third Law of Thermodynamics'''，它指出'''绝对零度 Absolute Zero'''下完美晶体的 ''S'' = 0。<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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对于任意不可逆过程，由于熵是一个状态函数，我们总是可以将初始状态和最终状态与一个虚拟的可逆过程联系起来，并在这条路径上积分以计算熵的差值。<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the [[Clausius inequality]] on this loop,<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the Clausius inequality on this loop,<br />
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现在把可逆过程逆过来，将其与上述不可逆过程结合。把克劳修斯不等式应用到这个循环,<br />
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: <math>-\Delta S+\int\frac{\delta Q}{T}=\oint\frac{\delta Q}{T}< 0</math><br />
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Thus,<br />
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Thus,<br />
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故,<br />
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: <math>\Delta S \ge \int \frac{\delta Q}{T} \,\!</math><br />
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where the equality holds if the transformation is reversible.<br />
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where the equality holds if the transformation is reversible.<br />
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如果变换可逆，等号成立。<br />
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Notice that if the process is an [[adiabatic process]], then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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Notice that if the process is an adiabatic process, then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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注意，若该过程是一个'''绝热过程 Adiabatic Process'''，则<math>\delta Q=0</math>，故<math>\Delta S\ge 0</math>。<br />
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===Energy, available useful work===<br />
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===Energy, available useful work===<br />
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能量，有用的工作<br />
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{{See also|Exergy}}<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an ''unlimited'' heat reservoir at temperature ''T<sub>R</sub>'' and pressure ''P<sub>R</sub>'' {{snd}}so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain ''T<sub>R</sub>''; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain ''P<sub>R</sub>''.<br />
<br />
An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an unlimited heat reservoir at temperature T<sub>R</sub> and pressure P<sub>R</sub> so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain T<sub>R</sub>; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain P<sub>R</sub>.<br />
<br />
考虑将第二定律应用于孤立系统(称为整体系统或宇宙)的情形，是一个重要的、具有启发性的理想情形。孤立系统由两部分组成: 感兴趣的子系统和子系统的环境。这些环境被想象为如此之大，以至于它们可以被认为是一个温度为 t 子 r / sub 和压力为 p 子 r / sub 的无限热源，因此无论有多少热量被转移到(或来自)子系统，周围环境的温度将保持 t 子 r / sub; 无论子系统的体积扩大(或收缩)多少，周围环境的压力将保持 p 子 r / sub。<br />
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Whatever changes to ''dS'' and ''dS<sub>R</sub>'' occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy ''S<sub>tot</sub>'' of the isolated total system must not decrease:<br />
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Whatever changes to dS and dS<sub>R</sub> occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy S<sub>tot</sub> of the isolated total system must not decrease:<br />
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根据第二定律，孤立总体系统的熵 s 子系统 / 子系统的熵不能减小，无论子系统和周围环境的熵对 dS 和 dS 子系统 r / 子系统发生什么变化:<br />
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: <math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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<math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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数学 dS + dS r ge 0 / math<br />
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According to the [[first law of thermodynamics]], the change ''dU'' in the internal energy of the sub-system is the sum of the heat ''δq'' added to the sub-system, ''less'' any work ''δw'' done ''by'' the sub-system, ''plus'' any net chemical energy entering the sub-system ''d ∑μ<sub>iR</sub>N<sub>i</sub>'', so that:<br />
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According to the first law of thermodynamics, the change dU in the internal energy of the sub-system is the sum of the heat δq added to the sub-system, less any work δw done by the sub-system, plus any net chemical energy entering the sub-system d ∑μ<sub>iR</sub>N<sub>i</sub>, so that:<br />
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根据能量守恒定律，子系统内部能量的变化 dU 是子系统内部能量 q 的总和，减去子系统所做的任何功，再加上进入子系统 d ∑ sub iR / sub n sub i / sub 的任何净化学能，因此:<br />
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: <math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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<math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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数学 dU delta q- delta w + d ( sum mu iR } n i) ，/ math<br />
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where μ<sub>iR</sub> are the [[chemical potential]]s of chemical species in the external surroundings.<br />
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where μ<sub>iR</sub> are the chemical potentials of chemical species in the external surroundings.<br />
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其中亚 iR / 亚是外部环境中化学物种的化学势。<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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现在热量离开储存器进入子系统是<br />
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: <math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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<math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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数学△ qtr (- dS r) le tr dS / math<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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其中我们首先使用了经典热力学中熵的定义(在统计热力学中，熵变、温度和吸收热量之间的关系可以推导出来) ，然后从上面推导出第二定律的不等式。<br />
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It therefore follows that any net work ''δw'' done by the sub-system must obey<br />
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It therefore follows that any net work δw done by the sub-system must obey<br />
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因此，子系统所做的任何净功都必须服从<br />
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: <math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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<math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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数学增量 w le-dU + tds + sum mu { iR } dN i，/ math<br />
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It is useful to separate the work ''δw'' done by the subsystem into the ''useful'' work ''δw<sub>u</sub>'' that can be done ''by'' the sub-system, over and beyond the work ''p<sub>R</sub> dV'' done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
<br />
It is useful to separate the work δw done by the subsystem into the useful work δw<sub>u</sub> that can be done by the sub-system, over and beyond the work p<sub>R</sub> dV done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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将子系统所做的功与子系统所能做的功分离为子系统所能做的有用功，超过了子系统对外界压力所做的功，给出了子系统所能做的有用功(火用)的下列关系式:<br />
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: <math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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<math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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D (u-t r s + p r v-sum { iR } n i) ，/ math<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the ''availability'' or ''[[exergy]]'' ''E'' of the subsystem,<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the availability or exergy E of the subsystem,<br />
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可以很方便地将子系统的右侧定义为热动力位能的精确衍生物，称为子系统的可用性或火用 e,<br />
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: <math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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<math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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数学 e u-t r s + p r v- sum { iR } n i / math<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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因此，第二定律意味着，对于任何可以简单地被视为分为一个子系统和一个与之接触的无限温度和压力储存器的过程,<br />
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: <math> dE + \delta w_u \le 0 \, </math><br />
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<math> dE + \delta w_u \le 0 \, </math><br />
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数学德 delta u le 0，/ math<br />
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i.e. the change in the subsystem's exergy plus the useful work done ''by'' the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done ''on'' the system) must be less than or equal to zero.<br />
<br />
i.e. the change in the subsystem's exergy plus the useful work done by the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done on the system) must be less than or equal to zero.<br />
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也就是。子系统(火用)的变化加上子系统所做的有用功(或者，子系统(火用)的变化不包括任何功，除了压力贮存器所做的功之外，在系统中所做的功)必须小于或等于零。<br />
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In sum, if a proper ''infinite-reservoir-like'' reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in ''E'' for an irreversible process and no change for a reversible process.<br />
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In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in E for an irreversible process and no change for a reversible process.<br />
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总之，如果选择一个合适的类似于无限库的参考状态作为现实世界中的系统环境，那么第二定律预测一个不可逆性的 e 值会减少，而一个可逆过程的 e 值不会变化。<br />
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: <math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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<math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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数学 d = 0 = 0 = 0 = 0 = 0<br />
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This expression together with the associated reference state permits a [[design engineer]] working at the macroscopic scale (above the [[thermodynamic limit]]) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (''Also, see [[process engineer]]''). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (''See [[Exergy efficiency|second law efficiency]]''.)<br />
<br />
This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (Also, see process engineer). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (See second law efficiency.)<br />
<br />
这个表达式连同相关的参考状态允许在宏观的设计工程师利用第二定律，而不需要直接测量或考虑整个孤立系统的熵变。(另外，请参阅流程工程师)。这些变化已经被考虑，假设被考虑的系统可以达到平衡的参考状态而不改变参考状态。还可以找到一个过程或过程集合的效率，将其与可逆的理想状态相比较<br />
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This approach to the Second Law is widely utilized in [[engineering]] practice, [[environmental accounting]], [[systems ecology]], and other disciplines.<br />
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This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines.<br />
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第二定律的这种方法被广泛应用于工程实践、环境会计、系统生态学和其他学科。<br />
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===The second law in chemical thermodynamics===<br />
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===The second law in chemical thermodynamics===<br />
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化学热力学的第二定律<br />
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For a [[spontaneous process|spontaneous chemical process]] in a closed system at constant temperature and pressure without non-''PV'' work, the Clausius inequality Δ''S > Q/T<sub>surr</sub>'' transforms into a condition for the change in [[Gibbs free energy]] <br />
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For a spontaneous chemical process in a closed system at constant temperature and pressure without non-PV work, the Clausius inequality ΔS > Q/T<sub>surr</sub> transforms into a condition for the change in Gibbs free energy <br />
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对于一个封闭系统中的自发化学过程，在没有非 pv 功的情况下，在恒定温度和压力下，将克劳修斯不等式 s q / t sub surr / sub 转化为吉布斯自由能变化的条件<br />
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: <math>\Delta G < 0 </math><br />
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<math>\Delta G < 0 </math><br />
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数学 Delta g 0 / 数学<br />
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or d''G'' < 0. For a similar process at constant temperature and volume, the change in [[Helmholtz free energy]] must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products.<ref name="Oxtoby8th"/> <ref name="MortimerBook"></ref> The chemical equilibrium condition at constant ''T'' and ''p'' without electrical work is d''G'' = 0.<br />
<br />
or dG < 0. For a similar process at constant temperature and volume, the change in Helmholtz free energy must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products. The chemical equilibrium condition at constant T and p without electrical work is dG = 0.<br />
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或者 dG 0。对于一个相似的过程，在恒定的温度和体积下，亥姆霍兹自由能的变化一定是负的，这是 Delta a 0 / math。因此，自由能(g 或 a)变化的负值是过程自发的必要条件。这是热力学第二定律在化学中最有用的形式，其中自由能的变化可以通过列表的生成焓和反应物和产物的标准摩尔熵来计算。在 t 和 p 不变的情况下，化学平衡条件是 dG 0。<br />
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==History==<br />
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==History==<br />
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历史<br />
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{{See also|History of entropy}}<br />
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[[File:Sadi Carnot.jpeg|thumb|upright|Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique]].]]<br />
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Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique.]]<br />
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尼古拉·莱昂纳尔·萨迪·卡诺穿着传统的巴黎综合理工学院制服<br />
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The first theory of the conversion of heat into mechanical work is due to [[Nicolas Léonard Sadi Carnot]] in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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The first theory of the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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第一个把热转化为机械功的理论是在1824年由尼古拉·莱昂纳尔·萨迪·卡诺提出的。他是第一个正确认识到这种转换的效率取决于发动机和环境之间的温差。<br />
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Recognizing the significance of [[James Prescott Joule]]'s work on the conservation of energy, [[Rudolf Clausius]] was the first to formulate the second law during 1850, in this form: heat does not flow ''spontaneously'' from cold to hot bodies. While common knowledge now, this was contrary to the [[caloric theory]] of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law during 1850, in this form: heat does not flow spontaneously from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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认识到詹姆斯·普雷斯科特·焦耳在能量守恒方面工作的重要性，Rudolf Clausius 在1850年第一个提出了第二定律: 热不会自发地从冷物体流向热物体。虽然现在的常识是，这与当时流行的热量理论相反，热量理论认为热量是一种流体。从那里他能够推断出萨迪卡诺定律和熵的定义(1865年)。<br />
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Established during the 19th century, the [[Kelvin-Planck statement|Kelvin-Planck statement of the Second Law]] says, "It is impossible for any device that operates on a [[cyclic process|cycle]] to receive heat from a single [[heat reservoir|reservoir]] and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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Established during the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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建立于19世纪的开尔文-普朗克第二定律(Kelvin-Planck)说，“任何循环运行的设备都不可能从单个蓄热体接收热量，并产生净功。”这被证明相当于克劳修斯的陈述。<br />
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The [[ergodic hypothesis]] is also important for the [[Boltzmann]] approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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遍历假设对玻尔兹曼方法也很重要。它说，在很长一段时间内，在具有相同能量的微观态的相空间的某些区域所花费的时间与这个区域的体积成正比，即。在很长一段时间内，所有可访问的微观状态的可能性都是一样的。同样的，它表明时间平均值和系综平均值是相同的。<br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a [[Macroscopic bodies|macroscopic system]]. This doctrine is obsolescent.<ref>Denbigh, K.G., Denbigh, J.S. (1985). ''Entropy in Relation to Incomplete Knowledge'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-25677-1}}, pp. 43–44.</ref><ref>Grandy, W.T., Jr (2008). ''Entropy and the Time Evolution of Macroscopic Systems'', Oxford University Press, Oxford, {{ISBN|978-0-19-954617-6}}, pp. 55–58.</ref><ref name=Lambert>[http://entropysite.oxy.edu Entropy Sites — A Guide] Content selected by [[Frank L. Lambert]]</ref><br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a macroscopic system. This doctrine is obsolescent.<br />
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有一种传统的学说，从克劳修斯开始，认为熵可以被理解为宏观系统中的分子“无序”。这种学说已经过时了。<br />
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===Account given by Clausius===<br />
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===Account given by Clausius===<br />
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作者: 克劳修斯<br />
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[[File:Clausius-1.jpg|thumb|upright|Rudolf Clausius]]<br />
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Rudolf Clausius<br />
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鲁道夫 · 克劳修斯<br />
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In 1856, the German physicist [[Rudolf Clausius]] stated what he called the "second fundamental theorem in the [[mechanical theory of heat]]" in the following form:{{sfnp|Clausius|1867}}<br />
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In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:<br />
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1856年，德国物理学家鲁道夫 · 克劳修斯阐述了他所谓的“热力学理论中的第二个基本定理” ，其形式如下:<br />
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: <math>\int \frac{\delta Q}{T} = -N</math><br />
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<math>\int \frac{\delta Q}{T} = -N</math><br />
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- n / math<br />
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where ''Q'' is heat, ''T'' is temperature and ''N'' is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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其中 q 是热，t 是温度，n 是一个循环过程中所有未补偿的相变的“等价值”。后来，在1865年，克劳修斯将“等价值”定义为熵。紧接着这个定义，也就是在同一年，第二定律最著名的版本在4月24日苏黎世哲学学会的一次演讲中被宣读，在演讲的最后，克劳修斯总结道:<br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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宇宙的熵趋于最大<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. [[universe]], as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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这句话是第二定律中最广为人知的措辞。由于其语言的松散性，例如:。这个问题的答案是肯定的，因为这个问题的答案是肯定的。开放，封闭，或孤立，许多人认为这个简单的陈述意味着热力学第二定律适用于几乎所有可以想象的主题。这是不正确的; 这个陈述只是一个更加扩展和精确描述的简化版本。<br />
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In terms of time variation, the mathematical statement of the second law for an [[isolated system]] undergoing an arbitrary transformation is:<br />
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In terms of time variation, the mathematical statement of the second law for an isolated system undergoing an arbitrary transformation is:<br />
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就时间变化而言，经历任意变换的孤立系统第二定律的数学表述是:<br />
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: <math>\frac{dS}{dt} \ge 0</math><br />
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<math>\frac{dS}{dt} \ge 0</math><br />
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0 / math<br />
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where<br />
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where<br />
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: ''S'' is the entropy of the system and<br />
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S is the entropy of the system and<br />
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S 是系统的熵<br />
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: ''t'' is [[time]].<br />
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t is time.<br />
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时间到了。<br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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!-- 可逆过程需要与周围环境保持平衡。这对于一个孤立的系统是不可能的。因此，关于可逆过程的讨论已经转移到对封闭系统的分析----<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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平衡后应用等式符号。建立孤立系统第二定律的另一种方法是:<br />
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: <math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math<br />
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with <math> \dot S_{i}</math> the sum of the rate of [[entropy production]] by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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with <math> \dot S_{i}</math> the sum of the rate of entropy production by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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用数学 s { i } / 数学表示系统内所有进程的产生熵之和。这种配方的优点是它显示了产生熵的效果。产生熵是一个非常重要的概念，因为它决定(限制)热机器的效率。与环境温度数学 t { a } / math 相乘，得到所谓的耗散能量数学 p { diss } t { a } dot s { i } / math。<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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封闭系统的第二定律(允许热量交换和移动边界，但不允许物质交换)的表达式是:<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math 来计算数学点 q } + 点 s { i } / math<br />
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Here<br />
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给你<br />
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: <math>\dot Q</math> is the heat flow into the system<br />
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<math>\dot Q</math> is the heat flow into the system<br />
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数学 q / math 是进入系统的热流<br />
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: <math>T</math> is the temperature at the point where the heat enters the system.<br />
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<math>T</math> is the temperature at the point where the heat enters the system.<br />
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Math 是热量进入系统时的温度。<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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等式符号在只有可逆过程在系统内发生的情况下成立。如果发生不可逆过程(在实际操作系统中就是这种情况) ,-符号保持不变。如果系统有多处供热，我们必须求相应项的代数和。<br />
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For open systems (also allowing exchange of matter):<br />
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For open systems (also allowing exchange of matter):<br />
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对于开放系统(也允许物质交换) :<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math 表示数学点 q } + 点 s + 点 s { i } / math<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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这里的数学 s / math 是熵流进入系统，与物质流进入系统有关。它不应该与熵的时间导数混淆。如果物质在几个地方提供，我们必须取这些贡献的代数和。<br />
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==Statistical mechanics==<br />
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==Statistical mechanics==<br />
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统计力学<br />
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[[Statistical mechanics]] gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a [[microstate (statistical mechanics)|microstate]] of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/{{radic|''N''}} where ''N'' is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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Statistical mechanics gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a microstate of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/ where N is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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通过假设物质是由不断运动的原子和分子组成的，统计力学对第二定律给出了解释。系统中每个粒子的一组特定的位置和速度称为系统的微观状态，由于系统的不断运动，系统不断地改变其微观状态。统计力学假设，在平衡状态下，系统可能处于的每个微观状态发生的可能性是相等的，当这个假设被提出时，它直接导致了第二定律必须在统计意义上成立的结论。也就是说，第二定律平均成立，其中 n 是系统中粒子数的1 / 数量级的统计变化。在日常(宏观)情况下，违反第二定律的概率几乎为零。然而，对于粒子数量很少的系统，热力学参数，包括熵，可能显示出与第二定律预测的显著的统计偏差。经典热力学理论没有处理这些统计变量。<br />
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（by Jie XIANG) 在通过假定一种物质由不断运动的原子和分子组成的前提下，统计力学对热力学第二定律进行了解释。系统中每个粒子其特定的位置和速度组成为系统的微观状态，由于（原子和分子的）不断运动，系统也持续改变其微观状态。统计力学假设，在系统力平衡的情况下，其微观状态其实是等可能发生的，进而能够得出结论：热力学第二定律必须在统计学意义上成立。也就是说，第二定律将保持均值不变， 其统计变化为1 /√N，N为系统中的粒子数。在日常(宏观)情况下，违反热力学第二定律的概率几乎为零，但是，对于粒子数量非常少的系统，其热力学参数（包括熵）可能会显示出与第二定律所预测的明显统计偏差。然而经典热力学理论并不处理这些统计变量。<br />
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==Derivation from statistical mechanics==<br />
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==Derivation from statistical mechanics==<br />
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从统计力学导出<br />
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{{Further information|H-theorem}}<br />
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The first mechanical argument of the [[Kinetic theory of gases]] that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to [[James Clerk Maxwell]] in 1860;<ref>{{Cite journal | last1 = Gyenis | first1 = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G }}</ref> [[Ludwig Boltzmann]] with his [[H-theorem]] of 1872 also argued that due to collisions gases should over time tend toward the [[Maxwell–Boltzmann distribution]].<br />
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The first mechanical argument of the Kinetic theory of gases that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to James Clerk Maxwell in 1860; Ludwig Boltzmann with his H-theorem of 1872 also argued that due to collisions gases should over time tend toward the Maxwell–Boltzmann distribution.<br />
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气体动力学理论的第一个力学论证由詹姆斯·克拉克·麦克斯韦在1860年给出，指出分子碰撞引起温度均衡，因此整体趋向于平衡； 路德维希·玻尔兹曼在1872年提出的 H 定理也认为，气体由于碰撞应该随着时间的推移趋向于麦克斯韦-波兹曼分布。<br />
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Due to [[Loschmidt's paradox]], derivations of the Second Law have to make an assumption regarding the past, namely that the system is [[Correlation and dependence|uncorrelated]] at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a [[boundary condition]], and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the [[Big Bang]]), though [[Boltzmann brain|other scenarios]] have also been suggested.<ref name="Hawking AOT">{{cite journal|last=Hawking|first=SW|title=Arrow of time in cosmology|journal=Phys. Rev. D|year=1985|volume=32|issue=10|pages=2489–2495|doi=10.1103/PhysRevD.32.2489|pmid=9956019|bibcode = 1985PhRvD..32.2489H }}</ref><ref>{{cite book | last = Greene | first = Brian | authorlink = Brian Greene | title = The Fabric of the Cosmos | url = https://archive.org/details/fabricofcosmossp00gree | url-access = registration | publisher = Alfred A. Knopf | year = 2004 | page = [https://archive.org/details/fabricofcosmossp00gree/page/171 171] | isbn = 978-0-375-41288-2}}</ref><ref name=Lebowitz>{{cite journal|last=Lebowitz|first=Joel L.|title= Boltzmann's Entropy and Time's Arrow|journal=Physics Today|date=September 1993|volume=46|issue=9|pages=32–38|url=http://users.df.uba.ar/ariel/materias/FT3_2008_1C/papers_pdf/lebowitz_370.pdf|accessdate=2013-02-22|doi=10.1063/1.881363|bibcode = 1993PhT....46i..32L }}</ref><br />
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Due to Loschmidt's paradox, derivations of the Second Law have to make an assumption regarding the past, namely that the system is uncorrelated at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a boundary condition, and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the Big Bang), though other scenarios have also been suggested.<br />
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由于洛施密特悖论，第二定律的导出必须对过去做出一个假设，即系统在过去的某个时刻是不相关的；这样的假设允许进行简单的概率处理。这个假设通常被认为是一个边界条件，因此热力学第二定律最终是过去某个地方的初始条件的结果，可能是在宇宙的开始(大爆炸) 。也有人提出了其他假设。<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the [[Statistical mechanics#Fundamental postulate|fundamental postulate]], also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy.{{citation needed|date=August 2012}} The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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考虑到这些假设，在统计力学中，第二定律不是一个假设，而是统计力学基本假设的一个结果，也被称为等先验概率假设。这个基本假设表明，只要一个人清楚地知道，简单的概率论证只适用于未来，而对于过去，有辅助的信息来源告诉我们，它是低熵的。热力学第二定律的第一部分指出，热孤立系统的熵只能增加。如果我们把熵的概念限制在热平衡系统中，那么热力学第二定律的第一部分是等先验概率假设的一个显然结果。处于热平衡状态的孤立系统包含能量<math>E</math>的熵表示为：<br />
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: <math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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<math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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其中<math>\Omega\left(E\right)</math> 是处于 <math>E</math>和<math>E +\delta E</math这个小区间内的量子态数目。这里的 <math>\delta E</math> 是一个宏观上很小的固定能量区间。严格地说，这意味着熵取决于对<math>\delta E</math>的选择。然而在热力学极限下（例如无穷大系统的极限），狭义的熵(单位体积或单位质量的熵)不依赖于 <math>\delta E</math>。<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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假设我们有一个孤立系统，其宏观状态由许多变量描述。这些宏观变量可以是总体积、活塞在系统中的位置等。从而<math>\Omega</math>将取决于这些变量的值。如果某个变量不是固定的（我们不会在某个位置夹住活塞) ，那么因为在平衡状态下所有可到达状态的等几率到达的，平衡状态下的自由变量会使 <math>\Omega</math> 最大，因为这是平衡状态下最可能的情况。<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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如果该变量最初固定到某个值然后释放，当达到新的平衡时，变量将自我调整使得 <math>\Omega</math> 最大，这意味着熵将增加或保持不变(如果变量初始固定的值恰好是平衡值)。<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's [[H-theorem]], however, proves that the quantity {{math|''H''}} increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the quantity increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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假设我们初始位于一个平衡状态，突然移除了对一个变量的约束。我们做完这件事的时候，可达到的微观状态的数为<math>\Omega</math>，但是系统还没有达到平衡，所以系统处于某些可达到的状态的实际概率还不等于先验概率 <math>1/\Omega</math>。我们已经知道，最终的平衡状态相对于之前的平衡状态，熵会增加或者保持不变。然而，玻耳兹曼的H定理证明系统在不处于平衡态的期间，那个量作为时间的函数单调增加。<br />
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===Derivation of the entropy change for reversible processes===<br />
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===Derivation of the entropy change for reversible processes===<br />
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可逆过程熵变的推导<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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第二定律的第二部分指出，经历可逆过程的系统的熵变是由以下因素给出的:<br />
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: <math>dS =\frac{\delta Q}{T}</math><br />
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<math>dS =\frac{\delta Q}{T}</math><br />
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数学 dS frac delta q }{ t } / math<br />
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where the temperature is defined as:<br />
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where the temperature is defined as:<br />
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其中温度定义为:<br />
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:<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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[数学][[欧米茄][左(右)][右][数学][数学]<br />
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See [[Microcanonical ensemble|here]] for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the [[adiabatic theorem]] of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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请参阅此处对此定义的正当理由。假设系统有一些可以改变的外部参数 x。一般来说，系统的能量本征态将依赖于 x。根据量子力学的绝热定理，在系统哈密顿量无限缓慢变化的极限下，系统将保持在相同的定态，因此系统的能量会随着其所在定态的能量变化而变化。<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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广义力，x，对应于外部变量 x 的定义是，如果 x 增加一个量 dx，那么 math x dx / math 就是系统所做的功。例如，如果 x 是体积，那么 x 是压强。一个已知的系统的广义力在定态数学 e { r } / math 中给出:<br />
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: <math>X = -\frac{dE_{r}}{dx}</math><br />
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<math>X = -\frac{dE_{r}}{dx}</math><br />
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数学 x- frac { dE { r }{ dx } / math<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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由于系统可以在一个数学△ e / math 区间内的任意定态内，我们将系统的广义力定义为上述表达式的期望值:<br />
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: <math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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<math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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数学 x- 左 langle frac { dE { dx }右 rangle，/ math<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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为了计算平均值，我们通过计算在数学 y / math 和数学 y + delta y / math 之间的范围内，其中有多少数学 y / math 能量本征态有一个值来划分 ω 左(e 右) / math 能量本征态。把这个数字命名为 math Omega { y } left (e right) / math，我们有:<br />
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: <math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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<math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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Math Omega left (e right) sum { y } Omega { y } left (e right) ，/ math<br />
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The average defining the generalized force can now be written:<br />
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The average defining the generalized force can now be written:<br />
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定义广义力的平均值现在可以写成:<br />
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: <math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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<math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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数学 x frac {1} Omega 左(e 右) sum { y Omega { y }左(e 右) ，/ math<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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我们可以把它和熵对 x 在恒定能量 e 下的导数联系起来，如下所示。假设我们把 x 改成 x + dx。然后 math Omega left (e right) / math 将会改变，因为能量本征态依赖于 x，导致能量本征态进入或超出数学 e / math 和数学 e + delta e / math 之间的范围。让我们再次关注数学 y / math 和数学 y + delta y / math 之间的能量本征态。由于这些能量本征态的能量增加了 y dx，所有这些能量本征态在 e-y dx 到 e 之间的区间内从 e 以下移动到 e 以上。 有<br />
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: <math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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<math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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数学 n { y }左(e 右) frac { y }左(e 右)}{ delta e } y dx ，/ math<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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这样的能量本征态。如果数学 y dx leq delta e / math，所有这些能量本征态将移动到数学 e / math 和数学 e + delta e / math 之间的范围内，有助于数学 ω / math 的增加。从数学 e + △ e / math 下移到数学 e + △ e / math 上的能量本征态的个数是由数学 n { y }左(e + △ e 右) / math 给出的。区别在于<br />
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: <math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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<math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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数学 n { y }左(e 右)-n { y }左(e + delta e 右) ，/ math<br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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因此是数学增长的净贡献欧米茄 / 数学。请注意，如果 y dx 大于 math delta e / math，那么将会有能量本征态从 e 下面移动到 e + delta e / math 上面。它们在数学 n { y } left (e right) / math 和数学 n { y } left (e + delta e right) / math 中都有计算，因此上述表达式在这种情况下也是有效的。<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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将上面的表达式表示为对 e 的导数，对 y 的求和得到表达式:<br />
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: <math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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数学左(部分欧米茄左(部分欧米茄右){ e }-和 y 左(部分欧米茄右){ x 左(部分欧米茄左(部分欧米茄右)}{ x 右) ，/ 数学<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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关于 x 的欧米茄 / 数学对数导数由以下方式给出:<br />
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: <math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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Math 左( frac { partial ln 左( Omega 右)}{部分 x 右){ e } beta x + 左( frac { partial x 右){ x } ，/ math<br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
<br />
The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
<br />
第一个术语是密集型的，即。它不能根据系统大小进行缩放。相比之下，最后一项的规模与逆系统的规模一样，因此将在热力学极限中消失。因此，我们发现:<br />
<br />
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: <math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
<br />
<math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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数学左(部分 s }右){ e } frac { x }{ t } ，/ math<br />
<br />
<br />
<br />
<br />
<br />
Combining this with<br />
<br />
Combining this with<br />
<br />
结合这一点<br />
<br />
<br />
<br />
<br />
<br />
: <math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
<br />
<math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
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数学左(部分 s }右){ x } frac {1}{ t } ，/ math<br />
<br />
<br />
<br />
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Gives:<br />
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Gives:<br />
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给予:<br />
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: <math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
<br />
<math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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数学 dS 左(部分 s }部分 e }右){ x } dE + 左(部分 s }右){ e } dx frac { t } + frac { t }{ delta q }{ t } ，/ math<br />
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===Derivation for systems described by the canonical ensemble===<br />
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===Derivation for systems described by the canonical ensemble===<br />
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正则系系统的推导<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the [[canonical ensemble]]:<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:<br />
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如果一个系统与某个温度为T的热浴热接触，那么在平衡状态下，关于能量本征值的概率分布值由正则系综给出:<br />
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: <math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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<math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the [[Partition function (statistical mechanics)|partition function]]. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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这里Z是一个使所有概率之和归一化的因子，这个函数被称为配分函数。现在我们考虑对温度和能级所依赖的外部参数的无限小的可逆改变。它遵循熵的一般公式:<br />
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: <math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
<br />
<math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
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<br />
<br />
<br />
<br />
<br />
that<br />
<br />
that<br />
<br />
以及<br />
<br />
<br />
<br />
<br />
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: <math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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<math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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<br />
<br />
<br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
<br />
Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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把正则系综的数学形式 <math>P_{j}</math> 插入，可以得到:<br />
<br />
<br />
<br />
<br />
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: <math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
<br />
<math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
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==Living organisms==<br />
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==Living organisms==<br />
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生命体<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as [[What is Life?|Erwin Schrödinger]], [[Léon Brillouin]]<ref>Léon Brillouin [https://books.google.com/books?hl=en&lr=&id=tPXVbiw_1P0C ''Science and Information Theory''] (Academic Press, 1962) (Dover, 2004)</ref> and [[Life and Energy|Isaac Asimov]]. It is also the topic of current{{When|date=July 2019}} research.{{Citation needed|date=July 2019}}<br />
<br />
There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as Erwin Schrödinger, Léon Brillouin and Isaac Asimov. It is also the topic of current research.<br />
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表述热力学主要有两种方式：(a)描述从一种热力学平衡状态到另一种状态的路径，(b)通过使系统保持不变，同时增加周围环境的总熵的循环过程。这两种方法有助于理解生命过程。这个话题大大超出了本文的范围，但是已经有一些研究者考虑过这个问题，比如埃尔温·薛定谔、莱昂·布里渊和艾萨克.阿西莫夫。这也是当前研究的主题。<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of [[metabolism]], give out breakdown products and heat. Plants [[Photosynthesis|take in radiative energy]] from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
<br />
To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of metabolism, give out breakdown products and heat. Plants take in radiative energy from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
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从近似的角度来看，生命体可以被认为是(b)的一个例子。近似地，一只动物的身体状态每天循环，使得它几乎没有什么变化。动物吸收食物、水和氧气，经过新陈代谢，输出分解的产物和热量。植物吸收来自太阳的辐射能量，这可以认为是热量，以及二氧化碳和水，然后它们释放氧气。它们就是这样生长的，最终会死亡，尸体腐烂，大部分重新变成二氧化碳和水。这可以看作是一个循环过程。总的来说，阳光来自一个高温的源——太阳，它的能量被传递到一个较低的温度汇，例如向太空辐射。这个过程使得植物周围环境的熵增加。因此从循环过程的角度来看，动物和植物服从热力学第二定律。简单的热机效率概念很难适用于这个问题，因为它们假定系统是封闭的。<br />
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From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to [[Non-equilibrium thermodynamics|unsettled current debate or research]]. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
<br />
From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to unsettled current debate or research. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
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从(a)的热力学观点考虑 ，即考虑从一个平衡状态到另一个平衡状态的路径，只会产生一个粗略近似的图像，因为生命体从来不会处于热力学平衡状态。生物体经常必须被认为是开放的系统，因为它们吸收营养物质并排出废弃物。开放系统热力学目前通常从一个热力学平衡状态到另一个状态的路径的角度来考虑，或者是考虑局部热力学平衡近似下的流。生命体的问题可以通过假定一个流不变的稳态来进一步简化。对于这样的近似，熵产生的一般原理在目前有争论和研究的不确定性。尽管如此，从热力学第二定律的这个角度出发衍生出来的想法对生物是有启发意义的。<br />
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==Gravitational systems==<br />
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==Gravitational systems==<br />
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重力系统<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive [[heat capacity]], meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it.<ref name="Grandy 151">Grandy, W.T. (Jr) (2008), p. 151.</ref> Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
<br />
In systems that do not require for their descriptions the general theory of relativity, bodies always have positive heat capacity, meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it. Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
<br />
在不需要描述广义相对论的系统中，物体总是具有正的热容，这意味着温度随着能量的增加而升高。因此，当能量从高温物体流向低温物体时，源温度降低，而汇温度升高; 因此温差随时间而减小。这种情况并不总是发生在那些需要引力和广义相对论的系统中。这样的系统可以自发地朝着质量和能量不均匀分布的方向变化。这在大尺度上适用于宇宙，因此可能很难或不可能将第二定律应用于宇宙。除此之外，广义相对论所描述的系统热力学超出了本文的范围。<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
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{{main article|Non-equilibrium thermodynamics}}<br />
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The theory of classical or [[thermodynamic equilibrium|equilibrium thermodynamics]] is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<ref name="Grandy 151"/><ref>Callen, H.B. (1960/1985), p. 15.</ref><br />
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The theory of classical or equilibrium thermodynamics is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<br />
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经典热力学或平衡热力学的理想化。一个主要的假设或假设，往往甚至没有明确说明，是系统的存在，在他们自己的内部状态的热力学平衡。一般来说，在给定时间内包含一个物理系统的空间区域---- 可能在自然界中找到---- 不是在热力学平衡中，用最严格的术语来解释。从更宽松的角度来说，整个宇宙中没有任何东西是或者曾经是完全精确的热力学平衡。<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of [[thermodynamic equilibrium]]. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable [[thermal fluctuations|fluctuations]]. There is an exception, the case of [[critical point (thermodynamics)|critical states]], which exhibit to the naked eye the phenomenon of [[critical opalescence]]. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of thermodynamic equilibrium. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable fluctuations. There is an exception, the case of critical states, which exhibit to the naked eye the phenomenon of critical opalescence. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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为了物理分析的目的，假设热力学平衡是非常方便的。这种假设可能依赖于试错来证明其正当性。如果这个假设是正确的，它通常是非常有价值和有用的，因为它提供了热力学理论。平衡假设的要素是，一个系统被观察到在无限长的时间内是不变的，而且一个系统中有如此多的粒子，以至于它的微粒性质可以被完全忽略。在这样的平衡假设下，一般来说，没有宏观上可以探测到的波动。有一个例外，即临界状态的例子，它们用肉眼就可以看到临界乳光现象。对于临界状态的实验室研究，需要非常长的观察时间。<br />
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In all cases, the assumption of [[thermodynamic equilibrium]], once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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In all cases, the assumption of thermodynamic equilibrium, once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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在所有情况下，热力学平衡的假设，一旦做出，就意味着没有假定的候选者“涨落”改变系统的熵。<br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<ref>Lieb, E.H., Yngvason, J. (2003), p. 190.</ref><br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<br />
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一个物理系统很容易发生内部的宏观变化，其速度之快足以使熵的恒定性假设失效。或者一个物理系统的粒子太少，以至于微粒的本质在可观测的波动中是显而易见的。那么热力学平衡的假设就会被放弃。对于非平衡态，熵没有绝对的一般定义。<br />
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There are intermediate cases, in which the assumption of local [[thermodynamic equilibrium]] is a very good approximation,<ref>Gyarmati, I. (1967/1970), pp. 4-14.</ref><ref>Glansdorff, P., Prigogine, I. (1971).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003).</ref> but strictly speaking it is still an approximation, not theoretically ideal.<br />
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There are intermediate cases, in which the assumption of local thermodynamic equilibrium is a very good approximation, but strictly speaking it is still an approximation, not theoretically ideal.<br />
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还有一些中间情况，其中局部热力学平衡的假设是一个非常好的近似值，但严格地说它仍然是一个近似值，不是理论上的理想值。<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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对于一般的非平衡状态，考虑其他量的统计力学定义可能是有用的，这些量可以方便地称为‘熵’ ，但是它们不应该与第二定律正确定义的熵混淆或混淆。这些其他的量确实属于统计力学，而不是热力学---- 第二定律的主要领域。<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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宏观观测波动的物理学超出了本文的范围。<br />
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==Arrow of time==<br />
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==Arrow of time==<br />
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时间之箭<br />
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{{Further information|Entropy (arrow of time)}}Thermodynamic Asymmetry in Time<br />
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Thermodynamic Asymmetry in Time<br />
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时间中的热力学不对称性<br />
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{{See also|Arrow of time}}<br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely [[CPT symmetry]], since the second law applies statistically, it is hypothesized, on time-asymmetric ''boundary conditions''.<ref>{{cite web|first=Craig|last=Callender|url=https://plato.stanford.edu/archives/fall2011/entries/time-thermo/|title=Thermodynamic Asymmetry in Time|encyclopedia=Stanford Encyclopedia of Philosophy|date=29 July 2011}}</ref><br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely CPT symmetry, since the second law applies statistically, it is hypothesized, on time-asymmetric boundary conditions.<br />
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热力学第二定律是一个物理定律，它不对称于时间方向的反转。这与已经观察到的物理学基本定律，即 CPT 对称性的概念并不冲突，因为第二定律在统计上适用，它是假设的，是关于时间非对称的边界条件。<br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect ([[Arrow of time#The causal arrow of time|the causal arrow of time]]).<ref>{{cite book | first = J.J.| last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1|display-authors=etal}} chapter 6</ref><br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect (the causal arrow of time).<br />
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第二定律被提出来部分解释时间向前和向后移动的区别，例如为什么原因先于结果(因果关系的时间箭头)。<br />
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==Irreversibility==<br />
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==Irreversibility==<br />
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不可逆性<br />
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Irreversibility in [[thermodynamic process]]es is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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Irreversibility in thermodynamic processes is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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热力学过程的不可逆性是热力学作用不对称性的结果，而不是物体内部不可逆的微观性质的结果。热力学操作是施加在参与机构上的宏观外部干预，而不是从它们的内部性质中得到的。由于没有认识到这一点，产生了被称为“悖论”的说法。<br />
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===Loschmidt's paradox===<br />
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===Loschmidt's paradox===<br />
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洛斯密特悖论<br />
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{{main article|Loschmidt's paradox}}<br />
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[[Loschmidt's paradox]], also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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洛斯密特悖论，也被称为可逆性悖论，其目的在于不可能从描述宏观系统微观演化的时间对称动力学推导出不可逆性。<br />
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In the opinion of [[Erwin Schrödinger|Schrödinger]], "It is now quite obvious in what manner you have to reformulate the law of entropy{{snd}}or for that matter, all other irreversible statements{{snd}}so that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other."<ref>[[Erwin Schrödinger|Schrödinger, E.]] (1950), p.&nbsp;192.</ref> The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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In the opinion of Schrödinger, "It is now quite obvious in what manner you have to reformulate the law of entropyor for that matter, all other irreversible statementsso that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other." The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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根据薛定谔的观点，“现在很明显，你必须用什么方式来重新定义该物质的熵定律，所有其他不可逆的状态，这样它们才能从可逆模型中导出。你不能只谈论一个孤立的系统，而应该至少谈论两个，你可以暂时认为它们与世界其它地方是孤立的，但并不总是相互孤立的。”这两个系统被隔离墙隔离开来，直到按照法律的设想被热力学操作拆除。热力学操作是外部强加的，不受支配系统组成部分的可逆微观动力学定律的制约。它是不可逆转性的原因。本文中的法律陈述符合薛定谔的建议。因果关系在逻辑上优先于第二定律，而不是从第二定律派生出来的。<br />
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===Poincaré recurrence theorem===<br />
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===Poincaré recurrence theorem===<br />
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庞加莱始态复现定理<br />
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{{Main article|Poincaré recurrence theorem}}<br />
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The [[Poincaré recurrence theorem]] considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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The Poincaré recurrence theorem considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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庞加莱始态复现定理认为这是一个孤立的物理系统的理论微观描述。这可以被认为是一个热力学系统的模型后，一个热力学操作已经移除了一个内墙。在足够长的时间之后，系统将返回到一个非常接近初始状态的显微定义状态。庞加莱回归时间是指回归前经过的时间长度。它极其漫长，可能比宇宙的寿命还要长，并且敏感地依赖于被热力学操作拆除的墙体的几何形状。递归定理可能被认为是明显与热力学第二定律相矛盾的。然而，更明显的是，它只不过是一个微观模型，热力学平衡在一个孤立的系统中，通过移除两个系统之间的一道墙而形成。对于一个典型的热力学系统，重现时间是如此之大(比宇宙的生命周期长许多倍) ，以至于在所有的实际目的中，人们都不能观察到重现。然而，人们可能希望想象一下，可以等待庞加莱复发，然后重新插入被热力学操作移除的壁。然后很明显，不可逆性的出现是由于庞加莱循环的完全不可预测性，只要初始状态是热力学平衡，就像宏观热力学的情况一样。即使可以等待，也没有实际可能性选择合适的时间重新插入墙壁。庞加莱始态复现定理为 Loschmidt 悖论提供了一个解决方案。如果一个孤立的热力学系统可以被监测到在越来越多的平均 poincar 递归时间的倍数上，系统的热力学行为将在时间反转下变得不变。<br />
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[[File:James-clerk-maxwell3.jpg|thumb|upright|James Clerk Maxwell]]<br />
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James Clerk Maxwell<br />
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詹姆斯·克拉克·麦克斯韦<br />
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===Maxwell's demon===<br />
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===Maxwell's demon===<br />
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麦克斯韦的魔鬼<br />
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{{unreferenced|section|date=August 2018}}<br />
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{{main article|Maxwell's demon}}<br />
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[[James Clerk Maxwell]] imagined one container divided into two parts, ''A'' and ''B''. Both parts are filled with the same [[gas]] at equal temperatures and placed next to each other, separated by a wall. Observing the [[molecule]]s on both sides, an imaginary [[demon]] guards a microscopic trapdoor in the wall. When a faster-than-average molecule from ''A'' flies towards the trapdoor, the demon opens it, and the molecule will fly from ''A'' to ''B''. The average [[speed]] of the molecules in ''B'' will have increased while in ''A'' they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in ''A'' and increases in ''B'', contrary to the second law of thermodynamics.<br />
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James Clerk Maxwell imagined one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other, separated by a wall. Observing the molecules on both sides, an imaginary demon guards a microscopic trapdoor in the wall. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics.<br />
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詹姆斯·克拉克·麦克斯韦想象一个容器分成两部分，a 和 b。两个部分在相同的温度下充入相同的气体，并被一堵墙隔开，挨着放置。通过观察两边的分子，一个想象中的恶魔守卫着墙上的一个微型暗门。当一个来自 a 的比平均速度快的分子飞向活板门时，魔鬼打开了它，分子就会从 a 飞到 b。B 中分子的平均速度会增加，而 a 中分子的平均速度会减慢。由于平均分子速度与温度相对应，因此 a 中的温度降低，b 中的温度升高，这与热力学第二定律相反。<br />
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One response to this question was suggested in 1929 by [[Leó Szilárd]] and later by [[Léon Brillouin]]. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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One response to this question was suggested in 1929 by Leó Szilárd and later by Léon Brillouin. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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1929年，Leó Szilárd 和后来的 l on brilouin 对这个问题提出了一个答案。Szil rd 指出，现实生活中的麦克斯韦恶魔需要一些测量分子速度的方法，而获取信息需要消耗大量能量。<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between ''A'' and ''B''. It is therefore performing [[thermodynamic operation]]s on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between A and B. It is therefore performing thermodynamic operations on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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麦克斯韦的“魔鬼”不断地改变 a 和 b 之间墙体的渗透性。因此，它是在微观尺度上执行热力学操作，而不仅仅是观察普通的自发或自然的宏观热力学过程。<br />
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==Quotations==<br />
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==Quotations==<br />
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语录<br />
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{{Wikiquote}}<br />
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{{quote|The law that entropy always increases holds, I think, the supreme position among the [[Laws of science|laws of Nature]]. If someone points out to you that your pet theory of the [[universe]] is in disagreement with [[Maxwell's equations]] – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.|Sir [[Arthur Stanley Eddington]], ''The Nature of the Physical World'' (1927)}}<br />
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{{quote|There have been nearly as many formulations of the second law as there have been discussions of it.|Philosopher / Physicist [[Percy Williams Bridgman|P.W. Bridgman]], (1941)}}<br />
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{{quote|Clausius is the author of the sibyllic utterance, "The energy of the universe is constant; the entropy of the universe tends to a maximum." The objectives of continuum thermomechanics stop far short of explaining the "universe", but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size.|[[Clifford Truesdell|Truesdell, C.]], Muncaster, R. G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 17.}}<br />
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==See also==<br />
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参见<br />
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*[[Clausius–Duhem inequality]]<br />
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*[[Fluctuation theorem]]<br />
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*[[Heat death of the universe]]<br />
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*[[History of thermodynamics]]<br />
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*[[Jarzynski equality]]<br />
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*[[Laws of thermodynamics]]<br />
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*[[Maximum entropy thermodynamics]]<br />
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*[[Quantum thermodynamics]]<br />
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*[[Reflections on the Motive Power of Fire]]<br />
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*[[Relativistic heat conduction]]<br />
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*[[Thermal diode]]<br />
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*[[Thermodynamic equilibrium]]<br />
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{{colend}}<br />
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==References==<br />
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==References==<br />
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参考资料<br />
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===Sources===<br />
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资料来源<br />
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{{refbegin|50em}}<br />
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*Adkins, C.J. (1968/1983). ''Equilibrium Thermodynamics'', (1st edition 1968), third edition 1983, Cambridge University Press, Cambridge UK, {{ISBN|0-521-25445-0}}.<br />
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*[[Peter Atkins|Atkins, P.W.]], de Paula, J. (2006). ''Atkins' Physical Chemistry'', eighth edition, W.H. Freeman, New York, {{ISBN|978-0-7167-8759-4}}.<br />
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*Attard, P. (2012). ''Non-equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications'', Oxford University Press, Oxford UK, {{ISBN|978-0-19-966276-0}}.<br />
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*Baierlein, R. (1999). ''Thermal Physics'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-59082-5}}.<br />
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*Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics, New York, {{ISBN|0-88318-797-3}}.<br />
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*{{Cite book|title=Concepts in thermal physics|last=Blundell|first=Stephen J.|last2=Blundell|authorlink1=Stephen Blundell|first2=Katherine M.|authorlink2=Katherine Blundell|publisher=[[Oxford University Press]]|year=2010|isbn=9780199562107|edition=2nd|location=Oxford|pages=|oclc=607907330|doi=10.1093/acprof:oso/9780199562091.001.0001 |url=http://cds.cern.ch/record/1235139}}<br />
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*[[Ludwig Boltzmann|Boltzmann, L.]] (1896/1964). ''Lectures on Gas Theory'', translated by S.G. Brush, University of California Press, Berkeley.<br />
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*Borgnakke, C., Sonntag., R.E. (2009). ''Fundamentals of Thermodynamics'', seventh edition, Wiley, {{ISBN|978-0-470-04192-5}}.<br />
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*Buchdahl, H.A. (1966). ''The Concepts of Classical Thermodynamics'', Cambridge University Press, Cambridge UK.<br />
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*[[Percy Williams Bridgman|Bridgman, P.W.]] (1943). ''The Nature of Thermodynamics'', Harvard University Press, Cambridge MA.<br />
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*[[Herbert Callen|Callen, H.B.]] (1960/1985). ''Thermodynamics and an Introduction to Thermostatistics'', (1st edition 1960) 2nd edition 1985, Wiley, New York, {{ISBN|0-471-86256-8}}.<br />
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*{{cite journal|author=C. Carathéodory|author1-link=Constantin Carathéodory<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
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2010年3月15日 | 第355-386页 | http://gdz.sub.uni-goettingen.de/index.php?id=11&ppn=ppn235181684_0067&dmdid=dmdlog_0033&l=1<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). ''The Second Law of Thermodynamics'', Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können.(p. 363) | doi 10.1007 / bf01450409}.译文可以在这里找到[ http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf ]。同时，在 Kestin，j。(1976).热力学第二定律，Dowden，Hutchinson & Ross，Stroudsburg PA. 。<br />
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*[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986). [http://www.worldcat.org/title/reflections-on-the-motive-power-of-fire-a-critical-edition-with-the-surviving-scientific-manuscripts-translated-and-edited-by-fox-robert/oclc/812944517&referer=brief_results ''Reflections on the motive power of fire''], Manchester University Press, Manchester UK, {{ISBN|0-7190-1741-6}}. [https://archive.org/stream/reflectionsonmot00carnrich#page/n7/mode/2up Also here.]<br />
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*[[Sydney Chapman (mathematician)|Chapman, S.]], [[Thomas George Cowling|Cowling, T.G.]] (1939/1970). ''The Mathematical Theory of Non-uniform gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases'', third edition 1970, Cambridge University Press, London.<br />
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*{{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Ueber Die Bewegende Kraft Der Wärme Und Die Gesetze, Welche Sich Daraus Für Die Wärmelehre Selbst Ableiten Lassen|journal=Annalen der Physik|year=1850|volume=79|issue=4|pages=368–397, 500–524|url=http://gallica.bnf.fr/ark:/12148/bpt6k15164w/f518.image|accessdate=26 June 2012|doi=10.1002/andp.18501550403|bibcode = 1850AnP...155..500C |hdl=2027/uc1.$b242250|hdl-access=free}} Translated into English: {{cite journal|last=Clausius|first=R.|title=On the Moving Force of Heat, and the Laws regarding the Nature of Heat itself which are deducible therefrom|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1851| volume=2|series=4th|issue=VIII|pages=1–21; 102–119 |url=https://archive.org/stream/londonedinburghd02lond#page/1/mode/1up|accessdate=26 June 2012<br />
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|doi=10.1080/14786445108646819}}<br />
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10.1080 / 14786445108646819}<br />
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* {{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie|year=1854|journal=Annalen der Physik|volume=xciii|issue=12|pages=481–506|url=http://zfbb.thulb.uni-jena.de/servlets/MCRFileNodeServlet/jportal_derivate_00140956/18541691202_ftp.pdf|accessdate=24 March 2014|doi=10.1002/andp.18541691202<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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| bibcode 1854AnP... 169. 1856年7月 | 第2卷 | 第4页 | 第86页 | https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=242014年3月<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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热力学理论及其在蒸汽机和物体物理性质中的应用 | 1867年 | 出版商 John van Voorst | 位置: 伦敦 | https://archive.org/details/mechanicaltheor04claugoog|quote=editions:pwr_sbkwa8ic。 2012年6月19日<br />
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*Denbigh, K. (1954/1981). ''The Principles of Chemical Equilibrium. With Applications in Chemistry and Chemical Engineering'', fourth edition, Cambridge University Press, Cambridge UK, {{ISBN|0-521-23682-7}}.<br />
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*Eu, B.C. (2002). ''Generalized Thermodynamics. The Thermodynamics of Irreversible Processes and Generalized Hydrodynamics'', Kluwer Academic Publishers, Dordrecht, {{ISBN|1-4020-0788-4}}.<br />
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*[[Josiah Willard Gibbs|Gibbs, J.W.]] (1876/1878). On the equilibrium of heterogeneous substances, ''Trans. Conn. Acad.'', '''3''': 108-248, 343-524, reprinted in ''The Collected Works of J. Willard Gibbs, Ph.D, LL. D.'', edited by W.R. Longley, R.G. Van Name, Longmans, Green & Co., New York, 1928, volume 1, pp.&nbsp;55–353.<br />
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*Griem, H.R. (2005). ''Principles of Plasma Spectroscopy (Cambridge Monographs on Plasma Physics)'', Cambridge University Press, New York {{ISBN|0-521-61941-6}}.<br />
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*Glansdorff, P., Prigogine, I. (1971). ''Thermodynamic Theory of Structure, Stability, and Fluctuations'', Wiley-Interscience, London, 1971, {{ISBN|0-471-30280-5}}.<br />
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*Grandy, W.T., Jr (2008). [http://global.oup.com/academic/product/entropy-and-the-time-evolution-of-macroscopic-systems-9780199546176?cc=au&lang=en& ''Entropy and the Time Evolution of Macroscopic Systems'']. Oxford University Press. {{ISBN|978-0-19-954617-6}}.<br />
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*Greven, A., Keller, G., Warnecke (editors) (2003). ''Entropy'', Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[Edward A. Guggenheim|Guggenheim, E.A.]] (1949). 'Statistical basis of thermodynamics', ''Research'', '''2''': 450–454.<br />
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*[[Edward A. Guggenheim|Guggenheim, E.A.]] (1967). ''Thermodynamics. An Advanced Treatment for Chemists and Physicists'', fifth revised edition, North Holland, Amsterdam.<br />
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*Gyarmati, I. (1967/1970) ''Non-equilibrium Thermodynamics. Field Theory and Variational Principles'', translated by E. Gyarmati and W.F. Heinz, Springer, New York.<br />
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*[[Charles Kittel|Kittel, C.]], [[Herbert Kroemer|Kroemer, H.]] (1969/1980). ''Thermal Physics'', second edition, Freeman, San Francisco CA, {{ISBN|0-7167-1088-9}}.<br />
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*Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998). ''Modern Thermodynamics: From Heat Engines to Dissipative Structures'', John Wiley & Sons, Chichester, {{ISBN|0-471-97393-9}}.<br />
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*Lebon, G., Jou, D., Casas-Vázquez, J. (2008). ''Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers'', Springer-Verlag, Berlin, {{ISBN|978-3-540-74252-4}}.<br />
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*{{cite journal|ref=harv|title=The Physics and Mathematics of the Second Law of Thermodynamics|last1=Lieb |first1=E. H. |last2=Yngvason |first2=J.|journal=Physics Reports|volume=310|issue=1|pages=1–96 |year=1999|doi=10.1016/S0370-1573(98)00082-9|arxiv = cond-mat/9708200 |bibcode = 1999PhR...310....1L}}<br />
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*Lieb, E.H., Yngvason, J. (2003). The Entropy of Classical Thermodynamics, pp.&nbsp;147–195, Chapter 8 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*{{cite book|ref=harv|first=F.|last=Mandl|title=Statistical physics|edition=second|year=1988|publisher=[[Wiley & Sons]]|isbn=978-0-471-91533-1}}<br />
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*{{cite journal | last1 = Maxwell | first1 = J.C. | authorlink = James Clerk Maxwell | year = 1867 | title = On the dynamical theory of gases | url = | journal = Phil. Trans. R. Soc. Lond. | volume = 157 | issue = | pages = 49–88 | doi = 10.1098/rstl.1867.0004 }}<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985). ''Thermodynamics'', Pitman, London, {{ISBN|0-273-08577-8}}.<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003). Entropy in Nonequilibrium, pp.&nbsp;79–109, Chapter 5 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*Münster, A. (1970), ''Classical Thermodynamics'', translated by E.S. Halberstadt, Wiley–Interscience, London, {{ISBN|0-471-62430-6}}.<br />
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*[[Brian Pippard|Pippard, A.B.]] (1957/1966). ''Elements of Classical Thermodynamics for Advanced Students of Physics'', original publication 1957, reprint 1966, Cambridge University Press, Cambridge UK.<br />
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*[[Max Planck|Planck, M.]] (1897/1903). [https://archive.org/stream/treatiseonthermo00planrich#page/100/mode/2up ''Treatise on Thermodynamics'', translated by A. Ogg, Longmans Green, London, p. 100.]<br />
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*[[Max Planck|Planck. M.]] (1914). [https://archive.org/details/theoryofheatradi00planrich ''The Theory of Heat Radiation''], a translation by Masius, M. of the second German edition, P. Blakiston's Son & Co., Philadelphia.<br />
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*[[Max Planck|Planck, M.]] (1926). Über die Begründung des zweiten Hauptsatzes der Thermodynamik, ''Sitzungsberichte der Preussischen Akademie der Wissenschaften: Physikalisch-mathematische Klasse'': 453–463.<br />
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* Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769-781.<br />
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* {{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}<br />
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*Quinn, T.J. (1983). ''Temperature'', Academic Press, London, {{ISBN|0-12-569680-9}}.<br />
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*{{cite book|ref=harv|first=Y.V.C.|last=Rao|title=An Introduction to thermodynamics|url=https://books.google.com/books?id=iYWiCXziWsEC&pg=PA213|date=2004|publisher=Universities Press|isbn=978-81-7371-461-0|page=213}}<br />
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*Roberts, J.K., Miller, A.R. (1928/1960). ''Heat and Thermodynamics'', (first edition 1928), fifth edition, Blackie & Son Limited, Glasgow.<br />
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*[[Erwin Schrödinger|Schrödinger, E.]] (1950). Irreversibility, ''Proc. R. Ir. Acad.'', '''A53''': 189–195.<br />
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*[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966). ''Elements of Thermodynamics'', Addison-Wesley Publishing, Reading MA.<br />
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*{{cite journal|ref=harv|last=Thomson|first=W.|author-link=William Thomson, 1st Baron Kelvin|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Transactions of the Royal Society of Edinburgh|year=1851|volume=XX|issue=part II|pages=261–268; 289–298|url=https://www.biodiversitylibrary.org/item/126047#page/295/mode/1up}} Also published in {{cite journal|last=Thomson|first=W.|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Philos. Mag. |date=December 1852 |volume=IV |series=4 |issue=22 |page=13|url=https://archive.org/stream/londonedinburghp04maga#page/12/mode/2up |accessdate=25 June 2012}}<br />
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*[[William Thomson, 1st Baron Kelvin|Thomson, W.]] (1852). ''On the universal tendency in nature to the dissipation of mechanical energy'' Philosophical Magazine, Ser. 4, p.&nbsp;304.<br />
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*[[László Tisza|Tisza, L.]] (1966). ''Generalized Thermodynamics'', M.I.T Press, Cambridge MA.<br />
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*[[Clifford Truesdell|Truesdell, C.]] (1980). ''The Tragicomical History of Thermodynamics 1822–1854'', Springer, New York, {{ISBN|0-387-90403-4}}.<br />
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*Uffink, J. (2001). Bluff your way in the second law of thermodynamics, ''Stud. Hist. Phil. Mod. Phys.'', '''32'''(3): 305–394.<br />
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*Uffink, J. (2003). Irreversibility and the Second Law of Thermodynamics, Chapter 7 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003), Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963). ''Lectures in Statistical Mechanics'', American Mathematical Society, Providence RI.<br />
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*[[Mark Zemansky|Zemansky, M.W.]] (1968). ''Heat and Thermodynamics. An Intermediate Textbook'', fifth edition, McGraw-Hill Book Company, New York.<br />
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==Further reading==<br />
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==Further reading==<br />
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进一步阅读<br />
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*Goldstein, Martin, and Inge F., 1993. ''The Refrigerator and the Universe''. Harvard Univ. Press. Chpts. 4–9 contain an introduction to the Second Law, one a bit less technical than this entry. {{ISBN|978-0-674-75324-2}}<br />
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*Leff, Harvey S., and Rex, Andrew F. (eds.) 2003. ''Maxwell's Demon 2 : Entropy, classical and quantum information, computing''. Bristol UK; Philadelphia PA: [[Institute of Physics]]. {{ISBN|978-0-585-49237-7}}<br />
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*{{Cite book | first = J.J. | last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1}}(technical).<br />
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*{{cite book |title=Reflections on the Motive Power of Heat and on Machines Fitted to Develop That Power |last=Carnot |first=Sadi |editor=[[Robert Henry Thurston|Thurston, Robert Henry]] |year=1890 |publisher=J. Wiley & Sons |location=New York |isbn= |pages= }} ([https://books.google.com/books?id=tgdJAAAAIAAJ full text of 1897 ed.]) ([http://www.history.rochester.edu/steam/carnot/1943/ html])<br />
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*Stephen Jay Kline (1999). ''The Low-Down on Entropy and Interpretive Thermodynamics'', La Cañada, CA: DCW Industries. {{ISBN|1-928729-01-0}}.<br />
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*{{cite book | last1 = Kostic | first1 = M | year = 2011 | title = Revisiting The Second Law of Energy Degradation and Entropy Generation: From Sadi Carnot's Ingenious Reasoning to Holistic Generalization | journal = AIP Conf. Proc. | volume = 1411 | issue = 1| pages = 327–350 | doi = 10.1063/1.3665247 | isbn = 978-0-7354-0985-9 | bibcode = 2011AIPC.1411..327K | series = AIP Conference Proceedings | citeseerx = 10.1.1.405.1945 }} also at [https://web.archive.org/web/20130420222450/http://www.kostic.niu.edu/2ndLaw/Revisiting%20The%20Second%20Law%20of%20Energy%20Degradation%20and%20Entropy%20Generation%20-%20From%20Carnot%20to%20Holistic%20Generalization-4.pdf].<br />
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==External links==<br />
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==External links==<br />
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外部链接<br />
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*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics]" – by Lawrence Sklar.<br />
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*[http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node30.html ''Second law of thermodynamics''] in the MIT Course [http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/notes.html ''Unified Thermodynamics and Propulsion''] from Prof. Z. S. Spakovszky<br />
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*[[E.T. Jaynes]], 1988, "[http://bayes.wustl.edu/etj/articles/ccarnot.pdf The evolution of Carnot's principle]," in G. J. Erickson and C. R. Smith (eds.)''Maximum-Entropy and Bayesian Methods in Science and Engineering, Vol'' 1: p.&nbsp;267.<br />
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*[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf Caratheodory, C., "Examination of the foundations of thermodynamics," trans. by D. H. Delphenich]<br />
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* [https://www.bbc.co.uk/programmes/p004y2bm The Second Law of Thermodynamics], BBC Radio 4 discussion with John Gribbin, Peter Atkins & Monica Grady (''In Our Time'', Dec. 16, 2004)<br />
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* [http://mdpi.org/entropy/papers/e6010001.pdf [[Entropy (journal)]], 2004]<br />
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* [https://www.journals.uchicago.edu/doi/abs/10.1086/663835 The Journal of the International Society for the History of Philosophy of Science, 2012]<br />
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{{DEFAULTSORT:Second Law Of Thermodynamics}}<br />
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[[Category:Concepts in physics]]<br />
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Category:Concepts in physics<br />
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分类: 物理概念<br />
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[[Category:Laws of thermodynamics|2]]<br />
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2<br />
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[[Category:Non-equilibrium thermodynamics]]<br />
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Category:Non-equilibrium thermodynamics<br />
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类别: 非平衡态热力学<br />
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[[Category:Philosophy of thermal and statistical physics]]<br />
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Category:Philosophy of thermal and statistical physics<br />
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类别: 热力学和统计物理学哲学<br />
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<small>This page was moved from [[wikipedia:en:Second law of thermodynamics]]. Its edit history can be viewed at [[热力学第二定律/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E7%83%AD%E5%8A%9B%E5%AD%A6%E7%AC%AC%E4%BA%8C%E5%AE%9A%E5%BE%8B&diff=10359热力学第二定律2020-07-16T09:37:04Z<p>Dorr：/* Entropy 熵 */</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Law of physics}}<br />
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{{Thermodynamics|laws}}<br />
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The '''second law of thermodynamics''' states that the total [[entropy]] of an [[isolated system]] can never decrease over time, and is constant if and only if all processes are reversible.<ref>http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node38.html#SECTION05224000000000000000</ref> Isolated systems spontaneously evolve towards [[thermodynamic equilibrium]], the state with maximum entropy. <br />
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The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, and is constant if and only if all processes are reversible. Isolated systems spontaneously evolve towards thermodynamic equilibrium, the state with maximum entropy. <br />
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热力学第二定律指出，孤立系统的总熵永远不会随着时间而减少，而且只有当且仅当所有过程都是可逆的时，总熵才是恒定的。孤立的系统自发地进化到热力学平衡状态，即熵最大的状态。<br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in [[thermodynamic equilibrium]], or is undergoing a (fictive) [[Reversible process (thermodynamics)|reversible process]]. In all processes that occur, including [[spontaneous process]]es,<ref>Atkins and de Paula, p.78</ref> the total entropy of the system and its surroundings increases and the process is [[Irreversible process|irreversible in the thermodynamic sense]]. The increase in entropy accounts for the irreversibility of natural processes, and the [[Arrow of time|asymmetry between future and past]].<ref>{{cite book|last=Zohuri|first=Bahman|title=Dimensional Analysis Beyond the Pi Theorem|url=https://books.google.com/books?id=pRVuDQAAQBAJ|year=2016|publisher=Springer|isbn=978-3-319-45726-0|page=[https://books.google.com.ph/books?id=pRVuDQAAQBAJ&pg=PA111&dq=%22increase+in+entropy+accounts+for+the+irreversibility+of+natural+processes+and+the+asymmetry+between+future+and+past.%22&hl=en&sa=X&ved=0ahUKEwj6spb61tbaAhUFS7wKHfftDtIQ6AEIKDAA#v=onepage&q=%22increase%20in%20entropy%20accounts%20for%20the%20irreversibility%20of%20natural%20processes%20and%20the%20asymmetry%20between%20future%20and%20past.%22 111]}}</ref><br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in thermodynamic equilibrium, or is undergoing a (fictive) reversible process. In all processes that occur, including spontaneous processes, the total entropy of the system and its surroundings increases and the process is irreversible in the thermodynamic sense. The increase in entropy accounts for the irreversibility of natural processes, and the asymmetry between future and past.<br />
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系统及其周围环境的总熵在理想情况下可以保持不变，在这种情况下，系统处于热力学平衡期，或者正在经历一个(虚拟的)可逆过程期。在所有发生的过程中，包括自发过程，系统及其周围环境的总熵增加，这一过程在热力学意义上是不可逆的。熵的增加解释了自然过程的不可逆性，以及未来和过去之间的不对称性。<br />
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Historically, the second law was an [[Empirical evidence|empirical finding]] that was accepted as an axiom of [[thermodynamics|thermodynamic theory]]. [[Statistical mechanics]], classical or [[quantum thermodynamics|quantum]], explains the microscopic origin of the law.<br />
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Historically, the second law was an empirical finding that was accepted as an axiom of thermodynamic theory. Statistical mechanics, classical or quantum, explains the microscopic origin of the law.<br />
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从历史上看，第二定律是一个经验性的发现，被公认为热力学理论的公理。统计力学，无论是经典的还是量子的，都可以解释这个定律的微观起源。<br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist [[Nicolas Léonard Sadi Carnot|Sadi Carnot]], who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as [[Carnot's theorem (thermodynamics)|Carnot's rule]] or limit.<ref>Jaffe, R.L., Taylor, W. (2018). ''The Physics of Energy'', Cambridge University Press, Cambridge UK, pages 150, 151, 259, 772, 743.</ref><br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist Sadi Carnot, who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as Carnot's rule or limit.<br />
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第二定律已经用许多方式表达过了。法国科学家萨迪 · 卡诺在1824年证明了热能在热机中转化为功的效率是有上限的。这方面的第二定律也被称为卡诺的规则或限制。<br />
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==Introduction==<br />
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==Introduction==<br />
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引言<br />
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[[File:Heat flow hot to cold.png|thumb|upright|Heat flow from hot water to cold water.]]<br />
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Heat flow from hot water to cold water.<br />
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热量从热水流向冷水。<br />
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The [[first law of thermodynamics]] provides the definition of the [[internal energy]] of a [[thermodynamic system]], and expresses the law of [[conservation of energy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 40–41.</ref><ref>Munster A. (1970), pp. 8–9, 50–51.</ref> The second law is concerned with the direction of natural processes.<ref>{{harvnb|Mandl|1988}}</ref> It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such [[Phenomenon|phenomena]] are accounted for in terms of [[entropy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 79–107.</ref><ref>Bailyn, M. (1994), Section 71, pp. 113–154.</ref> If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, {{math|''S''}}, increases.<br />
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The first law of thermodynamics provides the definition of the internal energy of a thermodynamic system, and expresses the law of conservation of energy. The second law is concerned with the direction of natural processes. It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such phenomena are accounted for in terms of entropy. If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, , increases.<br />
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能量守恒定律提供了热力学系统内能的定义，并表达了能量守恒定律。第二定律与自然过程的方向有关。它断言自然过程只在一种意义上运行，而且是不可逆的。例如，当有了传导和辐射的路径时，热总是自发地从一个较热的物体流向一个较冷的物体。这种现象可以用熵来解释。如果一个孤立的系统最初是通过内部分隔不可渗透的墙来保持在内部热力学平衡，然后一些操作使墙更具渗透性，那么系统自发地进化到最终达到一个新的内部热力学平衡，其总熵增加。<br />
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In a fictive reversible process, an infinitesimal increment in the entropy ({{math|d''S''}}) of a system is defined to result from an infinitesimal transfer of heat ({{math|δ''Q''}}) to a [[closed system]] (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature ({{math|''T''}}) of the system in equilibrium and the surroundings which supply the heat:<ref>Bailyn, M. (1994), p. 120.</ref><br />
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In a fictive reversible process, an infinitesimal increment in the entropy () of a system is defined to result from an infinitesimal transfer of heat () to a closed system (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature () of the system in equilibrium and the surroundings which supply the heat:<br />
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在虚构的可逆过程中，系统熵的无穷小增量被定义为由无穷小的热量传递()到一个封闭系统(允许能量进入或出去，但不允许物质传递)除以平衡系统和提供热量的环境的共同温度() :<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, idealized fictive reversible process)}.</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, idealized fictive reversible process)}.</math><br />
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(闭合系统，理想化的可逆过程) / math<br />
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Different notations are used for infinitesimal amounts of heat ({{mvar|δ}}) and infinitesimal amounts of entropy ({{mvar|d}}) because entropy is a [[function of state]], while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the [[Clausius Theorem|inequality]] <ref name="MortimerBook"> Mortimer, R. G. ''Physical Chemistry''. 3rd ed., p. 120, Academic Press, 2008. </ref><ref name="FermiBook"> Fermi, E. ''Thermodynamics'', footnote on p. 48, Dover Publications,1956 (still in print). </ref> <br />
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Different notations are used for infinitesimal amounts of heat () and infinitesimal amounts of entropy () because entropy is a function of state, while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the inequality <br />
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用不同的符号表示无穷小量的热量()和无穷小量的熵() ，因为熵是状态的函数，而热量和功一样不是。第二定律要求系统熵的增量满足不等式，对于实际上可能存在的不与环境发生质量交换的无穷小过程，系统熵增量满足不等式<br />
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: <math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process).}</math><br />
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<math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process).}</math><br />
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(实际上可能是封闭系统，不可逆性)。数学<br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature ({{math|''T''}}) and the temperature of the surroundings ({{math|''T''}}<sub>''surr''</sub>).<ref name=":0">Adkins, C.J. (1968/1983), p. 75.</ref><ref name="Munster 45"/> Note that the equality still applies for pure heat flow,<ref name="Schmidt-Rohr 14"> Schmidt-Rohr, K. (2014). "Expansion Work without the External Pressure, and Thermodynamics in Terms of Quasistatic Irreversible Processes" ''J. Chem. Educ.'' '''91''': 402-409. https://dx.doi.org/10.1021/ed3008704 </ref><br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature () and the temperature of the surroundings (<sub>surr</sub>).<br />
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这是因为这种情况下的一般过程可能包括周围环境对系统所做的功，这可能在系统内部产生摩擦或粘滞效应，因为一个化学反应可能正在进行，或者因为热传递实际上只是不可逆地发生，由系统温度()和周围环境温度(sub surr / sub)之间的差分驱动。<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change).}</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change).}</math><br />
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(实际上可能是不改变成分的准静态不可逆性)。数学<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry.<ref name="Oxtoby8th">Oxtoby, D. W; Gillis, H.P., [[Laurie Butler|Butler, L. J.]] (2015).''Principles of Modern Chemistry'', Brooks Cole. p. 617. {{ISBN|978-1305079113}}</ref> <ref name="MortimerBook"></ref> Introducing a set of internal variables <math>\xi</math> to describe the deviation of a thermodynamic system in physical equilibrium (with the required well-defined uniform pressure ''P'' and temperature ''T'')<ref name="Schmidt-Rohr 14"></ref> from the chemical equilibrium state, one can record the equality<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry. from the chemical equilibrium state, one can record the equality<br />
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这是根据测得的热容曲线和相变时的熵变精确确定纯物质的绝对熵的基础。用量热法。从化学平衡来看，人们可以记录平等<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process).}</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process).}</math><br />
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- frac { t }-frac {1}{ j } ， Xi { j } ， delta Xi j， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ,.数学<br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of ''the second principle of thermodynamics'' – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<ref> Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769–781.</ref><ref>{{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}</ref><br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of the second principle of thermodynamics – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<br />
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第二项代表内部变量的工作，这些内部变量可以受到外部影响的干扰，但是系统不能通过内部变量执行任何正功。这种说法介绍了热力学系统在时间上不可能逆转演化的事实，并且可以被认为是热力学第二原理的表述---- 这个表述，当然，相当于熵原理的表述。<br />
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The [[zeroth law of thermodynamics]] in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body.<ref name=dugdale>{{cite book|author=J. S. Dugdale|title=Entropy and its Physical Meaning|publisher=Taylor & Francis|year=1996|isbn=978-0-7484-0569-5|page=13|quote=This law is the basis of temperature.}}</ref> For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows{{how?|date=August 2018}} a distinguished temperature scale, which defines an absolute, [[thermodynamic temperature]], independent of the properties of any particular reference thermometric body.<ref>[[Mark Zemansky|Zemansky, M.W.]] (1968), pp. 207–209.</ref><ref>Quinn, T.J. (1983), p. 8.</ref><br />
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The zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body. For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows a distinguished temperature scale, which defines an absolute, thermodynamic temperature, independent of the properties of any particular reference thermometric body.<br />
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美国热力学第零定律协会在其通常的简短声明中允许承认，相对于热平衡的两个物体具有相同的温度，特别是测试物体具有与参考温度测量物体相同的温度。对于一个物体和另一个物体之间的热平衡，存在着无限多的经验温度标度，通常分别取决于特定参考温度标度的性质。第二定律允许区分温度标度，它定义了一个绝对的热力学温度，与任何特定的参考温度体的性质无关。<br />
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==Various statements of the law==<br />
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==Various statements of the law==<br />
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各种各样的法律陈述<br />
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The second law of thermodynamics may be expressed in many specific ways,<ref name=MIT>{{cite web|title=Concept and Statements of the Second Law|url=http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node37.html|accessdate=2010-10-07 |publisher=web.mit.edu}}</ref> the most prominent classical statements{{sfnp|Lieb|Yngvason|1999}} being the statement by [[Rudolf Clausius]] (1854), the statement by [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] (1851), and the statement in axiomatic thermodynamics by [[Constantin Carathéodory]] (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.{{sfnp|Rao|2004|p=213}}<br />
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The second law of thermodynamics may be expressed in many specific ways, the most prominent classical statements being the statement by Rudolf Clausius (1854), the statement by Lord Kelvin (1851), and the statement in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.<br />
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热力学第二定律可以用许多特定的方式来表达，最突出的经典陈述是 Rudolf Clausius (1854)的陈述，Kelvin 勋爵(1851)的陈述，以及康斯坦丁·卡拉西奥多里(1909)在公理化热力学中的陈述。这些陈述用一般的物理术语描述法律，引用某些过程的不可能性。克劳修斯和开尔文陈述被证明是等价的。<br />
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===Carnot's principle===<br />
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===Carnot's principle===<br />
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卡诺原理<br />
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The historical origin<ref>[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986).</ref> of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a [[Carnot heat engine]], fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the [[caloric theory]] of heat was seriously considered, before the recognition of the [[first law of thermodynamics]], and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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The historical origin of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a Carnot heat engine, fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the caloric theory of heat was seriously considered, before the recognition of the first law of thermodynamics, and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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热力学第二定律的历史起源源于卡诺的原则。它指的是一个卡诺热机周期，虚拟地以被称为准静态的极慢极限模式运行，因此热量和功量在子系统之间传递，这些子系统始终处于它们自己的内部热力学平衡状态。卡诺发动机是热机效率工程师特别感兴趣的理想设备。卡诺认识到卡诺原理的时候，正是热量热理论被认真考虑的时候，正是能量守恒定律理论被认识之前，正是熵概念的数学表达之前。根据第一定律的解释，它在物理上等同于热力学第二定律，并且在今天仍然有效。卡诺最初的论点是从卡路里理论的观点提出的，直到能量守恒定律的发现。以下是他书中的一些例子:<br />
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::...''wherever there exists a difference of temperature, motive power can be produced.''<ref>Carnot, S. (1824/1986), p. 51.</ref><br />
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...wherever there exists a difference of temperature, motive power can be produced.<br />
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只要有温差，就能产生动力。<br />
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::The production of motive power is then due in steam engines not to an actual consumption of caloric, but ''to its transportation from a warm body to a cold body ...''<ref>Carnot, S. (1824/1986), p. 46.</ref><br />
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The production of motive power is then due in steam engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body ...<br />
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动力的产生不是由于蒸汽机实际消耗的热量，而是由于它从一个温暖的物体运输到一个寒冷的物体..。<br />
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::''The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.''<ref>Carnot, S. (1824/1986), p. 68.</ref><br />
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The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.<br />
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热的动力与实现热的媒介无关，热的数量完全取决于两个物体之间的温度，最后是热量的传递。<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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用现代术语来说，卡诺的原则可能更为准确:<br />
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::The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<ref>[[Clifford Truesdell|Truesdell, C.]] (1980), Chapter 5.</ref><ref>Adkins, C.J. (1968/1983), pp. 56–58.</ref><ref>Münster, A. (1970), p. 11.</ref><ref>Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998), pp.67–75.</ref><ref>Lebon, G., Jou, D., Casas-Vázquez, J. (2008), p. 10.</ref><ref>Eu, B.C. (2002), pp. 32–35.</ref><br />
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The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<br />
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准静态或可逆卡诺循环的效率只取决于两个热源的温度，而且无论工作物质是什么，效率是相同的。以这种方式运行的卡诺发动机是使用这两种温度的最有效的热机。<br />
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===Clausius statement===<br />
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===Clausius statement===<br />
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克劳修斯声明<br />
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The German scientist [[Rudolf Clausius]] laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work.{{sfnp|Clausius|1850}} His formulation of the second law, which was published in German in 1854, is known as the ''Clausius statement'':<br />
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The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work. His formulation of the second law, which was published in German in 1854, is known as the Clausius statement:<br />
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1850年，德国科学家 Rudolf Clausius 通过研究热传递和功之间的关系，为热力学第二定律实验室奠定了基础。他在1854年用德语发表的第二定律的提法被称为克劳修斯声明:<br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.{{sfnp|Clausius|1854|p=86}}</blockquote><br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.</blockquote><br />
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热量不可能从一个较冷的物体传递到一个较热的物体，除非同时发生一些其他的变化。 / blockquote<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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克劳修斯的陈述使用了“热量通道”的概念。正如通常在热力学的讨论，这意味着“能量作为热量的净转移” ，并没有提到促成转移的一种方式和其他。<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of [[refrigeration]], for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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如果不对系统进行外部功，热就不能自发地从冷区流向热区，这一点从制冷的普通经验中可以看出。在冰箱中，热量从冷到热，但只有在外部介质——制冷系统的强制作用下才会发生变化。<br />
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===Kelvin statements===<br />
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===Kelvin statements===<br />
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开尔文陈述<br />
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[[William Thomson, 1st Baron Kelvin|Lord Kelvin]] expressed the second law in several wordings.<br />
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Lord Kelvin expressed the second law in several wordings.<br />
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开尔文勋爵用几个字表达了第二定律。<br />
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::It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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没有任何外部机构的帮助，自动机器是不可能把热量从一个物体传递到另一个物体的较高温度。<br />
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::It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.{{sfnp|Thomson|1851}}<br />
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It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.<br />
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通过无生命物质的作用，把物质冷却到周围最冷的物体的温度以下，从物质的任何部分获得机械效应是不可能的。<br />
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===Equivalence of the Clausius and the Kelvin statements===<br />
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===Equivalence of the Clausius and the Kelvin statements===<br />
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克劳修斯和开尔文陈述的等价性<br />
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[[Image:Deriving Kelvin Statement from Clausius Statement.svg|thumb|Derive Kelvin Statement from Clausius Statement]]<br />
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Derive Kelvin Statement from Clausius Statement<br />
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从克劳修斯陈述推导出开尔文陈述<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed [[Carnot engine]] as shown by the figure. The [[Heat engine#Efficiency|efficiency]] of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the [[first law of thermodynamics]], as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the figure. The efficiency of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the first law of thermodynamics, as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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假设有一个引擎违反了开尔文定理: 也就是说，这个引擎以循环的方式吸收热量并将其转化为功，没有任何其他结果。现在配对与反向卡诺发动机如图所示。普通热机的效率为，反向热机的效率为1 / 。这对组合发动机的净效应和唯一效应是将热量从较冷的贮存器转移到较热的贮存器，这违反了克劳修斯的说法。(这是能量守恒定律的结果，因为系统的总能量保持不变，数学文本{输入} + 文本{输出}0暗示 q-frac { eta }-qc / math，所以数学 qc 向左(frac { eta-1 right) / math)。因此，违反开尔文陈述意味着违反克劳修斯陈述，即。克劳修斯的说法暗示了凯尔文的说法。我们可以用类似的方式证明开尔文陈述暗示了克劳修斯陈述，因此两者是等价的。<br />
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===Planck's proposition===<br />
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===Planck's proposition===<br />
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普朗克的命题<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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普朗克提出了直接来自经验的如下命题。这有时被认为是他对第二定律的陈述，但他认为这是第二定律推导的起点。<br />
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::It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 86.</ref><ref>Roberts, J.K., Miller, A.R. (1928/1960), p. 319.</ref><br />
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It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<br />
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建造一台发动机是不可能的，这台发动机将在一个完整的循环中工作，并且除了提高重量和冷却热储以外不会产生任何效果。<br />
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===Relation between Kelvin's statement and Planck's proposition 开尔文表述与普朗克命题的关系===<br />
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It is almost customary in textbooks to speak of the "[[Kelvin-Planck statement]]" of the law, as for example in the text by [[Dirk ter Haar|ter Haar]] and [[Harald Wergeland|Wergeland]].<ref>[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966), p. 17.</ref><br />
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It is almost customary in textbooks to speak of the "Kelvin-Planck statement" of the law, as for example in the text by ter Haar and Wergeland.<br />
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教科书中几乎惯常提及该定律的“'''开尔文-普朗克表述 Kelvin-Planck Statement'''” ，例如'''德克·特哈尔 Diek ter Haar''' 和'''哈拉尔德·沃格兰 Harald Wergeland''' 在文中就是这样表述的。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）Diek ter Haar 中间是小写的<br />
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The '''Kelvin–Planck statement''' (or the ''heat engine statement'') of the second law of thermodynamics states that <br />
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The Kelvin–Planck statement (or the heat engine statement) of the second law of thermodynamics states that <br />
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热力学第二定律的开尔文-普朗克表述（或称“'''热机表述 Heat Engine Statement'''”）指出<br />
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::It is impossible to devise a [[thermodynamic cycle|cyclically]] operating device, the sole effect of which is to absorb energy in the form of heat from a single [[heat reservoir|thermal reservoir]] and to deliver an equivalent amount of [[Work (physics)|work]].<ref name="Rao">{{cite book|last=Rao|first=Y. V. C.|title=Chemical Engineering Thermodynamics|publisher=Universities Press|isbn=978-81-7371-048-3|page=158|year=1997}}</ref><br />
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It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.<br />
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设计一种唯一效果是从单一热源吸收热量并提供等量的功的循环运行装置是不可能的。<br />
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===Planck's statement 普朗克表述===<br />
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Planck stated the second law as follows.<br />
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Planck stated the second law as follows.<br />
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普朗克表述第二定律如下。<br />
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::Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<ref name="Planck 100">[[Max Planck|Planck, M.]] (1897/1903), p. 100.</ref><ref name="Planck 463">[[Max Planck|Planck, M.]] (1926), p. 463, translation by Uffink, J. (2003), p. 131.</ref><ref name="Roberts & Miller 382">Roberts, J.K., Miller, A.R. (1928/1960), p. 382. This source is partly verbatim from Planck's statement, but does not cite Planck. This source calls the statement the principle of the increase of entropy.</ref><br />
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Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<br />
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自然界中发生的任一过程都是沿参与其中的所有物体的熵之和增加的方向进行的。在某些限制下——即对于可逆过程，熵的总和保持不变。<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for ''irreversible phenomena''.<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for irreversible phenomena.<br />
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与普朗克表述非常相似的是'''乌伦贝克 Uhlenbeck'''和'''福特 Ford'''关于不可逆现象的表述。<br />
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::... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<ref>[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963), p. 16.</ref><br />
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... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<br />
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在从一个平衡态到另一个平衡态的不可逆或自发的变化中（例如，当两个物体 A 和 B 接触时的温度平衡过程），熵总是增加。<br />
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===Principle of Carathéodory 卡拉西奥多里原理===<br />
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<!-- [[Caratheodory's principle]] redirects here --><br />
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—— caratheodory 的原理在这里重新定向——<br />
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[[Constantin Carathéodory]] formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<ref>[[Constantin Carathéodory|Carathéodory, C.]] (1909).</ref><br />
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Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<br />
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'''康斯坦丁·卡拉西奥多里 Constantin Carathéodory'''在纯数学公理的基础上进行了热力学公理化。他对第二定律的陈述被称为'''卡拉西奥多里原理 Principle of Carathéodory'''，可以这样表述:<br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.<ref>Buchdahl, H.A. (1966), p. 68.</ref></blockquote><br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.</blockquote><br />
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在绝热封闭系统的任意状态 S 附近，总有从 S 出发不可达的状态。<br />
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With this formulation, he described the concept of [[adiabatic accessibility]] for the first time and provided the foundation for a new subfield of classical thermodynamics, often called [[Ruppeiner geometry|geometrical thermodynamics]]. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic [[process function]], in other words, <math>\delta Q=TdS</math>.<ref name="Sychev1991">{{cite book |last=Sychev |first=V. V. |title=The Differential Equations of Thermodynamics |year=1991 |publisher=Taylor & Francis |isbn=978-1-56032-121-7}}</ref> {{clarify|date=February 2014}}<br />
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With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, <math>\delta Q=TdS</math>. <br />
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公理化处理后，他第一次描述了'''绝热可达性 Adiabatic Accessibility'''的概念，并为经典热力学的一个新的子领域，即通常所说的'''几何热力学 Geometrical Thermodynamics'''奠定了基础。由卡拉西奥多里原理可以推出，准静态转移的热量值是一个可积过程函数，即<math>\delta Q=TdS</math>。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）准静态转移的热量值是一个可积过程函数 存疑<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.<ref name="Munster 45">Münster, A. (1970), p. 45.</ref>{{sfnp|Lieb|Yngvason|1999|p=49}}<ref name="Planck 1926">[[Max Planck|Planck, M.]] (1926).</ref><ref>Buchdahl, H.A. (1966), p. 69.</ref> {{clarify|date=February 2014}}<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium. <br />
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尽管在教科书几乎惯称卡拉西奥多里原理表述了第二定律，并认为其与克劳修斯表述或开尔文-普朗克表述等价，但事实并非如此。为了得到第二定律的所有内容，需要对卡拉西奥多里原理补充普朗克表述，即等量功总是增加一个最初处于自身内部热力学平衡的封闭系统的内部能量。<br />
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===Planck's principle 普朗克原理===<br />
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In 1926, [[Max Planck]] wrote an important paper on the basics of thermodynamics.<ref name="Planck 1926"/><ref>Uffink, J. (2003), pp. 129–132.</ref> He indicated the principle<br />
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In 1926, Max Planck wrote an important paper on the basics of thermodynamics. He indicated the principle<br />
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1926年，'''马克斯·普朗克 Max Planck'''写了一篇关于热力学基础的重要论文。他指出了以下原理<br />
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::The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<ref name="Munster 45"/>{{sfnp |Lieb|Yngvason|1999|p=49}}<br />
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The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<br />
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一个封闭系统的内部能量因绝热过程增加，在整个过程中，系统的体积保持不变。<br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work."<ref>[[Clifford Truesdell|Truesdell, C.]], Muncaster, R.G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 15.</ref> Planck wrote: "The production of heat by friction is irreversible."<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 81.</ref><ref>[[Max Planck|Planck, M.]] (1926), p. 457, Wikipedia editor's translation.</ref><br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work." Planck wrote: "The production of heat by friction is irreversible."<br />
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这个公式没有提到热，没有提到温度，甚至没有提到熵，也不一定隐含地依赖于这些概念，但它暗示了第二定律的内容。一个密切相关的表述为，“摩擦力从来不做正功。”普朗克写道: “摩擦生热是不可逆的。”<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above.<ref>Lieb, E.H., Yngvason, J. (2003), p. 149.</ref> It is relevant that for a system at constant volume and [[Mole (unit)|mole numbers]], the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above. It is relevant that for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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这个普朗克定理没有提到熵，是用物理术语来表述的。它与上面给出的开尔文表述密切相关。相关的是，对于恒定体积和摩尔数的系统，熵是内能的单调函数。然而，普朗克的这个原理实际上并不是普朗克对第二定律的首选表述（见前面小节），而是依赖于熵的概念。<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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Borgnakke 和 Sonntag 提出的以下表述在某种意义上是对普朗克原理的补充。他们没有将其作为第二定律的完整表述:<br />
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::... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<ref>Borgnakke, C., Sonntag., R.E. (2009), p. 304.</ref><br />
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... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<br />
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只有一种方法可以减少[闭合]系统的熵——从系统中转移热量。<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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与普朗克之前提出的原理不同，这一原理明确地用熵的变化来表示。从系统中去除物质也可以减少系统的熵。<br />
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===Statement for a system that has a known expression of its internal energy as a function of its extensive state variables 一个其内能有已知表达式（其扩展状态变量的函数）的系统的表述===<br />
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The second law has been shown to be equivalent to the [[internal energy]] ''U'' being a weakly [[convex function]], when written as a function of extensive properties (mass, volume, entropy, ...).<ref>{{cite book |last1=van Gool |first1=W. |last2=Bruggink |first2=J.J.C. (Eds) |url= |title=Energy and time in the economic and physical sciences |publisher=North-Holland |year=1985 |pages=41–56 |quote= |isbn=978-0-444-87748-2}}</ref><ref>{{Cite journal | last1 = Grubbström | first1 = Robert W. | doi = 10.1016/j.apenergy.2007.01.003 | title = An Attempt to Introduce Dynamics Into Generalised Exergy Considerations| journal = Applied Energy| volume = 84| issue = 7–8 | pages = 701–718 | year = 2007}}</ref> {{clarify|date=February 2014}}<br />
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The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...). <br />
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第二定律被证明等价于弱凸函数内能 U （写成广泛性质（质量，体积，熵，...）的函数时）。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）存疑<br />
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==Corollaries 推论==<br />
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===Perpetual motion of the second kind 第二类永动机===<br />
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{{main article|Perpetual motion}}<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of [[first law of thermodynamics]] by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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在第二定律建立之前，许多志在发明永动机的人试图通过提取环境的巨大内能作为机器动力，以突破'''热力学第一定律 First Law of Thermodynamics''' 的限制。这种机器被称为“'''第二类永动机 Perpetual Motion Machine of the Second Kind'''”。第二定律表明这种机器是不可能的。<br />
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===Carnot theorem 卡诺定理===<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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'''卡诺定理 Carnot's theorem'''（1824）是一条限制任何可能的发动机的最大效率的原理。效率完全取决于热库和冷库之间的温差。卡诺定理指出:<br />
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*All irreversible heat engines between two heat reservoirs are less efficient than a [[Carnot engine]] operating between the same reservoirs.<br />
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*All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs.<br />
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* 所有不可逆热机的效率低于在两相同热源之间工作的'''卡诺机 Carnot engine'''。<br />
* 所有可逆热机的效率与在两相同热源之间工作的卡诺机相等。<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as [[thermodynamic reversibility]]. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the [[Carnot cycle]]'s reversibility and was condemned to be less efficient.<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the Carnot cycle's reversibility and was condemned to be less efficient.<br />
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在他的理想模型中，热转换成功可以通过逆转循环的运动而恢复，这个概念后来被称为'''热力学可逆性 Thermodynamic Reversibility'''。然而，卡诺进一步假定，一些热量损失了，并没有转化为机械功。因此，没有一个真实的热机能够实现'''卡诺循环 Carnot cycle'''的可逆性，并且被认为效率较低。<br />
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Though formulated in terms of caloric (see the obsolete [[caloric theory]]), rather than [[entropy]], this was an early insight into the second law.<br />
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Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law.<br />
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该理论尽管是用热量表述的（见被取代的'''热质说 Caloric Theory'''），而不是熵，但是它是对第二定律的早期认识。<br />
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===Clausius inequality 克劳修斯不等式===<br />
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The [[Clausius theorem]] (1854) states that in a cyclic process<br />
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The Clausius theorem (1854) states that in a cyclic process<br />
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'''克劳修斯定理 Clausius theorem'''（1854）指出，在一个循环的过程中<br />
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: <math>\oint \frac{\delta Q}{T} \leq 0.</math><br />
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The equality holds in the reversible case<ref>[http://scienceworld.wolfram.com/physics/ClausiusTheorem.html ''Clausius theorem''] at [[Wolfram Research]]</ref> and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function [[entropy]]. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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The equality holds in the reversible case and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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等号在可逆情况下成立，严格不等号在不可逆情况下成立。可逆情况下引入状态函数熵。这是因为在循环过程中，状态函数的变化为零。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）末句存疑<br />
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===Thermodynamic temperature 热力学温度===<br />
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{{main article|Thermodynamic temperature}}<br />
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For an arbitrary heat engine, the efficiency is:<br />
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For an arbitrary heat engine, the efficiency is:<br />
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对于任意热机，效率为:<br />
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: <math>\eta = \frac {W_n}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H} \qquad (1)</math><br />
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where ''W''<sub>n</sub> is for the net work done per cycle. Thus the efficiency depends only on q<sub>C</sub>/q<sub>H</sub>.<br />
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其中 ''W''<sub>n</sub> 表示每个循环所做的净功。因此效率只取决于 q<sub>C</sub>/q<sub>H</sub>。<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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卡诺定理指出，所有在相同热源间运行的可逆发动机的效率是相同的。<br />
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Thus, any reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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Thus, any reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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因此，任何在温度''T''<sub>1</sub>和''T''<sub>2</sub>之间运行的可逆热机必须具有相同的效率，也就是说，效率只是温度的函数:<br />
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<math>\frac{q_C}{q_H} = f(T_H,T_C)\qquad (2).</math><br />
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In addition, a reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between ''T''<sub>1</sub> and another (intermediate) temperature ''T''<sub>2</sub>, and the second between ''T''<sub>2</sub> and''T''<sub>3</sub>. This can only be the case if<br />
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In addition, a reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between T<sub>1</sub> and another (intermediate) temperature T<sub>2</sub>, and the second between T<sub>2</sub> andT<sub>3</sub>. This can only be the case if<br />
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另外，在温度 ''T''<sub>1</sub> 和 ''T''<sub>3</sub> 之间工作的可逆热机必须具有与由分别在温度''T''<sub>1</sub> 和（中间）温度 ''T''<sub>2</sub>之间、在 ''T''<sub>2</sub> 和 ''T''<sub>3</sub> 之间的两个循环组成的系统效率相同。只有下式成立才会出现这种情况:<br />
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: <math>f(T_1,T_3) = \frac{q_3}{q_1} = \frac{q_2 q_3} {q_1 q_2} = f(T_1,T_2)f(T_2,T_3).</math><br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the [[triple point]] of water. Then for any ''T''<sub>2</sub> and ''T''<sub>3</sub>,<br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the triple point of water. Then for any T<sub>2</sub> and T<sub>3</sub>,<br />
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现在考虑如下情形，<math>T_1</math> 是一个固定的参考温度: 水的三相点的温度。则对于任意 T<sub>2</sub> 和 T<sub>3</sub>,<br />
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: <math>f(T_2,T_3) = \frac{f(T_1,T_3)}{f(T_1,T_2)} = \frac{273.16 \cdot f(T_1,T_3)}{273.16 \cdot f(T_1,T_2)}.</math><br />
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Therefore, if thermodynamic temperature is defined by<br />
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Therefore, if thermodynamic temperature is defined by<br />
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因此，如果热力学温度被定义为<br />
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: <math>T = 273.16 \cdot f(T_1,T) \,</math><br />
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then the function ''f'', viewed as a function of thermodynamic temperature, is simply<br />
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then the function f, viewed as a function of thermodynamic temperature, is simply<br />
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那么函数 ''f'' 作为热力学温度的函数，为<br />
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: <math>f(T_2,T_3) = \frac{T_3}{T_2},</math><br />
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and the reference temperature ''T''<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be used{{snd}}the choice here corresponds to the [[Kelvin]] scale.)<br />
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and the reference temperature T<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be usedthe choice here corresponds to the Kelvin scale.)<br />
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参考温度 ''T''<sub>1</sub> 的值为273.16。（任何参考温度和任何正值均可用——此处的选择对应开尔文标度。）<br />
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===Entropy 熵===<br />
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{{main article|Entropy (classical thermodynamics)}}<br />
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According to the [[Clausius theorem|Clausius equality]], for a reversible process<br />
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According to the Clausius equality, for a reversible process<br />
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根据克劳修斯定理，对可逆过程有<br />
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: <math>\oint \frac{\delta Q}{T}=0</math><br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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这意味着线积分 <math>\int_L \frac{\delta Q}{T}</math> 对于可逆过程是路径无关的。<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer<ref name="Schmidt-Rohr 14" /> satisfies<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer satisfies<br />
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所以我们可以定义一个叫做熵的状态函数 S，对于可逆过程或者纯热传递满足<br />
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: <math>dS = \frac{\delta Q}{T} \!</math><br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the [[third law of thermodynamics]], which states that ''S'' = 0 at [[absolute zero]] for perfect crystals.<br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the third law of thermodynamics, which states that S = 0 at absolute zero for perfect crystals.<br />
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据此，只有对上述公式进行积分，才能得到熵的差值。为了获得绝对值，我们需要'''热力学第三定律 Third Law of Thermodynamics'''，它指出'''绝对零度 Absolute Zero'''下完美晶体的 ''S'' = 0。<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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对于任意不可逆过程，由于熵是一个状态函数，我们总是可以将初始状态和最终状态与一个虚拟的可逆过程联系起来，并在这条路径上积分以计算熵的差值。<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the [[Clausius inequality]] on this loop,<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the Clausius inequality on this loop,<br />
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现在把可逆过程逆过来，将其与上述不可逆过程结合。把克劳修斯不等式应用到这个循环,<br />
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: <math>-\Delta S+\int\frac{\delta Q}{T}=\oint\frac{\delta Q}{T}< 0</math><br />
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Thus,<br />
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Thus,<br />
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故,<br />
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: <math>\Delta S \ge \int \frac{\delta Q}{T} \,\!</math><br />
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where the equality holds if the transformation is reversible.<br />
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where the equality holds if the transformation is reversible.<br />
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如果变换可逆，等号成立。<br />
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Notice that if the process is an [[adiabatic process]], then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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Notice that if the process is an adiabatic process, then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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注意，若该过程是一个'''绝热过程 Adiabatic Process'''，则<math>\delta Q=0</math>，故<math>\Delta S\ge 0</math>。<br />
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===Energy, available useful work===<br />
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===Energy, available useful work===<br />
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能量，有用的工作<br />
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{{See also|Exergy}}<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an ''unlimited'' heat reservoir at temperature ''T<sub>R</sub>'' and pressure ''P<sub>R</sub>'' {{snd}}so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain ''T<sub>R</sub>''; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain ''P<sub>R</sub>''.<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an unlimited heat reservoir at temperature T<sub>R</sub> and pressure P<sub>R</sub> so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain T<sub>R</sub>; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain P<sub>R</sub>.<br />
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考虑将第二定律应用于孤立系统(称为整体系统或宇宙)的情形，是一个重要的、具有启发性的理想情形。孤立系统由两部分组成: 感兴趣的子系统和子系统的环境。这些环境被想象为如此之大，以至于它们可以被认为是一个温度为 t 子 r / sub 和压力为 p 子 r / sub 的无限热源，因此无论有多少热量被转移到(或来自)子系统，周围环境的温度将保持 t 子 r / sub; 无论子系统的体积扩大(或收缩)多少，周围环境的压力将保持 p 子 r / sub。<br />
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Whatever changes to ''dS'' and ''dS<sub>R</sub>'' occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy ''S<sub>tot</sub>'' of the isolated total system must not decrease:<br />
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Whatever changes to dS and dS<sub>R</sub> occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy S<sub>tot</sub> of the isolated total system must not decrease:<br />
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根据第二定律，孤立总体系统的熵 s 子系统 / 子系统的熵不能减小，无论子系统和周围环境的熵对 dS 和 dS 子系统 r / 子系统发生什么变化:<br />
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: <math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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<math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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数学 dS + dS r ge 0 / math<br />
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According to the [[first law of thermodynamics]], the change ''dU'' in the internal energy of the sub-system is the sum of the heat ''δq'' added to the sub-system, ''less'' any work ''δw'' done ''by'' the sub-system, ''plus'' any net chemical energy entering the sub-system ''d ∑μ<sub>iR</sub>N<sub>i</sub>'', so that:<br />
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According to the first law of thermodynamics, the change dU in the internal energy of the sub-system is the sum of the heat δq added to the sub-system, less any work δw done by the sub-system, plus any net chemical energy entering the sub-system d ∑μ<sub>iR</sub>N<sub>i</sub>, so that:<br />
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根据能量守恒定律，子系统内部能量的变化 dU 是子系统内部能量 q 的总和，减去子系统所做的任何功，再加上进入子系统 d ∑ sub iR / sub n sub i / sub 的任何净化学能，因此:<br />
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: <math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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<math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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数学 dU delta q- delta w + d ( sum mu iR } n i) ，/ math<br />
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where μ<sub>iR</sub> are the [[chemical potential]]s of chemical species in the external surroundings.<br />
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where μ<sub>iR</sub> are the chemical potentials of chemical species in the external surroundings.<br />
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其中亚 iR / 亚是外部环境中化学物种的化学势。<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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现在热量离开储存器进入子系统是<br />
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: <math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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<math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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数学△ qtr (- dS r) le tr dS / math<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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其中我们首先使用了经典热力学中熵的定义(在统计热力学中，熵变、温度和吸收热量之间的关系可以推导出来) ，然后从上面推导出第二定律的不等式。<br />
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It therefore follows that any net work ''δw'' done by the sub-system must obey<br />
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It therefore follows that any net work δw done by the sub-system must obey<br />
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因此，子系统所做的任何净功都必须服从<br />
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: <math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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<math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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数学增量 w le-dU + tds + sum mu { iR } dN i，/ math<br />
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It is useful to separate the work ''δw'' done by the subsystem into the ''useful'' work ''δw<sub>u</sub>'' that can be done ''by'' the sub-system, over and beyond the work ''p<sub>R</sub> dV'' done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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It is useful to separate the work δw done by the subsystem into the useful work δw<sub>u</sub> that can be done by the sub-system, over and beyond the work p<sub>R</sub> dV done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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将子系统所做的功与子系统所能做的功分离为子系统所能做的有用功，超过了子系统对外界压力所做的功，给出了子系统所能做的有用功(火用)的下列关系式:<br />
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: <math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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<math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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D (u-t r s + p r v-sum { iR } n i) ，/ math<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the ''availability'' or ''[[exergy]]'' ''E'' of the subsystem,<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the availability or exergy E of the subsystem,<br />
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可以很方便地将子系统的右侧定义为热动力位能的精确衍生物，称为子系统的可用性或火用 e,<br />
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: <math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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<math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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数学 e u-t r s + p r v- sum { iR } n i / math<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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因此，第二定律意味着，对于任何可以简单地被视为分为一个子系统和一个与之接触的无限温度和压力储存器的过程,<br />
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: <math> dE + \delta w_u \le 0 \, </math><br />
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<math> dE + \delta w_u \le 0 \, </math><br />
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数学德 delta u le 0，/ math<br />
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i.e. the change in the subsystem's exergy plus the useful work done ''by'' the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done ''on'' the system) must be less than or equal to zero.<br />
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i.e. the change in the subsystem's exergy plus the useful work done by the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done on the system) must be less than or equal to zero.<br />
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也就是。子系统(火用)的变化加上子系统所做的有用功(或者，子系统(火用)的变化不包括任何功，除了压力贮存器所做的功之外，在系统中所做的功)必须小于或等于零。<br />
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In sum, if a proper ''infinite-reservoir-like'' reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in ''E'' for an irreversible process and no change for a reversible process.<br />
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In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in E for an irreversible process and no change for a reversible process.<br />
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总之，如果选择一个合适的类似于无限库的参考状态作为现实世界中的系统环境，那么第二定律预测一个不可逆性的 e 值会减少，而一个可逆过程的 e 值不会变化。<br />
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: <math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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<math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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数学 d = 0 = 0 = 0 = 0 = 0<br />
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This expression together with the associated reference state permits a [[design engineer]] working at the macroscopic scale (above the [[thermodynamic limit]]) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (''Also, see [[process engineer]]''). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (''See [[Exergy efficiency|second law efficiency]]''.)<br />
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This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (Also, see process engineer). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (See second law efficiency.)<br />
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这个表达式连同相关的参考状态允许在宏观的设计工程师利用第二定律，而不需要直接测量或考虑整个孤立系统的熵变。(另外，请参阅流程工程师)。这些变化已经被考虑，假设被考虑的系统可以达到平衡的参考状态而不改变参考状态。还可以找到一个过程或过程集合的效率，将其与可逆的理想状态相比较<br />
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This approach to the Second Law is widely utilized in [[engineering]] practice, [[environmental accounting]], [[systems ecology]], and other disciplines.<br />
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This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines.<br />
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第二定律的这种方法被广泛应用于工程实践、环境会计、系统生态学和其他学科。<br />
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===The second law in chemical thermodynamics===<br />
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===The second law in chemical thermodynamics===<br />
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化学热力学的第二定律<br />
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For a [[spontaneous process|spontaneous chemical process]] in a closed system at constant temperature and pressure without non-''PV'' work, the Clausius inequality Δ''S > Q/T<sub>surr</sub>'' transforms into a condition for the change in [[Gibbs free energy]] <br />
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For a spontaneous chemical process in a closed system at constant temperature and pressure without non-PV work, the Clausius inequality ΔS > Q/T<sub>surr</sub> transforms into a condition for the change in Gibbs free energy <br />
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对于一个封闭系统中的自发化学过程，在没有非 pv 功的情况下，在恒定温度和压力下，将克劳修斯不等式 s q / t sub surr / sub 转化为吉布斯自由能变化的条件<br />
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: <math>\Delta G < 0 </math><br />
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数学 Delta g 0 / 数学<br />
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or d''G'' < 0. For a similar process at constant temperature and volume, the change in [[Helmholtz free energy]] must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products.<ref name="Oxtoby8th"/> <ref name="MortimerBook"></ref> The chemical equilibrium condition at constant ''T'' and ''p'' without electrical work is d''G'' = 0.<br />
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or dG < 0. For a similar process at constant temperature and volume, the change in Helmholtz free energy must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products. The chemical equilibrium condition at constant T and p without electrical work is dG = 0.<br />
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或者 dG 0。对于一个相似的过程，在恒定的温度和体积下，亥姆霍兹自由能的变化一定是负的，这是 Delta a 0 / math。因此，自由能(g 或 a)变化的负值是过程自发的必要条件。这是热力学第二定律在化学中最有用的形式，其中自由能的变化可以通过列表的生成焓和反应物和产物的标准摩尔熵来计算。在 t 和 p 不变的情况下，化学平衡条件是 dG 0。<br />
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==History==<br />
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==History==<br />
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历史<br />
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{{See also|History of entropy}}<br />
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[[File:Sadi Carnot.jpeg|thumb|upright|Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique]].]]<br />
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Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique.]]<br />
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尼古拉·莱昂纳尔·萨迪·卡诺穿着传统的巴黎综合理工学院制服<br />
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The first theory of the conversion of heat into mechanical work is due to [[Nicolas Léonard Sadi Carnot]] in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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The first theory of the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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第一个把热转化为机械功的理论是在1824年由尼古拉·莱昂纳尔·萨迪·卡诺提出的。他是第一个正确认识到这种转换的效率取决于发动机和环境之间的温差。<br />
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Recognizing the significance of [[James Prescott Joule]]'s work on the conservation of energy, [[Rudolf Clausius]] was the first to formulate the second law during 1850, in this form: heat does not flow ''spontaneously'' from cold to hot bodies. While common knowledge now, this was contrary to the [[caloric theory]] of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law during 1850, in this form: heat does not flow spontaneously from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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认识到詹姆斯·普雷斯科特·焦耳在能量守恒方面工作的重要性，Rudolf Clausius 在1850年第一个提出了第二定律: 热不会自发地从冷物体流向热物体。虽然现在的常识是，这与当时流行的热量理论相反，热量理论认为热量是一种流体。从那里他能够推断出萨迪卡诺定律和熵的定义(1865年)。<br />
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Established during the 19th century, the [[Kelvin-Planck statement|Kelvin-Planck statement of the Second Law]] says, "It is impossible for any device that operates on a [[cyclic process|cycle]] to receive heat from a single [[heat reservoir|reservoir]] and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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Established during the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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建立于19世纪的开尔文-普朗克第二定律(Kelvin-Planck)说，“任何循环运行的设备都不可能从单个蓄热体接收热量，并产生净功。”这被证明相当于克劳修斯的陈述。<br />
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The [[ergodic hypothesis]] is also important for the [[Boltzmann]] approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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遍历假设对玻尔兹曼方法也很重要。它说，在很长一段时间内，在具有相同能量的微观态的相空间的某些区域所花费的时间与这个区域的体积成正比，即。在很长一段时间内，所有可访问的微观状态的可能性都是一样的。同样的，它表明时间平均值和系综平均值是相同的。<br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a [[Macroscopic bodies|macroscopic system]]. This doctrine is obsolescent.<ref>Denbigh, K.G., Denbigh, J.S. (1985). ''Entropy in Relation to Incomplete Knowledge'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-25677-1}}, pp. 43–44.</ref><ref>Grandy, W.T., Jr (2008). ''Entropy and the Time Evolution of Macroscopic Systems'', Oxford University Press, Oxford, {{ISBN|978-0-19-954617-6}}, pp. 55–58.</ref><ref name=Lambert>[http://entropysite.oxy.edu Entropy Sites — A Guide] Content selected by [[Frank L. Lambert]]</ref><br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a macroscopic system. This doctrine is obsolescent.<br />
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有一种传统的学说，从克劳修斯开始，认为熵可以被理解为宏观系统中的分子“无序”。这种学说已经过时了。<br />
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===Account given by Clausius===<br />
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===Account given by Clausius===<br />
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作者: 克劳修斯<br />
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[[File:Clausius-1.jpg|thumb|upright|Rudolf Clausius]]<br />
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Rudolf Clausius<br />
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鲁道夫 · 克劳修斯<br />
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In 1856, the German physicist [[Rudolf Clausius]] stated what he called the "second fundamental theorem in the [[mechanical theory of heat]]" in the following form:{{sfnp|Clausius|1867}}<br />
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In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:<br />
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1856年，德国物理学家鲁道夫 · 克劳修斯阐述了他所谓的“热力学理论中的第二个基本定理” ，其形式如下:<br />
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: <math>\int \frac{\delta Q}{T} = -N</math><br />
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<math>\int \frac{\delta Q}{T} = -N</math><br />
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- n / math<br />
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where ''Q'' is heat, ''T'' is temperature and ''N'' is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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其中 q 是热，t 是温度，n 是一个循环过程中所有未补偿的相变的“等价值”。后来，在1865年，克劳修斯将“等价值”定义为熵。紧接着这个定义，也就是在同一年，第二定律最著名的版本在4月24日苏黎世哲学学会的一次演讲中被宣读，在演讲的最后，克劳修斯总结道:<br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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宇宙的熵趋于最大<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. [[universe]], as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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这句话是第二定律中最广为人知的措辞。由于其语言的松散性，例如:。这个问题的答案是肯定的，因为这个问题的答案是肯定的。开放，封闭，或孤立，许多人认为这个简单的陈述意味着热力学第二定律适用于几乎所有可以想象的主题。这是不正确的; 这个陈述只是一个更加扩展和精确描述的简化版本。<br />
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In terms of time variation, the mathematical statement of the second law for an [[isolated system]] undergoing an arbitrary transformation is:<br />
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In terms of time variation, the mathematical statement of the second law for an isolated system undergoing an arbitrary transformation is:<br />
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就时间变化而言，经历任意变换的孤立系统第二定律的数学表述是:<br />
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: <math>\frac{dS}{dt} \ge 0</math><br />
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<math>\frac{dS}{dt} \ge 0</math><br />
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0 / math<br />
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where<br />
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where<br />
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在哪里<br />
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: ''S'' is the entropy of the system and<br />
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S is the entropy of the system and<br />
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S 是系统的熵<br />
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: ''t'' is [[time]].<br />
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t is time.<br />
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时间到了。<br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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!-- 可逆过程需要与周围环境保持平衡。这对于一个孤立的系统是不可能的。因此，关于可逆过程的讨论已经转移到对封闭系统的分析----<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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平衡后应用等式符号。建立孤立系统第二定律的另一种方法是:<br />
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: <math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math<br />
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with <math> \dot S_{i}</math> the sum of the rate of [[entropy production]] by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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with <math> \dot S_{i}</math> the sum of the rate of entropy production by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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用数学 s { i } / 数学表示系统内所有进程的产生熵之和。这种配方的优点是它显示了产生熵的效果。产生熵是一个非常重要的概念，因为它决定(限制)热机器的效率。与环境温度数学 t { a } / math 相乘，得到所谓的耗散能量数学 p { diss } t { a } dot s { i } / math。<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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封闭系统的第二定律(允许热量交换和移动边界，但不允许物质交换)的表达式是:<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math 来计算数学点 q } + 点 s { i } / math<br />
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Here<br />
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Here<br />
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给你<br />
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: <math>\dot Q</math> is the heat flow into the system<br />
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<math>\dot Q</math> is the heat flow into the system<br />
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数学 q / math 是进入系统的热流<br />
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: <math>T</math> is the temperature at the point where the heat enters the system.<br />
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<math>T</math> is the temperature at the point where the heat enters the system.<br />
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Math 是热量进入系统时的温度。<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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等式符号在只有可逆过程在系统内发生的情况下成立。如果发生不可逆过程(在实际操作系统中就是这种情况) ,-符号保持不变。如果系统有多处供热，我们必须求相应项的代数和。<br />
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For open systems (also allowing exchange of matter):<br />
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For open systems (also allowing exchange of matter):<br />
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对于开放系统(也允许物质交换) :<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math 表示数学点 q } + 点 s + 点 s { i } / math<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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这里的数学 s / math 是熵流进入系统，与物质流进入系统有关。它不应该与熵的时间导数混淆。如果物质在几个地方提供，我们必须取这些贡献的代数和。<br />
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==Statistical mechanics==<br />
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==Statistical mechanics==<br />
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统计力学<br />
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[[Statistical mechanics]] gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a [[microstate (statistical mechanics)|microstate]] of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/{{radic|''N''}} where ''N'' is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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Statistical mechanics gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a microstate of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/ where N is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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通过假设物质是由不断运动的原子和分子组成的，统计力学对第二定律给出了解释。系统中每个粒子的一组特定的位置和速度称为系统的微观状态，由于系统的不断运动，系统不断地改变其微观状态。统计力学假设，在平衡状态下，系统可能处于的每个微观状态发生的可能性是相等的，当这个假设被提出时，它直接导致了第二定律必须在统计意义上成立的结论。也就是说，第二定律平均成立，其中 n 是系统中粒子数的1 / 数量级的统计变化。在日常(宏观)情况下，违反第二定律的概率几乎为零。然而，对于粒子数量很少的系统，热力学参数，包括熵，可能显示出与第二定律预测的显著的统计偏差。经典热力学理论没有处理这些统计变量。<br />
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（by Jie XIANG) 在通过假定一种物质由不断运动的原子和分子组成的前提下，统计力学对热力学第二定律进行了解释。系统中每个粒子其特定的位置和速度组成为系统的微观状态，由于（原子和分子的）不断运动，系统也持续改变其微观状态。统计力学假设，在系统力平衡的情况下，其微观状态其实是等可能发生的，进而能够得出结论：热力学第二定律必须在统计学意义上成立。也就是说，第二定律将保持均值不变， 其统计变化为1 /√N，N为系统中的粒子数。在日常(宏观)情况下，违反热力学第二定律的概率几乎为零，但是，对于粒子数量非常少的系统，其热力学参数（包括熵）可能会显示出与第二定律所预测的明显统计偏差。然而经典热力学理论并不处理这些统计变量。<br />
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==Derivation from statistical mechanics==<br />
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==Derivation from statistical mechanics==<br />
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从统计力学导出<br />
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{{Further information|H-theorem}}<br />
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The first mechanical argument of the [[Kinetic theory of gases]] that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to [[James Clerk Maxwell]] in 1860;<ref>{{Cite journal | last1 = Gyenis | first1 = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G }}</ref> [[Ludwig Boltzmann]] with his [[H-theorem]] of 1872 also argued that due to collisions gases should over time tend toward the [[Maxwell–Boltzmann distribution]].<br />
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The first mechanical argument of the Kinetic theory of gases that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to James Clerk Maxwell in 1860; Ludwig Boltzmann with his H-theorem of 1872 also argued that due to collisions gases should over time tend toward the Maxwell–Boltzmann distribution.<br />
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气体动力学理论的第一个力学论证由詹姆斯·克拉克·麦克斯韦在1860年给出，指出分子碰撞引起温度均衡，因此整体趋向于平衡； 路德维希·玻尔兹曼在1872年提出的 H 定理也认为，气体由于碰撞应该随着时间的推移趋向于麦克斯韦-波兹曼分布。<br />
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Due to [[Loschmidt's paradox]], derivations of the Second Law have to make an assumption regarding the past, namely that the system is [[Correlation and dependence|uncorrelated]] at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a [[boundary condition]], and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the [[Big Bang]]), though [[Boltzmann brain|other scenarios]] have also been suggested.<ref name="Hawking AOT">{{cite journal|last=Hawking|first=SW|title=Arrow of time in cosmology|journal=Phys. Rev. D|year=1985|volume=32|issue=10|pages=2489–2495|doi=10.1103/PhysRevD.32.2489|pmid=9956019|bibcode = 1985PhRvD..32.2489H }}</ref><ref>{{cite book | last = Greene | first = Brian | authorlink = Brian Greene | title = The Fabric of the Cosmos | url = https://archive.org/details/fabricofcosmossp00gree | url-access = registration | publisher = Alfred A. Knopf | year = 2004 | page = [https://archive.org/details/fabricofcosmossp00gree/page/171 171] | isbn = 978-0-375-41288-2}}</ref><ref name=Lebowitz>{{cite journal|last=Lebowitz|first=Joel L.|title= Boltzmann's Entropy and Time's Arrow|journal=Physics Today|date=September 1993|volume=46|issue=9|pages=32–38|url=http://users.df.uba.ar/ariel/materias/FT3_2008_1C/papers_pdf/lebowitz_370.pdf|accessdate=2013-02-22|doi=10.1063/1.881363|bibcode = 1993PhT....46i..32L }}</ref><br />
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Due to Loschmidt's paradox, derivations of the Second Law have to make an assumption regarding the past, namely that the system is uncorrelated at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a boundary condition, and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the Big Bang), though other scenarios have also been suggested.<br />
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由于洛施密特悖论，第二定律的导出必须对过去做出一个假设，即系统在过去的某个时刻是不相关的；这样的假设允许进行简单的概率处理。这个假设通常被认为是一个边界条件，因此热力学第二定律最终是过去某个地方的初始条件的结果，可能是在宇宙的开始(大爆炸) 。也有人提出了其他假设。<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the [[Statistical mechanics#Fundamental postulate|fundamental postulate]], also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy.{{citation needed|date=August 2012}} The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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考虑到这些假设，在统计力学中，第二定律不是一个假设，而是统计力学基本假设的一个结果，也被称为等先验概率假设。这个基本假设表明，只要一个人清楚地知道，简单的概率论证只适用于未来，而对于过去，有辅助的信息来源告诉我们，它是低熵的。热力学第二定律的第一部分指出，热孤立系统的熵只能增加。如果我们把熵的概念限制在热平衡系统中，那么热力学第二定律的第一部分是等先验概率假设的一个显然结果。处于热平衡状态的孤立系统包含能量<math>E</math>的熵表示为：<br />
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: <math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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<math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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其中<math>\Omega\left(E\right)</math> 是处于 <math>E</math>和<math>E +\delta E</math这个小区间内的量子态数目。这里的 <math>\delta E</math> 是一个宏观上很小的固定能量区间。严格地说，这意味着熵取决于对<math>\delta E</math>的选择。然而在热力学极限下（例如无穷大系统的极限），狭义的熵(单位体积或单位质量的熵)不依赖于 <math>\delta E</math>。<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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假设我们有一个孤立系统，其宏观状态由许多变量描述。这些宏观变量可以是总体积、活塞在系统中的位置等。从而<math>\Omega</math>将取决于这些变量的值。如果某个变量不是固定的（我们不会在某个位置夹住活塞) ，那么因为在平衡状态下所有可到达状态的等几率到达的，平衡状态下的自由变量会使 <math>\Omega</math> 最大，因为这是平衡状态下最可能的情况。<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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如果该变量最初固定到某个值然后释放，当达到新的平衡时，变量将自我调整使得 <math>\Omega</math> 最大，这意味着熵将增加或保持不变(如果变量初始固定的值恰好是平衡值)。<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's [[H-theorem]], however, proves that the quantity {{math|''H''}} increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the quantity increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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假设我们初始位于一个平衡状态，突然移除了对一个变量的约束。我们做完这件事的时候，可达到的微观状态的数为<math>\Omega</math>，但是系统还没有达到平衡，所以系统处于某些可达到的状态的实际概率还不等于先验概率 <math>1/\Omega</math>。我们已经知道，最终的平衡状态相对于之前的平衡状态，熵会增加或者保持不变。然而，玻耳兹曼的H定理证明系统在不处于平衡态的期间，那个量作为时间的函数单调增加。<br />
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===Derivation of the entropy change for reversible processes===<br />
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===Derivation of the entropy change for reversible processes===<br />
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可逆过程熵变的推导<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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第二定律的第二部分指出，经历可逆过程的系统的熵变是由以下因素给出的:<br />
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: <math>dS =\frac{\delta Q}{T}</math><br />
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<math>dS =\frac{\delta Q}{T}</math><br />
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数学 dS frac delta q }{ t } / math<br />
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where the temperature is defined as:<br />
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where the temperature is defined as:<br />
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其中温度定义为:<br />
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:<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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[数学][[欧米茄][左(右)][右][数学][数学]<br />
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See [[Microcanonical ensemble|here]] for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the [[adiabatic theorem]] of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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请参阅此处对此定义的正当理由。假设系统有一些可以改变的外部参数 x。一般来说，系统的能量本征态将依赖于 x。根据量子力学的绝热定理，在系统哈密顿量无限缓慢变化的极限下，系统将保持在相同的定态，因此系统的能量会随着其所在定态的能量变化而变化。<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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广义力，x，对应于外部变量 x 的定义是，如果 x 增加一个量 dx，那么 math x dx / math 就是系统所做的功。例如，如果 x 是体积，那么 x 是压强。一个已知的系统的广义力在定态数学 e { r } / math 中给出:<br />
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: <math>X = -\frac{dE_{r}}{dx}</math><br />
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<math>X = -\frac{dE_{r}}{dx}</math><br />
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数学 x- frac { dE { r }{ dx } / math<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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由于系统可以在一个数学△ e / math 区间内的任意定态内，我们将系统的广义力定义为上述表达式的期望值:<br />
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: <math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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<math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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数学 x- 左 langle frac { dE { dx }右 rangle，/ math<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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为了计算平均值，我们通过计算在数学 y / math 和数学 y + delta y / math 之间的范围内，其中有多少数学 y / math 能量本征态有一个值来划分 ω 左(e 右) / math 能量本征态。把这个数字命名为 math Omega { y } left (e right) / math，我们有:<br />
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: <math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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<math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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Math Omega left (e right) sum { y } Omega { y } left (e right) ，/ math<br />
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The average defining the generalized force can now be written:<br />
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The average defining the generalized force can now be written:<br />
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定义广义力的平均值现在可以写成:<br />
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: <math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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<math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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数学 x frac {1} Omega 左(e 右) sum { y Omega { y }左(e 右) ，/ math<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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我们可以把它和熵对 x 在恒定能量 e 下的导数联系起来，如下所示。假设我们把 x 改成 x + dx。然后 math Omega left (e right) / math 将会改变，因为能量本征态依赖于 x，导致能量本征态进入或超出数学 e / math 和数学 e + delta e / math 之间的范围。让我们再次关注数学 y / math 和数学 y + delta y / math 之间的能量本征态。由于这些能量本征态的能量增加了 y dx，所有这些能量本征态在 e-y dx 到 e 之间的区间内从 e 以下移动到 e 以上。 有<br />
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: <math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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<math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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数学 n { y }左(e 右) frac { y }左(e 右)}{ delta e } y dx ，/ math<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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这样的能量本征态。如果数学 y dx leq delta e / math，所有这些能量本征态将移动到数学 e / math 和数学 e + delta e / math 之间的范围内，有助于数学 ω / math 的增加。从数学 e + △ e / math 下移到数学 e + △ e / math 上的能量本征态的个数是由数学 n { y }左(e + △ e 右) / math 给出的。区别在于<br />
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: <math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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<math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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数学 n { y }左(e 右)-n { y }左(e + delta e 右) ，/ math<br />
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<br />
is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
<br />
is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
<br />
因此是数学增长的净贡献欧米茄 / 数学。请注意，如果 y dx 大于 math delta e / math，那么将会有能量本征态从 e 下面移动到 e + delta e / math 上面。它们在数学 n { y } left (e right) / math 和数学 n { y } left (e + delta e right) / math 中都有计算，因此上述表达式在这种情况下也是有效的。<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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将上面的表达式表示为对 e 的导数，对 y 的求和得到表达式:<br />
<br />
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: <math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
<br />
<math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
<br />
数学左(部分欧米茄左(部分欧米茄右){ e }-和 y 左(部分欧米茄右){ x 左(部分欧米茄左(部分欧米茄右)}{ x 右) ，/ 数学<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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关于 x 的欧米茄 / 数学对数导数由以下方式给出:<br />
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: <math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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Math 左( frac { partial ln 左( Omega 右)}{部分 x 右){ e } beta x + 左( frac { partial x 右){ x } ，/ math<br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
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第一个术语是密集型的，即。它不能根据系统大小进行缩放。相比之下，最后一项的规模与逆系统的规模一样，因此将在热力学极限中消失。因此，我们发现:<br />
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: <math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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<math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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数学左(部分 s }右){ e } frac { x }{ t } ，/ math<br />
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<br />
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Combining this with<br />
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Combining this with<br />
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结合这一点<br />
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: <math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
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<math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
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数学左(部分 s }右){ x } frac {1}{ t } ，/ math<br />
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Gives:<br />
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Gives:<br />
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给予:<br />
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: <math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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<math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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数学 dS 左(部分 s }部分 e }右){ x } dE + 左(部分 s }右){ e } dx frac { t } + frac { t }{ delta q }{ t } ，/ math<br />
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===Derivation for systems described by the canonical ensemble===<br />
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===Derivation for systems described by the canonical ensemble===<br />
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正则系系统的推导<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the [[canonical ensemble]]:<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:<br />
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如果一个系统与某个温度为T的热浴热接触，那么在平衡状态下，关于能量本征值的概率分布值由正则系综给出:<br />
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: <math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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<math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the [[Partition function (statistical mechanics)|partition function]]. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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这里Z是一个使所有概率之和归一化的因子，这个函数被称为配分函数。现在我们考虑对温度和能级所依赖的外部参数的无限小的可逆改变。它遵循熵的一般公式:<br />
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: <math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
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<math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
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<br />
<br />
<br />
<br />
<br />
that<br />
<br />
that<br />
<br />
以及<br />
<br />
<br />
<br />
<br />
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: <math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
<br />
<math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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把正则系综的数学形式 <math>P_{j}</math> 插入，可以得到:<br />
<br />
<br />
<br />
<br />
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: <math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
<br />
<math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
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==Living organisms==<br />
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==Living organisms==<br />
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生命体<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as [[What is Life?|Erwin Schrödinger]], [[Léon Brillouin]]<ref>Léon Brillouin [https://books.google.com/books?hl=en&lr=&id=tPXVbiw_1P0C ''Science and Information Theory''] (Academic Press, 1962) (Dover, 2004)</ref> and [[Life and Energy|Isaac Asimov]]. It is also the topic of current{{When|date=July 2019}} research.{{Citation needed|date=July 2019}}<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as Erwin Schrödinger, Léon Brillouin and Isaac Asimov. It is also the topic of current research.<br />
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表述热力学主要有两种方式：(a)描述从一种热力学平衡状态到另一种状态的路径，(b)通过使系统保持不变，同时增加周围环境的总熵的循环过程。这两种方法有助于理解生命过程。这个话题大大超出了本文的范围，但是已经有一些研究者考虑过这个问题，比如埃尔温·薛定谔、莱昂·布里渊和艾萨克.阿西莫夫。这也是当前研究的主题。<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of [[metabolism]], give out breakdown products and heat. Plants [[Photosynthesis|take in radiative energy]] from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
<br />
To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of metabolism, give out breakdown products and heat. Plants take in radiative energy from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
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从近似的角度来看，生命体可以被认为是(b)的一个例子。近似地，一只动物的身体状态每天循环，使得它几乎没有什么变化。动物吸收食物、水和氧气，经过新陈代谢，输出分解的产物和热量。植物吸收来自太阳的辐射能量，这可以认为是热量，以及二氧化碳和水，然后它们释放氧气。它们就是这样生长的，最终会死亡，尸体腐烂，大部分重新变成二氧化碳和水。这可以看作是一个循环过程。总的来说，阳光来自一个高温的源——太阳，它的能量被传递到一个较低的温度汇，例如向太空辐射。这个过程使得植物周围环境的熵增加。因此从循环过程的角度来看，动物和植物服从热力学第二定律。简单的热机效率概念很难适用于这个问题，因为它们假定系统是封闭的。<br />
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From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to [[Non-equilibrium thermodynamics|unsettled current debate or research]]. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
<br />
From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to unsettled current debate or research. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
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从(a)的热力学观点考虑 ，即考虑从一个平衡状态到另一个平衡状态的路径，只会产生一个粗略近似的图像，因为生命体从来不会处于热力学平衡状态。生物体经常必须被认为是开放的系统，因为它们吸收营养物质并排出废弃物。开放系统热力学目前通常从一个热力学平衡状态到另一个状态的路径的角度来考虑，或者是考虑局部热力学平衡近似下的流。生命体的问题可以通过假定一个流不变的稳态来进一步简化。对于这样的近似，熵产生的一般原理在目前有争论和研究的不确定性。尽管如此，从热力学第二定律的这个角度出发衍生出来的想法对生物是有启发意义的。<br />
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==Gravitational systems==<br />
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==Gravitational systems==<br />
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重力系统<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive [[heat capacity]], meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it.<ref name="Grandy 151">Grandy, W.T. (Jr) (2008), p. 151.</ref> Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
<br />
In systems that do not require for their descriptions the general theory of relativity, bodies always have positive heat capacity, meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it. Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
<br />
在不需要描述广义相对论的系统中，物体总是具有正的热容，这意味着温度随着能量的增加而升高。因此，当能量从高温物体流向低温物体时，源温度降低，而汇温度升高; 因此温差随时间而减小。这种情况并不总是发生在那些需要引力和广义相对论的系统中。这样的系统可以自发地朝着质量和能量不均匀分布的方向变化。这在大尺度上适用于宇宙，因此可能很难或不可能将第二定律应用于宇宙。除此之外，广义相对论所描述的系统热力学超出了本文的范围。<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
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{{main article|Non-equilibrium thermodynamics}}<br />
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The theory of classical or [[thermodynamic equilibrium|equilibrium thermodynamics]] is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<ref name="Grandy 151"/><ref>Callen, H.B. (1960/1985), p. 15.</ref><br />
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The theory of classical or equilibrium thermodynamics is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<br />
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经典热力学或平衡热力学的理想化。一个主要的假设或假设，往往甚至没有明确说明，是系统的存在，在他们自己的内部状态的热力学平衡。一般来说，在给定时间内包含一个物理系统的空间区域---- 可能在自然界中找到---- 不是在热力学平衡中，用最严格的术语来解释。从更宽松的角度来说，整个宇宙中没有任何东西是或者曾经是完全精确的热力学平衡。<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of [[thermodynamic equilibrium]]. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable [[thermal fluctuations|fluctuations]]. There is an exception, the case of [[critical point (thermodynamics)|critical states]], which exhibit to the naked eye the phenomenon of [[critical opalescence]]. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of thermodynamic equilibrium. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable fluctuations. There is an exception, the case of critical states, which exhibit to the naked eye the phenomenon of critical opalescence. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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为了物理分析的目的，假设热力学平衡是非常方便的。这种假设可能依赖于试错来证明其正当性。如果这个假设是正确的，它通常是非常有价值和有用的，因为它提供了热力学理论。平衡假设的要素是，一个系统被观察到在无限长的时间内是不变的，而且一个系统中有如此多的粒子，以至于它的微粒性质可以被完全忽略。在这样的平衡假设下，一般来说，没有宏观上可以探测到的波动。有一个例外，即临界状态的例子，它们用肉眼就可以看到临界乳光现象。对于临界状态的实验室研究，需要非常长的观察时间。<br />
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In all cases, the assumption of [[thermodynamic equilibrium]], once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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In all cases, the assumption of thermodynamic equilibrium, once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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在所有情况下，热力学平衡的假设，一旦做出，就意味着没有假定的候选者“涨落”改变系统的熵。<br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<ref>Lieb, E.H., Yngvason, J. (2003), p. 190.</ref><br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<br />
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一个物理系统很容易发生内部的宏观变化，其速度之快足以使熵的恒定性假设失效。或者一个物理系统的粒子太少，以至于微粒的本质在可观测的波动中是显而易见的。那么热力学平衡的假设就会被放弃。对于非平衡态，熵没有绝对的一般定义。<br />
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There are intermediate cases, in which the assumption of local [[thermodynamic equilibrium]] is a very good approximation,<ref>Gyarmati, I. (1967/1970), pp. 4-14.</ref><ref>Glansdorff, P., Prigogine, I. (1971).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003).</ref> but strictly speaking it is still an approximation, not theoretically ideal.<br />
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There are intermediate cases, in which the assumption of local thermodynamic equilibrium is a very good approximation, but strictly speaking it is still an approximation, not theoretically ideal.<br />
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还有一些中间情况，其中局部热力学平衡的假设是一个非常好的近似值，但严格地说它仍然是一个近似值，不是理论上的理想值。<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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对于一般的非平衡状态，考虑其他量的统计力学定义可能是有用的，这些量可以方便地称为‘熵’ ，但是它们不应该与第二定律正确定义的熵混淆或混淆。这些其他的量确实属于统计力学，而不是热力学---- 第二定律的主要领域。<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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宏观观测波动的物理学超出了本文的范围。<br />
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==Arrow of time==<br />
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==Arrow of time==<br />
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时间之箭<br />
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{{Further information|Entropy (arrow of time)}}Thermodynamic Asymmetry in Time<br />
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Thermodynamic Asymmetry in Time<br />
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时间中的热力学不对称性<br />
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{{See also|Arrow of time}}<br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely [[CPT symmetry]], since the second law applies statistically, it is hypothesized, on time-asymmetric ''boundary conditions''.<ref>{{cite web|first=Craig|last=Callender|url=https://plato.stanford.edu/archives/fall2011/entries/time-thermo/|title=Thermodynamic Asymmetry in Time|encyclopedia=Stanford Encyclopedia of Philosophy|date=29 July 2011}}</ref><br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely CPT symmetry, since the second law applies statistically, it is hypothesized, on time-asymmetric boundary conditions.<br />
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热力学第二定律是一个物理定律，它不对称于时间方向的反转。这与已经观察到的物理学基本定律，即 CPT 对称性的概念并不冲突，因为第二定律在统计上适用，它是假设的，是关于时间非对称的边界条件。<br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect ([[Arrow of time#The causal arrow of time|the causal arrow of time]]).<ref>{{cite book | first = J.J.| last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1|display-authors=etal}} chapter 6</ref><br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect (the causal arrow of time).<br />
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第二定律被提出来部分解释时间向前和向后移动的区别，例如为什么原因先于结果(因果关系的时间箭头)。<br />
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==Irreversibility==<br />
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==Irreversibility==<br />
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不可逆性<br />
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Irreversibility in [[thermodynamic process]]es is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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Irreversibility in thermodynamic processes is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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热力学过程的不可逆性是热力学作用不对称性的结果，而不是物体内部不可逆的微观性质的结果。热力学操作是施加在参与机构上的宏观外部干预，而不是从它们的内部性质中得到的。由于没有认识到这一点，产生了被称为“悖论”的说法。<br />
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===Loschmidt's paradox===<br />
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===Loschmidt's paradox===<br />
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洛斯密特悖论<br />
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{{main article|Loschmidt's paradox}}<br />
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[[Loschmidt's paradox]], also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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洛斯密特悖论，也被称为可逆性悖论，其目的在于不可能从描述宏观系统微观演化的时间对称动力学推导出不可逆性。<br />
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In the opinion of [[Erwin Schrödinger|Schrödinger]], "It is now quite obvious in what manner you have to reformulate the law of entropy{{snd}}or for that matter, all other irreversible statements{{snd}}so that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other."<ref>[[Erwin Schrödinger|Schrödinger, E.]] (1950), p.&nbsp;192.</ref> The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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In the opinion of Schrödinger, "It is now quite obvious in what manner you have to reformulate the law of entropyor for that matter, all other irreversible statementsso that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other." The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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根据薛定谔的观点，“现在很明显，你必须用什么方式来重新定义该物质的熵定律，所有其他不可逆的状态，这样它们才能从可逆模型中导出。你不能只谈论一个孤立的系统，而应该至少谈论两个，你可以暂时认为它们与世界其它地方是孤立的，但并不总是相互孤立的。”这两个系统被隔离墙隔离开来，直到按照法律的设想被热力学操作拆除。热力学操作是外部强加的，不受支配系统组成部分的可逆微观动力学定律的制约。它是不可逆转性的原因。本文中的法律陈述符合薛定谔的建议。因果关系在逻辑上优先于第二定律，而不是从第二定律派生出来的。<br />
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===Poincaré recurrence theorem===<br />
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===Poincaré recurrence theorem===<br />
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庞加莱始态复现定理<br />
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{{Main article|Poincaré recurrence theorem}}<br />
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The [[Poincaré recurrence theorem]] considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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The Poincaré recurrence theorem considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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庞加莱始态复现定理认为这是一个孤立的物理系统的理论微观描述。这可以被认为是一个热力学系统的模型后，一个热力学操作已经移除了一个内墙。在足够长的时间之后，系统将返回到一个非常接近初始状态的显微定义状态。庞加莱回归时间是指回归前经过的时间长度。它极其漫长，可能比宇宙的寿命还要长，并且敏感地依赖于被热力学操作拆除的墙体的几何形状。递归定理可能被认为是明显与热力学第二定律相矛盾的。然而，更明显的是，它只不过是一个微观模型，热力学平衡在一个孤立的系统中，通过移除两个系统之间的一道墙而形成。对于一个典型的热力学系统，重现时间是如此之大(比宇宙的生命周期长许多倍) ，以至于在所有的实际目的中，人们都不能观察到重现。然而，人们可能希望想象一下，可以等待庞加莱复发，然后重新插入被热力学操作移除的壁。然后很明显，不可逆性的出现是由于庞加莱循环的完全不可预测性，只要初始状态是热力学平衡，就像宏观热力学的情况一样。即使可以等待，也没有实际可能性选择合适的时间重新插入墙壁。庞加莱始态复现定理为 Loschmidt 悖论提供了一个解决方案。如果一个孤立的热力学系统可以被监测到在越来越多的平均 poincar 递归时间的倍数上，系统的热力学行为将在时间反转下变得不变。<br />
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[[File:James-clerk-maxwell3.jpg|thumb|upright|James Clerk Maxwell]]<br />
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James Clerk Maxwell<br />
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詹姆斯·克拉克·麦克斯韦<br />
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===Maxwell's demon===<br />
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===Maxwell's demon===<br />
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麦克斯韦的魔鬼<br />
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{{unreferenced|section|date=August 2018}}<br />
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{{main article|Maxwell's demon}}<br />
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[[James Clerk Maxwell]] imagined one container divided into two parts, ''A'' and ''B''. Both parts are filled with the same [[gas]] at equal temperatures and placed next to each other, separated by a wall. Observing the [[molecule]]s on both sides, an imaginary [[demon]] guards a microscopic trapdoor in the wall. When a faster-than-average molecule from ''A'' flies towards the trapdoor, the demon opens it, and the molecule will fly from ''A'' to ''B''. The average [[speed]] of the molecules in ''B'' will have increased while in ''A'' they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in ''A'' and increases in ''B'', contrary to the second law of thermodynamics.<br />
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James Clerk Maxwell imagined one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other, separated by a wall. Observing the molecules on both sides, an imaginary demon guards a microscopic trapdoor in the wall. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics.<br />
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詹姆斯·克拉克·麦克斯韦想象一个容器分成两部分，a 和 b。两个部分在相同的温度下充入相同的气体，并被一堵墙隔开，挨着放置。通过观察两边的分子，一个想象中的恶魔守卫着墙上的一个微型暗门。当一个来自 a 的比平均速度快的分子飞向活板门时，魔鬼打开了它，分子就会从 a 飞到 b。B 中分子的平均速度会增加，而 a 中分子的平均速度会减慢。由于平均分子速度与温度相对应，因此 a 中的温度降低，b 中的温度升高，这与热力学第二定律相反。<br />
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One response to this question was suggested in 1929 by [[Leó Szilárd]] and later by [[Léon Brillouin]]. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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One response to this question was suggested in 1929 by Leó Szilárd and later by Léon Brillouin. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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1929年，Leó Szilárd 和后来的 l on brilouin 对这个问题提出了一个答案。Szil rd 指出，现实生活中的麦克斯韦恶魔需要一些测量分子速度的方法，而获取信息需要消耗大量能量。<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between ''A'' and ''B''. It is therefore performing [[thermodynamic operation]]s on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between A and B. It is therefore performing thermodynamic operations on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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麦克斯韦的“魔鬼”不断地改变 a 和 b 之间墙体的渗透性。因此，它是在微观尺度上执行热力学操作，而不仅仅是观察普通的自发或自然的宏观热力学过程。<br />
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==Quotations==<br />
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==Quotations==<br />
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语录<br />
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{{Wikiquote}}<br />
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{{quote|The law that entropy always increases holds, I think, the supreme position among the [[Laws of science|laws of Nature]]. If someone points out to you that your pet theory of the [[universe]] is in disagreement with [[Maxwell's equations]] – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.|Sir [[Arthur Stanley Eddington]], ''The Nature of the Physical World'' (1927)}}<br />
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{{quote|There have been nearly as many formulations of the second law as there have been discussions of it.|Philosopher / Physicist [[Percy Williams Bridgman|P.W. Bridgman]], (1941)}}<br />
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{{quote|Clausius is the author of the sibyllic utterance, "The energy of the universe is constant; the entropy of the universe tends to a maximum." The objectives of continuum thermomechanics stop far short of explaining the "universe", but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size.|[[Clifford Truesdell|Truesdell, C.]], Muncaster, R. G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 17.}}<br />
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==See also==<br />
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==See also==<br />
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参见<br />
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*[[Clausius–Duhem inequality]]<br />
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*[[Fluctuation theorem]]<br />
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*[[Heat death of the universe]]<br />
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*[[History of thermodynamics]]<br />
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*[[Jarzynski equality]]<br />
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*[[Laws of thermodynamics]]<br />
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*[[Maximum entropy thermodynamics]]<br />
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*[[Quantum thermodynamics]]<br />
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*[[Reflections on the Motive Power of Fire]]<br />
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*[[Relativistic heat conduction]]<br />
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*[[Thermal diode]]<br />
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*[[Thermodynamic equilibrium]]<br />
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{{colend}}<br />
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==References==<br />
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==References==<br />
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参考资料<br />
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===Sources===<br />
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资料来源<br />
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{{refbegin|50em}}<br />
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*Adkins, C.J. (1968/1983). ''Equilibrium Thermodynamics'', (1st edition 1968), third edition 1983, Cambridge University Press, Cambridge UK, {{ISBN|0-521-25445-0}}.<br />
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*[[Peter Atkins|Atkins, P.W.]], de Paula, J. (2006). ''Atkins' Physical Chemistry'', eighth edition, W.H. Freeman, New York, {{ISBN|978-0-7167-8759-4}}.<br />
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*Attard, P. (2012). ''Non-equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications'', Oxford University Press, Oxford UK, {{ISBN|978-0-19-966276-0}}.<br />
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*Baierlein, R. (1999). ''Thermal Physics'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-59082-5}}.<br />
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*Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics, New York, {{ISBN|0-88318-797-3}}.<br />
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*{{Cite book|title=Concepts in thermal physics|last=Blundell|first=Stephen J.|last2=Blundell|authorlink1=Stephen Blundell|first2=Katherine M.|authorlink2=Katherine Blundell|publisher=[[Oxford University Press]]|year=2010|isbn=9780199562107|edition=2nd|location=Oxford|pages=|oclc=607907330|doi=10.1093/acprof:oso/9780199562091.001.0001 |url=http://cds.cern.ch/record/1235139}}<br />
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*[[Ludwig Boltzmann|Boltzmann, L.]] (1896/1964). ''Lectures on Gas Theory'', translated by S.G. Brush, University of California Press, Berkeley.<br />
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*Borgnakke, C., Sonntag., R.E. (2009). ''Fundamentals of Thermodynamics'', seventh edition, Wiley, {{ISBN|978-0-470-04192-5}}.<br />
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*Buchdahl, H.A. (1966). ''The Concepts of Classical Thermodynamics'', Cambridge University Press, Cambridge UK.<br />
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*[[Percy Williams Bridgman|Bridgman, P.W.]] (1943). ''The Nature of Thermodynamics'', Harvard University Press, Cambridge MA.<br />
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*[[Herbert Callen|Callen, H.B.]] (1960/1985). ''Thermodynamics and an Introduction to Thermostatistics'', (1st edition 1960) 2nd edition 1985, Wiley, New York, {{ISBN|0-471-86256-8}}.<br />
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*{{cite journal|author=C. Carathéodory|author1-link=Constantin Carathéodory<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
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|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
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2010年3月15日 | 第355-386页 | http://gdz.sub.uni-goettingen.de/index.php?id=11&ppn=ppn235181684_0067&dmdid=dmdlog_0033&l=1<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). ''The Second Law of Thermodynamics'', Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können.(p. 363) | doi 10.1007 / bf01450409}.译文可以在这里找到[ http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf ]。同时，在 Kestin，j。(1976).热力学第二定律，Dowden，Hutchinson & Ross，Stroudsburg PA. 。<br />
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*[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986). [http://www.worldcat.org/title/reflections-on-the-motive-power-of-fire-a-critical-edition-with-the-surviving-scientific-manuscripts-translated-and-edited-by-fox-robert/oclc/812944517&referer=brief_results ''Reflections on the motive power of fire''], Manchester University Press, Manchester UK, {{ISBN|0-7190-1741-6}}. [https://archive.org/stream/reflectionsonmot00carnrich#page/n7/mode/2up Also here.]<br />
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*[[Sydney Chapman (mathematician)|Chapman, S.]], [[Thomas George Cowling|Cowling, T.G.]] (1939/1970). ''The Mathematical Theory of Non-uniform gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases'', third edition 1970, Cambridge University Press, London.<br />
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*{{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Ueber Die Bewegende Kraft Der Wärme Und Die Gesetze, Welche Sich Daraus Für Die Wärmelehre Selbst Ableiten Lassen|journal=Annalen der Physik|year=1850|volume=79|issue=4|pages=368–397, 500–524|url=http://gallica.bnf.fr/ark:/12148/bpt6k15164w/f518.image|accessdate=26 June 2012|doi=10.1002/andp.18501550403|bibcode = 1850AnP...155..500C |hdl=2027/uc1.$b242250|hdl-access=free}} Translated into English: {{cite journal|last=Clausius|first=R.|title=On the Moving Force of Heat, and the Laws regarding the Nature of Heat itself which are deducible therefrom|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1851| volume=2|series=4th|issue=VIII|pages=1–21; 102–119 |url=https://archive.org/stream/londonedinburghd02lond#page/1/mode/1up|accessdate=26 June 2012<br />
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|doi=10.1080/14786445108646819}}<br />
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|doi=10.1080/14786445108646819}}<br />
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10.1080 / 14786445108646819}<br />
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* {{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie|year=1854|journal=Annalen der Physik|volume=xciii|issue=12|pages=481–506|url=http://zfbb.thulb.uni-jena.de/servlets/MCRFileNodeServlet/jportal_derivate_00140956/18541691202_ftp.pdf|accessdate=24 March 2014|doi=10.1002/andp.18541691202<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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| bibcode 1854AnP... 169. 1856年7月 | 第2卷 | 第4页 | 第86页 | https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=242014年3月<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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热力学理论及其在蒸汽机和物体物理性质中的应用 | 1867年 | 出版商 John van Voorst | 位置: 伦敦 | https://archive.org/details/mechanicaltheor04claugoog|quote=editions:pwr_sbkwa8ic。 2012年6月19日<br />
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*Denbigh, K. (1954/1981). ''The Principles of Chemical Equilibrium. With Applications in Chemistry and Chemical Engineering'', fourth edition, Cambridge University Press, Cambridge UK, {{ISBN|0-521-23682-7}}.<br />
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*Eu, B.C. (2002). ''Generalized Thermodynamics. The Thermodynamics of Irreversible Processes and Generalized Hydrodynamics'', Kluwer Academic Publishers, Dordrecht, {{ISBN|1-4020-0788-4}}.<br />
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*[[Josiah Willard Gibbs|Gibbs, J.W.]] (1876/1878). On the equilibrium of heterogeneous substances, ''Trans. Conn. Acad.'', '''3''': 108-248, 343-524, reprinted in ''The Collected Works of J. Willard Gibbs, Ph.D, LL. D.'', edited by W.R. Longley, R.G. Van Name, Longmans, Green & Co., New York, 1928, volume 1, pp.&nbsp;55–353.<br />
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*Griem, H.R. (2005). ''Principles of Plasma Spectroscopy (Cambridge Monographs on Plasma Physics)'', Cambridge University Press, New York {{ISBN|0-521-61941-6}}.<br />
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*Glansdorff, P., Prigogine, I. (1971). ''Thermodynamic Theory of Structure, Stability, and Fluctuations'', Wiley-Interscience, London, 1971, {{ISBN|0-471-30280-5}}.<br />
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*Grandy, W.T., Jr (2008). [http://global.oup.com/academic/product/entropy-and-the-time-evolution-of-macroscopic-systems-9780199546176?cc=au&lang=en& ''Entropy and the Time Evolution of Macroscopic Systems'']. Oxford University Press. {{ISBN|978-0-19-954617-6}}.<br />
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*Greven, A., Keller, G., Warnecke (editors) (2003). ''Entropy'', Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[Edward A. Guggenheim|Guggenheim, E.A.]] (1949). 'Statistical basis of thermodynamics', ''Research'', '''2''': 450–454.<br />
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*[[Edward A. Guggenheim|Guggenheim, E.A.]] (1967). ''Thermodynamics. An Advanced Treatment for Chemists and Physicists'', fifth revised edition, North Holland, Amsterdam.<br />
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*Gyarmati, I. (1967/1970) ''Non-equilibrium Thermodynamics. Field Theory and Variational Principles'', translated by E. Gyarmati and W.F. Heinz, Springer, New York.<br />
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*[[Charles Kittel|Kittel, C.]], [[Herbert Kroemer|Kroemer, H.]] (1969/1980). ''Thermal Physics'', second edition, Freeman, San Francisco CA, {{ISBN|0-7167-1088-9}}.<br />
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*Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998). ''Modern Thermodynamics: From Heat Engines to Dissipative Structures'', John Wiley & Sons, Chichester, {{ISBN|0-471-97393-9}}.<br />
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*Lebon, G., Jou, D., Casas-Vázquez, J. (2008). ''Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers'', Springer-Verlag, Berlin, {{ISBN|978-3-540-74252-4}}.<br />
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*{{cite journal|ref=harv|title=The Physics and Mathematics of the Second Law of Thermodynamics|last1=Lieb |first1=E. H. |last2=Yngvason |first2=J.|journal=Physics Reports|volume=310|issue=1|pages=1–96 |year=1999|doi=10.1016/S0370-1573(98)00082-9|arxiv = cond-mat/9708200 |bibcode = 1999PhR...310....1L}}<br />
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*Lieb, E.H., Yngvason, J. (2003). The Entropy of Classical Thermodynamics, pp.&nbsp;147–195, Chapter 8 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*{{cite book|ref=harv|first=F.|last=Mandl|title=Statistical physics|edition=second|year=1988|publisher=[[Wiley & Sons]]|isbn=978-0-471-91533-1}}<br />
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*{{cite journal | last1 = Maxwell | first1 = J.C. | authorlink = James Clerk Maxwell | year = 1867 | title = On the dynamical theory of gases | url = | journal = Phil. Trans. R. Soc. Lond. | volume = 157 | issue = | pages = 49–88 | doi = 10.1098/rstl.1867.0004 }}<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985). ''Thermodynamics'', Pitman, London, {{ISBN|0-273-08577-8}}.<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003). Entropy in Nonequilibrium, pp.&nbsp;79–109, Chapter 5 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*Münster, A. (1970), ''Classical Thermodynamics'', translated by E.S. Halberstadt, Wiley–Interscience, London, {{ISBN|0-471-62430-6}}.<br />
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*[[Brian Pippard|Pippard, A.B.]] (1957/1966). ''Elements of Classical Thermodynamics for Advanced Students of Physics'', original publication 1957, reprint 1966, Cambridge University Press, Cambridge UK.<br />
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*[[Max Planck|Planck, M.]] (1897/1903). [https://archive.org/stream/treatiseonthermo00planrich#page/100/mode/2up ''Treatise on Thermodynamics'', translated by A. Ogg, Longmans Green, London, p. 100.]<br />
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*[[Max Planck|Planck. M.]] (1914). [https://archive.org/details/theoryofheatradi00planrich ''The Theory of Heat Radiation''], a translation by Masius, M. of the second German edition, P. Blakiston's Son & Co., Philadelphia.<br />
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*[[Max Planck|Planck, M.]] (1926). Über die Begründung des zweiten Hauptsatzes der Thermodynamik, ''Sitzungsberichte der Preussischen Akademie der Wissenschaften: Physikalisch-mathematische Klasse'': 453–463.<br />
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* Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769-781.<br />
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* {{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}<br />
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*Quinn, T.J. (1983). ''Temperature'', Academic Press, London, {{ISBN|0-12-569680-9}}.<br />
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*{{cite book|ref=harv|first=Y.V.C.|last=Rao|title=An Introduction to thermodynamics|url=https://books.google.com/books?id=iYWiCXziWsEC&pg=PA213|date=2004|publisher=Universities Press|isbn=978-81-7371-461-0|page=213}}<br />
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*Roberts, J.K., Miller, A.R. (1928/1960). ''Heat and Thermodynamics'', (first edition 1928), fifth edition, Blackie & Son Limited, Glasgow.<br />
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*[[Erwin Schrödinger|Schrödinger, E.]] (1950). Irreversibility, ''Proc. R. Ir. Acad.'', '''A53''': 189–195.<br />
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*[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966). ''Elements of Thermodynamics'', Addison-Wesley Publishing, Reading MA.<br />
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*{{cite journal|ref=harv|last=Thomson|first=W.|author-link=William Thomson, 1st Baron Kelvin|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Transactions of the Royal Society of Edinburgh|year=1851|volume=XX|issue=part II|pages=261–268; 289–298|url=https://www.biodiversitylibrary.org/item/126047#page/295/mode/1up}} Also published in {{cite journal|last=Thomson|first=W.|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Philos. Mag. |date=December 1852 |volume=IV |series=4 |issue=22 |page=13|url=https://archive.org/stream/londonedinburghp04maga#page/12/mode/2up |accessdate=25 June 2012}}<br />
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*[[William Thomson, 1st Baron Kelvin|Thomson, W.]] (1852). ''On the universal tendency in nature to the dissipation of mechanical energy'' Philosophical Magazine, Ser. 4, p.&nbsp;304.<br />
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*[[László Tisza|Tisza, L.]] (1966). ''Generalized Thermodynamics'', M.I.T Press, Cambridge MA.<br />
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*[[Clifford Truesdell|Truesdell, C.]] (1980). ''The Tragicomical History of Thermodynamics 1822–1854'', Springer, New York, {{ISBN|0-387-90403-4}}.<br />
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*Uffink, J. (2001). Bluff your way in the second law of thermodynamics, ''Stud. Hist. Phil. Mod. Phys.'', '''32'''(3): 305–394.<br />
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*Uffink, J. (2003). Irreversibility and the Second Law of Thermodynamics, Chapter 7 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003), Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963). ''Lectures in Statistical Mechanics'', American Mathematical Society, Providence RI.<br />
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*[[Mark Zemansky|Zemansky, M.W.]] (1968). ''Heat and Thermodynamics. An Intermediate Textbook'', fifth edition, McGraw-Hill Book Company, New York.<br />
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==Further reading==<br />
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==Further reading==<br />
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进一步阅读<br />
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*Goldstein, Martin, and Inge F., 1993. ''The Refrigerator and the Universe''. Harvard Univ. Press. Chpts. 4–9 contain an introduction to the Second Law, one a bit less technical than this entry. {{ISBN|978-0-674-75324-2}}<br />
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*Leff, Harvey S., and Rex, Andrew F. (eds.) 2003. ''Maxwell's Demon 2 : Entropy, classical and quantum information, computing''. Bristol UK; Philadelphia PA: [[Institute of Physics]]. {{ISBN|978-0-585-49237-7}}<br />
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*{{Cite book | first = J.J. | last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1}}(technical).<br />
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*{{cite book |title=Reflections on the Motive Power of Heat and on Machines Fitted to Develop That Power |last=Carnot |first=Sadi |editor=[[Robert Henry Thurston|Thurston, Robert Henry]] |year=1890 |publisher=J. Wiley & Sons |location=New York |isbn= |pages= }} ([https://books.google.com/books?id=tgdJAAAAIAAJ full text of 1897 ed.]) ([http://www.history.rochester.edu/steam/carnot/1943/ html])<br />
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*Stephen Jay Kline (1999). ''The Low-Down on Entropy and Interpretive Thermodynamics'', La Cañada, CA: DCW Industries. {{ISBN|1-928729-01-0}}.<br />
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*{{cite book | last1 = Kostic | first1 = M | year = 2011 | title = Revisiting The Second Law of Energy Degradation and Entropy Generation: From Sadi Carnot's Ingenious Reasoning to Holistic Generalization | journal = AIP Conf. Proc. | volume = 1411 | issue = 1| pages = 327–350 | doi = 10.1063/1.3665247 | isbn = 978-0-7354-0985-9 | bibcode = 2011AIPC.1411..327K | series = AIP Conference Proceedings | citeseerx = 10.1.1.405.1945 }} also at [https://web.archive.org/web/20130420222450/http://www.kostic.niu.edu/2ndLaw/Revisiting%20The%20Second%20Law%20of%20Energy%20Degradation%20and%20Entropy%20Generation%20-%20From%20Carnot%20to%20Holistic%20Generalization-4.pdf].<br />
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==External links==<br />
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==External links==<br />
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外部链接<br />
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*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics]" – by Lawrence Sklar.<br />
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*[http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node30.html ''Second law of thermodynamics''] in the MIT Course [http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/notes.html ''Unified Thermodynamics and Propulsion''] from Prof. Z. S. Spakovszky<br />
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*[[E.T. Jaynes]], 1988, "[http://bayes.wustl.edu/etj/articles/ccarnot.pdf The evolution of Carnot's principle]," in G. J. Erickson and C. R. Smith (eds.)''Maximum-Entropy and Bayesian Methods in Science and Engineering, Vol'' 1: p.&nbsp;267.<br />
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*[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf Caratheodory, C., "Examination of the foundations of thermodynamics," trans. by D. H. Delphenich]<br />
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* [https://www.bbc.co.uk/programmes/p004y2bm The Second Law of Thermodynamics], BBC Radio 4 discussion with John Gribbin, Peter Atkins & Monica Grady (''In Our Time'', Dec. 16, 2004)<br />
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* [http://mdpi.org/entropy/papers/e6010001.pdf [[Entropy (journal)]], 2004]<br />
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* [https://www.journals.uchicago.edu/doi/abs/10.1086/663835 The Journal of the International Society for the History of Philosophy of Science, 2012]<br />
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[[Category:Concepts in physics]]<br />
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Category:Concepts in physics<br />
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分类: 物理概念<br />
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[[Category:Laws of thermodynamics|2]]<br />
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[[Category:Non-equilibrium thermodynamics]]<br />
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Category:Non-equilibrium thermodynamics<br />
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类别: 非平衡态热力学<br />
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[[Category:Philosophy of thermal and statistical physics]]<br />
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Category:Philosophy of thermal and statistical physics<br />
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类别: 热力学和统计物理学哲学<br />
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<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Second law of thermodynamics]]. Its edit history can be viewed at [[热力学第二定律/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E7%83%AD%E5%8A%9B%E5%AD%A6%E7%AC%AC%E4%BA%8C%E5%AE%9A%E5%BE%8B&diff=10357热力学第二定律2020-07-16T09:35:04Z<p>Dorr：/* Clausius inequality 克劳修斯不等式 */</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Law of physics}}<br />
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{{Thermodynamics|laws}}<br />
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The '''second law of thermodynamics''' states that the total [[entropy]] of an [[isolated system]] can never decrease over time, and is constant if and only if all processes are reversible.<ref>http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node38.html#SECTION05224000000000000000</ref> Isolated systems spontaneously evolve towards [[thermodynamic equilibrium]], the state with maximum entropy. <br />
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The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, and is constant if and only if all processes are reversible. Isolated systems spontaneously evolve towards thermodynamic equilibrium, the state with maximum entropy. <br />
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热力学第二定律指出，孤立系统的总熵永远不会随着时间而减少，而且只有当且仅当所有过程都是可逆的时，总熵才是恒定的。孤立的系统自发地进化到热力学平衡状态，即熵最大的状态。<br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in [[thermodynamic equilibrium]], or is undergoing a (fictive) [[Reversible process (thermodynamics)|reversible process]]. In all processes that occur, including [[spontaneous process]]es,<ref>Atkins and de Paula, p.78</ref> the total entropy of the system and its surroundings increases and the process is [[Irreversible process|irreversible in the thermodynamic sense]]. The increase in entropy accounts for the irreversibility of natural processes, and the [[Arrow of time|asymmetry between future and past]].<ref>{{cite book|last=Zohuri|first=Bahman|title=Dimensional Analysis Beyond the Pi Theorem|url=https://books.google.com/books?id=pRVuDQAAQBAJ|year=2016|publisher=Springer|isbn=978-3-319-45726-0|page=[https://books.google.com.ph/books?id=pRVuDQAAQBAJ&pg=PA111&dq=%22increase+in+entropy+accounts+for+the+irreversibility+of+natural+processes+and+the+asymmetry+between+future+and+past.%22&hl=en&sa=X&ved=0ahUKEwj6spb61tbaAhUFS7wKHfftDtIQ6AEIKDAA#v=onepage&q=%22increase%20in%20entropy%20accounts%20for%20the%20irreversibility%20of%20natural%20processes%20and%20the%20asymmetry%20between%20future%20and%20past.%22 111]}}</ref><br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in thermodynamic equilibrium, or is undergoing a (fictive) reversible process. In all processes that occur, including spontaneous processes, the total entropy of the system and its surroundings increases and the process is irreversible in the thermodynamic sense. The increase in entropy accounts for the irreversibility of natural processes, and the asymmetry between future and past.<br />
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系统及其周围环境的总熵在理想情况下可以保持不变，在这种情况下，系统处于热力学平衡期，或者正在经历一个(虚拟的)可逆过程期。在所有发生的过程中，包括自发过程，系统及其周围环境的总熵增加，这一过程在热力学意义上是不可逆的。熵的增加解释了自然过程的不可逆性，以及未来和过去之间的不对称性。<br />
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Historically, the second law was an [[Empirical evidence|empirical finding]] that was accepted as an axiom of [[thermodynamics|thermodynamic theory]]. [[Statistical mechanics]], classical or [[quantum thermodynamics|quantum]], explains the microscopic origin of the law.<br />
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Historically, the second law was an empirical finding that was accepted as an axiom of thermodynamic theory. Statistical mechanics, classical or quantum, explains the microscopic origin of the law.<br />
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从历史上看，第二定律是一个经验性的发现，被公认为热力学理论的公理。统计力学，无论是经典的还是量子的，都可以解释这个定律的微观起源。<br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist [[Nicolas Léonard Sadi Carnot|Sadi Carnot]], who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as [[Carnot's theorem (thermodynamics)|Carnot's rule]] or limit.<ref>Jaffe, R.L., Taylor, W. (2018). ''The Physics of Energy'', Cambridge University Press, Cambridge UK, pages 150, 151, 259, 772, 743.</ref><br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist Sadi Carnot, who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as Carnot's rule or limit.<br />
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第二定律已经用许多方式表达过了。法国科学家萨迪 · 卡诺在1824年证明了热能在热机中转化为功的效率是有上限的。这方面的第二定律也被称为卡诺的规则或限制。<br />
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==Introduction==<br />
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==Introduction==<br />
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引言<br />
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[[File:Heat flow hot to cold.png|thumb|upright|Heat flow from hot water to cold water.]]<br />
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Heat flow from hot water to cold water.<br />
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热量从热水流向冷水。<br />
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The [[first law of thermodynamics]] provides the definition of the [[internal energy]] of a [[thermodynamic system]], and expresses the law of [[conservation of energy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 40–41.</ref><ref>Munster A. (1970), pp. 8–9, 50–51.</ref> The second law is concerned with the direction of natural processes.<ref>{{harvnb|Mandl|1988}}</ref> It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such [[Phenomenon|phenomena]] are accounted for in terms of [[entropy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 79–107.</ref><ref>Bailyn, M. (1994), Section 71, pp. 113–154.</ref> If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, {{math|''S''}}, increases.<br />
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The first law of thermodynamics provides the definition of the internal energy of a thermodynamic system, and expresses the law of conservation of energy. The second law is concerned with the direction of natural processes. It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such phenomena are accounted for in terms of entropy. If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, , increases.<br />
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能量守恒定律提供了热力学系统内能的定义，并表达了能量守恒定律。第二定律与自然过程的方向有关。它断言自然过程只在一种意义上运行，而且是不可逆的。例如，当有了传导和辐射的路径时，热总是自发地从一个较热的物体流向一个较冷的物体。这种现象可以用熵来解释。如果一个孤立的系统最初是通过内部分隔不可渗透的墙来保持在内部热力学平衡，然后一些操作使墙更具渗透性，那么系统自发地进化到最终达到一个新的内部热力学平衡，其总熵增加。<br />
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In a fictive reversible process, an infinitesimal increment in the entropy ({{math|d''S''}}) of a system is defined to result from an infinitesimal transfer of heat ({{math|δ''Q''}}) to a [[closed system]] (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature ({{math|''T''}}) of the system in equilibrium and the surroundings which supply the heat:<ref>Bailyn, M. (1994), p. 120.</ref><br />
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In a fictive reversible process, an infinitesimal increment in the entropy () of a system is defined to result from an infinitesimal transfer of heat () to a closed system (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature () of the system in equilibrium and the surroundings which supply the heat:<br />
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在虚构的可逆过程中，系统熵的无穷小增量被定义为由无穷小的热量传递()到一个封闭系统(允许能量进入或出去，但不允许物质传递)除以平衡系统和提供热量的环境的共同温度() :<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, idealized fictive reversible process)}.</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, idealized fictive reversible process)}.</math><br />
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(闭合系统，理想化的可逆过程) / math<br />
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Different notations are used for infinitesimal amounts of heat ({{mvar|δ}}) and infinitesimal amounts of entropy ({{mvar|d}}) because entropy is a [[function of state]], while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the [[Clausius Theorem|inequality]] <ref name="MortimerBook"> Mortimer, R. G. ''Physical Chemistry''. 3rd ed., p. 120, Academic Press, 2008. </ref><ref name="FermiBook"> Fermi, E. ''Thermodynamics'', footnote on p. 48, Dover Publications,1956 (still in print). </ref> <br />
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Different notations are used for infinitesimal amounts of heat () and infinitesimal amounts of entropy () because entropy is a function of state, while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the inequality <br />
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用不同的符号表示无穷小量的热量()和无穷小量的熵() ，因为熵是状态的函数，而热量和功一样不是。第二定律要求系统熵的增量满足不等式，对于实际上可能存在的不与环境发生质量交换的无穷小过程，系统熵增量满足不等式<br />
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: <math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process).}</math><br />
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<math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process).}</math><br />
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(实际上可能是封闭系统，不可逆性)。数学<br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature ({{math|''T''}}) and the temperature of the surroundings ({{math|''T''}}<sub>''surr''</sub>).<ref name=":0">Adkins, C.J. (1968/1983), p. 75.</ref><ref name="Munster 45"/> Note that the equality still applies for pure heat flow,<ref name="Schmidt-Rohr 14"> Schmidt-Rohr, K. (2014). "Expansion Work without the External Pressure, and Thermodynamics in Terms of Quasistatic Irreversible Processes" ''J. Chem. Educ.'' '''91''': 402-409. https://dx.doi.org/10.1021/ed3008704 </ref><br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature () and the temperature of the surroundings (<sub>surr</sub>).<br />
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这是因为这种情况下的一般过程可能包括周围环境对系统所做的功，这可能在系统内部产生摩擦或粘滞效应，因为一个化学反应可能正在进行，或者因为热传递实际上只是不可逆地发生，由系统温度()和周围环境温度(sub surr / sub)之间的差分驱动。<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change).}</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change).}</math><br />
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(实际上可能是不改变成分的准静态不可逆性)。数学<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry.<ref name="Oxtoby8th">Oxtoby, D. W; Gillis, H.P., [[Laurie Butler|Butler, L. J.]] (2015).''Principles of Modern Chemistry'', Brooks Cole. p. 617. {{ISBN|978-1305079113}}</ref> <ref name="MortimerBook"></ref> Introducing a set of internal variables <math>\xi</math> to describe the deviation of a thermodynamic system in physical equilibrium (with the required well-defined uniform pressure ''P'' and temperature ''T'')<ref name="Schmidt-Rohr 14"></ref> from the chemical equilibrium state, one can record the equality<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry. from the chemical equilibrium state, one can record the equality<br />
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这是根据测得的热容曲线和相变时的熵变精确确定纯物质的绝对熵的基础。用量热法。从化学平衡来看，人们可以记录平等<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process).}</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process).}</math><br />
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- frac { t }-frac {1}{ j } ， Xi { j } ， delta Xi j， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ,.数学<br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of ''the second principle of thermodynamics'' – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<ref> Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769–781.</ref><ref>{{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}</ref><br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of the second principle of thermodynamics – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<br />
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第二项代表内部变量的工作，这些内部变量可以受到外部影响的干扰，但是系统不能通过内部变量执行任何正功。这种说法介绍了热力学系统在时间上不可能逆转演化的事实，并且可以被认为是热力学第二原理的表述---- 这个表述，当然，相当于熵原理的表述。<br />
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The [[zeroth law of thermodynamics]] in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body.<ref name=dugdale>{{cite book|author=J. S. Dugdale|title=Entropy and its Physical Meaning|publisher=Taylor & Francis|year=1996|isbn=978-0-7484-0569-5|page=13|quote=This law is the basis of temperature.}}</ref> For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows{{how?|date=August 2018}} a distinguished temperature scale, which defines an absolute, [[thermodynamic temperature]], independent of the properties of any particular reference thermometric body.<ref>[[Mark Zemansky|Zemansky, M.W.]] (1968), pp. 207–209.</ref><ref>Quinn, T.J. (1983), p. 8.</ref><br />
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The zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body. For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows a distinguished temperature scale, which defines an absolute, thermodynamic temperature, independent of the properties of any particular reference thermometric body.<br />
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美国热力学第零定律协会在其通常的简短声明中允许承认，相对于热平衡的两个物体具有相同的温度，特别是测试物体具有与参考温度测量物体相同的温度。对于一个物体和另一个物体之间的热平衡，存在着无限多的经验温度标度，通常分别取决于特定参考温度标度的性质。第二定律允许区分温度标度，它定义了一个绝对的热力学温度，与任何特定的参考温度体的性质无关。<br />
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==Various statements of the law==<br />
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==Various statements of the law==<br />
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各种各样的法律陈述<br />
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The second law of thermodynamics may be expressed in many specific ways,<ref name=MIT>{{cite web|title=Concept and Statements of the Second Law|url=http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node37.html|accessdate=2010-10-07 |publisher=web.mit.edu}}</ref> the most prominent classical statements{{sfnp|Lieb|Yngvason|1999}} being the statement by [[Rudolf Clausius]] (1854), the statement by [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] (1851), and the statement in axiomatic thermodynamics by [[Constantin Carathéodory]] (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.{{sfnp|Rao|2004|p=213}}<br />
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The second law of thermodynamics may be expressed in many specific ways, the most prominent classical statements being the statement by Rudolf Clausius (1854), the statement by Lord Kelvin (1851), and the statement in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.<br />
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热力学第二定律可以用许多特定的方式来表达，最突出的经典陈述是 Rudolf Clausius (1854)的陈述，Kelvin 勋爵(1851)的陈述，以及康斯坦丁·卡拉西奥多里(1909)在公理化热力学中的陈述。这些陈述用一般的物理术语描述法律，引用某些过程的不可能性。克劳修斯和开尔文陈述被证明是等价的。<br />
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===Carnot's principle===<br />
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===Carnot's principle===<br />
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卡诺原理<br />
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The historical origin<ref>[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986).</ref> of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a [[Carnot heat engine]], fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the [[caloric theory]] of heat was seriously considered, before the recognition of the [[first law of thermodynamics]], and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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The historical origin of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a Carnot heat engine, fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the caloric theory of heat was seriously considered, before the recognition of the first law of thermodynamics, and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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热力学第二定律的历史起源源于卡诺的原则。它指的是一个卡诺热机周期，虚拟地以被称为准静态的极慢极限模式运行，因此热量和功量在子系统之间传递，这些子系统始终处于它们自己的内部热力学平衡状态。卡诺发动机是热机效率工程师特别感兴趣的理想设备。卡诺认识到卡诺原理的时候，正是热量热理论被认真考虑的时候，正是能量守恒定律理论被认识之前，正是熵概念的数学表达之前。根据第一定律的解释，它在物理上等同于热力学第二定律，并且在今天仍然有效。卡诺最初的论点是从卡路里理论的观点提出的，直到能量守恒定律的发现。以下是他书中的一些例子:<br />
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::...''wherever there exists a difference of temperature, motive power can be produced.''<ref>Carnot, S. (1824/1986), p. 51.</ref><br />
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...wherever there exists a difference of temperature, motive power can be produced.<br />
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只要有温差，就能产生动力。<br />
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::The production of motive power is then due in steam engines not to an actual consumption of caloric, but ''to its transportation from a warm body to a cold body ...''<ref>Carnot, S. (1824/1986), p. 46.</ref><br />
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The production of motive power is then due in steam engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body ...<br />
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动力的产生不是由于蒸汽机实际消耗的热量，而是由于它从一个温暖的物体运输到一个寒冷的物体..。<br />
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::''The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.''<ref>Carnot, S. (1824/1986), p. 68.</ref><br />
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The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.<br />
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热的动力与实现热的媒介无关，热的数量完全取决于两个物体之间的温度，最后是热量的传递。<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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用现代术语来说，卡诺的原则可能更为准确:<br />
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::The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<ref>[[Clifford Truesdell|Truesdell, C.]] (1980), Chapter 5.</ref><ref>Adkins, C.J. (1968/1983), pp. 56–58.</ref><ref>Münster, A. (1970), p. 11.</ref><ref>Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998), pp.67–75.</ref><ref>Lebon, G., Jou, D., Casas-Vázquez, J. (2008), p. 10.</ref><ref>Eu, B.C. (2002), pp. 32–35.</ref><br />
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The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<br />
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准静态或可逆卡诺循环的效率只取决于两个热源的温度，而且无论工作物质是什么，效率是相同的。以这种方式运行的卡诺发动机是使用这两种温度的最有效的热机。<br />
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===Clausius statement===<br />
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===Clausius statement===<br />
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克劳修斯声明<br />
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The German scientist [[Rudolf Clausius]] laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work.{{sfnp|Clausius|1850}} His formulation of the second law, which was published in German in 1854, is known as the ''Clausius statement'':<br />
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The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work. His formulation of the second law, which was published in German in 1854, is known as the Clausius statement:<br />
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1850年，德国科学家 Rudolf Clausius 通过研究热传递和功之间的关系，为热力学第二定律实验室奠定了基础。他在1854年用德语发表的第二定律的提法被称为克劳修斯声明:<br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.{{sfnp|Clausius|1854|p=86}}</blockquote><br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.</blockquote><br />
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热量不可能从一个较冷的物体传递到一个较热的物体，除非同时发生一些其他的变化。 / blockquote<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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克劳修斯的陈述使用了“热量通道”的概念。正如通常在热力学的讨论，这意味着“能量作为热量的净转移” ，并没有提到促成转移的一种方式和其他。<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of [[refrigeration]], for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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如果不对系统进行外部功，热就不能自发地从冷区流向热区，这一点从制冷的普通经验中可以看出。在冰箱中，热量从冷到热，但只有在外部介质——制冷系统的强制作用下才会发生变化。<br />
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===Kelvin statements===<br />
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===Kelvin statements===<br />
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开尔文陈述<br />
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[[William Thomson, 1st Baron Kelvin|Lord Kelvin]] expressed the second law in several wordings.<br />
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Lord Kelvin expressed the second law in several wordings.<br />
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开尔文勋爵用几个字表达了第二定律。<br />
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::It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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没有任何外部机构的帮助，自动机器是不可能把热量从一个物体传递到另一个物体的较高温度。<br />
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::It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.{{sfnp|Thomson|1851}}<br />
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It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.<br />
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通过无生命物质的作用，把物质冷却到周围最冷的物体的温度以下，从物质的任何部分获得机械效应是不可能的。<br />
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===Equivalence of the Clausius and the Kelvin statements===<br />
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===Equivalence of the Clausius and the Kelvin statements===<br />
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克劳修斯和开尔文陈述的等价性<br />
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[[Image:Deriving Kelvin Statement from Clausius Statement.svg|thumb|Derive Kelvin Statement from Clausius Statement]]<br />
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Derive Kelvin Statement from Clausius Statement<br />
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从克劳修斯陈述推导出开尔文陈述<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed [[Carnot engine]] as shown by the figure. The [[Heat engine#Efficiency|efficiency]] of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the [[first law of thermodynamics]], as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the figure. The efficiency of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the first law of thermodynamics, as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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假设有一个引擎违反了开尔文定理: 也就是说，这个引擎以循环的方式吸收热量并将其转化为功，没有任何其他结果。现在配对与反向卡诺发动机如图所示。普通热机的效率为，反向热机的效率为1 / 。这对组合发动机的净效应和唯一效应是将热量从较冷的贮存器转移到较热的贮存器，这违反了克劳修斯的说法。(这是能量守恒定律的结果，因为系统的总能量保持不变，数学文本{输入} + 文本{输出}0暗示 q-frac { eta }-qc / math，所以数学 qc 向左(frac { eta-1 right) / math)。因此，违反开尔文陈述意味着违反克劳修斯陈述，即。克劳修斯的说法暗示了凯尔文的说法。我们可以用类似的方式证明开尔文陈述暗示了克劳修斯陈述，因此两者是等价的。<br />
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===Planck's proposition===<br />
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===Planck's proposition===<br />
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普朗克的命题<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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普朗克提出了直接来自经验的如下命题。这有时被认为是他对第二定律的陈述，但他认为这是第二定律推导的起点。<br />
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::It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 86.</ref><ref>Roberts, J.K., Miller, A.R. (1928/1960), p. 319.</ref><br />
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It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<br />
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建造一台发动机是不可能的，这台发动机将在一个完整的循环中工作，并且除了提高重量和冷却热储以外不会产生任何效果。<br />
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===Relation between Kelvin's statement and Planck's proposition 开尔文表述与普朗克命题的关系===<br />
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It is almost customary in textbooks to speak of the "[[Kelvin-Planck statement]]" of the law, as for example in the text by [[Dirk ter Haar|ter Haar]] and [[Harald Wergeland|Wergeland]].<ref>[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966), p. 17.</ref><br />
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It is almost customary in textbooks to speak of the "Kelvin-Planck statement" of the law, as for example in the text by ter Haar and Wergeland.<br />
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教科书中几乎惯常提及该定律的“'''开尔文-普朗克表述 Kelvin-Planck Statement'''” ，例如'''德克·特哈尔 Diek ter Haar''' 和'''哈拉尔德·沃格兰 Harald Wergeland''' 在文中就是这样表述的。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）Diek ter Haar 中间是小写的<br />
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The '''Kelvin–Planck statement''' (or the ''heat engine statement'') of the second law of thermodynamics states that <br />
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The Kelvin–Planck statement (or the heat engine statement) of the second law of thermodynamics states that <br />
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热力学第二定律的开尔文-普朗克表述（或称“'''热机表述 Heat Engine Statement'''”）指出<br />
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::It is impossible to devise a [[thermodynamic cycle|cyclically]] operating device, the sole effect of which is to absorb energy in the form of heat from a single [[heat reservoir|thermal reservoir]] and to deliver an equivalent amount of [[Work (physics)|work]].<ref name="Rao">{{cite book|last=Rao|first=Y. V. C.|title=Chemical Engineering Thermodynamics|publisher=Universities Press|isbn=978-81-7371-048-3|page=158|year=1997}}</ref><br />
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It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.<br />
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设计一种唯一效果是从单一热源吸收热量并提供等量的功的循环运行装置是不可能的。<br />
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===Planck's statement 普朗克表述===<br />
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Planck stated the second law as follows.<br />
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Planck stated the second law as follows.<br />
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普朗克表述第二定律如下。<br />
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::Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<ref name="Planck 100">[[Max Planck|Planck, M.]] (1897/1903), p. 100.</ref><ref name="Planck 463">[[Max Planck|Planck, M.]] (1926), p. 463, translation by Uffink, J. (2003), p. 131.</ref><ref name="Roberts & Miller 382">Roberts, J.K., Miller, A.R. (1928/1960), p. 382. This source is partly verbatim from Planck's statement, but does not cite Planck. This source calls the statement the principle of the increase of entropy.</ref><br />
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Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<br />
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自然界中发生的任一过程都是沿参与其中的所有物体的熵之和增加的方向进行的。在某些限制下——即对于可逆过程，熵的总和保持不变。<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for ''irreversible phenomena''.<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for irreversible phenomena.<br />
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与普朗克表述非常相似的是'''乌伦贝克 Uhlenbeck'''和'''福特 Ford'''关于不可逆现象的表述。<br />
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::... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<ref>[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963), p. 16.</ref><br />
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... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<br />
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在从一个平衡态到另一个平衡态的不可逆或自发的变化中（例如，当两个物体 A 和 B 接触时的温度平衡过程），熵总是增加。<br />
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===Principle of Carathéodory 卡拉西奥多里原理===<br />
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<!-- [[Caratheodory's principle]] redirects here --><br />
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—— caratheodory 的原理在这里重新定向——<br />
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[[Constantin Carathéodory]] formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<ref>[[Constantin Carathéodory|Carathéodory, C.]] (1909).</ref><br />
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Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<br />
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'''康斯坦丁·卡拉西奥多里 Constantin Carathéodory'''在纯数学公理的基础上进行了热力学公理化。他对第二定律的陈述被称为'''卡拉西奥多里原理 Principle of Carathéodory'''，可以这样表述:<br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.<ref>Buchdahl, H.A. (1966), p. 68.</ref></blockquote><br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.</blockquote><br />
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在绝热封闭系统的任意状态 S 附近，总有从 S 出发不可达的状态。<br />
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With this formulation, he described the concept of [[adiabatic accessibility]] for the first time and provided the foundation for a new subfield of classical thermodynamics, often called [[Ruppeiner geometry|geometrical thermodynamics]]. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic [[process function]], in other words, <math>\delta Q=TdS</math>.<ref name="Sychev1991">{{cite book |last=Sychev |first=V. V. |title=The Differential Equations of Thermodynamics |year=1991 |publisher=Taylor & Francis |isbn=978-1-56032-121-7}}</ref> {{clarify|date=February 2014}}<br />
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With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, <math>\delta Q=TdS</math>. <br />
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公理化处理后，他第一次描述了'''绝热可达性 Adiabatic Accessibility'''的概念，并为经典热力学的一个新的子领域，即通常所说的'''几何热力学 Geometrical Thermodynamics'''奠定了基础。由卡拉西奥多里原理可以推出，准静态转移的热量值是一个可积过程函数，即<math>\delta Q=TdS</math>。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）准静态转移的热量值是一个可积过程函数 存疑<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.<ref name="Munster 45">Münster, A. (1970), p. 45.</ref>{{sfnp|Lieb|Yngvason|1999|p=49}}<ref name="Planck 1926">[[Max Planck|Planck, M.]] (1926).</ref><ref>Buchdahl, H.A. (1966), p. 69.</ref> {{clarify|date=February 2014}}<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium. <br />
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尽管在教科书几乎惯称卡拉西奥多里原理表述了第二定律，并认为其与克劳修斯表述或开尔文-普朗克表述等价，但事实并非如此。为了得到第二定律的所有内容，需要对卡拉西奥多里原理补充普朗克表述，即等量功总是增加一个最初处于自身内部热力学平衡的封闭系统的内部能量。<br />
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===Planck's principle 普朗克原理===<br />
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In 1926, [[Max Planck]] wrote an important paper on the basics of thermodynamics.<ref name="Planck 1926"/><ref>Uffink, J. (2003), pp. 129–132.</ref> He indicated the principle<br />
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In 1926, Max Planck wrote an important paper on the basics of thermodynamics. He indicated the principle<br />
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1926年，'''马克斯·普朗克 Max Planck'''写了一篇关于热力学基础的重要论文。他指出了以下原理<br />
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::The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<ref name="Munster 45"/>{{sfnp |Lieb|Yngvason|1999|p=49}}<br />
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The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<br />
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一个封闭系统的内部能量因绝热过程增加，在整个过程中，系统的体积保持不变。<br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work."<ref>[[Clifford Truesdell|Truesdell, C.]], Muncaster, R.G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 15.</ref> Planck wrote: "The production of heat by friction is irreversible."<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 81.</ref><ref>[[Max Planck|Planck, M.]] (1926), p. 457, Wikipedia editor's translation.</ref><br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work." Planck wrote: "The production of heat by friction is irreversible."<br />
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这个公式没有提到热，没有提到温度，甚至没有提到熵，也不一定隐含地依赖于这些概念，但它暗示了第二定律的内容。一个密切相关的表述为，“摩擦力从来不做正功。”普朗克写道: “摩擦生热是不可逆的。”<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above.<ref>Lieb, E.H., Yngvason, J. (2003), p. 149.</ref> It is relevant that for a system at constant volume and [[Mole (unit)|mole numbers]], the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above. It is relevant that for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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这个普朗克定理没有提到熵，是用物理术语来表述的。它与上面给出的开尔文表述密切相关。相关的是，对于恒定体积和摩尔数的系统，熵是内能的单调函数。然而，普朗克的这个原理实际上并不是普朗克对第二定律的首选表述（见前面小节），而是依赖于熵的概念。<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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Borgnakke 和 Sonntag 提出的以下表述在某种意义上是对普朗克原理的补充。他们没有将其作为第二定律的完整表述:<br />
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::... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<ref>Borgnakke, C., Sonntag., R.E. (2009), p. 304.</ref><br />
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... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<br />
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只有一种方法可以减少[闭合]系统的熵——从系统中转移热量。<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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与普朗克之前提出的原理不同，这一原理明确地用熵的变化来表示。从系统中去除物质也可以减少系统的熵。<br />
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===Statement for a system that has a known expression of its internal energy as a function of its extensive state variables 一个其内能有已知表达式（其扩展状态变量的函数）的系统的表述===<br />
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The second law has been shown to be equivalent to the [[internal energy]] ''U'' being a weakly [[convex function]], when written as a function of extensive properties (mass, volume, entropy, ...).<ref>{{cite book |last1=van Gool |first1=W. |last2=Bruggink |first2=J.J.C. (Eds) |url= |title=Energy and time in the economic and physical sciences |publisher=North-Holland |year=1985 |pages=41–56 |quote= |isbn=978-0-444-87748-2}}</ref><ref>{{Cite journal | last1 = Grubbström | first1 = Robert W. | doi = 10.1016/j.apenergy.2007.01.003 | title = An Attempt to Introduce Dynamics Into Generalised Exergy Considerations| journal = Applied Energy| volume = 84| issue = 7–8 | pages = 701–718 | year = 2007}}</ref> {{clarify|date=February 2014}}<br />
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The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...). <br />
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第二定律被证明等价于弱凸函数内能 U （写成广泛性质（质量，体积，熵，...）的函数时）。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）存疑<br />
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==Corollaries 推论==<br />
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===Perpetual motion of the second kind 第二类永动机===<br />
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{{main article|Perpetual motion}}<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of [[first law of thermodynamics]] by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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在第二定律建立之前，许多志在发明永动机的人试图通过提取环境的巨大内能作为机器动力，以突破'''热力学第一定律 First Law of Thermodynamics''' 的限制。这种机器被称为“'''第二类永动机 Perpetual Motion Machine of the Second Kind'''”。第二定律表明这种机器是不可能的。<br />
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===Carnot theorem 卡诺定理===<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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'''卡诺定理 Carnot's theorem'''（1824）是一条限制任何可能的发动机的最大效率的原理。效率完全取决于热库和冷库之间的温差。卡诺定理指出:<br />
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*All irreversible heat engines between two heat reservoirs are less efficient than a [[Carnot engine]] operating between the same reservoirs.<br />
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*All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs.<br />
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* 所有不可逆热机的效率低于在两相同热源之间工作的'''卡诺机 Carnot engine'''。<br />
* 所有可逆热机的效率与在两相同热源之间工作的卡诺机相等。<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as [[thermodynamic reversibility]]. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the [[Carnot cycle]]'s reversibility and was condemned to be less efficient.<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the Carnot cycle's reversibility and was condemned to be less efficient.<br />
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在他的理想模型中，热转换成功可以通过逆转循环的运动而恢复，这个概念后来被称为'''热力学可逆性 Thermodynamic Reversibility'''。然而，卡诺进一步假定，一些热量损失了，并没有转化为机械功。因此，没有一个真实的热机能够实现'''卡诺循环 Carnot cycle'''的可逆性，并且被认为效率较低。<br />
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Though formulated in terms of caloric (see the obsolete [[caloric theory]]), rather than [[entropy]], this was an early insight into the second law.<br />
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Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law.<br />
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该理论尽管是用热量表述的（见被取代的'''热质说 Caloric Theory'''），而不是熵，但是它是对第二定律的早期认识。<br />
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===Clausius inequality 克劳修斯不等式===<br />
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The [[Clausius theorem]] (1854) states that in a cyclic process<br />
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The Clausius theorem (1854) states that in a cyclic process<br />
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'''克劳修斯定理 Clausius theorem'''（1854）指出，在一个循环的过程中<br />
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: <math>\oint \frac{\delta Q}{T} \leq 0.</math><br />
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The equality holds in the reversible case<ref>[http://scienceworld.wolfram.com/physics/ClausiusTheorem.html ''Clausius theorem''] at [[Wolfram Research]]</ref> and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function [[entropy]]. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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The equality holds in the reversible case and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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等号在可逆情况下成立，严格不等号在不可逆情况下成立。可逆情况下引入状态函数熵。这是因为在循环过程中，状态函数的变化为零。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）末句存疑<br />
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===Thermodynamic temperature 热力学温度===<br />
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{{main article|Thermodynamic temperature}}<br />
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For an arbitrary heat engine, the efficiency is:<br />
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For an arbitrary heat engine, the efficiency is:<br />
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对于任意热机，效率为:<br />
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: <math>\eta = \frac {W_n}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H} \qquad (1)</math><br />
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where ''W''<sub>n</sub> is for the net work done per cycle. Thus the efficiency depends only on q<sub>C</sub>/q<sub>H</sub>.<br />
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其中 ''W''<sub>n</sub> 表示每个循环所做的净功。因此效率只取决于 q<sub>C</sub>/q<sub>H</sub>。<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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卡诺定理指出，所有在相同热源间运行的可逆发动机的效率是相同的。<br />
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Thus, any reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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Thus, any reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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因此，任何在温度''T''<sub>1</sub>和''T''<sub>2</sub>之间运行的可逆热机必须具有相同的效率，也就是说，效率只是温度的函数:<br />
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<math>\frac{q_C}{q_H} = f(T_H,T_C)\qquad (2).</math><br />
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In addition, a reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between ''T''<sub>1</sub> and another (intermediate) temperature ''T''<sub>2</sub>, and the second between ''T''<sub>2</sub> and''T''<sub>3</sub>. This can only be the case if<br />
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In addition, a reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between T<sub>1</sub> and another (intermediate) temperature T<sub>2</sub>, and the second between T<sub>2</sub> andT<sub>3</sub>. This can only be the case if<br />
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另外，在温度 ''T''<sub>1</sub> 和 ''T''<sub>3</sub> 之间工作的可逆热机必须具有与由分别在温度''T''<sub>1</sub> 和（中间）温度 ''T''<sub>2</sub>之间、在 ''T''<sub>2</sub> 和 ''T''<sub>3</sub> 之间的两个循环组成的系统效率相同。只有下式成立才会出现这种情况:<br />
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: <math>f(T_1,T_3) = \frac{q_3}{q_1} = \frac{q_2 q_3} {q_1 q_2} = f(T_1,T_2)f(T_2,T_3).</math><br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the [[triple point]] of water. Then for any ''T''<sub>2</sub> and ''T''<sub>3</sub>,<br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the triple point of water. Then for any T<sub>2</sub> and T<sub>3</sub>,<br />
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现在考虑如下情形，<math>T_1</math> 是一个固定的参考温度: 水的三相点的温度。则对于任意 T<sub>2</sub> 和 T<sub>3</sub>,<br />
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: <math>f(T_2,T_3) = \frac{f(T_1,T_3)}{f(T_1,T_2)} = \frac{273.16 \cdot f(T_1,T_3)}{273.16 \cdot f(T_1,T_2)}.</math><br />
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Therefore, if thermodynamic temperature is defined by<br />
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Therefore, if thermodynamic temperature is defined by<br />
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因此，如果热力学温度被定义为<br />
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: <math>T = 273.16 \cdot f(T_1,T) \,</math><br />
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then the function ''f'', viewed as a function of thermodynamic temperature, is simply<br />
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then the function f, viewed as a function of thermodynamic temperature, is simply<br />
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那么函数 ''f'' 作为热力学温度的函数，为<br />
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: <math>f(T_2,T_3) = \frac{T_3}{T_2},</math><br />
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and the reference temperature ''T''<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be used{{snd}}the choice here corresponds to the [[Kelvin]] scale.)<br />
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and the reference temperature T<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be usedthe choice here corresponds to the Kelvin scale.)<br />
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参考温度 ''T''<sub>1</sub> 的值为273.16。（任何参考温度和任何正值均可用——此处的选择对应开尔文标度。）<br />
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===Entropy 熵===<br />
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{{main article|Entropy (classical thermodynamics)}}<br />
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According to the [[Clausius theorem|Clausius equality]], for a reversible process<br />
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According to the Clausius equality, for a reversible process<br />
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根据克劳修斯定理，对可逆过程有<br />
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: <math>\oint \frac{\delta Q}{T}=0</math><br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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这意味着线积分 <math>\int_L \frac{\delta Q}{T}</math> 对于可逆过程是路径无关的。<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer<ref name="Schmidt-Rohr 14" /> satisfies<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer satisfies<br />
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所以我们可以定义一个叫做熵的状态函数 S，对于可逆过程或者纯热传递满足<br />
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: <math>dS = \frac{\delta Q}{T} \!</math><br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the [[third law of thermodynamics]], which states that ''S'' = 0 at [[absolute zero]] for perfect crystals.<br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the third law of thermodynamics, which states that S = 0 at absolute zero for perfect crystals.<br />
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据此，只有对上述公式进行积分，才能得到熵的差值。为了获得绝对值，我们需要'''热力学第三定律 Third Law of Thermodynamics'''，它指出绝对零度下完美晶体的 ''S'' = 0。<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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对于任意不可逆过程，由于熵是一个状态函数，我们总是可以将初始状态和最终状态与一个虚拟的可逆过程联系起来，并在这条路径上积分以计算熵的差值。<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the [[Clausius inequality]] on this loop,<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the Clausius inequality on this loop,<br />
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现在把可逆过程逆过来，将其与上述不可逆过程结合。把克劳修斯不等式应用到这个循环,<br />
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: <math>-\Delta S+\int\frac{\delta Q}{T}=\oint\frac{\delta Q}{T}< 0</math><br />
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Thus,<br />
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Thus,<br />
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故,<br />
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: <math>\Delta S \ge \int \frac{\delta Q}{T} \,\!</math><br />
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where the equality holds if the transformation is reversible.<br />
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where the equality holds if the transformation is reversible.<br />
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如果变换可逆，等号成立。<br />
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Notice that if the process is an [[adiabatic process]], then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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Notice that if the process is an adiabatic process, then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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注意，若该过程是一个'''绝热过程 Adiabatic Process'''，则<math>\delta Q=0</math>，故<math>\Delta S\ge 0</math>。<br />
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===Energy, available useful work===<br />
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===Energy, available useful work===<br />
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能量，有用的工作<br />
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{{See also|Exergy}}<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an ''unlimited'' heat reservoir at temperature ''T<sub>R</sub>'' and pressure ''P<sub>R</sub>'' {{snd}}so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain ''T<sub>R</sub>''; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain ''P<sub>R</sub>''.<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an unlimited heat reservoir at temperature T<sub>R</sub> and pressure P<sub>R</sub> so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain T<sub>R</sub>; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain P<sub>R</sub>.<br />
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考虑将第二定律应用于孤立系统(称为整体系统或宇宙)的情形，是一个重要的、具有启发性的理想情形。孤立系统由两部分组成: 感兴趣的子系统和子系统的环境。这些环境被想象为如此之大，以至于它们可以被认为是一个温度为 t 子 r / sub 和压力为 p 子 r / sub 的无限热源，因此无论有多少热量被转移到(或来自)子系统，周围环境的温度将保持 t 子 r / sub; 无论子系统的体积扩大(或收缩)多少，周围环境的压力将保持 p 子 r / sub。<br />
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Whatever changes to ''dS'' and ''dS<sub>R</sub>'' occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy ''S<sub>tot</sub>'' of the isolated total system must not decrease:<br />
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Whatever changes to dS and dS<sub>R</sub> occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy S<sub>tot</sub> of the isolated total system must not decrease:<br />
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根据第二定律，孤立总体系统的熵 s 子系统 / 子系统的熵不能减小，无论子系统和周围环境的熵对 dS 和 dS 子系统 r / 子系统发生什么变化:<br />
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: <math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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<math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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数学 dS + dS r ge 0 / math<br />
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According to the [[first law of thermodynamics]], the change ''dU'' in the internal energy of the sub-system is the sum of the heat ''δq'' added to the sub-system, ''less'' any work ''δw'' done ''by'' the sub-system, ''plus'' any net chemical energy entering the sub-system ''d ∑μ<sub>iR</sub>N<sub>i</sub>'', so that:<br />
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According to the first law of thermodynamics, the change dU in the internal energy of the sub-system is the sum of the heat δq added to the sub-system, less any work δw done by the sub-system, plus any net chemical energy entering the sub-system d ∑μ<sub>iR</sub>N<sub>i</sub>, so that:<br />
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根据能量守恒定律，子系统内部能量的变化 dU 是子系统内部能量 q 的总和，减去子系统所做的任何功，再加上进入子系统 d ∑ sub iR / sub n sub i / sub 的任何净化学能，因此:<br />
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: <math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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<math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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数学 dU delta q- delta w + d ( sum mu iR } n i) ，/ math<br />
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where μ<sub>iR</sub> are the [[chemical potential]]s of chemical species in the external surroundings.<br />
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where μ<sub>iR</sub> are the chemical potentials of chemical species in the external surroundings.<br />
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其中亚 iR / 亚是外部环境中化学物种的化学势。<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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现在热量离开储存器进入子系统是<br />
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: <math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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<math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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数学△ qtr (- dS r) le tr dS / math<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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其中我们首先使用了经典热力学中熵的定义(在统计热力学中，熵变、温度和吸收热量之间的关系可以推导出来) ，然后从上面推导出第二定律的不等式。<br />
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It therefore follows that any net work ''δw'' done by the sub-system must obey<br />
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It therefore follows that any net work δw done by the sub-system must obey<br />
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因此，子系统所做的任何净功都必须服从<br />
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: <math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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<math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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数学增量 w le-dU + tds + sum mu { iR } dN i，/ math<br />
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It is useful to separate the work ''δw'' done by the subsystem into the ''useful'' work ''δw<sub>u</sub>'' that can be done ''by'' the sub-system, over and beyond the work ''p<sub>R</sub> dV'' done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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It is useful to separate the work δw done by the subsystem into the useful work δw<sub>u</sub> that can be done by the sub-system, over and beyond the work p<sub>R</sub> dV done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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将子系统所做的功与子系统所能做的功分离为子系统所能做的有用功，超过了子系统对外界压力所做的功，给出了子系统所能做的有用功(火用)的下列关系式:<br />
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: <math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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<math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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D (u-t r s + p r v-sum { iR } n i) ，/ math<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the ''availability'' or ''[[exergy]]'' ''E'' of the subsystem,<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the availability or exergy E of the subsystem,<br />
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可以很方便地将子系统的右侧定义为热动力位能的精确衍生物，称为子系统的可用性或火用 e,<br />
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: <math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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<math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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数学 e u-t r s + p r v- sum { iR } n i / math<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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因此，第二定律意味着，对于任何可以简单地被视为分为一个子系统和一个与之接触的无限温度和压力储存器的过程,<br />
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: <math> dE + \delta w_u \le 0 \, </math><br />
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<math> dE + \delta w_u \le 0 \, </math><br />
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数学德 delta u le 0，/ math<br />
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i.e. the change in the subsystem's exergy plus the useful work done ''by'' the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done ''on'' the system) must be less than or equal to zero.<br />
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i.e. the change in the subsystem's exergy plus the useful work done by the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done on the system) must be less than or equal to zero.<br />
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也就是。子系统(火用)的变化加上子系统所做的有用功(或者，子系统(火用)的变化不包括任何功，除了压力贮存器所做的功之外，在系统中所做的功)必须小于或等于零。<br />
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In sum, if a proper ''infinite-reservoir-like'' reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in ''E'' for an irreversible process and no change for a reversible process.<br />
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In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in E for an irreversible process and no change for a reversible process.<br />
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总之，如果选择一个合适的类似于无限库的参考状态作为现实世界中的系统环境，那么第二定律预测一个不可逆性的 e 值会减少，而一个可逆过程的 e 值不会变化。<br />
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: <math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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<math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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数学 d = 0 = 0 = 0 = 0 = 0<br />
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This expression together with the associated reference state permits a [[design engineer]] working at the macroscopic scale (above the [[thermodynamic limit]]) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (''Also, see [[process engineer]]''). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (''See [[Exergy efficiency|second law efficiency]]''.)<br />
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This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (Also, see process engineer). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (See second law efficiency.)<br />
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这个表达式连同相关的参考状态允许在宏观的设计工程师利用第二定律，而不需要直接测量或考虑整个孤立系统的熵变。(另外，请参阅流程工程师)。这些变化已经被考虑，假设被考虑的系统可以达到平衡的参考状态而不改变参考状态。还可以找到一个过程或过程集合的效率，将其与可逆的理想状态相比较<br />
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This approach to the Second Law is widely utilized in [[engineering]] practice, [[environmental accounting]], [[systems ecology]], and other disciplines.<br />
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This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines.<br />
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第二定律的这种方法被广泛应用于工程实践、环境会计、系统生态学和其他学科。<br />
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===The second law in chemical thermodynamics===<br />
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===The second law in chemical thermodynamics===<br />
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化学热力学的第二定律<br />
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For a [[spontaneous process|spontaneous chemical process]] in a closed system at constant temperature and pressure without non-''PV'' work, the Clausius inequality Δ''S > Q/T<sub>surr</sub>'' transforms into a condition for the change in [[Gibbs free energy]] <br />
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For a spontaneous chemical process in a closed system at constant temperature and pressure without non-PV work, the Clausius inequality ΔS > Q/T<sub>surr</sub> transforms into a condition for the change in Gibbs free energy <br />
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对于一个封闭系统中的自发化学过程，在没有非 pv 功的情况下，在恒定温度和压力下，将克劳修斯不等式 s q / t sub surr / sub 转化为吉布斯自由能变化的条件<br />
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: <math>\Delta G < 0 </math><br />
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数学 Delta g 0 / 数学<br />
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or d''G'' < 0. For a similar process at constant temperature and volume, the change in [[Helmholtz free energy]] must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products.<ref name="Oxtoby8th"/> <ref name="MortimerBook"></ref> The chemical equilibrium condition at constant ''T'' and ''p'' without electrical work is d''G'' = 0.<br />
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or dG < 0. For a similar process at constant temperature and volume, the change in Helmholtz free energy must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products. The chemical equilibrium condition at constant T and p without electrical work is dG = 0.<br />
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或者 dG 0。对于一个相似的过程，在恒定的温度和体积下，亥姆霍兹自由能的变化一定是负的，这是 Delta a 0 / math。因此，自由能(g 或 a)变化的负值是过程自发的必要条件。这是热力学第二定律在化学中最有用的形式，其中自由能的变化可以通过列表的生成焓和反应物和产物的标准摩尔熵来计算。在 t 和 p 不变的情况下，化学平衡条件是 dG 0。<br />
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==History==<br />
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==History==<br />
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历史<br />
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{{See also|History of entropy}}<br />
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[[File:Sadi Carnot.jpeg|thumb|upright|Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique]].]]<br />
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Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique.]]<br />
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尼古拉·莱昂纳尔·萨迪·卡诺穿着传统的巴黎综合理工学院制服<br />
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The first theory of the conversion of heat into mechanical work is due to [[Nicolas Léonard Sadi Carnot]] in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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The first theory of the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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第一个把热转化为机械功的理论是在1824年由尼古拉·莱昂纳尔·萨迪·卡诺提出的。他是第一个正确认识到这种转换的效率取决于发动机和环境之间的温差。<br />
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Recognizing the significance of [[James Prescott Joule]]'s work on the conservation of energy, [[Rudolf Clausius]] was the first to formulate the second law during 1850, in this form: heat does not flow ''spontaneously'' from cold to hot bodies. While common knowledge now, this was contrary to the [[caloric theory]] of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law during 1850, in this form: heat does not flow spontaneously from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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认识到詹姆斯·普雷斯科特·焦耳在能量守恒方面工作的重要性，Rudolf Clausius 在1850年第一个提出了第二定律: 热不会自发地从冷物体流向热物体。虽然现在的常识是，这与当时流行的热量理论相反，热量理论认为热量是一种流体。从那里他能够推断出萨迪卡诺定律和熵的定义(1865年)。<br />
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Established during the 19th century, the [[Kelvin-Planck statement|Kelvin-Planck statement of the Second Law]] says, "It is impossible for any device that operates on a [[cyclic process|cycle]] to receive heat from a single [[heat reservoir|reservoir]] and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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Established during the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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建立于19世纪的开尔文-普朗克第二定律(Kelvin-Planck)说，“任何循环运行的设备都不可能从单个蓄热体接收热量，并产生净功。”这被证明相当于克劳修斯的陈述。<br />
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The [[ergodic hypothesis]] is also important for the [[Boltzmann]] approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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遍历假设对玻尔兹曼方法也很重要。它说，在很长一段时间内，在具有相同能量的微观态的相空间的某些区域所花费的时间与这个区域的体积成正比，即。在很长一段时间内，所有可访问的微观状态的可能性都是一样的。同样的，它表明时间平均值和系综平均值是相同的。<br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a [[Macroscopic bodies|macroscopic system]]. This doctrine is obsolescent.<ref>Denbigh, K.G., Denbigh, J.S. (1985). ''Entropy in Relation to Incomplete Knowledge'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-25677-1}}, pp. 43–44.</ref><ref>Grandy, W.T., Jr (2008). ''Entropy and the Time Evolution of Macroscopic Systems'', Oxford University Press, Oxford, {{ISBN|978-0-19-954617-6}}, pp. 55–58.</ref><ref name=Lambert>[http://entropysite.oxy.edu Entropy Sites — A Guide] Content selected by [[Frank L. Lambert]]</ref><br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a macroscopic system. This doctrine is obsolescent.<br />
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有一种传统的学说，从克劳修斯开始，认为熵可以被理解为宏观系统中的分子“无序”。这种学说已经过时了。<br />
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===Account given by Clausius===<br />
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===Account given by Clausius===<br />
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作者: 克劳修斯<br />
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[[File:Clausius-1.jpg|thumb|upright|Rudolf Clausius]]<br />
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Rudolf Clausius<br />
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鲁道夫 · 克劳修斯<br />
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In 1856, the German physicist [[Rudolf Clausius]] stated what he called the "second fundamental theorem in the [[mechanical theory of heat]]" in the following form:{{sfnp|Clausius|1867}}<br />
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In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:<br />
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1856年，德国物理学家鲁道夫 · 克劳修斯阐述了他所谓的“热力学理论中的第二个基本定理” ，其形式如下:<br />
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: <math>\int \frac{\delta Q}{T} = -N</math><br />
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<math>\int \frac{\delta Q}{T} = -N</math><br />
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- n / math<br />
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where ''Q'' is heat, ''T'' is temperature and ''N'' is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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其中 q 是热，t 是温度，n 是一个循环过程中所有未补偿的相变的“等价值”。后来，在1865年，克劳修斯将“等价值”定义为熵。紧接着这个定义，也就是在同一年，第二定律最著名的版本在4月24日苏黎世哲学学会的一次演讲中被宣读，在演讲的最后，克劳修斯总结道:<br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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宇宙的熵趋于最大<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. [[universe]], as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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这句话是第二定律中最广为人知的措辞。由于其语言的松散性，例如:。这个问题的答案是肯定的，因为这个问题的答案是肯定的。开放，封闭，或孤立，许多人认为这个简单的陈述意味着热力学第二定律适用于几乎所有可以想象的主题。这是不正确的; 这个陈述只是一个更加扩展和精确描述的简化版本。<br />
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In terms of time variation, the mathematical statement of the second law for an [[isolated system]] undergoing an arbitrary transformation is:<br />
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In terms of time variation, the mathematical statement of the second law for an isolated system undergoing an arbitrary transformation is:<br />
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就时间变化而言，经历任意变换的孤立系统第二定律的数学表述是:<br />
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: <math>\frac{dS}{dt} \ge 0</math><br />
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<math>\frac{dS}{dt} \ge 0</math><br />
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0 / math<br />
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where<br />
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where<br />
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在哪里<br />
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: ''S'' is the entropy of the system and<br />
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S is the entropy of the system and<br />
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S 是系统的熵<br />
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: ''t'' is [[time]].<br />
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t is time.<br />
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时间到了。<br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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!-- 可逆过程需要与周围环境保持平衡。这对于一个孤立的系统是不可能的。因此，关于可逆过程的讨论已经转移到对封闭系统的分析----<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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平衡后应用等式符号。建立孤立系统第二定律的另一种方法是:<br />
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: <math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math<br />
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with <math> \dot S_{i}</math> the sum of the rate of [[entropy production]] by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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with <math> \dot S_{i}</math> the sum of the rate of entropy production by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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用数学 s { i } / 数学表示系统内所有进程的产生熵之和。这种配方的优点是它显示了产生熵的效果。产生熵是一个非常重要的概念，因为它决定(限制)热机器的效率。与环境温度数学 t { a } / math 相乘，得到所谓的耗散能量数学 p { diss } t { a } dot s { i } / math。<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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封闭系统的第二定律(允许热量交换和移动边界，但不允许物质交换)的表达式是:<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math 来计算数学点 q } + 点 s { i } / math<br />
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: <math>\dot Q</math> is the heat flow into the system<br />
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<math>\dot Q</math> is the heat flow into the system<br />
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数学 q / math 是进入系统的热流<br />
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: <math>T</math> is the temperature at the point where the heat enters the system.<br />
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<math>T</math> is the temperature at the point where the heat enters the system.<br />
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Math 是热量进入系统时的温度。<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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等式符号在只有可逆过程在系统内发生的情况下成立。如果发生不可逆过程(在实际操作系统中就是这种情况) ,-符号保持不变。如果系统有多处供热，我们必须求相应项的代数和。<br />
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For open systems (also allowing exchange of matter):<br />
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For open systems (also allowing exchange of matter):<br />
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对于开放系统(也允许物质交换) :<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math 表示数学点 q } + 点 s + 点 s { i } / math<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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这里的数学 s / math 是熵流进入系统，与物质流进入系统有关。它不应该与熵的时间导数混淆。如果物质在几个地方提供，我们必须取这些贡献的代数和。<br />
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==Statistical mechanics==<br />
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==Statistical mechanics==<br />
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统计力学<br />
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[[Statistical mechanics]] gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a [[microstate (statistical mechanics)|microstate]] of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/{{radic|''N''}} where ''N'' is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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Statistical mechanics gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a microstate of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/ where N is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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通过假设物质是由不断运动的原子和分子组成的，统计力学对第二定律给出了解释。系统中每个粒子的一组特定的位置和速度称为系统的微观状态，由于系统的不断运动，系统不断地改变其微观状态。统计力学假设，在平衡状态下，系统可能处于的每个微观状态发生的可能性是相等的，当这个假设被提出时，它直接导致了第二定律必须在统计意义上成立的结论。也就是说，第二定律平均成立，其中 n 是系统中粒子数的1 / 数量级的统计变化。在日常(宏观)情况下，违反第二定律的概率几乎为零。然而，对于粒子数量很少的系统，热力学参数，包括熵，可能显示出与第二定律预测的显著的统计偏差。经典热力学理论没有处理这些统计变量。<br />
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（by Jie XIANG) 在通过假定一种物质由不断运动的原子和分子组成的前提下，统计力学对热力学第二定律进行了解释。系统中每个粒子其特定的位置和速度组成为系统的微观状态，由于（原子和分子的）不断运动，系统也持续改变其微观状态。统计力学假设，在系统力平衡的情况下，其微观状态其实是等可能发生的，进而能够得出结论：热力学第二定律必须在统计学意义上成立。也就是说，第二定律将保持均值不变， 其统计变化为1 /√N，N为系统中的粒子数。在日常(宏观)情况下，违反热力学第二定律的概率几乎为零，但是，对于粒子数量非常少的系统，其热力学参数（包括熵）可能会显示出与第二定律所预测的明显统计偏差。然而经典热力学理论并不处理这些统计变量。<br />
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==Derivation from statistical mechanics==<br />
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==Derivation from statistical mechanics==<br />
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从统计力学导出<br />
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{{Further information|H-theorem}}<br />
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The first mechanical argument of the [[Kinetic theory of gases]] that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to [[James Clerk Maxwell]] in 1860;<ref>{{Cite journal | last1 = Gyenis | first1 = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G }}</ref> [[Ludwig Boltzmann]] with his [[H-theorem]] of 1872 also argued that due to collisions gases should over time tend toward the [[Maxwell–Boltzmann distribution]].<br />
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The first mechanical argument of the Kinetic theory of gases that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to James Clerk Maxwell in 1860; Ludwig Boltzmann with his H-theorem of 1872 also argued that due to collisions gases should over time tend toward the Maxwell–Boltzmann distribution.<br />
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气体动力学理论的第一个力学论证由詹姆斯·克拉克·麦克斯韦在1860年给出，指出分子碰撞引起温度均衡，因此整体趋向于平衡； 路德维希·玻尔兹曼在1872年提出的 H 定理也认为，气体由于碰撞应该随着时间的推移趋向于麦克斯韦-波兹曼分布。<br />
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Due to [[Loschmidt's paradox]], derivations of the Second Law have to make an assumption regarding the past, namely that the system is [[Correlation and dependence|uncorrelated]] at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a [[boundary condition]], and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the [[Big Bang]]), though [[Boltzmann brain|other scenarios]] have also been suggested.<ref name="Hawking AOT">{{cite journal|last=Hawking|first=SW|title=Arrow of time in cosmology|journal=Phys. Rev. D|year=1985|volume=32|issue=10|pages=2489–2495|doi=10.1103/PhysRevD.32.2489|pmid=9956019|bibcode = 1985PhRvD..32.2489H }}</ref><ref>{{cite book | last = Greene | first = Brian | authorlink = Brian Greene | title = The Fabric of the Cosmos | url = https://archive.org/details/fabricofcosmossp00gree | url-access = registration | publisher = Alfred A. Knopf | year = 2004 | page = [https://archive.org/details/fabricofcosmossp00gree/page/171 171] | isbn = 978-0-375-41288-2}}</ref><ref name=Lebowitz>{{cite journal|last=Lebowitz|first=Joel L.|title= Boltzmann's Entropy and Time's Arrow|journal=Physics Today|date=September 1993|volume=46|issue=9|pages=32–38|url=http://users.df.uba.ar/ariel/materias/FT3_2008_1C/papers_pdf/lebowitz_370.pdf|accessdate=2013-02-22|doi=10.1063/1.881363|bibcode = 1993PhT....46i..32L }}</ref><br />
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Due to Loschmidt's paradox, derivations of the Second Law have to make an assumption regarding the past, namely that the system is uncorrelated at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a boundary condition, and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the Big Bang), though other scenarios have also been suggested.<br />
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由于洛施密特悖论，第二定律的导出必须对过去做出一个假设，即系统在过去的某个时刻是不相关的；这样的假设允许进行简单的概率处理。这个假设通常被认为是一个边界条件，因此热力学第二定律最终是过去某个地方的初始条件的结果，可能是在宇宙的开始(大爆炸) 。也有人提出了其他假设。<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the [[Statistical mechanics#Fundamental postulate|fundamental postulate]], also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy.{{citation needed|date=August 2012}} The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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考虑到这些假设，在统计力学中，第二定律不是一个假设，而是统计力学基本假设的一个结果，也被称为等先验概率假设。这个基本假设表明，只要一个人清楚地知道，简单的概率论证只适用于未来，而对于过去，有辅助的信息来源告诉我们，它是低熵的。热力学第二定律的第一部分指出，热孤立系统的熵只能增加。如果我们把熵的概念限制在热平衡系统中，那么热力学第二定律的第一部分是等先验概率假设的一个显然结果。处于热平衡状态的孤立系统包含能量<math>E</math>的熵表示为：<br />
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: <math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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<math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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其中<math>\Omega\left(E\right)</math> 是处于 <math>E</math>和<math>E +\delta E</math这个小区间内的量子态数目。这里的 <math>\delta E</math> 是一个宏观上很小的固定能量区间。严格地说，这意味着熵取决于对<math>\delta E</math>的选择。然而在热力学极限下（例如无穷大系统的极限），狭义的熵(单位体积或单位质量的熵)不依赖于 <math>\delta E</math>。<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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假设我们有一个孤立系统，其宏观状态由许多变量描述。这些宏观变量可以是总体积、活塞在系统中的位置等。从而<math>\Omega</math>将取决于这些变量的值。如果某个变量不是固定的（我们不会在某个位置夹住活塞) ，那么因为在平衡状态下所有可到达状态的等几率到达的，平衡状态下的自由变量会使 <math>\Omega</math> 最大，因为这是平衡状态下最可能的情况。<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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如果该变量最初固定到某个值然后释放，当达到新的平衡时，变量将自我调整使得 <math>\Omega</math> 最大，这意味着熵将增加或保持不变(如果变量初始固定的值恰好是平衡值)。<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's [[H-theorem]], however, proves that the quantity {{math|''H''}} increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the quantity increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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假设我们初始位于一个平衡状态，突然移除了对一个变量的约束。我们做完这件事的时候，可达到的微观状态的数为<math>\Omega</math>，但是系统还没有达到平衡，所以系统处于某些可达到的状态的实际概率还不等于先验概率 <math>1/\Omega</math>。我们已经知道，最终的平衡状态相对于之前的平衡状态，熵会增加或者保持不变。然而，玻耳兹曼的H定理证明系统在不处于平衡态的期间，那个量作为时间的函数单调增加。<br />
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===Derivation of the entropy change for reversible processes===<br />
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===Derivation of the entropy change for reversible processes===<br />
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可逆过程熵变的推导<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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第二定律的第二部分指出，经历可逆过程的系统的熵变是由以下因素给出的:<br />
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: <math>dS =\frac{\delta Q}{T}</math><br />
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<math>dS =\frac{\delta Q}{T}</math><br />
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数学 dS frac delta q }{ t } / math<br />
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where the temperature is defined as:<br />
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where the temperature is defined as:<br />
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其中温度定义为:<br />
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:<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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[数学][[欧米茄][左(右)][右][数学][数学]<br />
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See [[Microcanonical ensemble|here]] for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the [[adiabatic theorem]] of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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请参阅此处对此定义的正当理由。假设系统有一些可以改变的外部参数 x。一般来说，系统的能量本征态将依赖于 x。根据量子力学的绝热定理，在系统哈密顿量无限缓慢变化的极限下，系统将保持在相同的定态，因此系统的能量会随着其所在定态的能量变化而变化。<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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广义力，x，对应于外部变量 x 的定义是，如果 x 增加一个量 dx，那么 math x dx / math 就是系统所做的功。例如，如果 x 是体积，那么 x 是压强。一个已知的系统的广义力在定态数学 e { r } / math 中给出:<br />
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: <math>X = -\frac{dE_{r}}{dx}</math><br />
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<math>X = -\frac{dE_{r}}{dx}</math><br />
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数学 x- frac { dE { r }{ dx } / math<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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由于系统可以在一个数学△ e / math 区间内的任意定态内，我们将系统的广义力定义为上述表达式的期望值:<br />
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: <math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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<math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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数学 x- 左 langle frac { dE { dx }右 rangle，/ math<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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为了计算平均值，我们通过计算在数学 y / math 和数学 y + delta y / math 之间的范围内，其中有多少数学 y / math 能量本征态有一个值来划分 ω 左(e 右) / math 能量本征态。把这个数字命名为 math Omega { y } left (e right) / math，我们有:<br />
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: <math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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<math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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Math Omega left (e right) sum { y } Omega { y } left (e right) ，/ math<br />
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<br />
<br />
<br />
<br />
The average defining the generalized force can now be written:<br />
<br />
The average defining the generalized force can now be written:<br />
<br />
定义广义力的平均值现在可以写成:<br />
<br />
<br />
<br />
<br />
<br />
: <math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
<br />
<math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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数学 x frac {1} Omega 左(e 右) sum { y Omega { y }左(e 右) ，/ math<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
<br />
We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
<br />
我们可以把它和熵对 x 在恒定能量 e 下的导数联系起来，如下所示。假设我们把 x 改成 x + dx。然后 math Omega left (e right) / math 将会改变，因为能量本征态依赖于 x，导致能量本征态进入或超出数学 e / math 和数学 e + delta e / math 之间的范围。让我们再次关注数学 y / math 和数学 y + delta y / math 之间的能量本征态。由于这些能量本征态的能量增加了 y dx，所有这些能量本征态在 e-y dx 到 e 之间的区间内从 e 以下移动到 e 以上。 有<br />
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<br />
<br />
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: <math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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<math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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数学 n { y }左(e 右) frac { y }左(e 右)}{ delta e } y dx ，/ math<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
<br />
such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
<br />
这样的能量本征态。如果数学 y dx leq delta e / math，所有这些能量本征态将移动到数学 e / math 和数学 e + delta e / math 之间的范围内，有助于数学 ω / math 的增加。从数学 e + △ e / math 下移到数学 e + △ e / math 上的能量本征态的个数是由数学 n { y }左(e + △ e 右) / math 给出的。区别在于<br />
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: <math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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<math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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数学 n { y }左(e 右)-n { y }左(e + delta e 右) ，/ math<br />
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<br />
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<br />
is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
<br />
is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
<br />
因此是数学增长的净贡献欧米茄 / 数学。请注意，如果 y dx 大于 math delta e / math，那么将会有能量本征态从 e 下面移动到 e + delta e / math 上面。它们在数学 n { y } left (e right) / math 和数学 n { y } left (e + delta e right) / math 中都有计算，因此上述表达式在这种情况下也是有效的。<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
<br />
Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
<br />
将上面的表达式表示为对 e 的导数，对 y 的求和得到表达式:<br />
<br />
<br />
<br />
<br />
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: <math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
<br />
<math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
<br />
数学左(部分欧米茄左(部分欧米茄右){ e }-和 y 左(部分欧米茄右){ x 左(部分欧米茄左(部分欧米茄右)}{ x 右) ，/ 数学<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
<br />
The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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关于 x 的欧米茄 / 数学对数导数由以下方式给出:<br />
<br />
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<br />
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: <math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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Math 左( frac { partial ln 左( Omega 右)}{部分 x 右){ e } beta x + 左( frac { partial x 右){ x } ，/ math<br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
<br />
The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
<br />
第一个术语是密集型的，即。它不能根据系统大小进行缩放。相比之下，最后一项的规模与逆系统的规模一样，因此将在热力学极限中消失。因此，我们发现:<br />
<br />
<br />
<br />
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: <math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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<math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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数学左(部分 s }右){ e } frac { x }{ t } ，/ math<br />
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<br />
<br />
<br />
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Combining this with<br />
<br />
Combining this with<br />
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结合这一点<br />
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<br />
<br />
<br />
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: <math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
<br />
<math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
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数学左(部分 s }右){ x } frac {1}{ t } ，/ math<br />
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<br />
<br />
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<br />
Gives:<br />
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Gives:<br />
<br />
给予:<br />
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<br />
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: <math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
<br />
<math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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数学 dS 左(部分 s }部分 e }右){ x } dE + 左(部分 s }右){ e } dx frac { t } + frac { t }{ delta q }{ t } ，/ math<br />
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===Derivation for systems described by the canonical ensemble===<br />
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===Derivation for systems described by the canonical ensemble===<br />
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正则系系统的推导<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the [[canonical ensemble]]:<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:<br />
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如果一个系统与某个温度为T的热浴热接触，那么在平衡状态下，关于能量本征值的概率分布值由正则系综给出:<br />
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: <math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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<math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the [[Partition function (statistical mechanics)|partition function]]. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
<br />
这里Z是一个使所有概率之和归一化的因子，这个函数被称为配分函数。现在我们考虑对温度和能级所依赖的外部参数的无限小的可逆改变。它遵循熵的一般公式:<br />
<br />
<br />
<br />
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: <math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
<br />
<math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
<br />
<br />
<br />
<br />
<br />
<br />
that<br />
<br />
that<br />
<br />
以及<br />
<br />
<br />
<br />
<br />
<br />
: <math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
<br />
<math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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<br />
<br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
<br />
Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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把正则系综的数学形式 <math>P_{j}</math> 插入，可以得到:<br />
<br />
<br />
<br />
<br />
<br />
: <math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
<br />
<math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
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==Living organisms==<br />
<br />
==Living organisms==<br />
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生命体<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as [[What is Life?|Erwin Schrödinger]], [[Léon Brillouin]]<ref>Léon Brillouin [https://books.google.com/books?hl=en&lr=&id=tPXVbiw_1P0C ''Science and Information Theory''] (Academic Press, 1962) (Dover, 2004)</ref> and [[Life and Energy|Isaac Asimov]]. It is also the topic of current{{When|date=July 2019}} research.{{Citation needed|date=July 2019}}<br />
<br />
There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as Erwin Schrödinger, Léon Brillouin and Isaac Asimov. It is also the topic of current research.<br />
<br />
表述热力学主要有两种方式：(a)描述从一种热力学平衡状态到另一种状态的路径，(b)通过使系统保持不变，同时增加周围环境的总熵的循环过程。这两种方法有助于理解生命过程。这个话题大大超出了本文的范围，但是已经有一些研究者考虑过这个问题，比如埃尔温·薛定谔、莱昂·布里渊和艾萨克.阿西莫夫。这也是当前研究的主题。<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of [[metabolism]], give out breakdown products and heat. Plants [[Photosynthesis|take in radiative energy]] from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
<br />
To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of metabolism, give out breakdown products and heat. Plants take in radiative energy from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
<br />
从近似的角度来看，生命体可以被认为是(b)的一个例子。近似地，一只动物的身体状态每天循环，使得它几乎没有什么变化。动物吸收食物、水和氧气，经过新陈代谢，输出分解的产物和热量。植物吸收来自太阳的辐射能量，这可以认为是热量，以及二氧化碳和水，然后它们释放氧气。它们就是这样生长的，最终会死亡，尸体腐烂，大部分重新变成二氧化碳和水。这可以看作是一个循环过程。总的来说，阳光来自一个高温的源——太阳，它的能量被传递到一个较低的温度汇，例如向太空辐射。这个过程使得植物周围环境的熵增加。因此从循环过程的角度来看，动物和植物服从热力学第二定律。简单的热机效率概念很难适用于这个问题，因为它们假定系统是封闭的。<br />
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From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to [[Non-equilibrium thermodynamics|unsettled current debate or research]]. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
<br />
From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to unsettled current debate or research. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
<br />
从(a)的热力学观点考虑 ，即考虑从一个平衡状态到另一个平衡状态的路径，只会产生一个粗略近似的图像，因为生命体从来不会处于热力学平衡状态。生物体经常必须被认为是开放的系统，因为它们吸收营养物质并排出废弃物。开放系统热力学目前通常从一个热力学平衡状态到另一个状态的路径的角度来考虑，或者是考虑局部热力学平衡近似下的流。生命体的问题可以通过假定一个流不变的稳态来进一步简化。对于这样的近似，熵产生的一般原理在目前有争论和研究的不确定性。尽管如此，从热力学第二定律的这个角度出发衍生出来的想法对生物是有启发意义的。<br />
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==Gravitational systems==<br />
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==Gravitational systems==<br />
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重力系统<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive [[heat capacity]], meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it.<ref name="Grandy 151">Grandy, W.T. (Jr) (2008), p. 151.</ref> Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
<br />
In systems that do not require for their descriptions the general theory of relativity, bodies always have positive heat capacity, meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it. Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
<br />
在不需要描述广义相对论的系统中，物体总是具有正的热容，这意味着温度随着能量的增加而升高。因此，当能量从高温物体流向低温物体时，源温度降低，而汇温度升高; 因此温差随时间而减小。这种情况并不总是发生在那些需要引力和广义相对论的系统中。这样的系统可以自发地朝着质量和能量不均匀分布的方向变化。这在大尺度上适用于宇宙，因此可能很难或不可能将第二定律应用于宇宙。除此之外，广义相对论所描述的系统热力学超出了本文的范围。<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
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{{main article|Non-equilibrium thermodynamics}}<br />
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The theory of classical or [[thermodynamic equilibrium|equilibrium thermodynamics]] is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<ref name="Grandy 151"/><ref>Callen, H.B. (1960/1985), p. 15.</ref><br />
<br />
The theory of classical or equilibrium thermodynamics is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<br />
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经典热力学或平衡热力学的理想化。一个主要的假设或假设，往往甚至没有明确说明，是系统的存在，在他们自己的内部状态的热力学平衡。一般来说，在给定时间内包含一个物理系统的空间区域---- 可能在自然界中找到---- 不是在热力学平衡中，用最严格的术语来解释。从更宽松的角度来说，整个宇宙中没有任何东西是或者曾经是完全精确的热力学平衡。<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of [[thermodynamic equilibrium]]. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable [[thermal fluctuations|fluctuations]]. There is an exception, the case of [[critical point (thermodynamics)|critical states]], which exhibit to the naked eye the phenomenon of [[critical opalescence]]. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of thermodynamic equilibrium. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable fluctuations. There is an exception, the case of critical states, which exhibit to the naked eye the phenomenon of critical opalescence. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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为了物理分析的目的，假设热力学平衡是非常方便的。这种假设可能依赖于试错来证明其正当性。如果这个假设是正确的，它通常是非常有价值和有用的，因为它提供了热力学理论。平衡假设的要素是，一个系统被观察到在无限长的时间内是不变的，而且一个系统中有如此多的粒子，以至于它的微粒性质可以被完全忽略。在这样的平衡假设下，一般来说，没有宏观上可以探测到的波动。有一个例外，即临界状态的例子，它们用肉眼就可以看到临界乳光现象。对于临界状态的实验室研究，需要非常长的观察时间。<br />
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In all cases, the assumption of [[thermodynamic equilibrium]], once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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In all cases, the assumption of thermodynamic equilibrium, once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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在所有情况下，热力学平衡的假设，一旦做出，就意味着没有假定的候选者“涨落”改变系统的熵。<br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<ref>Lieb, E.H., Yngvason, J. (2003), p. 190.</ref><br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<br />
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一个物理系统很容易发生内部的宏观变化，其速度之快足以使熵的恒定性假设失效。或者一个物理系统的粒子太少，以至于微粒的本质在可观测的波动中是显而易见的。那么热力学平衡的假设就会被放弃。对于非平衡态，熵没有绝对的一般定义。<br />
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There are intermediate cases, in which the assumption of local [[thermodynamic equilibrium]] is a very good approximation,<ref>Gyarmati, I. (1967/1970), pp. 4-14.</ref><ref>Glansdorff, P., Prigogine, I. (1971).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003).</ref> but strictly speaking it is still an approximation, not theoretically ideal.<br />
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There are intermediate cases, in which the assumption of local thermodynamic equilibrium is a very good approximation, but strictly speaking it is still an approximation, not theoretically ideal.<br />
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还有一些中间情况，其中局部热力学平衡的假设是一个非常好的近似值，但严格地说它仍然是一个近似值，不是理论上的理想值。<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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对于一般的非平衡状态，考虑其他量的统计力学定义可能是有用的，这些量可以方便地称为‘熵’ ，但是它们不应该与第二定律正确定义的熵混淆或混淆。这些其他的量确实属于统计力学，而不是热力学---- 第二定律的主要领域。<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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宏观观测波动的物理学超出了本文的范围。<br />
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==Arrow of time==<br />
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==Arrow of time==<br />
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时间之箭<br />
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{{Further information|Entropy (arrow of time)}}Thermodynamic Asymmetry in Time<br />
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Thermodynamic Asymmetry in Time<br />
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时间中的热力学不对称性<br />
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{{See also|Arrow of time}}<br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely [[CPT symmetry]], since the second law applies statistically, it is hypothesized, on time-asymmetric ''boundary conditions''.<ref>{{cite web|first=Craig|last=Callender|url=https://plato.stanford.edu/archives/fall2011/entries/time-thermo/|title=Thermodynamic Asymmetry in Time|encyclopedia=Stanford Encyclopedia of Philosophy|date=29 July 2011}}</ref><br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely CPT symmetry, since the second law applies statistically, it is hypothesized, on time-asymmetric boundary conditions.<br />
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热力学第二定律是一个物理定律，它不对称于时间方向的反转。这与已经观察到的物理学基本定律，即 CPT 对称性的概念并不冲突，因为第二定律在统计上适用，它是假设的，是关于时间非对称的边界条件。<br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect ([[Arrow of time#The causal arrow of time|the causal arrow of time]]).<ref>{{cite book | first = J.J.| last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1|display-authors=etal}} chapter 6</ref><br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect (the causal arrow of time).<br />
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第二定律被提出来部分解释时间向前和向后移动的区别，例如为什么原因先于结果(因果关系的时间箭头)。<br />
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==Irreversibility==<br />
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==Irreversibility==<br />
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不可逆性<br />
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Irreversibility in [[thermodynamic process]]es is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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Irreversibility in thermodynamic processes is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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热力学过程的不可逆性是热力学作用不对称性的结果，而不是物体内部不可逆的微观性质的结果。热力学操作是施加在参与机构上的宏观外部干预，而不是从它们的内部性质中得到的。由于没有认识到这一点，产生了被称为“悖论”的说法。<br />
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===Loschmidt's paradox===<br />
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===Loschmidt's paradox===<br />
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洛斯密特悖论<br />
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{{main article|Loschmidt's paradox}}<br />
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[[Loschmidt's paradox]], also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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洛斯密特悖论，也被称为可逆性悖论，其目的在于不可能从描述宏观系统微观演化的时间对称动力学推导出不可逆性。<br />
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In the opinion of [[Erwin Schrödinger|Schrödinger]], "It is now quite obvious in what manner you have to reformulate the law of entropy{{snd}}or for that matter, all other irreversible statements{{snd}}so that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other."<ref>[[Erwin Schrödinger|Schrödinger, E.]] (1950), p.&nbsp;192.</ref> The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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In the opinion of Schrödinger, "It is now quite obvious in what manner you have to reformulate the law of entropyor for that matter, all other irreversible statementsso that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other." The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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根据薛定谔的观点，“现在很明显，你必须用什么方式来重新定义该物质的熵定律，所有其他不可逆的状态，这样它们才能从可逆模型中导出。你不能只谈论一个孤立的系统，而应该至少谈论两个，你可以暂时认为它们与世界其它地方是孤立的，但并不总是相互孤立的。”这两个系统被隔离墙隔离开来，直到按照法律的设想被热力学操作拆除。热力学操作是外部强加的，不受支配系统组成部分的可逆微观动力学定律的制约。它是不可逆转性的原因。本文中的法律陈述符合薛定谔的建议。因果关系在逻辑上优先于第二定律，而不是从第二定律派生出来的。<br />
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===Poincaré recurrence theorem===<br />
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===Poincaré recurrence theorem===<br />
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庞加莱始态复现定理<br />
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{{Main article|Poincaré recurrence theorem}}<br />
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The [[Poincaré recurrence theorem]] considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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The Poincaré recurrence theorem considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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庞加莱始态复现定理认为这是一个孤立的物理系统的理论微观描述。这可以被认为是一个热力学系统的模型后，一个热力学操作已经移除了一个内墙。在足够长的时间之后，系统将返回到一个非常接近初始状态的显微定义状态。庞加莱回归时间是指回归前经过的时间长度。它极其漫长，可能比宇宙的寿命还要长，并且敏感地依赖于被热力学操作拆除的墙体的几何形状。递归定理可能被认为是明显与热力学第二定律相矛盾的。然而，更明显的是，它只不过是一个微观模型，热力学平衡在一个孤立的系统中，通过移除两个系统之间的一道墙而形成。对于一个典型的热力学系统，重现时间是如此之大(比宇宙的生命周期长许多倍) ，以至于在所有的实际目的中，人们都不能观察到重现。然而，人们可能希望想象一下，可以等待庞加莱复发，然后重新插入被热力学操作移除的壁。然后很明显，不可逆性的出现是由于庞加莱循环的完全不可预测性，只要初始状态是热力学平衡，就像宏观热力学的情况一样。即使可以等待，也没有实际可能性选择合适的时间重新插入墙壁。庞加莱始态复现定理为 Loschmidt 悖论提供了一个解决方案。如果一个孤立的热力学系统可以被监测到在越来越多的平均 poincar 递归时间的倍数上，系统的热力学行为将在时间反转下变得不变。<br />
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[[File:James-clerk-maxwell3.jpg|thumb|upright|James Clerk Maxwell]]<br />
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James Clerk Maxwell<br />
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詹姆斯·克拉克·麦克斯韦<br />
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===Maxwell's demon===<br />
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===Maxwell's demon===<br />
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麦克斯韦的魔鬼<br />
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{{unreferenced|section|date=August 2018}}<br />
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{{main article|Maxwell's demon}}<br />
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[[James Clerk Maxwell]] imagined one container divided into two parts, ''A'' and ''B''. Both parts are filled with the same [[gas]] at equal temperatures and placed next to each other, separated by a wall. Observing the [[molecule]]s on both sides, an imaginary [[demon]] guards a microscopic trapdoor in the wall. When a faster-than-average molecule from ''A'' flies towards the trapdoor, the demon opens it, and the molecule will fly from ''A'' to ''B''. The average [[speed]] of the molecules in ''B'' will have increased while in ''A'' they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in ''A'' and increases in ''B'', contrary to the second law of thermodynamics.<br />
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James Clerk Maxwell imagined one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other, separated by a wall. Observing the molecules on both sides, an imaginary demon guards a microscopic trapdoor in the wall. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics.<br />
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詹姆斯·克拉克·麦克斯韦想象一个容器分成两部分，a 和 b。两个部分在相同的温度下充入相同的气体，并被一堵墙隔开，挨着放置。通过观察两边的分子，一个想象中的恶魔守卫着墙上的一个微型暗门。当一个来自 a 的比平均速度快的分子飞向活板门时，魔鬼打开了它，分子就会从 a 飞到 b。B 中分子的平均速度会增加，而 a 中分子的平均速度会减慢。由于平均分子速度与温度相对应，因此 a 中的温度降低，b 中的温度升高，这与热力学第二定律相反。<br />
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One response to this question was suggested in 1929 by [[Leó Szilárd]] and later by [[Léon Brillouin]]. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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One response to this question was suggested in 1929 by Leó Szilárd and later by Léon Brillouin. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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1929年，Leó Szilárd 和后来的 l on brilouin 对这个问题提出了一个答案。Szil rd 指出，现实生活中的麦克斯韦恶魔需要一些测量分子速度的方法，而获取信息需要消耗大量能量。<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between ''A'' and ''B''. It is therefore performing [[thermodynamic operation]]s on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between A and B. It is therefore performing thermodynamic operations on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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麦克斯韦的“魔鬼”不断地改变 a 和 b 之间墙体的渗透性。因此，它是在微观尺度上执行热力学操作，而不仅仅是观察普通的自发或自然的宏观热力学过程。<br />
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==Quotations==<br />
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==Quotations==<br />
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语录<br />
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{{Wikiquote}}<br />
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{{quote|The law that entropy always increases holds, I think, the supreme position among the [[Laws of science|laws of Nature]]. If someone points out to you that your pet theory of the [[universe]] is in disagreement with [[Maxwell's equations]] – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.|Sir [[Arthur Stanley Eddington]], ''The Nature of the Physical World'' (1927)}}<br />
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{{quote|There have been nearly as many formulations of the second law as there have been discussions of it.|Philosopher / Physicist [[Percy Williams Bridgman|P.W. Bridgman]], (1941)}}<br />
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{{quote|Clausius is the author of the sibyllic utterance, "The energy of the universe is constant; the entropy of the universe tends to a maximum." The objectives of continuum thermomechanics stop far short of explaining the "universe", but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size.|[[Clifford Truesdell|Truesdell, C.]], Muncaster, R. G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 17.}}<br />
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==See also==<br />
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参见<br />
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*[[Clausius–Duhem inequality]]<br />
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*[[Fluctuation theorem]]<br />
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*[[Heat death of the universe]]<br />
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*[[History of thermodynamics]]<br />
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*[[Jarzynski equality]]<br />
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*[[Laws of thermodynamics]]<br />
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*[[Maximum entropy thermodynamics]]<br />
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*[[Quantum thermodynamics]]<br />
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*[[Reflections on the Motive Power of Fire]]<br />
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*[[Relativistic heat conduction]]<br />
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*[[Thermal diode]]<br />
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*[[Thermodynamic equilibrium]]<br />
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{{colend}}<br />
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==References==<br />
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==References==<br />
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参考资料<br />
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{{reflist|20em}}<br />
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===Sources===<br />
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===Sources===<br />
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资料来源<br />
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{{refbegin|50em}}<br />
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*Adkins, C.J. (1968/1983). ''Equilibrium Thermodynamics'', (1st edition 1968), third edition 1983, Cambridge University Press, Cambridge UK, {{ISBN|0-521-25445-0}}.<br />
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*[[Peter Atkins|Atkins, P.W.]], de Paula, J. (2006). ''Atkins' Physical Chemistry'', eighth edition, W.H. Freeman, New York, {{ISBN|978-0-7167-8759-4}}.<br />
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*Attard, P. (2012). ''Non-equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications'', Oxford University Press, Oxford UK, {{ISBN|978-0-19-966276-0}}.<br />
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*Baierlein, R. (1999). ''Thermal Physics'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-59082-5}}.<br />
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*Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics, New York, {{ISBN|0-88318-797-3}}.<br />
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*{{Cite book|title=Concepts in thermal physics|last=Blundell|first=Stephen J.|last2=Blundell|authorlink1=Stephen Blundell|first2=Katherine M.|authorlink2=Katherine Blundell|publisher=[[Oxford University Press]]|year=2010|isbn=9780199562107|edition=2nd|location=Oxford|pages=|oclc=607907330|doi=10.1093/acprof:oso/9780199562091.001.0001 |url=http://cds.cern.ch/record/1235139}}<br />
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*[[Ludwig Boltzmann|Boltzmann, L.]] (1896/1964). ''Lectures on Gas Theory'', translated by S.G. Brush, University of California Press, Berkeley.<br />
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*Borgnakke, C., Sonntag., R.E. (2009). ''Fundamentals of Thermodynamics'', seventh edition, Wiley, {{ISBN|978-0-470-04192-5}}.<br />
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*Buchdahl, H.A. (1966). ''The Concepts of Classical Thermodynamics'', Cambridge University Press, Cambridge UK.<br />
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*[[Percy Williams Bridgman|Bridgman, P.W.]] (1943). ''The Nature of Thermodynamics'', Harvard University Press, Cambridge MA.<br />
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*[[Herbert Callen|Callen, H.B.]] (1960/1985). ''Thermodynamics and an Introduction to Thermostatistics'', (1st edition 1960) 2nd edition 1985, Wiley, New York, {{ISBN|0-471-86256-8}}.<br />
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*{{cite journal|author=C. Carathéodory|author1-link=Constantin Carathéodory<br />
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<br />
|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
<br />
|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
<br />
|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
<br />
2010年3月15日 | 第355-386页 | http://gdz.sub.uni-goettingen.de/index.php?id=11&ppn=ppn235181684_0067&dmdid=dmdlog_0033&l=1<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). ''The Second Law of Thermodynamics'', Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
<br />
|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
<br />
|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können.(p. 363) | doi 10.1007 / bf01450409}.译文可以在这里找到[ http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf ]。同时，在 Kestin，j。(1976).热力学第二定律，Dowden，Hutchinson & Ross，Stroudsburg PA. 。<br />
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*[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986). [http://www.worldcat.org/title/reflections-on-the-motive-power-of-fire-a-critical-edition-with-the-surviving-scientific-manuscripts-translated-and-edited-by-fox-robert/oclc/812944517&referer=brief_results ''Reflections on the motive power of fire''], Manchester University Press, Manchester UK, {{ISBN|0-7190-1741-6}}. [https://archive.org/stream/reflectionsonmot00carnrich#page/n7/mode/2up Also here.]<br />
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*[[Sydney Chapman (mathematician)|Chapman, S.]], [[Thomas George Cowling|Cowling, T.G.]] (1939/1970). ''The Mathematical Theory of Non-uniform gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases'', third edition 1970, Cambridge University Press, London.<br />
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*{{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Ueber Die Bewegende Kraft Der Wärme Und Die Gesetze, Welche Sich Daraus Für Die Wärmelehre Selbst Ableiten Lassen|journal=Annalen der Physik|year=1850|volume=79|issue=4|pages=368–397, 500–524|url=http://gallica.bnf.fr/ark:/12148/bpt6k15164w/f518.image|accessdate=26 June 2012|doi=10.1002/andp.18501550403|bibcode = 1850AnP...155..500C |hdl=2027/uc1.$b242250|hdl-access=free}} Translated into English: {{cite journal|last=Clausius|first=R.|title=On the Moving Force of Heat, and the Laws regarding the Nature of Heat itself which are deducible therefrom|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1851| volume=2|series=4th|issue=VIII|pages=1–21; 102–119 |url=https://archive.org/stream/londonedinburghd02lond#page/1/mode/1up|accessdate=26 June 2012<br />
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|doi=10.1080/14786445108646819}}<br />
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|doi=10.1080/14786445108646819}}<br />
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10.1080 / 14786445108646819}<br />
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* {{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie|year=1854|journal=Annalen der Physik|volume=xciii|issue=12|pages=481–506|url=http://zfbb.thulb.uni-jena.de/servlets/MCRFileNodeServlet/jportal_derivate_00140956/18541691202_ftp.pdf|accessdate=24 March 2014|doi=10.1002/andp.18541691202<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
<br />
| bibcode 1854AnP... 169. 1856年7月 | 第2卷 | 第4页 | 第86页 | https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=242014年3月<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
<br />
热力学理论及其在蒸汽机和物体物理性质中的应用 | 1867年 | 出版商 John van Voorst | 位置: 伦敦 | https://archive.org/details/mechanicaltheor04claugoog|quote=editions:pwr_sbkwa8ic。 2012年6月19日<br />
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}}<br />
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*Denbigh, K. (1954/1981). ''The Principles of Chemical Equilibrium. With Applications in Chemistry and Chemical Engineering'', fourth edition, Cambridge University Press, Cambridge UK, {{ISBN|0-521-23682-7}}.<br />
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*Eu, B.C. (2002). ''Generalized Thermodynamics. The Thermodynamics of Irreversible Processes and Generalized Hydrodynamics'', Kluwer Academic Publishers, Dordrecht, {{ISBN|1-4020-0788-4}}.<br />
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*[[Josiah Willard Gibbs|Gibbs, J.W.]] (1876/1878). On the equilibrium of heterogeneous substances, ''Trans. Conn. Acad.'', '''3''': 108-248, 343-524, reprinted in ''The Collected Works of J. Willard Gibbs, Ph.D, LL. D.'', edited by W.R. Longley, R.G. Van Name, Longmans, Green & Co., New York, 1928, volume 1, pp.&nbsp;55–353.<br />
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*Griem, H.R. (2005). ''Principles of Plasma Spectroscopy (Cambridge Monographs on Plasma Physics)'', Cambridge University Press, New York {{ISBN|0-521-61941-6}}.<br />
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*Glansdorff, P., Prigogine, I. (1971). ''Thermodynamic Theory of Structure, Stability, and Fluctuations'', Wiley-Interscience, London, 1971, {{ISBN|0-471-30280-5}}.<br />
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*Grandy, W.T., Jr (2008). [http://global.oup.com/academic/product/entropy-and-the-time-evolution-of-macroscopic-systems-9780199546176?cc=au&lang=en& ''Entropy and the Time Evolution of Macroscopic Systems'']. Oxford University Press. {{ISBN|978-0-19-954617-6}}.<br />
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*Greven, A., Keller, G., Warnecke (editors) (2003). ''Entropy'', Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[Edward A. Guggenheim|Guggenheim, E.A.]] (1949). 'Statistical basis of thermodynamics', ''Research'', '''2''': 450–454.<br />
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*[[Edward A. Guggenheim|Guggenheim, E.A.]] (1967). ''Thermodynamics. An Advanced Treatment for Chemists and Physicists'', fifth revised edition, North Holland, Amsterdam.<br />
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*Gyarmati, I. (1967/1970) ''Non-equilibrium Thermodynamics. Field Theory and Variational Principles'', translated by E. Gyarmati and W.F. Heinz, Springer, New York.<br />
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*[[Charles Kittel|Kittel, C.]], [[Herbert Kroemer|Kroemer, H.]] (1969/1980). ''Thermal Physics'', second edition, Freeman, San Francisco CA, {{ISBN|0-7167-1088-9}}.<br />
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*Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998). ''Modern Thermodynamics: From Heat Engines to Dissipative Structures'', John Wiley & Sons, Chichester, {{ISBN|0-471-97393-9}}.<br />
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*Lebon, G., Jou, D., Casas-Vázquez, J. (2008). ''Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers'', Springer-Verlag, Berlin, {{ISBN|978-3-540-74252-4}}.<br />
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*{{cite journal|ref=harv|title=The Physics and Mathematics of the Second Law of Thermodynamics|last1=Lieb |first1=E. H. |last2=Yngvason |first2=J.|journal=Physics Reports|volume=310|issue=1|pages=1–96 |year=1999|doi=10.1016/S0370-1573(98)00082-9|arxiv = cond-mat/9708200 |bibcode = 1999PhR...310....1L}}<br />
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*Lieb, E.H., Yngvason, J. (2003). The Entropy of Classical Thermodynamics, pp.&nbsp;147–195, Chapter 8 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*{{cite book|ref=harv|first=F.|last=Mandl|title=Statistical physics|edition=second|year=1988|publisher=[[Wiley & Sons]]|isbn=978-0-471-91533-1}}<br />
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*{{cite journal | last1 = Maxwell | first1 = J.C. | authorlink = James Clerk Maxwell | year = 1867 | title = On the dynamical theory of gases | url = | journal = Phil. Trans. R. Soc. Lond. | volume = 157 | issue = | pages = 49–88 | doi = 10.1098/rstl.1867.0004 }}<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985). ''Thermodynamics'', Pitman, London, {{ISBN|0-273-08577-8}}.<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003). Entropy in Nonequilibrium, pp.&nbsp;79–109, Chapter 5 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*Münster, A. (1970), ''Classical Thermodynamics'', translated by E.S. Halberstadt, Wiley–Interscience, London, {{ISBN|0-471-62430-6}}.<br />
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*[[Brian Pippard|Pippard, A.B.]] (1957/1966). ''Elements of Classical Thermodynamics for Advanced Students of Physics'', original publication 1957, reprint 1966, Cambridge University Press, Cambridge UK.<br />
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*[[Max Planck|Planck, M.]] (1897/1903). [https://archive.org/stream/treatiseonthermo00planrich#page/100/mode/2up ''Treatise on Thermodynamics'', translated by A. Ogg, Longmans Green, London, p. 100.]<br />
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*[[Max Planck|Planck. M.]] (1914). [https://archive.org/details/theoryofheatradi00planrich ''The Theory of Heat Radiation''], a translation by Masius, M. of the second German edition, P. Blakiston's Son & Co., Philadelphia.<br />
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*[[Max Planck|Planck, M.]] (1926). Über die Begründung des zweiten Hauptsatzes der Thermodynamik, ''Sitzungsberichte der Preussischen Akademie der Wissenschaften: Physikalisch-mathematische Klasse'': 453–463.<br />
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* Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769-781.<br />
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* {{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}<br />
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*Quinn, T.J. (1983). ''Temperature'', Academic Press, London, {{ISBN|0-12-569680-9}}.<br />
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*{{cite book|ref=harv|first=Y.V.C.|last=Rao|title=An Introduction to thermodynamics|url=https://books.google.com/books?id=iYWiCXziWsEC&pg=PA213|date=2004|publisher=Universities Press|isbn=978-81-7371-461-0|page=213}}<br />
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*Roberts, J.K., Miller, A.R. (1928/1960). ''Heat and Thermodynamics'', (first edition 1928), fifth edition, Blackie & Son Limited, Glasgow.<br />
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*[[Erwin Schrödinger|Schrödinger, E.]] (1950). Irreversibility, ''Proc. R. Ir. Acad.'', '''A53''': 189–195.<br />
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*[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966). ''Elements of Thermodynamics'', Addison-Wesley Publishing, Reading MA.<br />
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*{{cite journal|ref=harv|last=Thomson|first=W.|author-link=William Thomson, 1st Baron Kelvin|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Transactions of the Royal Society of Edinburgh|year=1851|volume=XX|issue=part II|pages=261–268; 289–298|url=https://www.biodiversitylibrary.org/item/126047#page/295/mode/1up}} Also published in {{cite journal|last=Thomson|first=W.|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Philos. Mag. |date=December 1852 |volume=IV |series=4 |issue=22 |page=13|url=https://archive.org/stream/londonedinburghp04maga#page/12/mode/2up |accessdate=25 June 2012}}<br />
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*[[William Thomson, 1st Baron Kelvin|Thomson, W.]] (1852). ''On the universal tendency in nature to the dissipation of mechanical energy'' Philosophical Magazine, Ser. 4, p.&nbsp;304.<br />
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*[[László Tisza|Tisza, L.]] (1966). ''Generalized Thermodynamics'', M.I.T Press, Cambridge MA.<br />
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*[[Clifford Truesdell|Truesdell, C.]] (1980). ''The Tragicomical History of Thermodynamics 1822–1854'', Springer, New York, {{ISBN|0-387-90403-4}}.<br />
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*Uffink, J. (2001). Bluff your way in the second law of thermodynamics, ''Stud. Hist. Phil. Mod. Phys.'', '''32'''(3): 305–394.<br />
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*Uffink, J. (2003). Irreversibility and the Second Law of Thermodynamics, Chapter 7 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003), Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963). ''Lectures in Statistical Mechanics'', American Mathematical Society, Providence RI.<br />
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*[[Mark Zemansky|Zemansky, M.W.]] (1968). ''Heat and Thermodynamics. An Intermediate Textbook'', fifth edition, McGraw-Hill Book Company, New York.<br />
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==Further reading==<br />
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==Further reading==<br />
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进一步阅读<br />
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*Goldstein, Martin, and Inge F., 1993. ''The Refrigerator and the Universe''. Harvard Univ. Press. Chpts. 4–9 contain an introduction to the Second Law, one a bit less technical than this entry. {{ISBN|978-0-674-75324-2}}<br />
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*Leff, Harvey S., and Rex, Andrew F. (eds.) 2003. ''Maxwell's Demon 2 : Entropy, classical and quantum information, computing''. Bristol UK; Philadelphia PA: [[Institute of Physics]]. {{ISBN|978-0-585-49237-7}}<br />
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*{{Cite book | first = J.J. | last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1}}(technical).<br />
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*{{cite book |title=Reflections on the Motive Power of Heat and on Machines Fitted to Develop That Power |last=Carnot |first=Sadi |editor=[[Robert Henry Thurston|Thurston, Robert Henry]] |year=1890 |publisher=J. Wiley & Sons |location=New York |isbn= |pages= }} ([https://books.google.com/books?id=tgdJAAAAIAAJ full text of 1897 ed.]) ([http://www.history.rochester.edu/steam/carnot/1943/ html])<br />
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*Stephen Jay Kline (1999). ''The Low-Down on Entropy and Interpretive Thermodynamics'', La Cañada, CA: DCW Industries. {{ISBN|1-928729-01-0}}.<br />
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*{{cite book | last1 = Kostic | first1 = M | year = 2011 | title = Revisiting The Second Law of Energy Degradation and Entropy Generation: From Sadi Carnot's Ingenious Reasoning to Holistic Generalization | journal = AIP Conf. Proc. | volume = 1411 | issue = 1| pages = 327–350 | doi = 10.1063/1.3665247 | isbn = 978-0-7354-0985-9 | bibcode = 2011AIPC.1411..327K | series = AIP Conference Proceedings | citeseerx = 10.1.1.405.1945 }} also at [https://web.archive.org/web/20130420222450/http://www.kostic.niu.edu/2ndLaw/Revisiting%20The%20Second%20Law%20of%20Energy%20Degradation%20and%20Entropy%20Generation%20-%20From%20Carnot%20to%20Holistic%20Generalization-4.pdf].<br />
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==External links==<br />
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==External links==<br />
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外部链接<br />
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*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics]" – by Lawrence Sklar.<br />
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*[http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node30.html ''Second law of thermodynamics''] in the MIT Course [http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/notes.html ''Unified Thermodynamics and Propulsion''] from Prof. Z. S. Spakovszky<br />
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*[[E.T. Jaynes]], 1988, "[http://bayes.wustl.edu/etj/articles/ccarnot.pdf The evolution of Carnot's principle]," in G. J. Erickson and C. R. Smith (eds.)''Maximum-Entropy and Bayesian Methods in Science and Engineering, Vol'' 1: p.&nbsp;267.<br />
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*[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf Caratheodory, C., "Examination of the foundations of thermodynamics," trans. by D. H. Delphenich]<br />
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* [https://www.bbc.co.uk/programmes/p004y2bm The Second Law of Thermodynamics], BBC Radio 4 discussion with John Gribbin, Peter Atkins & Monica Grady (''In Our Time'', Dec. 16, 2004)<br />
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* [http://mdpi.org/entropy/papers/e6010001.pdf [[Entropy (journal)]], 2004]<br />
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* [https://www.journals.uchicago.edu/doi/abs/10.1086/663835 The Journal of the International Society for the History of Philosophy of Science, 2012]<br />
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{{DEFAULTSORT:Second Law Of Thermodynamics}}<br />
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[[Category:Concepts in physics]]<br />
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Category:Concepts in physics<br />
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分类: 物理概念<br />
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[[Category:Laws of thermodynamics|2]]<br />
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[[Category:Non-equilibrium thermodynamics]]<br />
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Category:Non-equilibrium thermodynamics<br />
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类别: 非平衡态热力学<br />
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[[Category:Philosophy of thermal and statistical physics]]<br />
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Category:Philosophy of thermal and statistical physics<br />
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类别: 热力学和统计物理学哲学<br />
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<small>This page was moved from [[wikipedia:en:Second law of thermodynamics]]. Its edit history can be viewed at [[热力学第二定律/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E7%83%AD%E5%8A%9B%E5%AD%A6%E7%AC%AC%E4%BA%8C%E5%AE%9A%E5%BE%8B&diff=10356热力学第二定律2020-07-16T09:33:39Z<p>Dorr：/* Carnot theorem 卡诺定理 */</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Law of physics}}<br />
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{{Thermodynamics|laws}}<br />
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The '''second law of thermodynamics''' states that the total [[entropy]] of an [[isolated system]] can never decrease over time, and is constant if and only if all processes are reversible.<ref>http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node38.html#SECTION05224000000000000000</ref> Isolated systems spontaneously evolve towards [[thermodynamic equilibrium]], the state with maximum entropy. <br />
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The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, and is constant if and only if all processes are reversible. Isolated systems spontaneously evolve towards thermodynamic equilibrium, the state with maximum entropy. <br />
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热力学第二定律指出，孤立系统的总熵永远不会随着时间而减少，而且只有当且仅当所有过程都是可逆的时，总熵才是恒定的。孤立的系统自发地进化到热力学平衡状态，即熵最大的状态。<br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in [[thermodynamic equilibrium]], or is undergoing a (fictive) [[Reversible process (thermodynamics)|reversible process]]. In all processes that occur, including [[spontaneous process]]es,<ref>Atkins and de Paula, p.78</ref> the total entropy of the system and its surroundings increases and the process is [[Irreversible process|irreversible in the thermodynamic sense]]. The increase in entropy accounts for the irreversibility of natural processes, and the [[Arrow of time|asymmetry between future and past]].<ref>{{cite book|last=Zohuri|first=Bahman|title=Dimensional Analysis Beyond the Pi Theorem|url=https://books.google.com/books?id=pRVuDQAAQBAJ|year=2016|publisher=Springer|isbn=978-3-319-45726-0|page=[https://books.google.com.ph/books?id=pRVuDQAAQBAJ&pg=PA111&dq=%22increase+in+entropy+accounts+for+the+irreversibility+of+natural+processes+and+the+asymmetry+between+future+and+past.%22&hl=en&sa=X&ved=0ahUKEwj6spb61tbaAhUFS7wKHfftDtIQ6AEIKDAA#v=onepage&q=%22increase%20in%20entropy%20accounts%20for%20the%20irreversibility%20of%20natural%20processes%20and%20the%20asymmetry%20between%20future%20and%20past.%22 111]}}</ref><br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in thermodynamic equilibrium, or is undergoing a (fictive) reversible process. In all processes that occur, including spontaneous processes, the total entropy of the system and its surroundings increases and the process is irreversible in the thermodynamic sense. The increase in entropy accounts for the irreversibility of natural processes, and the asymmetry between future and past.<br />
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系统及其周围环境的总熵在理想情况下可以保持不变，在这种情况下，系统处于热力学平衡期，或者正在经历一个(虚拟的)可逆过程期。在所有发生的过程中，包括自发过程，系统及其周围环境的总熵增加，这一过程在热力学意义上是不可逆的。熵的增加解释了自然过程的不可逆性，以及未来和过去之间的不对称性。<br />
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Historically, the second law was an [[Empirical evidence|empirical finding]] that was accepted as an axiom of [[thermodynamics|thermodynamic theory]]. [[Statistical mechanics]], classical or [[quantum thermodynamics|quantum]], explains the microscopic origin of the law.<br />
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Historically, the second law was an empirical finding that was accepted as an axiom of thermodynamic theory. Statistical mechanics, classical or quantum, explains the microscopic origin of the law.<br />
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从历史上看，第二定律是一个经验性的发现，被公认为热力学理论的公理。统计力学，无论是经典的还是量子的，都可以解释这个定律的微观起源。<br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist [[Nicolas Léonard Sadi Carnot|Sadi Carnot]], who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as [[Carnot's theorem (thermodynamics)|Carnot's rule]] or limit.<ref>Jaffe, R.L., Taylor, W. (2018). ''The Physics of Energy'', Cambridge University Press, Cambridge UK, pages 150, 151, 259, 772, 743.</ref><br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist Sadi Carnot, who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as Carnot's rule or limit.<br />
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第二定律已经用许多方式表达过了。法国科学家萨迪 · 卡诺在1824年证明了热能在热机中转化为功的效率是有上限的。这方面的第二定律也被称为卡诺的规则或限制。<br />
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==Introduction==<br />
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==Introduction==<br />
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引言<br />
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[[File:Heat flow hot to cold.png|thumb|upright|Heat flow from hot water to cold water.]]<br />
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Heat flow from hot water to cold water.<br />
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热量从热水流向冷水。<br />
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The [[first law of thermodynamics]] provides the definition of the [[internal energy]] of a [[thermodynamic system]], and expresses the law of [[conservation of energy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 40–41.</ref><ref>Munster A. (1970), pp. 8–9, 50–51.</ref> The second law is concerned with the direction of natural processes.<ref>{{harvnb|Mandl|1988}}</ref> It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such [[Phenomenon|phenomena]] are accounted for in terms of [[entropy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 79–107.</ref><ref>Bailyn, M. (1994), Section 71, pp. 113–154.</ref> If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, {{math|''S''}}, increases.<br />
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The first law of thermodynamics provides the definition of the internal energy of a thermodynamic system, and expresses the law of conservation of energy. The second law is concerned with the direction of natural processes. It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such phenomena are accounted for in terms of entropy. If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, , increases.<br />
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能量守恒定律提供了热力学系统内能的定义，并表达了能量守恒定律。第二定律与自然过程的方向有关。它断言自然过程只在一种意义上运行，而且是不可逆的。例如，当有了传导和辐射的路径时，热总是自发地从一个较热的物体流向一个较冷的物体。这种现象可以用熵来解释。如果一个孤立的系统最初是通过内部分隔不可渗透的墙来保持在内部热力学平衡，然后一些操作使墙更具渗透性，那么系统自发地进化到最终达到一个新的内部热力学平衡，其总熵增加。<br />
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In a fictive reversible process, an infinitesimal increment in the entropy ({{math|d''S''}}) of a system is defined to result from an infinitesimal transfer of heat ({{math|δ''Q''}}) to a [[closed system]] (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature ({{math|''T''}}) of the system in equilibrium and the surroundings which supply the heat:<ref>Bailyn, M. (1994), p. 120.</ref><br />
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In a fictive reversible process, an infinitesimal increment in the entropy () of a system is defined to result from an infinitesimal transfer of heat () to a closed system (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature () of the system in equilibrium and the surroundings which supply the heat:<br />
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在虚构的可逆过程中，系统熵的无穷小增量被定义为由无穷小的热量传递()到一个封闭系统(允许能量进入或出去，但不允许物质传递)除以平衡系统和提供热量的环境的共同温度() :<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, idealized fictive reversible process)}.</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, idealized fictive reversible process)}.</math><br />
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(闭合系统，理想化的可逆过程) / math<br />
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Different notations are used for infinitesimal amounts of heat ({{mvar|δ}}) and infinitesimal amounts of entropy ({{mvar|d}}) because entropy is a [[function of state]], while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the [[Clausius Theorem|inequality]] <ref name="MortimerBook"> Mortimer, R. G. ''Physical Chemistry''. 3rd ed., p. 120, Academic Press, 2008. </ref><ref name="FermiBook"> Fermi, E. ''Thermodynamics'', footnote on p. 48, Dover Publications,1956 (still in print). </ref> <br />
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Different notations are used for infinitesimal amounts of heat () and infinitesimal amounts of entropy () because entropy is a function of state, while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the inequality <br />
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用不同的符号表示无穷小量的热量()和无穷小量的熵() ，因为熵是状态的函数，而热量和功一样不是。第二定律要求系统熵的增量满足不等式，对于实际上可能存在的不与环境发生质量交换的无穷小过程，系统熵增量满足不等式<br />
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: <math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process).}</math><br />
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<math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process).}</math><br />
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(实际上可能是封闭系统，不可逆性)。数学<br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature ({{math|''T''}}) and the temperature of the surroundings ({{math|''T''}}<sub>''surr''</sub>).<ref name=":0">Adkins, C.J. (1968/1983), p. 75.</ref><ref name="Munster 45"/> Note that the equality still applies for pure heat flow,<ref name="Schmidt-Rohr 14"> Schmidt-Rohr, K. (2014). "Expansion Work without the External Pressure, and Thermodynamics in Terms of Quasistatic Irreversible Processes" ''J. Chem. Educ.'' '''91''': 402-409. https://dx.doi.org/10.1021/ed3008704 </ref><br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature () and the temperature of the surroundings (<sub>surr</sub>).<br />
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这是因为这种情况下的一般过程可能包括周围环境对系统所做的功，这可能在系统内部产生摩擦或粘滞效应，因为一个化学反应可能正在进行，或者因为热传递实际上只是不可逆地发生，由系统温度()和周围环境温度(sub surr / sub)之间的差分驱动。<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change).}</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change).}</math><br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry.<ref name="Oxtoby8th">Oxtoby, D. W; Gillis, H.P., [[Laurie Butler|Butler, L. J.]] (2015).''Principles of Modern Chemistry'', Brooks Cole. p. 617. {{ISBN|978-1305079113}}</ref> <ref name="MortimerBook"></ref> Introducing a set of internal variables <math>\xi</math> to describe the deviation of a thermodynamic system in physical equilibrium (with the required well-defined uniform pressure ''P'' and temperature ''T'')<ref name="Schmidt-Rohr 14"></ref> from the chemical equilibrium state, one can record the equality<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry. from the chemical equilibrium state, one can record the equality<br />
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这是根据测得的热容曲线和相变时的熵变精确确定纯物质的绝对熵的基础。用量热法。从化学平衡来看，人们可以记录平等<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process).}</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process).}</math><br />
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- frac { t }-frac {1}{ j } ， Xi { j } ， delta Xi j， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ,.数学<br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of ''the second principle of thermodynamics'' – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<ref> Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769–781.</ref><ref>{{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}</ref><br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of the second principle of thermodynamics – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<br />
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第二项代表内部变量的工作，这些内部变量可以受到外部影响的干扰，但是系统不能通过内部变量执行任何正功。这种说法介绍了热力学系统在时间上不可能逆转演化的事实，并且可以被认为是热力学第二原理的表述---- 这个表述，当然，相当于熵原理的表述。<br />
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The [[zeroth law of thermodynamics]] in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body.<ref name=dugdale>{{cite book|author=J. S. Dugdale|title=Entropy and its Physical Meaning|publisher=Taylor & Francis|year=1996|isbn=978-0-7484-0569-5|page=13|quote=This law is the basis of temperature.}}</ref> For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows{{how?|date=August 2018}} a distinguished temperature scale, which defines an absolute, [[thermodynamic temperature]], independent of the properties of any particular reference thermometric body.<ref>[[Mark Zemansky|Zemansky, M.W.]] (1968), pp. 207–209.</ref><ref>Quinn, T.J. (1983), p. 8.</ref><br />
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The zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body. For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows a distinguished temperature scale, which defines an absolute, thermodynamic temperature, independent of the properties of any particular reference thermometric body.<br />
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美国热力学第零定律协会在其通常的简短声明中允许承认，相对于热平衡的两个物体具有相同的温度，特别是测试物体具有与参考温度测量物体相同的温度。对于一个物体和另一个物体之间的热平衡，存在着无限多的经验温度标度，通常分别取决于特定参考温度标度的性质。第二定律允许区分温度标度，它定义了一个绝对的热力学温度，与任何特定的参考温度体的性质无关。<br />
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==Various statements of the law==<br />
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==Various statements of the law==<br />
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各种各样的法律陈述<br />
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The second law of thermodynamics may be expressed in many specific ways,<ref name=MIT>{{cite web|title=Concept and Statements of the Second Law|url=http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node37.html|accessdate=2010-10-07 |publisher=web.mit.edu}}</ref> the most prominent classical statements{{sfnp|Lieb|Yngvason|1999}} being the statement by [[Rudolf Clausius]] (1854), the statement by [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] (1851), and the statement in axiomatic thermodynamics by [[Constantin Carathéodory]] (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.{{sfnp|Rao|2004|p=213}}<br />
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The second law of thermodynamics may be expressed in many specific ways, the most prominent classical statements being the statement by Rudolf Clausius (1854), the statement by Lord Kelvin (1851), and the statement in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.<br />
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热力学第二定律可以用许多特定的方式来表达，最突出的经典陈述是 Rudolf Clausius (1854)的陈述，Kelvin 勋爵(1851)的陈述，以及康斯坦丁·卡拉西奥多里(1909)在公理化热力学中的陈述。这些陈述用一般的物理术语描述法律，引用某些过程的不可能性。克劳修斯和开尔文陈述被证明是等价的。<br />
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===Carnot's principle===<br />
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===Carnot's principle===<br />
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卡诺原理<br />
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The historical origin<ref>[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986).</ref> of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a [[Carnot heat engine]], fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the [[caloric theory]] of heat was seriously considered, before the recognition of the [[first law of thermodynamics]], and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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The historical origin of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a Carnot heat engine, fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the caloric theory of heat was seriously considered, before the recognition of the first law of thermodynamics, and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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热力学第二定律的历史起源源于卡诺的原则。它指的是一个卡诺热机周期，虚拟地以被称为准静态的极慢极限模式运行，因此热量和功量在子系统之间传递，这些子系统始终处于它们自己的内部热力学平衡状态。卡诺发动机是热机效率工程师特别感兴趣的理想设备。卡诺认识到卡诺原理的时候，正是热量热理论被认真考虑的时候，正是能量守恒定律理论被认识之前，正是熵概念的数学表达之前。根据第一定律的解释，它在物理上等同于热力学第二定律，并且在今天仍然有效。卡诺最初的论点是从卡路里理论的观点提出的，直到能量守恒定律的发现。以下是他书中的一些例子:<br />
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::...''wherever there exists a difference of temperature, motive power can be produced.''<ref>Carnot, S. (1824/1986), p. 51.</ref><br />
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...wherever there exists a difference of temperature, motive power can be produced.<br />
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只要有温差，就能产生动力。<br />
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::The production of motive power is then due in steam engines not to an actual consumption of caloric, but ''to its transportation from a warm body to a cold body ...''<ref>Carnot, S. (1824/1986), p. 46.</ref><br />
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The production of motive power is then due in steam engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body ...<br />
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动力的产生不是由于蒸汽机实际消耗的热量，而是由于它从一个温暖的物体运输到一个寒冷的物体..。<br />
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::''The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.''<ref>Carnot, S. (1824/1986), p. 68.</ref><br />
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The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.<br />
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热的动力与实现热的媒介无关，热的数量完全取决于两个物体之间的温度，最后是热量的传递。<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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用现代术语来说，卡诺的原则可能更为准确:<br />
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::The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<ref>[[Clifford Truesdell|Truesdell, C.]] (1980), Chapter 5.</ref><ref>Adkins, C.J. (1968/1983), pp. 56–58.</ref><ref>Münster, A. (1970), p. 11.</ref><ref>Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998), pp.67–75.</ref><ref>Lebon, G., Jou, D., Casas-Vázquez, J. (2008), p. 10.</ref><ref>Eu, B.C. (2002), pp. 32–35.</ref><br />
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The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<br />
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准静态或可逆卡诺循环的效率只取决于两个热源的温度，而且无论工作物质是什么，效率是相同的。以这种方式运行的卡诺发动机是使用这两种温度的最有效的热机。<br />
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===Clausius statement===<br />
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===Clausius statement===<br />
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克劳修斯声明<br />
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The German scientist [[Rudolf Clausius]] laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work.{{sfnp|Clausius|1850}} His formulation of the second law, which was published in German in 1854, is known as the ''Clausius statement'':<br />
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The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work. His formulation of the second law, which was published in German in 1854, is known as the Clausius statement:<br />
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1850年，德国科学家 Rudolf Clausius 通过研究热传递和功之间的关系，为热力学第二定律实验室奠定了基础。他在1854年用德语发表的第二定律的提法被称为克劳修斯声明:<br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.{{sfnp|Clausius|1854|p=86}}</blockquote><br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.</blockquote><br />
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热量不可能从一个较冷的物体传递到一个较热的物体，除非同时发生一些其他的变化。 / blockquote<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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克劳修斯的陈述使用了“热量通道”的概念。正如通常在热力学的讨论，这意味着“能量作为热量的净转移” ，并没有提到促成转移的一种方式和其他。<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of [[refrigeration]], for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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如果不对系统进行外部功，热就不能自发地从冷区流向热区，这一点从制冷的普通经验中可以看出。在冰箱中，热量从冷到热，但只有在外部介质——制冷系统的强制作用下才会发生变化。<br />
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===Kelvin statements===<br />
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===Kelvin statements===<br />
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开尔文陈述<br />
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[[William Thomson, 1st Baron Kelvin|Lord Kelvin]] expressed the second law in several wordings.<br />
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Lord Kelvin expressed the second law in several wordings.<br />
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开尔文勋爵用几个字表达了第二定律。<br />
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::It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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没有任何外部机构的帮助，自动机器是不可能把热量从一个物体传递到另一个物体的较高温度。<br />
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::It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.{{sfnp|Thomson|1851}}<br />
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It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.<br />
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通过无生命物质的作用，把物质冷却到周围最冷的物体的温度以下，从物质的任何部分获得机械效应是不可能的。<br />
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===Equivalence of the Clausius and the Kelvin statements===<br />
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===Equivalence of the Clausius and the Kelvin statements===<br />
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克劳修斯和开尔文陈述的等价性<br />
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[[Image:Deriving Kelvin Statement from Clausius Statement.svg|thumb|Derive Kelvin Statement from Clausius Statement]]<br />
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Derive Kelvin Statement from Clausius Statement<br />
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从克劳修斯陈述推导出开尔文陈述<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed [[Carnot engine]] as shown by the figure. The [[Heat engine#Efficiency|efficiency]] of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the [[first law of thermodynamics]], as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the figure. The efficiency of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the first law of thermodynamics, as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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假设有一个引擎违反了开尔文定理: 也就是说，这个引擎以循环的方式吸收热量并将其转化为功，没有任何其他结果。现在配对与反向卡诺发动机如图所示。普通热机的效率为，反向热机的效率为1 / 。这对组合发动机的净效应和唯一效应是将热量从较冷的贮存器转移到较热的贮存器，这违反了克劳修斯的说法。(这是能量守恒定律的结果，因为系统的总能量保持不变，数学文本{输入} + 文本{输出}0暗示 q-frac { eta }-qc / math，所以数学 qc 向左(frac { eta-1 right) / math)。因此，违反开尔文陈述意味着违反克劳修斯陈述，即。克劳修斯的说法暗示了凯尔文的说法。我们可以用类似的方式证明开尔文陈述暗示了克劳修斯陈述，因此两者是等价的。<br />
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===Planck's proposition===<br />
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===Planck's proposition===<br />
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普朗克的命题<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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普朗克提出了直接来自经验的如下命题。这有时被认为是他对第二定律的陈述，但他认为这是第二定律推导的起点。<br />
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::It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 86.</ref><ref>Roberts, J.K., Miller, A.R. (1928/1960), p. 319.</ref><br />
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It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<br />
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建造一台发动机是不可能的，这台发动机将在一个完整的循环中工作，并且除了提高重量和冷却热储以外不会产生任何效果。<br />
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===Relation between Kelvin's statement and Planck's proposition 开尔文表述与普朗克命题的关系===<br />
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It is almost customary in textbooks to speak of the "[[Kelvin-Planck statement]]" of the law, as for example in the text by [[Dirk ter Haar|ter Haar]] and [[Harald Wergeland|Wergeland]].<ref>[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966), p. 17.</ref><br />
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It is almost customary in textbooks to speak of the "Kelvin-Planck statement" of the law, as for example in the text by ter Haar and Wergeland.<br />
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教科书中几乎惯常提及该定律的“'''开尔文-普朗克表述 Kelvin-Planck Statement'''” ，例如'''德克·特哈尔 Diek ter Haar''' 和'''哈拉尔德·沃格兰 Harald Wergeland''' 在文中就是这样表述的。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）Diek ter Haar 中间是小写的<br />
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The '''Kelvin–Planck statement''' (or the ''heat engine statement'') of the second law of thermodynamics states that <br />
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The Kelvin–Planck statement (or the heat engine statement) of the second law of thermodynamics states that <br />
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热力学第二定律的开尔文-普朗克表述（或称“'''热机表述 Heat Engine Statement'''”）指出<br />
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::It is impossible to devise a [[thermodynamic cycle|cyclically]] operating device, the sole effect of which is to absorb energy in the form of heat from a single [[heat reservoir|thermal reservoir]] and to deliver an equivalent amount of [[Work (physics)|work]].<ref name="Rao">{{cite book|last=Rao|first=Y. V. C.|title=Chemical Engineering Thermodynamics|publisher=Universities Press|isbn=978-81-7371-048-3|page=158|year=1997}}</ref><br />
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It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.<br />
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设计一种唯一效果是从单一热源吸收热量并提供等量的功的循环运行装置是不可能的。<br />
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===Planck's statement 普朗克表述===<br />
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Planck stated the second law as follows.<br />
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Planck stated the second law as follows.<br />
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普朗克表述第二定律如下。<br />
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::Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<ref name="Planck 100">[[Max Planck|Planck, M.]] (1897/1903), p. 100.</ref><ref name="Planck 463">[[Max Planck|Planck, M.]] (1926), p. 463, translation by Uffink, J. (2003), p. 131.</ref><ref name="Roberts & Miller 382">Roberts, J.K., Miller, A.R. (1928/1960), p. 382. This source is partly verbatim from Planck's statement, but does not cite Planck. This source calls the statement the principle of the increase of entropy.</ref><br />
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Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<br />
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自然界中发生的任一过程都是沿参与其中的所有物体的熵之和增加的方向进行的。在某些限制下——即对于可逆过程，熵的总和保持不变。<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for ''irreversible phenomena''.<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for irreversible phenomena.<br />
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与普朗克表述非常相似的是'''乌伦贝克 Uhlenbeck'''和'''福特 Ford'''关于不可逆现象的表述。<br />
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::... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<ref>[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963), p. 16.</ref><br />
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... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<br />
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在从一个平衡态到另一个平衡态的不可逆或自发的变化中（例如，当两个物体 A 和 B 接触时的温度平衡过程），熵总是增加。<br />
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===Principle of Carathéodory 卡拉西奥多里原理===<br />
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—— caratheodory 的原理在这里重新定向——<br />
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[[Constantin Carathéodory]] formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<ref>[[Constantin Carathéodory|Carathéodory, C.]] (1909).</ref><br />
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Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<br />
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'''康斯坦丁·卡拉西奥多里 Constantin Carathéodory'''在纯数学公理的基础上进行了热力学公理化。他对第二定律的陈述被称为'''卡拉西奥多里原理 Principle of Carathéodory'''，可以这样表述:<br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.<ref>Buchdahl, H.A. (1966), p. 68.</ref></blockquote><br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.</blockquote><br />
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在绝热封闭系统的任意状态 S 附近，总有从 S 出发不可达的状态。<br />
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With this formulation, he described the concept of [[adiabatic accessibility]] for the first time and provided the foundation for a new subfield of classical thermodynamics, often called [[Ruppeiner geometry|geometrical thermodynamics]]. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic [[process function]], in other words, <math>\delta Q=TdS</math>.<ref name="Sychev1991">{{cite book |last=Sychev |first=V. V. |title=The Differential Equations of Thermodynamics |year=1991 |publisher=Taylor & Francis |isbn=978-1-56032-121-7}}</ref> {{clarify|date=February 2014}}<br />
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With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, <math>\delta Q=TdS</math>. <br />
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公理化处理后，他第一次描述了'''绝热可达性 Adiabatic Accessibility'''的概念，并为经典热力学的一个新的子领域，即通常所说的'''几何热力学 Geometrical Thermodynamics'''奠定了基础。由卡拉西奥多里原理可以推出，准静态转移的热量值是一个可积过程函数，即<math>\delta Q=TdS</math>。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）准静态转移的热量值是一个可积过程函数 存疑<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.<ref name="Munster 45">Münster, A. (1970), p. 45.</ref>{{sfnp|Lieb|Yngvason|1999|p=49}}<ref name="Planck 1926">[[Max Planck|Planck, M.]] (1926).</ref><ref>Buchdahl, H.A. (1966), p. 69.</ref> {{clarify|date=February 2014}}<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium. <br />
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尽管在教科书几乎惯称卡拉西奥多里原理表述了第二定律，并认为其与克劳修斯表述或开尔文-普朗克表述等价，但事实并非如此。为了得到第二定律的所有内容，需要对卡拉西奥多里原理补充普朗克表述，即等量功总是增加一个最初处于自身内部热力学平衡的封闭系统的内部能量。<br />
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===Planck's principle 普朗克原理===<br />
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In 1926, [[Max Planck]] wrote an important paper on the basics of thermodynamics.<ref name="Planck 1926"/><ref>Uffink, J. (2003), pp. 129–132.</ref> He indicated the principle<br />
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In 1926, Max Planck wrote an important paper on the basics of thermodynamics. He indicated the principle<br />
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1926年，'''马克斯·普朗克 Max Planck'''写了一篇关于热力学基础的重要论文。他指出了以下原理<br />
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::The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<ref name="Munster 45"/>{{sfnp |Lieb|Yngvason|1999|p=49}}<br />
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The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<br />
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一个封闭系统的内部能量因绝热过程增加，在整个过程中，系统的体积保持不变。<br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work."<ref>[[Clifford Truesdell|Truesdell, C.]], Muncaster, R.G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 15.</ref> Planck wrote: "The production of heat by friction is irreversible."<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 81.</ref><ref>[[Max Planck|Planck, M.]] (1926), p. 457, Wikipedia editor's translation.</ref><br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work." Planck wrote: "The production of heat by friction is irreversible."<br />
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这个公式没有提到热，没有提到温度，甚至没有提到熵，也不一定隐含地依赖于这些概念，但它暗示了第二定律的内容。一个密切相关的表述为，“摩擦力从来不做正功。”普朗克写道: “摩擦生热是不可逆的。”<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above.<ref>Lieb, E.H., Yngvason, J. (2003), p. 149.</ref> It is relevant that for a system at constant volume and [[Mole (unit)|mole numbers]], the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above. It is relevant that for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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这个普朗克定理没有提到熵，是用物理术语来表述的。它与上面给出的开尔文表述密切相关。相关的是，对于恒定体积和摩尔数的系统，熵是内能的单调函数。然而，普朗克的这个原理实际上并不是普朗克对第二定律的首选表述（见前面小节），而是依赖于熵的概念。<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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Borgnakke 和 Sonntag 提出的以下表述在某种意义上是对普朗克原理的补充。他们没有将其作为第二定律的完整表述:<br />
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::... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<ref>Borgnakke, C., Sonntag., R.E. (2009), p. 304.</ref><br />
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... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<br />
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只有一种方法可以减少[闭合]系统的熵——从系统中转移热量。<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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与普朗克之前提出的原理不同，这一原理明确地用熵的变化来表示。从系统中去除物质也可以减少系统的熵。<br />
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===Statement for a system that has a known expression of its internal energy as a function of its extensive state variables 一个其内能有已知表达式（其扩展状态变量的函数）的系统的表述===<br />
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The second law has been shown to be equivalent to the [[internal energy]] ''U'' being a weakly [[convex function]], when written as a function of extensive properties (mass, volume, entropy, ...).<ref>{{cite book |last1=van Gool |first1=W. |last2=Bruggink |first2=J.J.C. (Eds) |url= |title=Energy and time in the economic and physical sciences |publisher=North-Holland |year=1985 |pages=41–56 |quote= |isbn=978-0-444-87748-2}}</ref><ref>{{Cite journal | last1 = Grubbström | first1 = Robert W. | doi = 10.1016/j.apenergy.2007.01.003 | title = An Attempt to Introduce Dynamics Into Generalised Exergy Considerations| journal = Applied Energy| volume = 84| issue = 7–8 | pages = 701–718 | year = 2007}}</ref> {{clarify|date=February 2014}}<br />
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The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...). <br />
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第二定律被证明等价于弱凸函数内能 U （写成广泛性质（质量，体积，熵，...）的函数时）。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）存疑<br />
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==Corollaries 推论==<br />
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===Perpetual motion of the second kind 第二类永动机===<br />
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{{main article|Perpetual motion}}<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of [[first law of thermodynamics]] by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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在第二定律建立之前，许多志在发明永动机的人试图通过提取环境的巨大内能作为机器动力，以突破'''热力学第一定律 First Law of Thermodynamics''' 的限制。这种机器被称为“'''第二类永动机 Perpetual Motion Machine of the Second Kind'''”。第二定律表明这种机器是不可能的。<br />
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===Carnot theorem 卡诺定理===<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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'''卡诺定理 Carnot's theorem'''（1824）是一条限制任何可能的发动机的最大效率的原理。效率完全取决于热库和冷库之间的温差。卡诺定理指出:<br />
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*All irreversible heat engines between two heat reservoirs are less efficient than a [[Carnot engine]] operating between the same reservoirs.<br />
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*All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs.<br />
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* 所有不可逆热机的效率低于在两相同热源之间工作的'''卡诺机 Carnot engine'''。<br />
* 所有可逆热机的效率与在两相同热源之间工作的卡诺机相等。<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as [[thermodynamic reversibility]]. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the [[Carnot cycle]]'s reversibility and was condemned to be less efficient.<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the Carnot cycle's reversibility and was condemned to be less efficient.<br />
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在他的理想模型中，热转换成功可以通过逆转循环的运动而恢复，这个概念后来被称为'''热力学可逆性 Thermodynamic Reversibility'''。然而，卡诺进一步假定，一些热量损失了，并没有转化为机械功。因此，没有一个真实的热机能够实现'''卡诺循环 Carnot cycle'''的可逆性，并且被认为效率较低。<br />
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Though formulated in terms of caloric (see the obsolete [[caloric theory]]), rather than [[entropy]], this was an early insight into the second law.<br />
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Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law.<br />
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该理论尽管是用热量表述的（见被取代的'''热质说 Caloric Theory'''），而不是熵，但是它是对第二定律的早期认识。<br />
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===Clausius inequality 克劳修斯不等式===<br />
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The [[Clausius theorem]] (1854) states that in a cyclic process<br />
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The Clausius theorem (1854) states that in a cyclic process<br />
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克劳修斯定理（1854）指出，在一个循环的过程中<br />
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: <math>\oint \frac{\delta Q}{T} \leq 0.</math><br />
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The equality holds in the reversible case<ref>[http://scienceworld.wolfram.com/physics/ClausiusTheorem.html ''Clausius theorem''] at [[Wolfram Research]]</ref> and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function [[entropy]]. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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The equality holds in the reversible case and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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等号在可逆情况下成立，严格不等号在不可逆情况下成立。可逆情况下引入状态函数熵。这是因为在循环过程中，状态函数的变化为零。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）末句存疑<br />
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===Thermodynamic temperature 热力学温度===<br />
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{{main article|Thermodynamic temperature}}<br />
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For an arbitrary heat engine, the efficiency is:<br />
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For an arbitrary heat engine, the efficiency is:<br />
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对于任意热机，效率为:<br />
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: <math>\eta = \frac {W_n}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H} \qquad (1)</math><br />
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where ''W''<sub>n</sub> is for the net work done per cycle. Thus the efficiency depends only on q<sub>C</sub>/q<sub>H</sub>.<br />
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其中 ''W''<sub>n</sub> 表示每个循环所做的净功。因此效率只取决于 q<sub>C</sub>/q<sub>H</sub>。<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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卡诺定理指出，所有在相同热源间运行的可逆发动机的效率是相同的。<br />
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Thus, any reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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Thus, any reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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因此，任何在温度''T''<sub>1</sub>和''T''<sub>2</sub>之间运行的可逆热机必须具有相同的效率，也就是说，效率只是温度的函数:<br />
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<math>\frac{q_C}{q_H} = f(T_H,T_C)\qquad (2).</math><br />
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In addition, a reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between ''T''<sub>1</sub> and another (intermediate) temperature ''T''<sub>2</sub>, and the second between ''T''<sub>2</sub> and''T''<sub>3</sub>. This can only be the case if<br />
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In addition, a reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between T<sub>1</sub> and another (intermediate) temperature T<sub>2</sub>, and the second between T<sub>2</sub> andT<sub>3</sub>. This can only be the case if<br />
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另外，在温度 ''T''<sub>1</sub> 和 ''T''<sub>3</sub> 之间工作的可逆热机必须具有与由分别在温度''T''<sub>1</sub> 和（中间）温度 ''T''<sub>2</sub>之间、在 ''T''<sub>2</sub> 和 ''T''<sub>3</sub> 之间的两个循环组成的系统效率相同。只有下式成立才会出现这种情况:<br />
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: <math>f(T_1,T_3) = \frac{q_3}{q_1} = \frac{q_2 q_3} {q_1 q_2} = f(T_1,T_2)f(T_2,T_3).</math><br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the [[triple point]] of water. Then for any ''T''<sub>2</sub> and ''T''<sub>3</sub>,<br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the triple point of water. Then for any T<sub>2</sub> and T<sub>3</sub>,<br />
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现在考虑如下情形，<math>T_1</math> 是一个固定的参考温度: 水的三相点的温度。则对于任意 T<sub>2</sub> 和 T<sub>3</sub>,<br />
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: <math>f(T_2,T_3) = \frac{f(T_1,T_3)}{f(T_1,T_2)} = \frac{273.16 \cdot f(T_1,T_3)}{273.16 \cdot f(T_1,T_2)}.</math><br />
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Therefore, if thermodynamic temperature is defined by<br />
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Therefore, if thermodynamic temperature is defined by<br />
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因此，如果热力学温度被定义为<br />
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: <math>T = 273.16 \cdot f(T_1,T) \,</math><br />
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then the function ''f'', viewed as a function of thermodynamic temperature, is simply<br />
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then the function f, viewed as a function of thermodynamic temperature, is simply<br />
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那么函数 ''f'' 作为热力学温度的函数，为<br />
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: <math>f(T_2,T_3) = \frac{T_3}{T_2},</math><br />
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and the reference temperature ''T''<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be used{{snd}}the choice here corresponds to the [[Kelvin]] scale.)<br />
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and the reference temperature T<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be usedthe choice here corresponds to the Kelvin scale.)<br />
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参考温度 ''T''<sub>1</sub> 的值为273.16。（任何参考温度和任何正值均可用——此处的选择对应开尔文标度。）<br />
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===Entropy 熵===<br />
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{{main article|Entropy (classical thermodynamics)}}<br />
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According to the [[Clausius theorem|Clausius equality]], for a reversible process<br />
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According to the Clausius equality, for a reversible process<br />
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根据克劳修斯定理，对可逆过程有<br />
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: <math>\oint \frac{\delta Q}{T}=0</math><br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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这意味着线积分 <math>\int_L \frac{\delta Q}{T}</math> 对于可逆过程是路径无关的。<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer<ref name="Schmidt-Rohr 14" /> satisfies<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer satisfies<br />
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所以我们可以定义一个叫做熵的状态函数 S，对于可逆过程或者纯热传递满足<br />
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: <math>dS = \frac{\delta Q}{T} \!</math><br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the [[third law of thermodynamics]], which states that ''S'' = 0 at [[absolute zero]] for perfect crystals.<br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the third law of thermodynamics, which states that S = 0 at absolute zero for perfect crystals.<br />
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据此，只有对上述公式进行积分，才能得到熵的差值。为了获得绝对值，我们需要'''热力学第三定律 Third Law of Thermodynamics'''，它指出绝对零度下完美晶体的 ''S'' = 0。<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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对于任意不可逆过程，由于熵是一个状态函数，我们总是可以将初始状态和最终状态与一个虚拟的可逆过程联系起来，并在这条路径上积分以计算熵的差值。<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the [[Clausius inequality]] on this loop,<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the Clausius inequality on this loop,<br />
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现在把可逆过程逆过来，将其与上述不可逆过程结合。把克劳修斯不等式应用到这个循环,<br />
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: <math>-\Delta S+\int\frac{\delta Q}{T}=\oint\frac{\delta Q}{T}< 0</math><br />
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Thus,<br />
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Thus,<br />
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故,<br />
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: <math>\Delta S \ge \int \frac{\delta Q}{T} \,\!</math><br />
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where the equality holds if the transformation is reversible.<br />
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where the equality holds if the transformation is reversible.<br />
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如果变换可逆，等号成立。<br />
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Notice that if the process is an [[adiabatic process]], then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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Notice that if the process is an adiabatic process, then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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注意，若该过程是一个'''绝热过程 Adiabatic Process'''，则<math>\delta Q=0</math>，故<math>\Delta S\ge 0</math>。<br />
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===Energy, available useful work===<br />
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===Energy, available useful work===<br />
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能量，有用的工作<br />
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{{See also|Exergy}}<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an ''unlimited'' heat reservoir at temperature ''T<sub>R</sub>'' and pressure ''P<sub>R</sub>'' {{snd}}so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain ''T<sub>R</sub>''; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain ''P<sub>R</sub>''.<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an unlimited heat reservoir at temperature T<sub>R</sub> and pressure P<sub>R</sub> so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain T<sub>R</sub>; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain P<sub>R</sub>.<br />
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考虑将第二定律应用于孤立系统(称为整体系统或宇宙)的情形，是一个重要的、具有启发性的理想情形。孤立系统由两部分组成: 感兴趣的子系统和子系统的环境。这些环境被想象为如此之大，以至于它们可以被认为是一个温度为 t 子 r / sub 和压力为 p 子 r / sub 的无限热源，因此无论有多少热量被转移到(或来自)子系统，周围环境的温度将保持 t 子 r / sub; 无论子系统的体积扩大(或收缩)多少，周围环境的压力将保持 p 子 r / sub。<br />
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Whatever changes to ''dS'' and ''dS<sub>R</sub>'' occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy ''S<sub>tot</sub>'' of the isolated total system must not decrease:<br />
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Whatever changes to dS and dS<sub>R</sub> occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy S<sub>tot</sub> of the isolated total system must not decrease:<br />
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根据第二定律，孤立总体系统的熵 s 子系统 / 子系统的熵不能减小，无论子系统和周围环境的熵对 dS 和 dS 子系统 r / 子系统发生什么变化:<br />
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: <math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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<math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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数学 dS + dS r ge 0 / math<br />
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According to the [[first law of thermodynamics]], the change ''dU'' in the internal energy of the sub-system is the sum of the heat ''δq'' added to the sub-system, ''less'' any work ''δw'' done ''by'' the sub-system, ''plus'' any net chemical energy entering the sub-system ''d ∑μ<sub>iR</sub>N<sub>i</sub>'', so that:<br />
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According to the first law of thermodynamics, the change dU in the internal energy of the sub-system is the sum of the heat δq added to the sub-system, less any work δw done by the sub-system, plus any net chemical energy entering the sub-system d ∑μ<sub>iR</sub>N<sub>i</sub>, so that:<br />
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根据能量守恒定律，子系统内部能量的变化 dU 是子系统内部能量 q 的总和，减去子系统所做的任何功，再加上进入子系统 d ∑ sub iR / sub n sub i / sub 的任何净化学能，因此:<br />
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: <math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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<math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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数学 dU delta q- delta w + d ( sum mu iR } n i) ，/ math<br />
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where μ<sub>iR</sub> are the [[chemical potential]]s of chemical species in the external surroundings.<br />
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where μ<sub>iR</sub> are the chemical potentials of chemical species in the external surroundings.<br />
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其中亚 iR / 亚是外部环境中化学物种的化学势。<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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现在热量离开储存器进入子系统是<br />
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: <math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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<math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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数学△ qtr (- dS r) le tr dS / math<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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其中我们首先使用了经典热力学中熵的定义(在统计热力学中，熵变、温度和吸收热量之间的关系可以推导出来) ，然后从上面推导出第二定律的不等式。<br />
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It therefore follows that any net work ''δw'' done by the sub-system must obey<br />
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It therefore follows that any net work δw done by the sub-system must obey<br />
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因此，子系统所做的任何净功都必须服从<br />
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: <math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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<math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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数学增量 w le-dU + tds + sum mu { iR } dN i，/ math<br />
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It is useful to separate the work ''δw'' done by the subsystem into the ''useful'' work ''δw<sub>u</sub>'' that can be done ''by'' the sub-system, over and beyond the work ''p<sub>R</sub> dV'' done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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It is useful to separate the work δw done by the subsystem into the useful work δw<sub>u</sub> that can be done by the sub-system, over and beyond the work p<sub>R</sub> dV done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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将子系统所做的功与子系统所能做的功分离为子系统所能做的有用功，超过了子系统对外界压力所做的功，给出了子系统所能做的有用功(火用)的下列关系式:<br />
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: <math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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<math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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D (u-t r s + p r v-sum { iR } n i) ，/ math<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the ''availability'' or ''[[exergy]]'' ''E'' of the subsystem,<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the availability or exergy E of the subsystem,<br />
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可以很方便地将子系统的右侧定义为热动力位能的精确衍生物，称为子系统的可用性或火用 e,<br />
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: <math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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<math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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数学 e u-t r s + p r v- sum { iR } n i / math<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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因此，第二定律意味着，对于任何可以简单地被视为分为一个子系统和一个与之接触的无限温度和压力储存器的过程,<br />
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: <math> dE + \delta w_u \le 0 \, </math><br />
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<math> dE + \delta w_u \le 0 \, </math><br />
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数学德 delta u le 0，/ math<br />
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i.e. the change in the subsystem's exergy plus the useful work done ''by'' the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done ''on'' the system) must be less than or equal to zero.<br />
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i.e. the change in the subsystem's exergy plus the useful work done by the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done on the system) must be less than or equal to zero.<br />
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也就是。子系统(火用)的变化加上子系统所做的有用功(或者，子系统(火用)的变化不包括任何功，除了压力贮存器所做的功之外，在系统中所做的功)必须小于或等于零。<br />
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In sum, if a proper ''infinite-reservoir-like'' reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in ''E'' for an irreversible process and no change for a reversible process.<br />
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In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in E for an irreversible process and no change for a reversible process.<br />
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总之，如果选择一个合适的类似于无限库的参考状态作为现实世界中的系统环境，那么第二定律预测一个不可逆性的 e 值会减少，而一个可逆过程的 e 值不会变化。<br />
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: <math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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<math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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数学 d = 0 = 0 = 0 = 0 = 0<br />
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This expression together with the associated reference state permits a [[design engineer]] working at the macroscopic scale (above the [[thermodynamic limit]]) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (''Also, see [[process engineer]]''). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (''See [[Exergy efficiency|second law efficiency]]''.)<br />
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This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (Also, see process engineer). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (See second law efficiency.)<br />
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这个表达式连同相关的参考状态允许在宏观的设计工程师利用第二定律，而不需要直接测量或考虑整个孤立系统的熵变。(另外，请参阅流程工程师)。这些变化已经被考虑，假设被考虑的系统可以达到平衡的参考状态而不改变参考状态。还可以找到一个过程或过程集合的效率，将其与可逆的理想状态相比较<br />
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This approach to the Second Law is widely utilized in [[engineering]] practice, [[environmental accounting]], [[systems ecology]], and other disciplines.<br />
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This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines.<br />
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第二定律的这种方法被广泛应用于工程实践、环境会计、系统生态学和其他学科。<br />
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===The second law in chemical thermodynamics===<br />
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===The second law in chemical thermodynamics===<br />
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化学热力学的第二定律<br />
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For a [[spontaneous process|spontaneous chemical process]] in a closed system at constant temperature and pressure without non-''PV'' work, the Clausius inequality Δ''S > Q/T<sub>surr</sub>'' transforms into a condition for the change in [[Gibbs free energy]] <br />
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For a spontaneous chemical process in a closed system at constant temperature and pressure without non-PV work, the Clausius inequality ΔS > Q/T<sub>surr</sub> transforms into a condition for the change in Gibbs free energy <br />
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对于一个封闭系统中的自发化学过程，在没有非 pv 功的情况下，在恒定温度和压力下，将克劳修斯不等式 s q / t sub surr / sub 转化为吉布斯自由能变化的条件<br />
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: <math>\Delta G < 0 </math><br />
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<math>\Delta G < 0 </math><br />
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数学 Delta g 0 / 数学<br />
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or d''G'' < 0. For a similar process at constant temperature and volume, the change in [[Helmholtz free energy]] must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products.<ref name="Oxtoby8th"/> <ref name="MortimerBook"></ref> The chemical equilibrium condition at constant ''T'' and ''p'' without electrical work is d''G'' = 0.<br />
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or dG < 0. For a similar process at constant temperature and volume, the change in Helmholtz free energy must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products. The chemical equilibrium condition at constant T and p without electrical work is dG = 0.<br />
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或者 dG 0。对于一个相似的过程，在恒定的温度和体积下，亥姆霍兹自由能的变化一定是负的，这是 Delta a 0 / math。因此，自由能(g 或 a)变化的负值是过程自发的必要条件。这是热力学第二定律在化学中最有用的形式，其中自由能的变化可以通过列表的生成焓和反应物和产物的标准摩尔熵来计算。在 t 和 p 不变的情况下，化学平衡条件是 dG 0。<br />
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==History==<br />
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==History==<br />
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历史<br />
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{{See also|History of entropy}}<br />
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[[File:Sadi Carnot.jpeg|thumb|upright|Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique]].]]<br />
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Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique.]]<br />
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尼古拉·莱昂纳尔·萨迪·卡诺穿着传统的巴黎综合理工学院制服<br />
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The first theory of the conversion of heat into mechanical work is due to [[Nicolas Léonard Sadi Carnot]] in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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The first theory of the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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第一个把热转化为机械功的理论是在1824年由尼古拉·莱昂纳尔·萨迪·卡诺提出的。他是第一个正确认识到这种转换的效率取决于发动机和环境之间的温差。<br />
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Recognizing the significance of [[James Prescott Joule]]'s work on the conservation of energy, [[Rudolf Clausius]] was the first to formulate the second law during 1850, in this form: heat does not flow ''spontaneously'' from cold to hot bodies. While common knowledge now, this was contrary to the [[caloric theory]] of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law during 1850, in this form: heat does not flow spontaneously from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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认识到詹姆斯·普雷斯科特·焦耳在能量守恒方面工作的重要性，Rudolf Clausius 在1850年第一个提出了第二定律: 热不会自发地从冷物体流向热物体。虽然现在的常识是，这与当时流行的热量理论相反，热量理论认为热量是一种流体。从那里他能够推断出萨迪卡诺定律和熵的定义(1865年)。<br />
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Established during the 19th century, the [[Kelvin-Planck statement|Kelvin-Planck statement of the Second Law]] says, "It is impossible for any device that operates on a [[cyclic process|cycle]] to receive heat from a single [[heat reservoir|reservoir]] and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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Established during the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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建立于19世纪的开尔文-普朗克第二定律(Kelvin-Planck)说，“任何循环运行的设备都不可能从单个蓄热体接收热量，并产生净功。”这被证明相当于克劳修斯的陈述。<br />
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The [[ergodic hypothesis]] is also important for the [[Boltzmann]] approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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遍历假设对玻尔兹曼方法也很重要。它说，在很长一段时间内，在具有相同能量的微观态的相空间的某些区域所花费的时间与这个区域的体积成正比，即。在很长一段时间内，所有可访问的微观状态的可能性都是一样的。同样的，它表明时间平均值和系综平均值是相同的。<br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a [[Macroscopic bodies|macroscopic system]]. This doctrine is obsolescent.<ref>Denbigh, K.G., Denbigh, J.S. (1985). ''Entropy in Relation to Incomplete Knowledge'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-25677-1}}, pp. 43–44.</ref><ref>Grandy, W.T., Jr (2008). ''Entropy and the Time Evolution of Macroscopic Systems'', Oxford University Press, Oxford, {{ISBN|978-0-19-954617-6}}, pp. 55–58.</ref><ref name=Lambert>[http://entropysite.oxy.edu Entropy Sites — A Guide] Content selected by [[Frank L. Lambert]]</ref><br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a macroscopic system. This doctrine is obsolescent.<br />
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有一种传统的学说，从克劳修斯开始，认为熵可以被理解为宏观系统中的分子“无序”。这种学说已经过时了。<br />
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===Account given by Clausius===<br />
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===Account given by Clausius===<br />
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作者: 克劳修斯<br />
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[[File:Clausius-1.jpg|thumb|upright|Rudolf Clausius]]<br />
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Rudolf Clausius<br />
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鲁道夫 · 克劳修斯<br />
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In 1856, the German physicist [[Rudolf Clausius]] stated what he called the "second fundamental theorem in the [[mechanical theory of heat]]" in the following form:{{sfnp|Clausius|1867}}<br />
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In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:<br />
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1856年，德国物理学家鲁道夫 · 克劳修斯阐述了他所谓的“热力学理论中的第二个基本定理” ，其形式如下:<br />
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: <math>\int \frac{\delta Q}{T} = -N</math><br />
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<math>\int \frac{\delta Q}{T} = -N</math><br />
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- n / math<br />
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where ''Q'' is heat, ''T'' is temperature and ''N'' is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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其中 q 是热，t 是温度，n 是一个循环过程中所有未补偿的相变的“等价值”。后来，在1865年，克劳修斯将“等价值”定义为熵。紧接着这个定义，也就是在同一年，第二定律最著名的版本在4月24日苏黎世哲学学会的一次演讲中被宣读，在演讲的最后，克劳修斯总结道:<br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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宇宙的熵趋于最大<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. [[universe]], as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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这句话是第二定律中最广为人知的措辞。由于其语言的松散性，例如:。这个问题的答案是肯定的，因为这个问题的答案是肯定的。开放，封闭，或孤立，许多人认为这个简单的陈述意味着热力学第二定律适用于几乎所有可以想象的主题。这是不正确的; 这个陈述只是一个更加扩展和精确描述的简化版本。<br />
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In terms of time variation, the mathematical statement of the second law for an [[isolated system]] undergoing an arbitrary transformation is:<br />
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In terms of time variation, the mathematical statement of the second law for an isolated system undergoing an arbitrary transformation is:<br />
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就时间变化而言，经历任意变换的孤立系统第二定律的数学表述是:<br />
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: <math>\frac{dS}{dt} \ge 0</math><br />
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<math>\frac{dS}{dt} \ge 0</math><br />
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0 / math<br />
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where<br />
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where<br />
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: ''S'' is the entropy of the system and<br />
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S is the entropy of the system and<br />
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S 是系统的熵<br />
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: ''t'' is [[time]].<br />
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t is time.<br />
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时间到了。<br />
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!-- 可逆过程需要与周围环境保持平衡。这对于一个孤立的系统是不可能的。因此，关于可逆过程的讨论已经转移到对封闭系统的分析----<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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平衡后应用等式符号。建立孤立系统第二定律的另一种方法是:<br />
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: <math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math<br />
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with <math> \dot S_{i}</math> the sum of the rate of [[entropy production]] by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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with <math> \dot S_{i}</math> the sum of the rate of entropy production by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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用数学 s { i } / 数学表示系统内所有进程的产生熵之和。这种配方的优点是它显示了产生熵的效果。产生熵是一个非常重要的概念，因为它决定(限制)热机器的效率。与环境温度数学 t { a } / math 相乘，得到所谓的耗散能量数学 p { diss } t { a } dot s { i } / math。<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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封闭系统的第二定律(允许热量交换和移动边界，但不允许物质交换)的表达式是:<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math 来计算数学点 q } + 点 s { i } / math<br />
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: <math>\dot Q</math> is the heat flow into the system<br />
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<math>\dot Q</math> is the heat flow into the system<br />
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数学 q / math 是进入系统的热流<br />
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: <math>T</math> is the temperature at the point where the heat enters the system.<br />
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<math>T</math> is the temperature at the point where the heat enters the system.<br />
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Math 是热量进入系统时的温度。<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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等式符号在只有可逆过程在系统内发生的情况下成立。如果发生不可逆过程(在实际操作系统中就是这种情况) ,-符号保持不变。如果系统有多处供热，我们必须求相应项的代数和。<br />
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For open systems (also allowing exchange of matter):<br />
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For open systems (also allowing exchange of matter):<br />
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对于开放系统(也允许物质交换) :<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math 表示数学点 q } + 点 s + 点 s { i } / math<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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这里的数学 s / math 是熵流进入系统，与物质流进入系统有关。它不应该与熵的时间导数混淆。如果物质在几个地方提供，我们必须取这些贡献的代数和。<br />
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==Statistical mechanics==<br />
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==Statistical mechanics==<br />
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统计力学<br />
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[[Statistical mechanics]] gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a [[microstate (statistical mechanics)|microstate]] of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/{{radic|''N''}} where ''N'' is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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Statistical mechanics gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a microstate of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/ where N is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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通过假设物质是由不断运动的原子和分子组成的，统计力学对第二定律给出了解释。系统中每个粒子的一组特定的位置和速度称为系统的微观状态，由于系统的不断运动，系统不断地改变其微观状态。统计力学假设，在平衡状态下，系统可能处于的每个微观状态发生的可能性是相等的，当这个假设被提出时，它直接导致了第二定律必须在统计意义上成立的结论。也就是说，第二定律平均成立，其中 n 是系统中粒子数的1 / 数量级的统计变化。在日常(宏观)情况下，违反第二定律的概率几乎为零。然而，对于粒子数量很少的系统，热力学参数，包括熵，可能显示出与第二定律预测的显著的统计偏差。经典热力学理论没有处理这些统计变量。<br />
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（by Jie XIANG) 在通过假定一种物质由不断运动的原子和分子组成的前提下，统计力学对热力学第二定律进行了解释。系统中每个粒子其特定的位置和速度组成为系统的微观状态，由于（原子和分子的）不断运动，系统也持续改变其微观状态。统计力学假设，在系统力平衡的情况下，其微观状态其实是等可能发生的，进而能够得出结论：热力学第二定律必须在统计学意义上成立。也就是说，第二定律将保持均值不变， 其统计变化为1 /√N，N为系统中的粒子数。在日常(宏观)情况下，违反热力学第二定律的概率几乎为零，但是，对于粒子数量非常少的系统，其热力学参数（包括熵）可能会显示出与第二定律所预测的明显统计偏差。然而经典热力学理论并不处理这些统计变量。<br />
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==Derivation from statistical mechanics==<br />
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==Derivation from statistical mechanics==<br />
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从统计力学导出<br />
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{{Further information|H-theorem}}<br />
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The first mechanical argument of the [[Kinetic theory of gases]] that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to [[James Clerk Maxwell]] in 1860;<ref>{{Cite journal | last1 = Gyenis | first1 = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G }}</ref> [[Ludwig Boltzmann]] with his [[H-theorem]] of 1872 also argued that due to collisions gases should over time tend toward the [[Maxwell–Boltzmann distribution]].<br />
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The first mechanical argument of the Kinetic theory of gases that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to James Clerk Maxwell in 1860; Ludwig Boltzmann with his H-theorem of 1872 also argued that due to collisions gases should over time tend toward the Maxwell–Boltzmann distribution.<br />
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气体动力学理论的第一个力学论证由詹姆斯·克拉克·麦克斯韦在1860年给出，指出分子碰撞引起温度均衡，因此整体趋向于平衡； 路德维希·玻尔兹曼在1872年提出的 H 定理也认为，气体由于碰撞应该随着时间的推移趋向于麦克斯韦-波兹曼分布。<br />
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Due to [[Loschmidt's paradox]], derivations of the Second Law have to make an assumption regarding the past, namely that the system is [[Correlation and dependence|uncorrelated]] at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a [[boundary condition]], and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the [[Big Bang]]), though [[Boltzmann brain|other scenarios]] have also been suggested.<ref name="Hawking AOT">{{cite journal|last=Hawking|first=SW|title=Arrow of time in cosmology|journal=Phys. Rev. D|year=1985|volume=32|issue=10|pages=2489–2495|doi=10.1103/PhysRevD.32.2489|pmid=9956019|bibcode = 1985PhRvD..32.2489H }}</ref><ref>{{cite book | last = Greene | first = Brian | authorlink = Brian Greene | title = The Fabric of the Cosmos | url = https://archive.org/details/fabricofcosmossp00gree | url-access = registration | publisher = Alfred A. Knopf | year = 2004 | page = [https://archive.org/details/fabricofcosmossp00gree/page/171 171] | isbn = 978-0-375-41288-2}}</ref><ref name=Lebowitz>{{cite journal|last=Lebowitz|first=Joel L.|title= Boltzmann's Entropy and Time's Arrow|journal=Physics Today|date=September 1993|volume=46|issue=9|pages=32–38|url=http://users.df.uba.ar/ariel/materias/FT3_2008_1C/papers_pdf/lebowitz_370.pdf|accessdate=2013-02-22|doi=10.1063/1.881363|bibcode = 1993PhT....46i..32L }}</ref><br />
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Due to Loschmidt's paradox, derivations of the Second Law have to make an assumption regarding the past, namely that the system is uncorrelated at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a boundary condition, and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the Big Bang), though other scenarios have also been suggested.<br />
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由于洛施密特悖论，第二定律的导出必须对过去做出一个假设，即系统在过去的某个时刻是不相关的；这样的假设允许进行简单的概率处理。这个假设通常被认为是一个边界条件，因此热力学第二定律最终是过去某个地方的初始条件的结果，可能是在宇宙的开始(大爆炸) 。也有人提出了其他假设。<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the [[Statistical mechanics#Fundamental postulate|fundamental postulate]], also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy.{{citation needed|date=August 2012}} The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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考虑到这些假设，在统计力学中，第二定律不是一个假设，而是统计力学基本假设的一个结果，也被称为等先验概率假设。这个基本假设表明，只要一个人清楚地知道，简单的概率论证只适用于未来，而对于过去，有辅助的信息来源告诉我们，它是低熵的。热力学第二定律的第一部分指出，热孤立系统的熵只能增加。如果我们把熵的概念限制在热平衡系统中，那么热力学第二定律的第一部分是等先验概率假设的一个显然结果。处于热平衡状态的孤立系统包含能量<math>E</math>的熵表示为：<br />
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: <math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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<math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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其中<math>\Omega\left(E\right)</math> 是处于 <math>E</math>和<math>E +\delta E</math这个小区间内的量子态数目。这里的 <math>\delta E</math> 是一个宏观上很小的固定能量区间。严格地说，这意味着熵取决于对<math>\delta E</math>的选择。然而在热力学极限下（例如无穷大系统的极限），狭义的熵(单位体积或单位质量的熵)不依赖于 <math>\delta E</math>。<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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假设我们有一个孤立系统，其宏观状态由许多变量描述。这些宏观变量可以是总体积、活塞在系统中的位置等。从而<math>\Omega</math>将取决于这些变量的值。如果某个变量不是固定的（我们不会在某个位置夹住活塞) ，那么因为在平衡状态下所有可到达状态的等几率到达的，平衡状态下的自由变量会使 <math>\Omega</math> 最大，因为这是平衡状态下最可能的情况。<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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如果该变量最初固定到某个值然后释放，当达到新的平衡时，变量将自我调整使得 <math>\Omega</math> 最大，这意味着熵将增加或保持不变(如果变量初始固定的值恰好是平衡值)。<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's [[H-theorem]], however, proves that the quantity {{math|''H''}} increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the quantity increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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假设我们初始位于一个平衡状态，突然移除了对一个变量的约束。我们做完这件事的时候，可达到的微观状态的数为<math>\Omega</math>，但是系统还没有达到平衡，所以系统处于某些可达到的状态的实际概率还不等于先验概率 <math>1/\Omega</math>。我们已经知道，最终的平衡状态相对于之前的平衡状态，熵会增加或者保持不变。然而，玻耳兹曼的H定理证明系统在不处于平衡态的期间，那个量作为时间的函数单调增加。<br />
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===Derivation of the entropy change for reversible processes===<br />
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===Derivation of the entropy change for reversible processes===<br />
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可逆过程熵变的推导<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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第二定律的第二部分指出，经历可逆过程的系统的熵变是由以下因素给出的:<br />
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: <math>dS =\frac{\delta Q}{T}</math><br />
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<math>dS =\frac{\delta Q}{T}</math><br />
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数学 dS frac delta q }{ t } / math<br />
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where the temperature is defined as:<br />
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where the temperature is defined as:<br />
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其中温度定义为:<br />
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:<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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[数学][[欧米茄][左(右)][右][数学][数学]<br />
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See [[Microcanonical ensemble|here]] for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the [[adiabatic theorem]] of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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请参阅此处对此定义的正当理由。假设系统有一些可以改变的外部参数 x。一般来说，系统的能量本征态将依赖于 x。根据量子力学的绝热定理，在系统哈密顿量无限缓慢变化的极限下，系统将保持在相同的定态，因此系统的能量会随着其所在定态的能量变化而变化。<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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广义力，x，对应于外部变量 x 的定义是，如果 x 增加一个量 dx，那么 math x dx / math 就是系统所做的功。例如，如果 x 是体积，那么 x 是压强。一个已知的系统的广义力在定态数学 e { r } / math 中给出:<br />
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: <math>X = -\frac{dE_{r}}{dx}</math><br />
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<math>X = -\frac{dE_{r}}{dx}</math><br />
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数学 x- frac { dE { r }{ dx } / math<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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由于系统可以在一个数学△ e / math 区间内的任意定态内，我们将系统的广义力定义为上述表达式的期望值:<br />
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: <math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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<math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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数学 x- 左 langle frac { dE { dx }右 rangle，/ math<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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为了计算平均值，我们通过计算在数学 y / math 和数学 y + delta y / math 之间的范围内，其中有多少数学 y / math 能量本征态有一个值来划分 ω 左(e 右) / math 能量本征态。把这个数字命名为 math Omega { y } left (e right) / math，我们有:<br />
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: <math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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<math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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Math Omega left (e right) sum { y } Omega { y } left (e right) ，/ math<br />
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The average defining the generalized force can now be written:<br />
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The average defining the generalized force can now be written:<br />
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定义广义力的平均值现在可以写成:<br />
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: <math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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<math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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数学 x frac {1} Omega 左(e 右) sum { y Omega { y }左(e 右) ，/ math<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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我们可以把它和熵对 x 在恒定能量 e 下的导数联系起来，如下所示。假设我们把 x 改成 x + dx。然后 math Omega left (e right) / math 将会改变，因为能量本征态依赖于 x，导致能量本征态进入或超出数学 e / math 和数学 e + delta e / math 之间的范围。让我们再次关注数学 y / math 和数学 y + delta y / math 之间的能量本征态。由于这些能量本征态的能量增加了 y dx，所有这些能量本征态在 e-y dx 到 e 之间的区间内从 e 以下移动到 e 以上。 有<br />
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: <math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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<math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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数学 n { y }左(e 右) frac { y }左(e 右)}{ delta e } y dx ，/ math<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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这样的能量本征态。如果数学 y dx leq delta e / math，所有这些能量本征态将移动到数学 e / math 和数学 e + delta e / math 之间的范围内，有助于数学 ω / math 的增加。从数学 e + △ e / math 下移到数学 e + △ e / math 上的能量本征态的个数是由数学 n { y }左(e + △ e 右) / math 给出的。区别在于<br />
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: <math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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<math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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数学 n { y }左(e 右)-n { y }左(e + delta e 右) ，/ math<br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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因此是数学增长的净贡献欧米茄 / 数学。请注意，如果 y dx 大于 math delta e / math，那么将会有能量本征态从 e 下面移动到 e + delta e / math 上面。它们在数学 n { y } left (e right) / math 和数学 n { y } left (e + delta e right) / math 中都有计算，因此上述表达式在这种情况下也是有效的。<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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将上面的表达式表示为对 e 的导数，对 y 的求和得到表达式:<br />
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: <math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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数学左(部分欧米茄左(部分欧米茄右){ e }-和 y 左(部分欧米茄右){ x 左(部分欧米茄左(部分欧米茄右)}{ x 右) ，/ 数学<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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关于 x 的欧米茄 / 数学对数导数由以下方式给出:<br />
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: <math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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Math 左( frac { partial ln 左( Omega 右)}{部分 x 右){ e } beta x + 左( frac { partial x 右){ x } ，/ math<br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
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第一个术语是密集型的，即。它不能根据系统大小进行缩放。相比之下，最后一项的规模与逆系统的规模一样，因此将在热力学极限中消失。因此，我们发现:<br />
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: <math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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<math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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数学左(部分 s }右){ e } frac { x }{ t } ，/ math<br />
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Combining this with<br />
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Combining this with<br />
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结合这一点<br />
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: <math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
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<math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
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数学左(部分 s }右){ x } frac {1}{ t } ，/ math<br />
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给予:<br />
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: <math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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<math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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数学 dS 左(部分 s }部分 e }右){ x } dE + 左(部分 s }右){ e } dx frac { t } + frac { t }{ delta q }{ t } ，/ math<br />
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===Derivation for systems described by the canonical ensemble===<br />
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===Derivation for systems described by the canonical ensemble===<br />
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正则系系统的推导<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the [[canonical ensemble]]:<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:<br />
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如果一个系统与某个温度为T的热浴热接触，那么在平衡状态下，关于能量本征值的概率分布值由正则系综给出:<br />
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: <math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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<math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the [[Partition function (statistical mechanics)|partition function]]. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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这里Z是一个使所有概率之和归一化的因子，这个函数被称为配分函数。现在我们考虑对温度和能级所依赖的外部参数的无限小的可逆改变。它遵循熵的一般公式:<br />
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: <math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
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<math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
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that<br />
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that<br />
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以及<br />
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: <math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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<math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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把正则系综的数学形式 <math>P_{j}</math> 插入，可以得到:<br />
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: <math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
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<math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
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==Living organisms==<br />
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==Living organisms==<br />
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生命体<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as [[What is Life?|Erwin Schrödinger]], [[Léon Brillouin]]<ref>Léon Brillouin [https://books.google.com/books?hl=en&lr=&id=tPXVbiw_1P0C ''Science and Information Theory''] (Academic Press, 1962) (Dover, 2004)</ref> and [[Life and Energy|Isaac Asimov]]. It is also the topic of current{{When|date=July 2019}} research.{{Citation needed|date=July 2019}}<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as Erwin Schrödinger, Léon Brillouin and Isaac Asimov. It is also the topic of current research.<br />
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表述热力学主要有两种方式：(a)描述从一种热力学平衡状态到另一种状态的路径，(b)通过使系统保持不变，同时增加周围环境的总熵的循环过程。这两种方法有助于理解生命过程。这个话题大大超出了本文的范围，但是已经有一些研究者考虑过这个问题，比如埃尔温·薛定谔、莱昂·布里渊和艾萨克.阿西莫夫。这也是当前研究的主题。<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of [[metabolism]], give out breakdown products and heat. Plants [[Photosynthesis|take in radiative energy]] from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of metabolism, give out breakdown products and heat. Plants take in radiative energy from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
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从近似的角度来看，生命体可以被认为是(b)的一个例子。近似地，一只动物的身体状态每天循环，使得它几乎没有什么变化。动物吸收食物、水和氧气，经过新陈代谢，输出分解的产物和热量。植物吸收来自太阳的辐射能量，这可以认为是热量，以及二氧化碳和水，然后它们释放氧气。它们就是这样生长的，最终会死亡，尸体腐烂，大部分重新变成二氧化碳和水。这可以看作是一个循环过程。总的来说，阳光来自一个高温的源——太阳，它的能量被传递到一个较低的温度汇，例如向太空辐射。这个过程使得植物周围环境的熵增加。因此从循环过程的角度来看，动物和植物服从热力学第二定律。简单的热机效率概念很难适用于这个问题，因为它们假定系统是封闭的。<br />
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From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to [[Non-equilibrium thermodynamics|unsettled current debate or research]]. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
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From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to unsettled current debate or research. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
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从(a)的热力学观点考虑 ，即考虑从一个平衡状态到另一个平衡状态的路径，只会产生一个粗略近似的图像，因为生命体从来不会处于热力学平衡状态。生物体经常必须被认为是开放的系统，因为它们吸收营养物质并排出废弃物。开放系统热力学目前通常从一个热力学平衡状态到另一个状态的路径的角度来考虑，或者是考虑局部热力学平衡近似下的流。生命体的问题可以通过假定一个流不变的稳态来进一步简化。对于这样的近似，熵产生的一般原理在目前有争论和研究的不确定性。尽管如此，从热力学第二定律的这个角度出发衍生出来的想法对生物是有启发意义的。<br />
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==Gravitational systems==<br />
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==Gravitational systems==<br />
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重力系统<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive [[heat capacity]], meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it.<ref name="Grandy 151">Grandy, W.T. (Jr) (2008), p. 151.</ref> Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive heat capacity, meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it. Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
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在不需要描述广义相对论的系统中，物体总是具有正的热容，这意味着温度随着能量的增加而升高。因此，当能量从高温物体流向低温物体时，源温度降低，而汇温度升高; 因此温差随时间而减小。这种情况并不总是发生在那些需要引力和广义相对论的系统中。这样的系统可以自发地朝着质量和能量不均匀分布的方向变化。这在大尺度上适用于宇宙，因此可能很难或不可能将第二定律应用于宇宙。除此之外，广义相对论所描述的系统热力学超出了本文的范围。<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
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{{main article|Non-equilibrium thermodynamics}}<br />
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The theory of classical or [[thermodynamic equilibrium|equilibrium thermodynamics]] is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<ref name="Grandy 151"/><ref>Callen, H.B. (1960/1985), p. 15.</ref><br />
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The theory of classical or equilibrium thermodynamics is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<br />
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经典热力学或平衡热力学的理想化。一个主要的假设或假设，往往甚至没有明确说明，是系统的存在，在他们自己的内部状态的热力学平衡。一般来说，在给定时间内包含一个物理系统的空间区域---- 可能在自然界中找到---- 不是在热力学平衡中，用最严格的术语来解释。从更宽松的角度来说，整个宇宙中没有任何东西是或者曾经是完全精确的热力学平衡。<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of [[thermodynamic equilibrium]]. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable [[thermal fluctuations|fluctuations]]. There is an exception, the case of [[critical point (thermodynamics)|critical states]], which exhibit to the naked eye the phenomenon of [[critical opalescence]]. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of thermodynamic equilibrium. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable fluctuations. There is an exception, the case of critical states, which exhibit to the naked eye the phenomenon of critical opalescence. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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为了物理分析的目的，假设热力学平衡是非常方便的。这种假设可能依赖于试错来证明其正当性。如果这个假设是正确的，它通常是非常有价值和有用的，因为它提供了热力学理论。平衡假设的要素是，一个系统被观察到在无限长的时间内是不变的，而且一个系统中有如此多的粒子，以至于它的微粒性质可以被完全忽略。在这样的平衡假设下，一般来说，没有宏观上可以探测到的波动。有一个例外，即临界状态的例子，它们用肉眼就可以看到临界乳光现象。对于临界状态的实验室研究，需要非常长的观察时间。<br />
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In all cases, the assumption of [[thermodynamic equilibrium]], once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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In all cases, the assumption of thermodynamic equilibrium, once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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在所有情况下，热力学平衡的假设，一旦做出，就意味着没有假定的候选者“涨落”改变系统的熵。<br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<ref>Lieb, E.H., Yngvason, J. (2003), p. 190.</ref><br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<br />
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一个物理系统很容易发生内部的宏观变化，其速度之快足以使熵的恒定性假设失效。或者一个物理系统的粒子太少，以至于微粒的本质在可观测的波动中是显而易见的。那么热力学平衡的假设就会被放弃。对于非平衡态，熵没有绝对的一般定义。<br />
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There are intermediate cases, in which the assumption of local [[thermodynamic equilibrium]] is a very good approximation,<ref>Gyarmati, I. (1967/1970), pp. 4-14.</ref><ref>Glansdorff, P., Prigogine, I. (1971).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003).</ref> but strictly speaking it is still an approximation, not theoretically ideal.<br />
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There are intermediate cases, in which the assumption of local thermodynamic equilibrium is a very good approximation, but strictly speaking it is still an approximation, not theoretically ideal.<br />
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还有一些中间情况，其中局部热力学平衡的假设是一个非常好的近似值，但严格地说它仍然是一个近似值，不是理论上的理想值。<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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对于一般的非平衡状态，考虑其他量的统计力学定义可能是有用的，这些量可以方便地称为‘熵’ ，但是它们不应该与第二定律正确定义的熵混淆或混淆。这些其他的量确实属于统计力学，而不是热力学---- 第二定律的主要领域。<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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宏观观测波动的物理学超出了本文的范围。<br />
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==Arrow of time==<br />
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==Arrow of time==<br />
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时间之箭<br />
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{{Further information|Entropy (arrow of time)}}Thermodynamic Asymmetry in Time<br />
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Thermodynamic Asymmetry in Time<br />
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时间中的热力学不对称性<br />
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{{See also|Arrow of time}}<br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely [[CPT symmetry]], since the second law applies statistically, it is hypothesized, on time-asymmetric ''boundary conditions''.<ref>{{cite web|first=Craig|last=Callender|url=https://plato.stanford.edu/archives/fall2011/entries/time-thermo/|title=Thermodynamic Asymmetry in Time|encyclopedia=Stanford Encyclopedia of Philosophy|date=29 July 2011}}</ref><br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely CPT symmetry, since the second law applies statistically, it is hypothesized, on time-asymmetric boundary conditions.<br />
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热力学第二定律是一个物理定律，它不对称于时间方向的反转。这与已经观察到的物理学基本定律，即 CPT 对称性的概念并不冲突，因为第二定律在统计上适用，它是假设的，是关于时间非对称的边界条件。<br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect ([[Arrow of time#The causal arrow of time|the causal arrow of time]]).<ref>{{cite book | first = J.J.| last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1|display-authors=etal}} chapter 6</ref><br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect (the causal arrow of time).<br />
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第二定律被提出来部分解释时间向前和向后移动的区别，例如为什么原因先于结果(因果关系的时间箭头)。<br />
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==Irreversibility==<br />
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==Irreversibility==<br />
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不可逆性<br />
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Irreversibility in [[thermodynamic process]]es is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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Irreversibility in thermodynamic processes is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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热力学过程的不可逆性是热力学作用不对称性的结果，而不是物体内部不可逆的微观性质的结果。热力学操作是施加在参与机构上的宏观外部干预，而不是从它们的内部性质中得到的。由于没有认识到这一点，产生了被称为“悖论”的说法。<br />
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===Loschmidt's paradox===<br />
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===Loschmidt's paradox===<br />
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洛斯密特悖论<br />
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{{main article|Loschmidt's paradox}}<br />
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[[Loschmidt's paradox]], also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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洛斯密特悖论，也被称为可逆性悖论，其目的在于不可能从描述宏观系统微观演化的时间对称动力学推导出不可逆性。<br />
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In the opinion of [[Erwin Schrödinger|Schrödinger]], "It is now quite obvious in what manner you have to reformulate the law of entropy{{snd}}or for that matter, all other irreversible statements{{snd}}so that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other."<ref>[[Erwin Schrödinger|Schrödinger, E.]] (1950), p.&nbsp;192.</ref> The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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In the opinion of Schrödinger, "It is now quite obvious in what manner you have to reformulate the law of entropyor for that matter, all other irreversible statementsso that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other." The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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根据薛定谔的观点，“现在很明显，你必须用什么方式来重新定义该物质的熵定律，所有其他不可逆的状态，这样它们才能从可逆模型中导出。你不能只谈论一个孤立的系统，而应该至少谈论两个，你可以暂时认为它们与世界其它地方是孤立的，但并不总是相互孤立的。”这两个系统被隔离墙隔离开来，直到按照法律的设想被热力学操作拆除。热力学操作是外部强加的，不受支配系统组成部分的可逆微观动力学定律的制约。它是不可逆转性的原因。本文中的法律陈述符合薛定谔的建议。因果关系在逻辑上优先于第二定律，而不是从第二定律派生出来的。<br />
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===Poincaré recurrence theorem===<br />
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===Poincaré recurrence theorem===<br />
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庞加莱始态复现定理<br />
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{{Main article|Poincaré recurrence theorem}}<br />
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The [[Poincaré recurrence theorem]] considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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The Poincaré recurrence theorem considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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庞加莱始态复现定理认为这是一个孤立的物理系统的理论微观描述。这可以被认为是一个热力学系统的模型后，一个热力学操作已经移除了一个内墙。在足够长的时间之后，系统将返回到一个非常接近初始状态的显微定义状态。庞加莱回归时间是指回归前经过的时间长度。它极其漫长，可能比宇宙的寿命还要长，并且敏感地依赖于被热力学操作拆除的墙体的几何形状。递归定理可能被认为是明显与热力学第二定律相矛盾的。然而，更明显的是，它只不过是一个微观模型，热力学平衡在一个孤立的系统中，通过移除两个系统之间的一道墙而形成。对于一个典型的热力学系统，重现时间是如此之大(比宇宙的生命周期长许多倍) ，以至于在所有的实际目的中，人们都不能观察到重现。然而，人们可能希望想象一下，可以等待庞加莱复发，然后重新插入被热力学操作移除的壁。然后很明显，不可逆性的出现是由于庞加莱循环的完全不可预测性，只要初始状态是热力学平衡，就像宏观热力学的情况一样。即使可以等待，也没有实际可能性选择合适的时间重新插入墙壁。庞加莱始态复现定理为 Loschmidt 悖论提供了一个解决方案。如果一个孤立的热力学系统可以被监测到在越来越多的平均 poincar 递归时间的倍数上，系统的热力学行为将在时间反转下变得不变。<br />
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[[File:James-clerk-maxwell3.jpg|thumb|upright|James Clerk Maxwell]]<br />
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James Clerk Maxwell<br />
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詹姆斯·克拉克·麦克斯韦<br />
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===Maxwell's demon===<br />
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===Maxwell's demon===<br />
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麦克斯韦的魔鬼<br />
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{{main article|Maxwell's demon}}<br />
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[[James Clerk Maxwell]] imagined one container divided into two parts, ''A'' and ''B''. Both parts are filled with the same [[gas]] at equal temperatures and placed next to each other, separated by a wall. Observing the [[molecule]]s on both sides, an imaginary [[demon]] guards a microscopic trapdoor in the wall. When a faster-than-average molecule from ''A'' flies towards the trapdoor, the demon opens it, and the molecule will fly from ''A'' to ''B''. The average [[speed]] of the molecules in ''B'' will have increased while in ''A'' they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in ''A'' and increases in ''B'', contrary to the second law of thermodynamics.<br />
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James Clerk Maxwell imagined one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other, separated by a wall. Observing the molecules on both sides, an imaginary demon guards a microscopic trapdoor in the wall. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics.<br />
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詹姆斯·克拉克·麦克斯韦想象一个容器分成两部分，a 和 b。两个部分在相同的温度下充入相同的气体，并被一堵墙隔开，挨着放置。通过观察两边的分子，一个想象中的恶魔守卫着墙上的一个微型暗门。当一个来自 a 的比平均速度快的分子飞向活板门时，魔鬼打开了它，分子就会从 a 飞到 b。B 中分子的平均速度会增加，而 a 中分子的平均速度会减慢。由于平均分子速度与温度相对应，因此 a 中的温度降低，b 中的温度升高，这与热力学第二定律相反。<br />
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One response to this question was suggested in 1929 by [[Leó Szilárd]] and later by [[Léon Brillouin]]. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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One response to this question was suggested in 1929 by Leó Szilárd and later by Léon Brillouin. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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1929年，Leó Szilárd 和后来的 l on brilouin 对这个问题提出了一个答案。Szil rd 指出，现实生活中的麦克斯韦恶魔需要一些测量分子速度的方法，而获取信息需要消耗大量能量。<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between ''A'' and ''B''. It is therefore performing [[thermodynamic operation]]s on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between A and B. It is therefore performing thermodynamic operations on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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麦克斯韦的“魔鬼”不断地改变 a 和 b 之间墙体的渗透性。因此，它是在微观尺度上执行热力学操作，而不仅仅是观察普通的自发或自然的宏观热力学过程。<br />
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==Quotations==<br />
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语录<br />
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{{Wikiquote}}<br />
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{{quote|The law that entropy always increases holds, I think, the supreme position among the [[Laws of science|laws of Nature]]. If someone points out to you that your pet theory of the [[universe]] is in disagreement with [[Maxwell's equations]] – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.|Sir [[Arthur Stanley Eddington]], ''The Nature of the Physical World'' (1927)}}<br />
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{{quote|There have been nearly as many formulations of the second law as there have been discussions of it.|Philosopher / Physicist [[Percy Williams Bridgman|P.W. Bridgman]], (1941)}}<br />
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{{quote|Clausius is the author of the sibyllic utterance, "The energy of the universe is constant; the entropy of the universe tends to a maximum." The objectives of continuum thermomechanics stop far short of explaining the "universe", but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size.|[[Clifford Truesdell|Truesdell, C.]], Muncaster, R. G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 17.}}<br />
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==See also==<br />
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==See also==<br />
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参见<br />
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*[[Clausius–Duhem inequality]]<br />
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*[[Fluctuation theorem]]<br />
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*[[Heat death of the universe]]<br />
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*[[History of thermodynamics]]<br />
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*[[Jarzynski equality]]<br />
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*[[Laws of thermodynamics]]<br />
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*[[Maximum entropy thermodynamics]]<br />
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*[[Quantum thermodynamics]]<br />
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*[[Reflections on the Motive Power of Fire]]<br />
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*[[Relativistic heat conduction]]<br />
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*[[Thermal diode]]<br />
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*[[Thermodynamic equilibrium]]<br />
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==References==<br />
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==References==<br />
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参考资料<br />
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===Sources===<br />
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===Sources===<br />
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资料来源<br />
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{{refbegin|50em}}<br />
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*Adkins, C.J. (1968/1983). ''Equilibrium Thermodynamics'', (1st edition 1968), third edition 1983, Cambridge University Press, Cambridge UK, {{ISBN|0-521-25445-0}}.<br />
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*[[Peter Atkins|Atkins, P.W.]], de Paula, J. (2006). ''Atkins' Physical Chemistry'', eighth edition, W.H. Freeman, New York, {{ISBN|978-0-7167-8759-4}}.<br />
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*Attard, P. (2012). ''Non-equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications'', Oxford University Press, Oxford UK, {{ISBN|978-0-19-966276-0}}.<br />
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*Baierlein, R. (1999). ''Thermal Physics'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-59082-5}}.<br />
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*Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics, New York, {{ISBN|0-88318-797-3}}.<br />
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*{{Cite book|title=Concepts in thermal physics|last=Blundell|first=Stephen J.|last2=Blundell|authorlink1=Stephen Blundell|first2=Katherine M.|authorlink2=Katherine Blundell|publisher=[[Oxford University Press]]|year=2010|isbn=9780199562107|edition=2nd|location=Oxford|pages=|oclc=607907330|doi=10.1093/acprof:oso/9780199562091.001.0001 |url=http://cds.cern.ch/record/1235139}}<br />
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*[[Ludwig Boltzmann|Boltzmann, L.]] (1896/1964). ''Lectures on Gas Theory'', translated by S.G. Brush, University of California Press, Berkeley.<br />
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*Borgnakke, C., Sonntag., R.E. (2009). ''Fundamentals of Thermodynamics'', seventh edition, Wiley, {{ISBN|978-0-470-04192-5}}.<br />
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*Buchdahl, H.A. (1966). ''The Concepts of Classical Thermodynamics'', Cambridge University Press, Cambridge UK.<br />
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*[[Percy Williams Bridgman|Bridgman, P.W.]] (1943). ''The Nature of Thermodynamics'', Harvard University Press, Cambridge MA.<br />
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*[[Herbert Callen|Callen, H.B.]] (1960/1985). ''Thermodynamics and an Introduction to Thermostatistics'', (1st edition 1960) 2nd edition 1985, Wiley, New York, {{ISBN|0-471-86256-8}}.<br />
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*{{cite journal|author=C. Carathéodory|author1-link=Constantin Carathéodory<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
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|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
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2010年3月15日 | 第355-386页 | http://gdz.sub.uni-goettingen.de/index.php?id=11&ppn=ppn235181684_0067&dmdid=dmdlog_0033&l=1<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). ''The Second Law of Thermodynamics'', Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können.(p. 363) | doi 10.1007 / bf01450409}.译文可以在这里找到[ http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf ]。同时，在 Kestin，j。(1976).热力学第二定律，Dowden，Hutchinson & Ross，Stroudsburg PA. 。<br />
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*[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986). [http://www.worldcat.org/title/reflections-on-the-motive-power-of-fire-a-critical-edition-with-the-surviving-scientific-manuscripts-translated-and-edited-by-fox-robert/oclc/812944517&referer=brief_results ''Reflections on the motive power of fire''], Manchester University Press, Manchester UK, {{ISBN|0-7190-1741-6}}. [https://archive.org/stream/reflectionsonmot00carnrich#page/n7/mode/2up Also here.]<br />
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*[[Sydney Chapman (mathematician)|Chapman, S.]], [[Thomas George Cowling|Cowling, T.G.]] (1939/1970). ''The Mathematical Theory of Non-uniform gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases'', third edition 1970, Cambridge University Press, London.<br />
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*{{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Ueber Die Bewegende Kraft Der Wärme Und Die Gesetze, Welche Sich Daraus Für Die Wärmelehre Selbst Ableiten Lassen|journal=Annalen der Physik|year=1850|volume=79|issue=4|pages=368–397, 500–524|url=http://gallica.bnf.fr/ark:/12148/bpt6k15164w/f518.image|accessdate=26 June 2012|doi=10.1002/andp.18501550403|bibcode = 1850AnP...155..500C |hdl=2027/uc1.$b242250|hdl-access=free}} Translated into English: {{cite journal|last=Clausius|first=R.|title=On the Moving Force of Heat, and the Laws regarding the Nature of Heat itself which are deducible therefrom|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1851| volume=2|series=4th|issue=VIII|pages=1–21; 102–119 |url=https://archive.org/stream/londonedinburghd02lond#page/1/mode/1up|accessdate=26 June 2012<br />
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|doi=10.1080/14786445108646819}}<br />
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|doi=10.1080/14786445108646819}}<br />
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10.1080 / 14786445108646819}<br />
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* {{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie|year=1854|journal=Annalen der Physik|volume=xciii|issue=12|pages=481–506|url=http://zfbb.thulb.uni-jena.de/servlets/MCRFileNodeServlet/jportal_derivate_00140956/18541691202_ftp.pdf|accessdate=24 March 2014|doi=10.1002/andp.18541691202<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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| bibcode 1854AnP... 169. 1856年7月 | 第2卷 | 第4页 | 第86页 | https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=242014年3月<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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热力学理论及其在蒸汽机和物体物理性质中的应用 | 1867年 | 出版商 John van Voorst | 位置: 伦敦 | https://archive.org/details/mechanicaltheor04claugoog|quote=editions:pwr_sbkwa8ic。 2012年6月19日<br />
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*{{cite journal|ref=harv|last=Thomson|first=W.|author-link=William Thomson, 1st Baron Kelvin|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Transactions of the Royal Society of Edinburgh|year=1851|volume=XX|issue=part II|pages=261–268; 289–298|url=https://www.biodiversitylibrary.org/item/126047#page/295/mode/1up}} Also published in {{cite journal|last=Thomson|first=W.|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Philos. Mag. |date=December 1852 |volume=IV |series=4 |issue=22 |page=13|url=https://archive.org/stream/londonedinburghp04maga#page/12/mode/2up |accessdate=25 June 2012}}<br />
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*[[William Thomson, 1st Baron Kelvin|Thomson, W.]] (1852). ''On the universal tendency in nature to the dissipation of mechanical energy'' Philosophical Magazine, Ser. 4, p.&nbsp;304.<br />
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*[[László Tisza|Tisza, L.]] (1966). ''Generalized Thermodynamics'', M.I.T Press, Cambridge MA.<br />
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*[[Clifford Truesdell|Truesdell, C.]] (1980). ''The Tragicomical History of Thermodynamics 1822–1854'', Springer, New York, {{ISBN|0-387-90403-4}}.<br />
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*Uffink, J. (2001). Bluff your way in the second law of thermodynamics, ''Stud. Hist. Phil. Mod. Phys.'', '''32'''(3): 305–394.<br />
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*Uffink, J. (2003). Irreversibility and the Second Law of Thermodynamics, Chapter 7 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003), Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963). ''Lectures in Statistical Mechanics'', American Mathematical Society, Providence RI.<br />
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*[[Mark Zemansky|Zemansky, M.W.]] (1968). ''Heat and Thermodynamics. An Intermediate Textbook'', fifth edition, McGraw-Hill Book Company, New York.<br />
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==Further reading==<br />
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==Further reading==<br />
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进一步阅读<br />
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*Goldstein, Martin, and Inge F., 1993. ''The Refrigerator and the Universe''. Harvard Univ. Press. Chpts. 4–9 contain an introduction to the Second Law, one a bit less technical than this entry. {{ISBN|978-0-674-75324-2}}<br />
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*Leff, Harvey S., and Rex, Andrew F. (eds.) 2003. ''Maxwell's Demon 2 : Entropy, classical and quantum information, computing''. Bristol UK; Philadelphia PA: [[Institute of Physics]]. {{ISBN|978-0-585-49237-7}}<br />
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*{{Cite book | first = J.J. | last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1}}(technical).<br />
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*{{cite book |title=Reflections on the Motive Power of Heat and on Machines Fitted to Develop That Power |last=Carnot |first=Sadi |editor=[[Robert Henry Thurston|Thurston, Robert Henry]] |year=1890 |publisher=J. Wiley & Sons |location=New York |isbn= |pages= }} ([https://books.google.com/books?id=tgdJAAAAIAAJ full text of 1897 ed.]) ([http://www.history.rochester.edu/steam/carnot/1943/ html])<br />
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*Stephen Jay Kline (1999). ''The Low-Down on Entropy and Interpretive Thermodynamics'', La Cañada, CA: DCW Industries. {{ISBN|1-928729-01-0}}.<br />
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*{{cite book | last1 = Kostic | first1 = M | year = 2011 | title = Revisiting The Second Law of Energy Degradation and Entropy Generation: From Sadi Carnot's Ingenious Reasoning to Holistic Generalization | journal = AIP Conf. Proc. | volume = 1411 | issue = 1| pages = 327–350 | doi = 10.1063/1.3665247 | isbn = 978-0-7354-0985-9 | bibcode = 2011AIPC.1411..327K | series = AIP Conference Proceedings | citeseerx = 10.1.1.405.1945 }} also at [https://web.archive.org/web/20130420222450/http://www.kostic.niu.edu/2ndLaw/Revisiting%20The%20Second%20Law%20of%20Energy%20Degradation%20and%20Entropy%20Generation%20-%20From%20Carnot%20to%20Holistic%20Generalization-4.pdf].<br />
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==External links==<br />
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==External links==<br />
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外部链接<br />
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*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics]" – by Lawrence Sklar.<br />
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*[http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node30.html ''Second law of thermodynamics''] in the MIT Course [http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/notes.html ''Unified Thermodynamics and Propulsion''] from Prof. Z. S. Spakovszky<br />
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*[[E.T. Jaynes]], 1988, "[http://bayes.wustl.edu/etj/articles/ccarnot.pdf The evolution of Carnot's principle]," in G. J. Erickson and C. R. Smith (eds.)''Maximum-Entropy and Bayesian Methods in Science and Engineering, Vol'' 1: p.&nbsp;267.<br />
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*[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf Caratheodory, C., "Examination of the foundations of thermodynamics," trans. by D. H. Delphenich]<br />
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* [https://www.bbc.co.uk/programmes/p004y2bm The Second Law of Thermodynamics], BBC Radio 4 discussion with John Gribbin, Peter Atkins & Monica Grady (''In Our Time'', Dec. 16, 2004)<br />
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* [http://mdpi.org/entropy/papers/e6010001.pdf [[Entropy (journal)]], 2004]<br />
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* [https://www.journals.uchicago.edu/doi/abs/10.1086/663835 The Journal of the International Society for the History of Philosophy of Science, 2012]<br />
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{{DEFAULTSORT:Second Law Of Thermodynamics}}<br />
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[[Category:Concepts in physics]]<br />
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Category:Concepts in physics<br />
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分类: 物理概念<br />
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[[Category:Laws of thermodynamics|2]]<br />
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2<br />
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[[Category:Non-equilibrium thermodynamics]]<br />
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Category:Non-equilibrium thermodynamics<br />
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类别: 非平衡态热力学<br />
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[[Category:Philosophy of thermal and statistical physics]]<br />
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Category:Philosophy of thermal and statistical physics<br />
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类别: 热力学和统计物理学哲学<br />
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<small>This page was moved from [[wikipedia:en:Second law of thermodynamics]]. Its edit history can be viewed at [[热力学第二定律/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E7%83%AD%E5%8A%9B%E5%AD%A6%E7%AC%AC%E4%BA%8C%E5%AE%9A%E5%BE%8B&diff=10355热力学第二定律2020-07-16T09:30:44Z<p>Dorr：/* Perpetual motion of the second kind 第二类永动机 */</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Law of physics}}<br />
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{{Thermodynamics|laws}}<br />
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The '''second law of thermodynamics''' states that the total [[entropy]] of an [[isolated system]] can never decrease over time, and is constant if and only if all processes are reversible.<ref>http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node38.html#SECTION05224000000000000000</ref> Isolated systems spontaneously evolve towards [[thermodynamic equilibrium]], the state with maximum entropy. <br />
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The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, and is constant if and only if all processes are reversible. Isolated systems spontaneously evolve towards thermodynamic equilibrium, the state with maximum entropy. <br />
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热力学第二定律指出，孤立系统的总熵永远不会随着时间而减少，而且只有当且仅当所有过程都是可逆的时，总熵才是恒定的。孤立的系统自发地进化到热力学平衡状态，即熵最大的状态。<br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in [[thermodynamic equilibrium]], or is undergoing a (fictive) [[Reversible process (thermodynamics)|reversible process]]. In all processes that occur, including [[spontaneous process]]es,<ref>Atkins and de Paula, p.78</ref> the total entropy of the system and its surroundings increases and the process is [[Irreversible process|irreversible in the thermodynamic sense]]. The increase in entropy accounts for the irreversibility of natural processes, and the [[Arrow of time|asymmetry between future and past]].<ref>{{cite book|last=Zohuri|first=Bahman|title=Dimensional Analysis Beyond the Pi Theorem|url=https://books.google.com/books?id=pRVuDQAAQBAJ|year=2016|publisher=Springer|isbn=978-3-319-45726-0|page=[https://books.google.com.ph/books?id=pRVuDQAAQBAJ&pg=PA111&dq=%22increase+in+entropy+accounts+for+the+irreversibility+of+natural+processes+and+the+asymmetry+between+future+and+past.%22&hl=en&sa=X&ved=0ahUKEwj6spb61tbaAhUFS7wKHfftDtIQ6AEIKDAA#v=onepage&q=%22increase%20in%20entropy%20accounts%20for%20the%20irreversibility%20of%20natural%20processes%20and%20the%20asymmetry%20between%20future%20and%20past.%22 111]}}</ref><br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in thermodynamic equilibrium, or is undergoing a (fictive) reversible process. In all processes that occur, including spontaneous processes, the total entropy of the system and its surroundings increases and the process is irreversible in the thermodynamic sense. The increase in entropy accounts for the irreversibility of natural processes, and the asymmetry between future and past.<br />
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系统及其周围环境的总熵在理想情况下可以保持不变，在这种情况下，系统处于热力学平衡期，或者正在经历一个(虚拟的)可逆过程期。在所有发生的过程中，包括自发过程，系统及其周围环境的总熵增加，这一过程在热力学意义上是不可逆的。熵的增加解释了自然过程的不可逆性，以及未来和过去之间的不对称性。<br />
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Historically, the second law was an [[Empirical evidence|empirical finding]] that was accepted as an axiom of [[thermodynamics|thermodynamic theory]]. [[Statistical mechanics]], classical or [[quantum thermodynamics|quantum]], explains the microscopic origin of the law.<br />
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Historically, the second law was an empirical finding that was accepted as an axiom of thermodynamic theory. Statistical mechanics, classical or quantum, explains the microscopic origin of the law.<br />
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从历史上看，第二定律是一个经验性的发现，被公认为热力学理论的公理。统计力学，无论是经典的还是量子的，都可以解释这个定律的微观起源。<br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist [[Nicolas Léonard Sadi Carnot|Sadi Carnot]], who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as [[Carnot's theorem (thermodynamics)|Carnot's rule]] or limit.<ref>Jaffe, R.L., Taylor, W. (2018). ''The Physics of Energy'', Cambridge University Press, Cambridge UK, pages 150, 151, 259, 772, 743.</ref><br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist Sadi Carnot, who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as Carnot's rule or limit.<br />
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第二定律已经用许多方式表达过了。法国科学家萨迪 · 卡诺在1824年证明了热能在热机中转化为功的效率是有上限的。这方面的第二定律也被称为卡诺的规则或限制。<br />
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==Introduction==<br />
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==Introduction==<br />
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引言<br />
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[[File:Heat flow hot to cold.png|thumb|upright|Heat flow from hot water to cold water.]]<br />
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Heat flow from hot water to cold water.<br />
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热量从热水流向冷水。<br />
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The [[first law of thermodynamics]] provides the definition of the [[internal energy]] of a [[thermodynamic system]], and expresses the law of [[conservation of energy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 40–41.</ref><ref>Munster A. (1970), pp. 8–9, 50–51.</ref> The second law is concerned with the direction of natural processes.<ref>{{harvnb|Mandl|1988}}</ref> It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such [[Phenomenon|phenomena]] are accounted for in terms of [[entropy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 79–107.</ref><ref>Bailyn, M. (1994), Section 71, pp. 113–154.</ref> If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, {{math|''S''}}, increases.<br />
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The first law of thermodynamics provides the definition of the internal energy of a thermodynamic system, and expresses the law of conservation of energy. The second law is concerned with the direction of natural processes. It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such phenomena are accounted for in terms of entropy. If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, , increases.<br />
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能量守恒定律提供了热力学系统内能的定义，并表达了能量守恒定律。第二定律与自然过程的方向有关。它断言自然过程只在一种意义上运行，而且是不可逆的。例如，当有了传导和辐射的路径时，热总是自发地从一个较热的物体流向一个较冷的物体。这种现象可以用熵来解释。如果一个孤立的系统最初是通过内部分隔不可渗透的墙来保持在内部热力学平衡，然后一些操作使墙更具渗透性，那么系统自发地进化到最终达到一个新的内部热力学平衡，其总熵增加。<br />
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In a fictive reversible process, an infinitesimal increment in the entropy ({{math|d''S''}}) of a system is defined to result from an infinitesimal transfer of heat ({{math|δ''Q''}}) to a [[closed system]] (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature ({{math|''T''}}) of the system in equilibrium and the surroundings which supply the heat:<ref>Bailyn, M. (1994), p. 120.</ref><br />
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In a fictive reversible process, an infinitesimal increment in the entropy () of a system is defined to result from an infinitesimal transfer of heat () to a closed system (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature () of the system in equilibrium and the surroundings which supply the heat:<br />
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在虚构的可逆过程中，系统熵的无穷小增量被定义为由无穷小的热量传递()到一个封闭系统(允许能量进入或出去，但不允许物质传递)除以平衡系统和提供热量的环境的共同温度() :<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, idealized fictive reversible process)}.</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, idealized fictive reversible process)}.</math><br />
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(闭合系统，理想化的可逆过程) / math<br />
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Different notations are used for infinitesimal amounts of heat ({{mvar|δ}}) and infinitesimal amounts of entropy ({{mvar|d}}) because entropy is a [[function of state]], while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the [[Clausius Theorem|inequality]] <ref name="MortimerBook"> Mortimer, R. G. ''Physical Chemistry''. 3rd ed., p. 120, Academic Press, 2008. </ref><ref name="FermiBook"> Fermi, E. ''Thermodynamics'', footnote on p. 48, Dover Publications,1956 (still in print). </ref> <br />
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Different notations are used for infinitesimal amounts of heat () and infinitesimal amounts of entropy () because entropy is a function of state, while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the inequality <br />
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用不同的符号表示无穷小量的热量()和无穷小量的熵() ，因为熵是状态的函数，而热量和功一样不是。第二定律要求系统熵的增量满足不等式，对于实际上可能存在的不与环境发生质量交换的无穷小过程，系统熵增量满足不等式<br />
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: <math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process).}</math><br />
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<math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process).}</math><br />
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(实际上可能是封闭系统，不可逆性)。数学<br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature ({{math|''T''}}) and the temperature of the surroundings ({{math|''T''}}<sub>''surr''</sub>).<ref name=":0">Adkins, C.J. (1968/1983), p. 75.</ref><ref name="Munster 45"/> Note that the equality still applies for pure heat flow,<ref name="Schmidt-Rohr 14"> Schmidt-Rohr, K. (2014). "Expansion Work without the External Pressure, and Thermodynamics in Terms of Quasistatic Irreversible Processes" ''J. Chem. Educ.'' '''91''': 402-409. https://dx.doi.org/10.1021/ed3008704 </ref><br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature () and the temperature of the surroundings (<sub>surr</sub>).<br />
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这是因为这种情况下的一般过程可能包括周围环境对系统所做的功，这可能在系统内部产生摩擦或粘滞效应，因为一个化学反应可能正在进行，或者因为热传递实际上只是不可逆地发生，由系统温度()和周围环境温度(sub surr / sub)之间的差分驱动。<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change).}</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change).}</math><br />
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(实际上可能是不改变成分的准静态不可逆性)。数学<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry.<ref name="Oxtoby8th">Oxtoby, D. W; Gillis, H.P., [[Laurie Butler|Butler, L. J.]] (2015).''Principles of Modern Chemistry'', Brooks Cole. p. 617. {{ISBN|978-1305079113}}</ref> <ref name="MortimerBook"></ref> Introducing a set of internal variables <math>\xi</math> to describe the deviation of a thermodynamic system in physical equilibrium (with the required well-defined uniform pressure ''P'' and temperature ''T'')<ref name="Schmidt-Rohr 14"></ref> from the chemical equilibrium state, one can record the equality<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry. from the chemical equilibrium state, one can record the equality<br />
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这是根据测得的热容曲线和相变时的熵变精确确定纯物质的绝对熵的基础。用量热法。从化学平衡来看，人们可以记录平等<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process).}</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process).}</math><br />
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- frac { t }-frac {1}{ j } ， Xi { j } ， delta Xi j， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ,.数学<br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of ''the second principle of thermodynamics'' – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<ref> Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769–781.</ref><ref>{{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}</ref><br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of the second principle of thermodynamics – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<br />
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第二项代表内部变量的工作，这些内部变量可以受到外部影响的干扰，但是系统不能通过内部变量执行任何正功。这种说法介绍了热力学系统在时间上不可能逆转演化的事实，并且可以被认为是热力学第二原理的表述---- 这个表述，当然，相当于熵原理的表述。<br />
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The [[zeroth law of thermodynamics]] in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body.<ref name=dugdale>{{cite book|author=J. S. Dugdale|title=Entropy and its Physical Meaning|publisher=Taylor & Francis|year=1996|isbn=978-0-7484-0569-5|page=13|quote=This law is the basis of temperature.}}</ref> For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows{{how?|date=August 2018}} a distinguished temperature scale, which defines an absolute, [[thermodynamic temperature]], independent of the properties of any particular reference thermometric body.<ref>[[Mark Zemansky|Zemansky, M.W.]] (1968), pp. 207–209.</ref><ref>Quinn, T.J. (1983), p. 8.</ref><br />
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The zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body. For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows a distinguished temperature scale, which defines an absolute, thermodynamic temperature, independent of the properties of any particular reference thermometric body.<br />
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美国热力学第零定律协会在其通常的简短声明中允许承认，相对于热平衡的两个物体具有相同的温度，特别是测试物体具有与参考温度测量物体相同的温度。对于一个物体和另一个物体之间的热平衡，存在着无限多的经验温度标度，通常分别取决于特定参考温度标度的性质。第二定律允许区分温度标度，它定义了一个绝对的热力学温度，与任何特定的参考温度体的性质无关。<br />
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==Various statements of the law==<br />
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==Various statements of the law==<br />
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各种各样的法律陈述<br />
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The second law of thermodynamics may be expressed in many specific ways,<ref name=MIT>{{cite web|title=Concept and Statements of the Second Law|url=http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node37.html|accessdate=2010-10-07 |publisher=web.mit.edu}}</ref> the most prominent classical statements{{sfnp|Lieb|Yngvason|1999}} being the statement by [[Rudolf Clausius]] (1854), the statement by [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] (1851), and the statement in axiomatic thermodynamics by [[Constantin Carathéodory]] (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.{{sfnp|Rao|2004|p=213}}<br />
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The second law of thermodynamics may be expressed in many specific ways, the most prominent classical statements being the statement by Rudolf Clausius (1854), the statement by Lord Kelvin (1851), and the statement in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.<br />
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热力学第二定律可以用许多特定的方式来表达，最突出的经典陈述是 Rudolf Clausius (1854)的陈述，Kelvin 勋爵(1851)的陈述，以及康斯坦丁·卡拉西奥多里(1909)在公理化热力学中的陈述。这些陈述用一般的物理术语描述法律，引用某些过程的不可能性。克劳修斯和开尔文陈述被证明是等价的。<br />
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===Carnot's principle===<br />
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===Carnot's principle===<br />
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卡诺原理<br />
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The historical origin<ref>[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986).</ref> of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a [[Carnot heat engine]], fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the [[caloric theory]] of heat was seriously considered, before the recognition of the [[first law of thermodynamics]], and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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The historical origin of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a Carnot heat engine, fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the caloric theory of heat was seriously considered, before the recognition of the first law of thermodynamics, and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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热力学第二定律的历史起源源于卡诺的原则。它指的是一个卡诺热机周期，虚拟地以被称为准静态的极慢极限模式运行，因此热量和功量在子系统之间传递，这些子系统始终处于它们自己的内部热力学平衡状态。卡诺发动机是热机效率工程师特别感兴趣的理想设备。卡诺认识到卡诺原理的时候，正是热量热理论被认真考虑的时候，正是能量守恒定律理论被认识之前，正是熵概念的数学表达之前。根据第一定律的解释，它在物理上等同于热力学第二定律，并且在今天仍然有效。卡诺最初的论点是从卡路里理论的观点提出的，直到能量守恒定律的发现。以下是他书中的一些例子:<br />
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::...''wherever there exists a difference of temperature, motive power can be produced.''<ref>Carnot, S. (1824/1986), p. 51.</ref><br />
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...wherever there exists a difference of temperature, motive power can be produced.<br />
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只要有温差，就能产生动力。<br />
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::The production of motive power is then due in steam engines not to an actual consumption of caloric, but ''to its transportation from a warm body to a cold body ...''<ref>Carnot, S. (1824/1986), p. 46.</ref><br />
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The production of motive power is then due in steam engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body ...<br />
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动力的产生不是由于蒸汽机实际消耗的热量，而是由于它从一个温暖的物体运输到一个寒冷的物体..。<br />
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::''The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.''<ref>Carnot, S. (1824/1986), p. 68.</ref><br />
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The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.<br />
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热的动力与实现热的媒介无关，热的数量完全取决于两个物体之间的温度，最后是热量的传递。<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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用现代术语来说，卡诺的原则可能更为准确:<br />
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::The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<ref>[[Clifford Truesdell|Truesdell, C.]] (1980), Chapter 5.</ref><ref>Adkins, C.J. (1968/1983), pp. 56–58.</ref><ref>Münster, A. (1970), p. 11.</ref><ref>Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998), pp.67–75.</ref><ref>Lebon, G., Jou, D., Casas-Vázquez, J. (2008), p. 10.</ref><ref>Eu, B.C. (2002), pp. 32–35.</ref><br />
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The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<br />
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准静态或可逆卡诺循环的效率只取决于两个热源的温度，而且无论工作物质是什么，效率是相同的。以这种方式运行的卡诺发动机是使用这两种温度的最有效的热机。<br />
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===Clausius statement===<br />
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===Clausius statement===<br />
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克劳修斯声明<br />
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The German scientist [[Rudolf Clausius]] laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work.{{sfnp|Clausius|1850}} His formulation of the second law, which was published in German in 1854, is known as the ''Clausius statement'':<br />
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The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work. His formulation of the second law, which was published in German in 1854, is known as the Clausius statement:<br />
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1850年，德国科学家 Rudolf Clausius 通过研究热传递和功之间的关系，为热力学第二定律实验室奠定了基础。他在1854年用德语发表的第二定律的提法被称为克劳修斯声明:<br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.{{sfnp|Clausius|1854|p=86}}</blockquote><br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.</blockquote><br />
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热量不可能从一个较冷的物体传递到一个较热的物体，除非同时发生一些其他的变化。 / blockquote<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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克劳修斯的陈述使用了“热量通道”的概念。正如通常在热力学的讨论，这意味着“能量作为热量的净转移” ，并没有提到促成转移的一种方式和其他。<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of [[refrigeration]], for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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如果不对系统进行外部功，热就不能自发地从冷区流向热区，这一点从制冷的普通经验中可以看出。在冰箱中，热量从冷到热，但只有在外部介质——制冷系统的强制作用下才会发生变化。<br />
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===Kelvin statements===<br />
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===Kelvin statements===<br />
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开尔文陈述<br />
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[[William Thomson, 1st Baron Kelvin|Lord Kelvin]] expressed the second law in several wordings.<br />
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Lord Kelvin expressed the second law in several wordings.<br />
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开尔文勋爵用几个字表达了第二定律。<br />
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::It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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没有任何外部机构的帮助，自动机器是不可能把热量从一个物体传递到另一个物体的较高温度。<br />
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::It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.{{sfnp|Thomson|1851}}<br />
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It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.<br />
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通过无生命物质的作用，把物质冷却到周围最冷的物体的温度以下，从物质的任何部分获得机械效应是不可能的。<br />
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===Equivalence of the Clausius and the Kelvin statements===<br />
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===Equivalence of the Clausius and the Kelvin statements===<br />
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克劳修斯和开尔文陈述的等价性<br />
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[[Image:Deriving Kelvin Statement from Clausius Statement.svg|thumb|Derive Kelvin Statement from Clausius Statement]]<br />
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Derive Kelvin Statement from Clausius Statement<br />
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从克劳修斯陈述推导出开尔文陈述<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed [[Carnot engine]] as shown by the figure. The [[Heat engine#Efficiency|efficiency]] of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the [[first law of thermodynamics]], as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the figure. The efficiency of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the first law of thermodynamics, as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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假设有一个引擎违反了开尔文定理: 也就是说，这个引擎以循环的方式吸收热量并将其转化为功，没有任何其他结果。现在配对与反向卡诺发动机如图所示。普通热机的效率为，反向热机的效率为1 / 。这对组合发动机的净效应和唯一效应是将热量从较冷的贮存器转移到较热的贮存器，这违反了克劳修斯的说法。(这是能量守恒定律的结果，因为系统的总能量保持不变，数学文本{输入} + 文本{输出}0暗示 q-frac { eta }-qc / math，所以数学 qc 向左(frac { eta-1 right) / math)。因此，违反开尔文陈述意味着违反克劳修斯陈述，即。克劳修斯的说法暗示了凯尔文的说法。我们可以用类似的方式证明开尔文陈述暗示了克劳修斯陈述，因此两者是等价的。<br />
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===Planck's proposition===<br />
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===Planck's proposition===<br />
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普朗克的命题<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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普朗克提出了直接来自经验的如下命题。这有时被认为是他对第二定律的陈述，但他认为这是第二定律推导的起点。<br />
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::It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 86.</ref><ref>Roberts, J.K., Miller, A.R. (1928/1960), p. 319.</ref><br />
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It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<br />
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建造一台发动机是不可能的，这台发动机将在一个完整的循环中工作，并且除了提高重量和冷却热储以外不会产生任何效果。<br />
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===Relation between Kelvin's statement and Planck's proposition 开尔文表述与普朗克命题的关系===<br />
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It is almost customary in textbooks to speak of the "[[Kelvin-Planck statement]]" of the law, as for example in the text by [[Dirk ter Haar|ter Haar]] and [[Harald Wergeland|Wergeland]].<ref>[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966), p. 17.</ref><br />
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It is almost customary in textbooks to speak of the "Kelvin-Planck statement" of the law, as for example in the text by ter Haar and Wergeland.<br />
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教科书中几乎惯常提及该定律的“'''开尔文-普朗克表述 Kelvin-Planck Statement'''” ，例如'''德克·特哈尔 Diek ter Haar''' 和'''哈拉尔德·沃格兰 Harald Wergeland''' 在文中就是这样表述的。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）Diek ter Haar 中间是小写的<br />
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The '''Kelvin–Planck statement''' (or the ''heat engine statement'') of the second law of thermodynamics states that <br />
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The Kelvin–Planck statement (or the heat engine statement) of the second law of thermodynamics states that <br />
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热力学第二定律的开尔文-普朗克表述（或称“'''热机表述 Heat Engine Statement'''”）指出<br />
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::It is impossible to devise a [[thermodynamic cycle|cyclically]] operating device, the sole effect of which is to absorb energy in the form of heat from a single [[heat reservoir|thermal reservoir]] and to deliver an equivalent amount of [[Work (physics)|work]].<ref name="Rao">{{cite book|last=Rao|first=Y. V. C.|title=Chemical Engineering Thermodynamics|publisher=Universities Press|isbn=978-81-7371-048-3|page=158|year=1997}}</ref><br />
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It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.<br />
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设计一种唯一效果是从单一热源吸收热量并提供等量的功的循环运行装置是不可能的。<br />
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===Planck's statement 普朗克表述===<br />
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Planck stated the second law as follows.<br />
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Planck stated the second law as follows.<br />
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普朗克表述第二定律如下。<br />
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::Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<ref name="Planck 100">[[Max Planck|Planck, M.]] (1897/1903), p. 100.</ref><ref name="Planck 463">[[Max Planck|Planck, M.]] (1926), p. 463, translation by Uffink, J. (2003), p. 131.</ref><ref name="Roberts & Miller 382">Roberts, J.K., Miller, A.R. (1928/1960), p. 382. This source is partly verbatim from Planck's statement, but does not cite Planck. This source calls the statement the principle of the increase of entropy.</ref><br />
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Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<br />
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自然界中发生的任一过程都是沿参与其中的所有物体的熵之和增加的方向进行的。在某些限制下——即对于可逆过程，熵的总和保持不变。<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for ''irreversible phenomena''.<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for irreversible phenomena.<br />
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与普朗克表述非常相似的是'''乌伦贝克 Uhlenbeck'''和'''福特 Ford'''关于不可逆现象的表述。<br />
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::... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<ref>[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963), p. 16.</ref><br />
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... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<br />
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在从一个平衡态到另一个平衡态的不可逆或自发的变化中（例如，当两个物体 A 和 B 接触时的温度平衡过程），熵总是增加。<br />
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===Principle of Carathéodory 卡拉西奥多里原理===<br />
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<!-- [[Caratheodory's principle]] redirects here --><br />
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—— caratheodory 的原理在这里重新定向——<br />
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[[Constantin Carathéodory]] formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<ref>[[Constantin Carathéodory|Carathéodory, C.]] (1909).</ref><br />
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Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<br />
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'''康斯坦丁·卡拉西奥多里 Constantin Carathéodory'''在纯数学公理的基础上进行了热力学公理化。他对第二定律的陈述被称为'''卡拉西奥多里原理 Principle of Carathéodory'''，可以这样表述:<br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.<ref>Buchdahl, H.A. (1966), p. 68.</ref></blockquote><br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.</blockquote><br />
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在绝热封闭系统的任意状态 S 附近，总有从 S 出发不可达的状态。<br />
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With this formulation, he described the concept of [[adiabatic accessibility]] for the first time and provided the foundation for a new subfield of classical thermodynamics, often called [[Ruppeiner geometry|geometrical thermodynamics]]. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic [[process function]], in other words, <math>\delta Q=TdS</math>.<ref name="Sychev1991">{{cite book |last=Sychev |first=V. V. |title=The Differential Equations of Thermodynamics |year=1991 |publisher=Taylor & Francis |isbn=978-1-56032-121-7}}</ref> {{clarify|date=February 2014}}<br />
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With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, <math>\delta Q=TdS</math>. <br />
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公理化处理后，他第一次描述了'''绝热可达性 Adiabatic Accessibility'''的概念，并为经典热力学的一个新的子领域，即通常所说的'''几何热力学 Geometrical Thermodynamics'''奠定了基础。由卡拉西奥多里原理可以推出，准静态转移的热量值是一个可积过程函数，即<math>\delta Q=TdS</math>。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）准静态转移的热量值是一个可积过程函数 存疑<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.<ref name="Munster 45">Münster, A. (1970), p. 45.</ref>{{sfnp|Lieb|Yngvason|1999|p=49}}<ref name="Planck 1926">[[Max Planck|Planck, M.]] (1926).</ref><ref>Buchdahl, H.A. (1966), p. 69.</ref> {{clarify|date=February 2014}}<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium. <br />
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尽管在教科书几乎惯称卡拉西奥多里原理表述了第二定律，并认为其与克劳修斯表述或开尔文-普朗克表述等价，但事实并非如此。为了得到第二定律的所有内容，需要对卡拉西奥多里原理补充普朗克表述，即等量功总是增加一个最初处于自身内部热力学平衡的封闭系统的内部能量。<br />
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===Planck's principle 普朗克原理===<br />
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In 1926, [[Max Planck]] wrote an important paper on the basics of thermodynamics.<ref name="Planck 1926"/><ref>Uffink, J. (2003), pp. 129–132.</ref> He indicated the principle<br />
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In 1926, Max Planck wrote an important paper on the basics of thermodynamics. He indicated the principle<br />
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1926年，'''马克斯·普朗克 Max Planck'''写了一篇关于热力学基础的重要论文。他指出了以下原理<br />
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::The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<ref name="Munster 45"/>{{sfnp |Lieb|Yngvason|1999|p=49}}<br />
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The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<br />
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一个封闭系统的内部能量因绝热过程增加，在整个过程中，系统的体积保持不变。<br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work."<ref>[[Clifford Truesdell|Truesdell, C.]], Muncaster, R.G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 15.</ref> Planck wrote: "The production of heat by friction is irreversible."<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 81.</ref><ref>[[Max Planck|Planck, M.]] (1926), p. 457, Wikipedia editor's translation.</ref><br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work." Planck wrote: "The production of heat by friction is irreversible."<br />
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这个公式没有提到热，没有提到温度，甚至没有提到熵，也不一定隐含地依赖于这些概念，但它暗示了第二定律的内容。一个密切相关的表述为，“摩擦力从来不做正功。”普朗克写道: “摩擦生热是不可逆的。”<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above.<ref>Lieb, E.H., Yngvason, J. (2003), p. 149.</ref> It is relevant that for a system at constant volume and [[Mole (unit)|mole numbers]], the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above. It is relevant that for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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这个普朗克定理没有提到熵，是用物理术语来表述的。它与上面给出的开尔文表述密切相关。相关的是，对于恒定体积和摩尔数的系统，熵是内能的单调函数。然而，普朗克的这个原理实际上并不是普朗克对第二定律的首选表述（见前面小节），而是依赖于熵的概念。<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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Borgnakke 和 Sonntag 提出的以下表述在某种意义上是对普朗克原理的补充。他们没有将其作为第二定律的完整表述:<br />
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::... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<ref>Borgnakke, C., Sonntag., R.E. (2009), p. 304.</ref><br />
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... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<br />
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只有一种方法可以减少[闭合]系统的熵——从系统中转移热量。<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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与普朗克之前提出的原理不同，这一原理明确地用熵的变化来表示。从系统中去除物质也可以减少系统的熵。<br />
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===Statement for a system that has a known expression of its internal energy as a function of its extensive state variables 一个其内能有已知表达式（其扩展状态变量的函数）的系统的表述===<br />
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The second law has been shown to be equivalent to the [[internal energy]] ''U'' being a weakly [[convex function]], when written as a function of extensive properties (mass, volume, entropy, ...).<ref>{{cite book |last1=van Gool |first1=W. |last2=Bruggink |first2=J.J.C. (Eds) |url= |title=Energy and time in the economic and physical sciences |publisher=North-Holland |year=1985 |pages=41–56 |quote= |isbn=978-0-444-87748-2}}</ref><ref>{{Cite journal | last1 = Grubbström | first1 = Robert W. | doi = 10.1016/j.apenergy.2007.01.003 | title = An Attempt to Introduce Dynamics Into Generalised Exergy Considerations| journal = Applied Energy| volume = 84| issue = 7–8 | pages = 701–718 | year = 2007}}</ref> {{clarify|date=February 2014}}<br />
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The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...). <br />
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第二定律被证明等价于弱凸函数内能 U （写成广泛性质（质量，体积，熵，...）的函数时）。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）存疑<br />
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==Corollaries 推论==<br />
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===Perpetual motion of the second kind 第二类永动机===<br />
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{{main article|Perpetual motion}}<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of [[first law of thermodynamics]] by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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在第二定律建立之前，许多志在发明永动机的人试图通过提取环境的巨大内能作为机器动力，以突破'''热力学第一定律 First Law of Thermodynamics''' 的限制。这种机器被称为“'''第二类永动机 Perpetual Motion Machine of the Second Kind'''”。第二定律表明这种机器是不可能的。<br />
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===Carnot theorem 卡诺定理===<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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卡诺定理（1824）是一条限制任何可能的发动机的最大效率的原理。效率完全取决于热库和冷库之间的温差。卡诺定理指出:<br />
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*All irreversible heat engines between two heat reservoirs are less efficient than a [[Carnot engine]] operating between the same reservoirs.<br />
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*All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs.<br />
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* 所有不可逆热机的效率低于在两相同热源之间工作的'''卡诺机 Carnot engine'''。<br />
* 所有可逆热机的效率与在两相同热源之间工作的卡诺机相等。<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as [[thermodynamic reversibility]]. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the [[Carnot cycle]]'s reversibility and was condemned to be less efficient.<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the Carnot cycle's reversibility and was condemned to be less efficient.<br />
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在他的理想模型中，热转换成功可以通过逆转循环的运动而恢复，这个概念后来被称为'''热力学可逆性 Thermodynamic Reversibility'''。然而，卡诺进一步假定，一些热量损失了，并没有转化为机械功。因此，没有一个真实的热机能够实现'''卡诺循环 Carnot cycle'''的可逆性，并且被认为效率较低。<br />
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Though formulated in terms of caloric (see the obsolete [[caloric theory]]), rather than [[entropy]], this was an early insight into the second law.<br />
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Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law.<br />
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该理论尽管是用热量表述的（见被取代的'''热质说 Caloric Theory'''），而不是熵，但是它是对第二定律的早期认识。<br />
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===Clausius inequality 克劳修斯不等式===<br />
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The [[Clausius theorem]] (1854) states that in a cyclic process<br />
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The Clausius theorem (1854) states that in a cyclic process<br />
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克劳修斯定理（1854）指出，在一个循环的过程中<br />
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: <math>\oint \frac{\delta Q}{T} \leq 0.</math><br />
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The equality holds in the reversible case<ref>[http://scienceworld.wolfram.com/physics/ClausiusTheorem.html ''Clausius theorem''] at [[Wolfram Research]]</ref> and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function [[entropy]]. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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The equality holds in the reversible case and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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等号在可逆情况下成立，严格不等号在不可逆情况下成立。可逆情况下引入状态函数熵。这是因为在循环过程中，状态函数的变化为零。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）末句存疑<br />
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===Thermodynamic temperature 热力学温度===<br />
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{{main article|Thermodynamic temperature}}<br />
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For an arbitrary heat engine, the efficiency is:<br />
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For an arbitrary heat engine, the efficiency is:<br />
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对于任意热机，效率为:<br />
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: <math>\eta = \frac {W_n}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H} \qquad (1)</math><br />
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where ''W''<sub>n</sub> is for the net work done per cycle. Thus the efficiency depends only on q<sub>C</sub>/q<sub>H</sub>.<br />
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其中 ''W''<sub>n</sub> 表示每个循环所做的净功。因此效率只取决于 q<sub>C</sub>/q<sub>H</sub>。<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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卡诺定理指出，所有在相同热源间运行的可逆发动机的效率是相同的。<br />
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Thus, any reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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Thus, any reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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因此，任何在温度''T''<sub>1</sub>和''T''<sub>2</sub>之间运行的可逆热机必须具有相同的效率，也就是说，效率只是温度的函数:<br />
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<math>\frac{q_C}{q_H} = f(T_H,T_C)\qquad (2).</math><br />
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In addition, a reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between ''T''<sub>1</sub> and another (intermediate) temperature ''T''<sub>2</sub>, and the second between ''T''<sub>2</sub> and''T''<sub>3</sub>. This can only be the case if<br />
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In addition, a reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between T<sub>1</sub> and another (intermediate) temperature T<sub>2</sub>, and the second between T<sub>2</sub> andT<sub>3</sub>. This can only be the case if<br />
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另外，在温度 ''T''<sub>1</sub> 和 ''T''<sub>3</sub> 之间工作的可逆热机必须具有与由分别在温度''T''<sub>1</sub> 和（中间）温度 ''T''<sub>2</sub>之间、在 ''T''<sub>2</sub> 和 ''T''<sub>3</sub> 之间的两个循环组成的系统效率相同。只有下式成立才会出现这种情况:<br />
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: <math>f(T_1,T_3) = \frac{q_3}{q_1} = \frac{q_2 q_3} {q_1 q_2} = f(T_1,T_2)f(T_2,T_3).</math><br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the [[triple point]] of water. Then for any ''T''<sub>2</sub> and ''T''<sub>3</sub>,<br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the triple point of water. Then for any T<sub>2</sub> and T<sub>3</sub>,<br />
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现在考虑如下情形，<math>T_1</math> 是一个固定的参考温度: 水的三相点的温度。则对于任意 T<sub>2</sub> 和 T<sub>3</sub>,<br />
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: <math>f(T_2,T_3) = \frac{f(T_1,T_3)}{f(T_1,T_2)} = \frac{273.16 \cdot f(T_1,T_3)}{273.16 \cdot f(T_1,T_2)}.</math><br />
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Therefore, if thermodynamic temperature is defined by<br />
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Therefore, if thermodynamic temperature is defined by<br />
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因此，如果热力学温度被定义为<br />
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: <math>T = 273.16 \cdot f(T_1,T) \,</math><br />
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then the function ''f'', viewed as a function of thermodynamic temperature, is simply<br />
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then the function f, viewed as a function of thermodynamic temperature, is simply<br />
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那么函数 ''f'' 作为热力学温度的函数，为<br />
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: <math>f(T_2,T_3) = \frac{T_3}{T_2},</math><br />
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and the reference temperature ''T''<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be used{{snd}}the choice here corresponds to the [[Kelvin]] scale.)<br />
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and the reference temperature T<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be usedthe choice here corresponds to the Kelvin scale.)<br />
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参考温度 ''T''<sub>1</sub> 的值为273.16。（任何参考温度和任何正值均可用——此处的选择对应开尔文标度。）<br />
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===Entropy 熵===<br />
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{{main article|Entropy (classical thermodynamics)}}<br />
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According to the [[Clausius theorem|Clausius equality]], for a reversible process<br />
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According to the Clausius equality, for a reversible process<br />
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根据克劳修斯定理，对可逆过程有<br />
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: <math>\oint \frac{\delta Q}{T}=0</math><br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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这意味着线积分 <math>\int_L \frac{\delta Q}{T}</math> 对于可逆过程是路径无关的。<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer<ref name="Schmidt-Rohr 14" /> satisfies<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer satisfies<br />
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所以我们可以定义一个叫做熵的状态函数 S，对于可逆过程或者纯热传递满足<br />
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: <math>dS = \frac{\delta Q}{T} \!</math><br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the [[third law of thermodynamics]], which states that ''S'' = 0 at [[absolute zero]] for perfect crystals.<br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the third law of thermodynamics, which states that S = 0 at absolute zero for perfect crystals.<br />
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据此，只有对上述公式进行积分，才能得到熵的差值。为了获得绝对值，我们需要'''热力学第三定律 Third Law of Thermodynamics'''，它指出绝对零度下完美晶体的 ''S'' = 0。<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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对于任意不可逆过程，由于熵是一个状态函数，我们总是可以将初始状态和最终状态与一个虚拟的可逆过程联系起来，并在这条路径上积分以计算熵的差值。<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the [[Clausius inequality]] on this loop,<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the Clausius inequality on this loop,<br />
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现在把可逆过程逆过来，将其与上述不可逆过程结合。把克劳修斯不等式应用到这个循环,<br />
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: <math>-\Delta S+\int\frac{\delta Q}{T}=\oint\frac{\delta Q}{T}< 0</math><br />
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Thus,<br />
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Thus,<br />
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故,<br />
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: <math>\Delta S \ge \int \frac{\delta Q}{T} \,\!</math><br />
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where the equality holds if the transformation is reversible.<br />
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where the equality holds if the transformation is reversible.<br />
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如果变换可逆，等号成立。<br />
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Notice that if the process is an [[adiabatic process]], then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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Notice that if the process is an adiabatic process, then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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注意，若该过程是一个'''绝热过程 Adiabatic Process'''，则<math>\delta Q=0</math>，故<math>\Delta S\ge 0</math>。<br />
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===Energy, available useful work===<br />
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===Energy, available useful work===<br />
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能量，有用的工作<br />
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{{See also|Exergy}}<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an ''unlimited'' heat reservoir at temperature ''T<sub>R</sub>'' and pressure ''P<sub>R</sub>'' {{snd}}so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain ''T<sub>R</sub>''; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain ''P<sub>R</sub>''.<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an unlimited heat reservoir at temperature T<sub>R</sub> and pressure P<sub>R</sub> so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain T<sub>R</sub>; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain P<sub>R</sub>.<br />
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考虑将第二定律应用于孤立系统(称为整体系统或宇宙)的情形，是一个重要的、具有启发性的理想情形。孤立系统由两部分组成: 感兴趣的子系统和子系统的环境。这些环境被想象为如此之大，以至于它们可以被认为是一个温度为 t 子 r / sub 和压力为 p 子 r / sub 的无限热源，因此无论有多少热量被转移到(或来自)子系统，周围环境的温度将保持 t 子 r / sub; 无论子系统的体积扩大(或收缩)多少，周围环境的压力将保持 p 子 r / sub。<br />
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Whatever changes to ''dS'' and ''dS<sub>R</sub>'' occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy ''S<sub>tot</sub>'' of the isolated total system must not decrease:<br />
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Whatever changes to dS and dS<sub>R</sub> occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy S<sub>tot</sub> of the isolated total system must not decrease:<br />
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根据第二定律，孤立总体系统的熵 s 子系统 / 子系统的熵不能减小，无论子系统和周围环境的熵对 dS 和 dS 子系统 r / 子系统发生什么变化:<br />
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: <math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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<math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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数学 dS + dS r ge 0 / math<br />
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According to the [[first law of thermodynamics]], the change ''dU'' in the internal energy of the sub-system is the sum of the heat ''δq'' added to the sub-system, ''less'' any work ''δw'' done ''by'' the sub-system, ''plus'' any net chemical energy entering the sub-system ''d ∑μ<sub>iR</sub>N<sub>i</sub>'', so that:<br />
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According to the first law of thermodynamics, the change dU in the internal energy of the sub-system is the sum of the heat δq added to the sub-system, less any work δw done by the sub-system, plus any net chemical energy entering the sub-system d ∑μ<sub>iR</sub>N<sub>i</sub>, so that:<br />
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根据能量守恒定律，子系统内部能量的变化 dU 是子系统内部能量 q 的总和，减去子系统所做的任何功，再加上进入子系统 d ∑ sub iR / sub n sub i / sub 的任何净化学能，因此:<br />
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: <math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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<math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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数学 dU delta q- delta w + d ( sum mu iR } n i) ，/ math<br />
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where μ<sub>iR</sub> are the [[chemical potential]]s of chemical species in the external surroundings.<br />
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where μ<sub>iR</sub> are the chemical potentials of chemical species in the external surroundings.<br />
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其中亚 iR / 亚是外部环境中化学物种的化学势。<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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现在热量离开储存器进入子系统是<br />
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: <math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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<math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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数学△ qtr (- dS r) le tr dS / math<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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其中我们首先使用了经典热力学中熵的定义(在统计热力学中，熵变、温度和吸收热量之间的关系可以推导出来) ，然后从上面推导出第二定律的不等式。<br />
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It therefore follows that any net work ''δw'' done by the sub-system must obey<br />
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It therefore follows that any net work δw done by the sub-system must obey<br />
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因此，子系统所做的任何净功都必须服从<br />
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: <math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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<math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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数学增量 w le-dU + tds + sum mu { iR } dN i，/ math<br />
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It is useful to separate the work ''δw'' done by the subsystem into the ''useful'' work ''δw<sub>u</sub>'' that can be done ''by'' the sub-system, over and beyond the work ''p<sub>R</sub> dV'' done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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It is useful to separate the work δw done by the subsystem into the useful work δw<sub>u</sub> that can be done by the sub-system, over and beyond the work p<sub>R</sub> dV done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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将子系统所做的功与子系统所能做的功分离为子系统所能做的有用功，超过了子系统对外界压力所做的功，给出了子系统所能做的有用功(火用)的下列关系式:<br />
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: <math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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<math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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D (u-t r s + p r v-sum { iR } n i) ，/ math<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the ''availability'' or ''[[exergy]]'' ''E'' of the subsystem,<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the availability or exergy E of the subsystem,<br />
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可以很方便地将子系统的右侧定义为热动力位能的精确衍生物，称为子系统的可用性或火用 e,<br />
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: <math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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<math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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数学 e u-t r s + p r v- sum { iR } n i / math<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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因此，第二定律意味着，对于任何可以简单地被视为分为一个子系统和一个与之接触的无限温度和压力储存器的过程,<br />
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: <math> dE + \delta w_u \le 0 \, </math><br />
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<math> dE + \delta w_u \le 0 \, </math><br />
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数学德 delta u le 0，/ math<br />
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i.e. the change in the subsystem's exergy plus the useful work done ''by'' the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done ''on'' the system) must be less than or equal to zero.<br />
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i.e. the change in the subsystem's exergy plus the useful work done by the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done on the system) must be less than or equal to zero.<br />
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也就是。子系统(火用)的变化加上子系统所做的有用功(或者，子系统(火用)的变化不包括任何功，除了压力贮存器所做的功之外，在系统中所做的功)必须小于或等于零。<br />
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In sum, if a proper ''infinite-reservoir-like'' reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in ''E'' for an irreversible process and no change for a reversible process.<br />
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In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in E for an irreversible process and no change for a reversible process.<br />
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总之，如果选择一个合适的类似于无限库的参考状态作为现实世界中的系统环境，那么第二定律预测一个不可逆性的 e 值会减少，而一个可逆过程的 e 值不会变化。<br />
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: <math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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<math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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数学 d = 0 = 0 = 0 = 0 = 0<br />
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This expression together with the associated reference state permits a [[design engineer]] working at the macroscopic scale (above the [[thermodynamic limit]]) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (''Also, see [[process engineer]]''). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (''See [[Exergy efficiency|second law efficiency]]''.)<br />
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This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (Also, see process engineer). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (See second law efficiency.)<br />
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这个表达式连同相关的参考状态允许在宏观的设计工程师利用第二定律，而不需要直接测量或考虑整个孤立系统的熵变。(另外，请参阅流程工程师)。这些变化已经被考虑，假设被考虑的系统可以达到平衡的参考状态而不改变参考状态。还可以找到一个过程或过程集合的效率，将其与可逆的理想状态相比较<br />
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This approach to the Second Law is widely utilized in [[engineering]] practice, [[environmental accounting]], [[systems ecology]], and other disciplines.<br />
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This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines.<br />
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第二定律的这种方法被广泛应用于工程实践、环境会计、系统生态学和其他学科。<br />
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===The second law in chemical thermodynamics===<br />
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===The second law in chemical thermodynamics===<br />
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化学热力学的第二定律<br />
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For a [[spontaneous process|spontaneous chemical process]] in a closed system at constant temperature and pressure without non-''PV'' work, the Clausius inequality Δ''S > Q/T<sub>surr</sub>'' transforms into a condition for the change in [[Gibbs free energy]] <br />
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For a spontaneous chemical process in a closed system at constant temperature and pressure without non-PV work, the Clausius inequality ΔS > Q/T<sub>surr</sub> transforms into a condition for the change in Gibbs free energy <br />
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对于一个封闭系统中的自发化学过程，在没有非 pv 功的情况下，在恒定温度和压力下，将克劳修斯不等式 s q / t sub surr / sub 转化为吉布斯自由能变化的条件<br />
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: <math>\Delta G < 0 </math><br />
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数学 Delta g 0 / 数学<br />
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or d''G'' < 0. For a similar process at constant temperature and volume, the change in [[Helmholtz free energy]] must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products.<ref name="Oxtoby8th"/> <ref name="MortimerBook"></ref> The chemical equilibrium condition at constant ''T'' and ''p'' without electrical work is d''G'' = 0.<br />
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or dG < 0. For a similar process at constant temperature and volume, the change in Helmholtz free energy must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products. The chemical equilibrium condition at constant T and p without electrical work is dG = 0.<br />
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或者 dG 0。对于一个相似的过程，在恒定的温度和体积下，亥姆霍兹自由能的变化一定是负的，这是 Delta a 0 / math。因此，自由能(g 或 a)变化的负值是过程自发的必要条件。这是热力学第二定律在化学中最有用的形式，其中自由能的变化可以通过列表的生成焓和反应物和产物的标准摩尔熵来计算。在 t 和 p 不变的情况下，化学平衡条件是 dG 0。<br />
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==History==<br />
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==History==<br />
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历史<br />
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{{See also|History of entropy}}<br />
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[[File:Sadi Carnot.jpeg|thumb|upright|Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique]].]]<br />
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Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique.]]<br />
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尼古拉·莱昂纳尔·萨迪·卡诺穿着传统的巴黎综合理工学院制服<br />
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The first theory of the conversion of heat into mechanical work is due to [[Nicolas Léonard Sadi Carnot]] in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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The first theory of the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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第一个把热转化为机械功的理论是在1824年由尼古拉·莱昂纳尔·萨迪·卡诺提出的。他是第一个正确认识到这种转换的效率取决于发动机和环境之间的温差。<br />
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Recognizing the significance of [[James Prescott Joule]]'s work on the conservation of energy, [[Rudolf Clausius]] was the first to formulate the second law during 1850, in this form: heat does not flow ''spontaneously'' from cold to hot bodies. While common knowledge now, this was contrary to the [[caloric theory]] of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law during 1850, in this form: heat does not flow spontaneously from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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认识到詹姆斯·普雷斯科特·焦耳在能量守恒方面工作的重要性，Rudolf Clausius 在1850年第一个提出了第二定律: 热不会自发地从冷物体流向热物体。虽然现在的常识是，这与当时流行的热量理论相反，热量理论认为热量是一种流体。从那里他能够推断出萨迪卡诺定律和熵的定义(1865年)。<br />
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Established during the 19th century, the [[Kelvin-Planck statement|Kelvin-Planck statement of the Second Law]] says, "It is impossible for any device that operates on a [[cyclic process|cycle]] to receive heat from a single [[heat reservoir|reservoir]] and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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Established during the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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建立于19世纪的开尔文-普朗克第二定律(Kelvin-Planck)说，“任何循环运行的设备都不可能从单个蓄热体接收热量，并产生净功。”这被证明相当于克劳修斯的陈述。<br />
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The [[ergodic hypothesis]] is also important for the [[Boltzmann]] approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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遍历假设对玻尔兹曼方法也很重要。它说，在很长一段时间内，在具有相同能量的微观态的相空间的某些区域所花费的时间与这个区域的体积成正比，即。在很长一段时间内，所有可访问的微观状态的可能性都是一样的。同样的，它表明时间平均值和系综平均值是相同的。<br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a [[Macroscopic bodies|macroscopic system]]. This doctrine is obsolescent.<ref>Denbigh, K.G., Denbigh, J.S. (1985). ''Entropy in Relation to Incomplete Knowledge'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-25677-1}}, pp. 43–44.</ref><ref>Grandy, W.T., Jr (2008). ''Entropy and the Time Evolution of Macroscopic Systems'', Oxford University Press, Oxford, {{ISBN|978-0-19-954617-6}}, pp. 55–58.</ref><ref name=Lambert>[http://entropysite.oxy.edu Entropy Sites — A Guide] Content selected by [[Frank L. Lambert]]</ref><br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a macroscopic system. This doctrine is obsolescent.<br />
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有一种传统的学说，从克劳修斯开始，认为熵可以被理解为宏观系统中的分子“无序”。这种学说已经过时了。<br />
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===Account given by Clausius===<br />
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===Account given by Clausius===<br />
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作者: 克劳修斯<br />
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[[File:Clausius-1.jpg|thumb|upright|Rudolf Clausius]]<br />
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Rudolf Clausius<br />
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鲁道夫 · 克劳修斯<br />
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In 1856, the German physicist [[Rudolf Clausius]] stated what he called the "second fundamental theorem in the [[mechanical theory of heat]]" in the following form:{{sfnp|Clausius|1867}}<br />
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In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:<br />
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1856年，德国物理学家鲁道夫 · 克劳修斯阐述了他所谓的“热力学理论中的第二个基本定理” ，其形式如下:<br />
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: <math>\int \frac{\delta Q}{T} = -N</math><br />
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<math>\int \frac{\delta Q}{T} = -N</math><br />
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- n / math<br />
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where ''Q'' is heat, ''T'' is temperature and ''N'' is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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其中 q 是热，t 是温度，n 是一个循环过程中所有未补偿的相变的“等价值”。后来，在1865年，克劳修斯将“等价值”定义为熵。紧接着这个定义，也就是在同一年，第二定律最著名的版本在4月24日苏黎世哲学学会的一次演讲中被宣读，在演讲的最后，克劳修斯总结道:<br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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宇宙的熵趋于最大<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. [[universe]], as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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这句话是第二定律中最广为人知的措辞。由于其语言的松散性，例如:。这个问题的答案是肯定的，因为这个问题的答案是肯定的。开放，封闭，或孤立，许多人认为这个简单的陈述意味着热力学第二定律适用于几乎所有可以想象的主题。这是不正确的; 这个陈述只是一个更加扩展和精确描述的简化版本。<br />
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In terms of time variation, the mathematical statement of the second law for an [[isolated system]] undergoing an arbitrary transformation is:<br />
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In terms of time variation, the mathematical statement of the second law for an isolated system undergoing an arbitrary transformation is:<br />
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就时间变化而言，经历任意变换的孤立系统第二定律的数学表述是:<br />
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: <math>\frac{dS}{dt} \ge 0</math><br />
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<math>\frac{dS}{dt} \ge 0</math><br />
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0 / math<br />
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where<br />
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where<br />
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: ''S'' is the entropy of the system and<br />
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S is the entropy of the system and<br />
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S 是系统的熵<br />
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: ''t'' is [[time]].<br />
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t is time.<br />
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时间到了。<br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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!-- 可逆过程需要与周围环境保持平衡。这对于一个孤立的系统是不可能的。因此，关于可逆过程的讨论已经转移到对封闭系统的分析----<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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平衡后应用等式符号。建立孤立系统第二定律的另一种方法是:<br />
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: <math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math<br />
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with <math> \dot S_{i}</math> the sum of the rate of [[entropy production]] by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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with <math> \dot S_{i}</math> the sum of the rate of entropy production by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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用数学 s { i } / 数学表示系统内所有进程的产生熵之和。这种配方的优点是它显示了产生熵的效果。产生熵是一个非常重要的概念，因为它决定(限制)热机器的效率。与环境温度数学 t { a } / math 相乘，得到所谓的耗散能量数学 p { diss } t { a } dot s { i } / math。<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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封闭系统的第二定律(允许热量交换和移动边界，但不允许物质交换)的表达式是:<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math 来计算数学点 q } + 点 s { i } / math<br />
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给你<br />
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: <math>\dot Q</math> is the heat flow into the system<br />
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<math>\dot Q</math> is the heat flow into the system<br />
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数学 q / math 是进入系统的热流<br />
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: <math>T</math> is the temperature at the point where the heat enters the system.<br />
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<math>T</math> is the temperature at the point where the heat enters the system.<br />
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Math 是热量进入系统时的温度。<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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等式符号在只有可逆过程在系统内发生的情况下成立。如果发生不可逆过程(在实际操作系统中就是这种情况) ,-符号保持不变。如果系统有多处供热，我们必须求相应项的代数和。<br />
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For open systems (also allowing exchange of matter):<br />
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For open systems (also allowing exchange of matter):<br />
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对于开放系统(也允许物质交换) :<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math 表示数学点 q } + 点 s + 点 s { i } / math<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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这里的数学 s / math 是熵流进入系统，与物质流进入系统有关。它不应该与熵的时间导数混淆。如果物质在几个地方提供，我们必须取这些贡献的代数和。<br />
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==Statistical mechanics==<br />
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==Statistical mechanics==<br />
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统计力学<br />
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[[Statistical mechanics]] gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a [[microstate (statistical mechanics)|microstate]] of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/{{radic|''N''}} where ''N'' is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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Statistical mechanics gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a microstate of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/ where N is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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通过假设物质是由不断运动的原子和分子组成的，统计力学对第二定律给出了解释。系统中每个粒子的一组特定的位置和速度称为系统的微观状态，由于系统的不断运动，系统不断地改变其微观状态。统计力学假设，在平衡状态下，系统可能处于的每个微观状态发生的可能性是相等的，当这个假设被提出时，它直接导致了第二定律必须在统计意义上成立的结论。也就是说，第二定律平均成立，其中 n 是系统中粒子数的1 / 数量级的统计变化。在日常(宏观)情况下，违反第二定律的概率几乎为零。然而，对于粒子数量很少的系统，热力学参数，包括熵，可能显示出与第二定律预测的显著的统计偏差。经典热力学理论没有处理这些统计变量。<br />
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（by Jie XIANG) 在通过假定一种物质由不断运动的原子和分子组成的前提下，统计力学对热力学第二定律进行了解释。系统中每个粒子其特定的位置和速度组成为系统的微观状态，由于（原子和分子的）不断运动，系统也持续改变其微观状态。统计力学假设，在系统力平衡的情况下，其微观状态其实是等可能发生的，进而能够得出结论：热力学第二定律必须在统计学意义上成立。也就是说，第二定律将保持均值不变， 其统计变化为1 /√N，N为系统中的粒子数。在日常(宏观)情况下，违反热力学第二定律的概率几乎为零，但是，对于粒子数量非常少的系统，其热力学参数（包括熵）可能会显示出与第二定律所预测的明显统计偏差。然而经典热力学理论并不处理这些统计变量。<br />
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==Derivation from statistical mechanics==<br />
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==Derivation from statistical mechanics==<br />
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从统计力学导出<br />
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{{Further information|H-theorem}}<br />
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The first mechanical argument of the [[Kinetic theory of gases]] that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to [[James Clerk Maxwell]] in 1860;<ref>{{Cite journal | last1 = Gyenis | first1 = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G }}</ref> [[Ludwig Boltzmann]] with his [[H-theorem]] of 1872 also argued that due to collisions gases should over time tend toward the [[Maxwell–Boltzmann distribution]].<br />
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The first mechanical argument of the Kinetic theory of gases that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to James Clerk Maxwell in 1860; Ludwig Boltzmann with his H-theorem of 1872 also argued that due to collisions gases should over time tend toward the Maxwell–Boltzmann distribution.<br />
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气体动力学理论的第一个力学论证由詹姆斯·克拉克·麦克斯韦在1860年给出，指出分子碰撞引起温度均衡，因此整体趋向于平衡； 路德维希·玻尔兹曼在1872年提出的 H 定理也认为，气体由于碰撞应该随着时间的推移趋向于麦克斯韦-波兹曼分布。<br />
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Due to [[Loschmidt's paradox]], derivations of the Second Law have to make an assumption regarding the past, namely that the system is [[Correlation and dependence|uncorrelated]] at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a [[boundary condition]], and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the [[Big Bang]]), though [[Boltzmann brain|other scenarios]] have also been suggested.<ref name="Hawking AOT">{{cite journal|last=Hawking|first=SW|title=Arrow of time in cosmology|journal=Phys. Rev. D|year=1985|volume=32|issue=10|pages=2489–2495|doi=10.1103/PhysRevD.32.2489|pmid=9956019|bibcode = 1985PhRvD..32.2489H }}</ref><ref>{{cite book | last = Greene | first = Brian | authorlink = Brian Greene | title = The Fabric of the Cosmos | url = https://archive.org/details/fabricofcosmossp00gree | url-access = registration | publisher = Alfred A. Knopf | year = 2004 | page = [https://archive.org/details/fabricofcosmossp00gree/page/171 171] | isbn = 978-0-375-41288-2}}</ref><ref name=Lebowitz>{{cite journal|last=Lebowitz|first=Joel L.|title= Boltzmann's Entropy and Time's Arrow|journal=Physics Today|date=September 1993|volume=46|issue=9|pages=32–38|url=http://users.df.uba.ar/ariel/materias/FT3_2008_1C/papers_pdf/lebowitz_370.pdf|accessdate=2013-02-22|doi=10.1063/1.881363|bibcode = 1993PhT....46i..32L }}</ref><br />
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Due to Loschmidt's paradox, derivations of the Second Law have to make an assumption regarding the past, namely that the system is uncorrelated at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a boundary condition, and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the Big Bang), though other scenarios have also been suggested.<br />
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由于洛施密特悖论，第二定律的导出必须对过去做出一个假设，即系统在过去的某个时刻是不相关的；这样的假设允许进行简单的概率处理。这个假设通常被认为是一个边界条件，因此热力学第二定律最终是过去某个地方的初始条件的结果，可能是在宇宙的开始(大爆炸) 。也有人提出了其他假设。<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the [[Statistical mechanics#Fundamental postulate|fundamental postulate]], also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy.{{citation needed|date=August 2012}} The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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考虑到这些假设，在统计力学中，第二定律不是一个假设，而是统计力学基本假设的一个结果，也被称为等先验概率假设。这个基本假设表明，只要一个人清楚地知道，简单的概率论证只适用于未来，而对于过去，有辅助的信息来源告诉我们，它是低熵的。热力学第二定律的第一部分指出，热孤立系统的熵只能增加。如果我们把熵的概念限制在热平衡系统中，那么热力学第二定律的第一部分是等先验概率假设的一个显然结果。处于热平衡状态的孤立系统包含能量<math>E</math>的熵表示为：<br />
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: <math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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<math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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其中<math>\Omega\left(E\right)</math> 是处于 <math>E</math>和<math>E +\delta E</math这个小区间内的量子态数目。这里的 <math>\delta E</math> 是一个宏观上很小的固定能量区间。严格地说，这意味着熵取决于对<math>\delta E</math>的选择。然而在热力学极限下（例如无穷大系统的极限），狭义的熵(单位体积或单位质量的熵)不依赖于 <math>\delta E</math>。<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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假设我们有一个孤立系统，其宏观状态由许多变量描述。这些宏观变量可以是总体积、活塞在系统中的位置等。从而<math>\Omega</math>将取决于这些变量的值。如果某个变量不是固定的（我们不会在某个位置夹住活塞) ，那么因为在平衡状态下所有可到达状态的等几率到达的，平衡状态下的自由变量会使 <math>\Omega</math> 最大，因为这是平衡状态下最可能的情况。<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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如果该变量最初固定到某个值然后释放，当达到新的平衡时，变量将自我调整使得 <math>\Omega</math> 最大，这意味着熵将增加或保持不变(如果变量初始固定的值恰好是平衡值)。<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's [[H-theorem]], however, proves that the quantity {{math|''H''}} increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the quantity increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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假设我们初始位于一个平衡状态，突然移除了对一个变量的约束。我们做完这件事的时候，可达到的微观状态的数为<math>\Omega</math>，但是系统还没有达到平衡，所以系统处于某些可达到的状态的实际概率还不等于先验概率 <math>1/\Omega</math>。我们已经知道，最终的平衡状态相对于之前的平衡状态，熵会增加或者保持不变。然而，玻耳兹曼的H定理证明系统在不处于平衡态的期间，那个量作为时间的函数单调增加。<br />
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===Derivation of the entropy change for reversible processes===<br />
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===Derivation of the entropy change for reversible processes===<br />
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可逆过程熵变的推导<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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第二定律的第二部分指出，经历可逆过程的系统的熵变是由以下因素给出的:<br />
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: <math>dS =\frac{\delta Q}{T}</math><br />
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<math>dS =\frac{\delta Q}{T}</math><br />
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数学 dS frac delta q }{ t } / math<br />
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where the temperature is defined as:<br />
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where the temperature is defined as:<br />
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其中温度定义为:<br />
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:<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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[数学][[欧米茄][左(右)][右][数学][数学]<br />
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See [[Microcanonical ensemble|here]] for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the [[adiabatic theorem]] of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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请参阅此处对此定义的正当理由。假设系统有一些可以改变的外部参数 x。一般来说，系统的能量本征态将依赖于 x。根据量子力学的绝热定理，在系统哈密顿量无限缓慢变化的极限下，系统将保持在相同的定态，因此系统的能量会随着其所在定态的能量变化而变化。<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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广义力，x，对应于外部变量 x 的定义是，如果 x 增加一个量 dx，那么 math x dx / math 就是系统所做的功。例如，如果 x 是体积，那么 x 是压强。一个已知的系统的广义力在定态数学 e { r } / math 中给出:<br />
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: <math>X = -\frac{dE_{r}}{dx}</math><br />
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<math>X = -\frac{dE_{r}}{dx}</math><br />
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数学 x- frac { dE { r }{ dx } / math<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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由于系统可以在一个数学△ e / math 区间内的任意定态内，我们将系统的广义力定义为上述表达式的期望值:<br />
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: <math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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<math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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数学 x- 左 langle frac { dE { dx }右 rangle，/ math<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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为了计算平均值，我们通过计算在数学 y / math 和数学 y + delta y / math 之间的范围内，其中有多少数学 y / math 能量本征态有一个值来划分 ω 左(e 右) / math 能量本征态。把这个数字命名为 math Omega { y } left (e right) / math，我们有:<br />
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: <math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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<math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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Math Omega left (e right) sum { y } Omega { y } left (e right) ，/ math<br />
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The average defining the generalized force can now be written:<br />
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The average defining the generalized force can now be written:<br />
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定义广义力的平均值现在可以写成:<br />
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: <math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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<math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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数学 x frac {1} Omega 左(e 右) sum { y Omega { y }左(e 右) ，/ math<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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我们可以把它和熵对 x 在恒定能量 e 下的导数联系起来，如下所示。假设我们把 x 改成 x + dx。然后 math Omega left (e right) / math 将会改变，因为能量本征态依赖于 x，导致能量本征态进入或超出数学 e / math 和数学 e + delta e / math 之间的范围。让我们再次关注数学 y / math 和数学 y + delta y / math 之间的能量本征态。由于这些能量本征态的能量增加了 y dx，所有这些能量本征态在 e-y dx 到 e 之间的区间内从 e 以下移动到 e 以上。 有<br />
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: <math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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<math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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数学 n { y }左(e 右) frac { y }左(e 右)}{ delta e } y dx ，/ math<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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这样的能量本征态。如果数学 y dx leq delta e / math，所有这些能量本征态将移动到数学 e / math 和数学 e + delta e / math 之间的范围内，有助于数学 ω / math 的增加。从数学 e + △ e / math 下移到数学 e + △ e / math 上的能量本征态的个数是由数学 n { y }左(e + △ e 右) / math 给出的。区别在于<br />
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: <math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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<math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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数学 n { y }左(e 右)-n { y }左(e + delta e 右) ，/ math<br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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因此是数学增长的净贡献欧米茄 / 数学。请注意，如果 y dx 大于 math delta e / math，那么将会有能量本征态从 e 下面移动到 e + delta e / math 上面。它们在数学 n { y } left (e right) / math 和数学 n { y } left (e + delta e right) / math 中都有计算，因此上述表达式在这种情况下也是有效的。<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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将上面的表达式表示为对 e 的导数，对 y 的求和得到表达式:<br />
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: <math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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数学左(部分欧米茄左(部分欧米茄右){ e }-和 y 左(部分欧米茄右){ x 左(部分欧米茄左(部分欧米茄右)}{ x 右) ，/ 数学<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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关于 x 的欧米茄 / 数学对数导数由以下方式给出:<br />
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: <math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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Math 左( frac { partial ln 左( Omega 右)}{部分 x 右){ e } beta x + 左( frac { partial x 右){ x } ，/ math<br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
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第一个术语是密集型的，即。它不能根据系统大小进行缩放。相比之下，最后一项的规模与逆系统的规模一样，因此将在热力学极限中消失。因此，我们发现:<br />
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: <math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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<math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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数学左(部分 s }右){ e } frac { x }{ t } ，/ math<br />
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Combining this with<br />
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Combining this with<br />
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结合这一点<br />
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: <math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
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<math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
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数学左(部分 s }右){ x } frac {1}{ t } ，/ math<br />
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给予:<br />
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: <math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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<math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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数学 dS 左(部分 s }部分 e }右){ x } dE + 左(部分 s }右){ e } dx frac { t } + frac { t }{ delta q }{ t } ，/ math<br />
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===Derivation for systems described by the canonical ensemble===<br />
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===Derivation for systems described by the canonical ensemble===<br />
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正则系系统的推导<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the [[canonical ensemble]]:<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:<br />
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如果一个系统与某个温度为T的热浴热接触，那么在平衡状态下，关于能量本征值的概率分布值由正则系综给出:<br />
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: <math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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<math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the [[Partition function (statistical mechanics)|partition function]]. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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这里Z是一个使所有概率之和归一化的因子，这个函数被称为配分函数。现在我们考虑对温度和能级所依赖的外部参数的无限小的可逆改变。它遵循熵的一般公式:<br />
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: <math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
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<math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
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that<br />
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that<br />
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: <math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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<math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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把正则系综的数学形式 <math>P_{j}</math> 插入，可以得到:<br />
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: <math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
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<math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
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==Living organisms==<br />
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==Living organisms==<br />
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生命体<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as [[What is Life?|Erwin Schrödinger]], [[Léon Brillouin]]<ref>Léon Brillouin [https://books.google.com/books?hl=en&lr=&id=tPXVbiw_1P0C ''Science and Information Theory''] (Academic Press, 1962) (Dover, 2004)</ref> and [[Life and Energy|Isaac Asimov]]. It is also the topic of current{{When|date=July 2019}} research.{{Citation needed|date=July 2019}}<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as Erwin Schrödinger, Léon Brillouin and Isaac Asimov. It is also the topic of current research.<br />
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表述热力学主要有两种方式：(a)描述从一种热力学平衡状态到另一种状态的路径，(b)通过使系统保持不变，同时增加周围环境的总熵的循环过程。这两种方法有助于理解生命过程。这个话题大大超出了本文的范围，但是已经有一些研究者考虑过这个问题，比如埃尔温·薛定谔、莱昂·布里渊和艾萨克.阿西莫夫。这也是当前研究的主题。<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of [[metabolism]], give out breakdown products and heat. Plants [[Photosynthesis|take in radiative energy]] from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of metabolism, give out breakdown products and heat. Plants take in radiative energy from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
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从近似的角度来看，生命体可以被认为是(b)的一个例子。近似地，一只动物的身体状态每天循环，使得它几乎没有什么变化。动物吸收食物、水和氧气，经过新陈代谢，输出分解的产物和热量。植物吸收来自太阳的辐射能量，这可以认为是热量，以及二氧化碳和水，然后它们释放氧气。它们就是这样生长的，最终会死亡，尸体腐烂，大部分重新变成二氧化碳和水。这可以看作是一个循环过程。总的来说，阳光来自一个高温的源——太阳，它的能量被传递到一个较低的温度汇，例如向太空辐射。这个过程使得植物周围环境的熵增加。因此从循环过程的角度来看，动物和植物服从热力学第二定律。简单的热机效率概念很难适用于这个问题，因为它们假定系统是封闭的。<br />
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From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to [[Non-equilibrium thermodynamics|unsettled current debate or research]]. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
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From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to unsettled current debate or research. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
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从(a)的热力学观点考虑 ，即考虑从一个平衡状态到另一个平衡状态的路径，只会产生一个粗略近似的图像，因为生命体从来不会处于热力学平衡状态。生物体经常必须被认为是开放的系统，因为它们吸收营养物质并排出废弃物。开放系统热力学目前通常从一个热力学平衡状态到另一个状态的路径的角度来考虑，或者是考虑局部热力学平衡近似下的流。生命体的问题可以通过假定一个流不变的稳态来进一步简化。对于这样的近似，熵产生的一般原理在目前有争论和研究的不确定性。尽管如此，从热力学第二定律的这个角度出发衍生出来的想法对生物是有启发意义的。<br />
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==Gravitational systems==<br />
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==Gravitational systems==<br />
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重力系统<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive [[heat capacity]], meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it.<ref name="Grandy 151">Grandy, W.T. (Jr) (2008), p. 151.</ref> Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive heat capacity, meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it. Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
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在不需要描述广义相对论的系统中，物体总是具有正的热容，这意味着温度随着能量的增加而升高。因此，当能量从高温物体流向低温物体时，源温度降低，而汇温度升高; 因此温差随时间而减小。这种情况并不总是发生在那些需要引力和广义相对论的系统中。这样的系统可以自发地朝着质量和能量不均匀分布的方向变化。这在大尺度上适用于宇宙，因此可能很难或不可能将第二定律应用于宇宙。除此之外，广义相对论所描述的系统热力学超出了本文的范围。<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
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{{main article|Non-equilibrium thermodynamics}}<br />
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The theory of classical or [[thermodynamic equilibrium|equilibrium thermodynamics]] is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<ref name="Grandy 151"/><ref>Callen, H.B. (1960/1985), p. 15.</ref><br />
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The theory of classical or equilibrium thermodynamics is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<br />
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经典热力学或平衡热力学的理想化。一个主要的假设或假设，往往甚至没有明确说明，是系统的存在，在他们自己的内部状态的热力学平衡。一般来说，在给定时间内包含一个物理系统的空间区域---- 可能在自然界中找到---- 不是在热力学平衡中，用最严格的术语来解释。从更宽松的角度来说，整个宇宙中没有任何东西是或者曾经是完全精确的热力学平衡。<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of [[thermodynamic equilibrium]]. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable [[thermal fluctuations|fluctuations]]. There is an exception, the case of [[critical point (thermodynamics)|critical states]], which exhibit to the naked eye the phenomenon of [[critical opalescence]]. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of thermodynamic equilibrium. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable fluctuations. There is an exception, the case of critical states, which exhibit to the naked eye the phenomenon of critical opalescence. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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为了物理分析的目的，假设热力学平衡是非常方便的。这种假设可能依赖于试错来证明其正当性。如果这个假设是正确的，它通常是非常有价值和有用的，因为它提供了热力学理论。平衡假设的要素是，一个系统被观察到在无限长的时间内是不变的，而且一个系统中有如此多的粒子，以至于它的微粒性质可以被完全忽略。在这样的平衡假设下，一般来说，没有宏观上可以探测到的波动。有一个例外，即临界状态的例子，它们用肉眼就可以看到临界乳光现象。对于临界状态的实验室研究，需要非常长的观察时间。<br />
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In all cases, the assumption of [[thermodynamic equilibrium]], once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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In all cases, the assumption of thermodynamic equilibrium, once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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在所有情况下，热力学平衡的假设，一旦做出，就意味着没有假定的候选者“涨落”改变系统的熵。<br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<ref>Lieb, E.H., Yngvason, J. (2003), p. 190.</ref><br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<br />
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一个物理系统很容易发生内部的宏观变化，其速度之快足以使熵的恒定性假设失效。或者一个物理系统的粒子太少，以至于微粒的本质在可观测的波动中是显而易见的。那么热力学平衡的假设就会被放弃。对于非平衡态，熵没有绝对的一般定义。<br />
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There are intermediate cases, in which the assumption of local [[thermodynamic equilibrium]] is a very good approximation,<ref>Gyarmati, I. (1967/1970), pp. 4-14.</ref><ref>Glansdorff, P., Prigogine, I. (1971).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003).</ref> but strictly speaking it is still an approximation, not theoretically ideal.<br />
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There are intermediate cases, in which the assumption of local thermodynamic equilibrium is a very good approximation, but strictly speaking it is still an approximation, not theoretically ideal.<br />
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还有一些中间情况，其中局部热力学平衡的假设是一个非常好的近似值，但严格地说它仍然是一个近似值，不是理论上的理想值。<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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对于一般的非平衡状态，考虑其他量的统计力学定义可能是有用的，这些量可以方便地称为‘熵’ ，但是它们不应该与第二定律正确定义的熵混淆或混淆。这些其他的量确实属于统计力学，而不是热力学---- 第二定律的主要领域。<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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宏观观测波动的物理学超出了本文的范围。<br />
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==Arrow of time==<br />
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==Arrow of time==<br />
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时间之箭<br />
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{{Further information|Entropy (arrow of time)}}Thermodynamic Asymmetry in Time<br />
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Thermodynamic Asymmetry in Time<br />
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时间中的热力学不对称性<br />
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{{See also|Arrow of time}}<br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely [[CPT symmetry]], since the second law applies statistically, it is hypothesized, on time-asymmetric ''boundary conditions''.<ref>{{cite web|first=Craig|last=Callender|url=https://plato.stanford.edu/archives/fall2011/entries/time-thermo/|title=Thermodynamic Asymmetry in Time|encyclopedia=Stanford Encyclopedia of Philosophy|date=29 July 2011}}</ref><br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely CPT symmetry, since the second law applies statistically, it is hypothesized, on time-asymmetric boundary conditions.<br />
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热力学第二定律是一个物理定律，它不对称于时间方向的反转。这与已经观察到的物理学基本定律，即 CPT 对称性的概念并不冲突，因为第二定律在统计上适用，它是假设的，是关于时间非对称的边界条件。<br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect ([[Arrow of time#The causal arrow of time|the causal arrow of time]]).<ref>{{cite book | first = J.J.| last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1|display-authors=etal}} chapter 6</ref><br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect (the causal arrow of time).<br />
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第二定律被提出来部分解释时间向前和向后移动的区别，例如为什么原因先于结果(因果关系的时间箭头)。<br />
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==Irreversibility==<br />
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==Irreversibility==<br />
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不可逆性<br />
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Irreversibility in [[thermodynamic process]]es is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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Irreversibility in thermodynamic processes is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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热力学过程的不可逆性是热力学作用不对称性的结果，而不是物体内部不可逆的微观性质的结果。热力学操作是施加在参与机构上的宏观外部干预，而不是从它们的内部性质中得到的。由于没有认识到这一点，产生了被称为“悖论”的说法。<br />
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===Loschmidt's paradox===<br />
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===Loschmidt's paradox===<br />
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洛斯密特悖论<br />
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{{main article|Loschmidt's paradox}}<br />
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[[Loschmidt's paradox]], also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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洛斯密特悖论，也被称为可逆性悖论，其目的在于不可能从描述宏观系统微观演化的时间对称动力学推导出不可逆性。<br />
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In the opinion of [[Erwin Schrödinger|Schrödinger]], "It is now quite obvious in what manner you have to reformulate the law of entropy{{snd}}or for that matter, all other irreversible statements{{snd}}so that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other."<ref>[[Erwin Schrödinger|Schrödinger, E.]] (1950), p.&nbsp;192.</ref> The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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In the opinion of Schrödinger, "It is now quite obvious in what manner you have to reformulate the law of entropyor for that matter, all other irreversible statementsso that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other." The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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根据薛定谔的观点，“现在很明显，你必须用什么方式来重新定义该物质的熵定律，所有其他不可逆的状态，这样它们才能从可逆模型中导出。你不能只谈论一个孤立的系统，而应该至少谈论两个，你可以暂时认为它们与世界其它地方是孤立的，但并不总是相互孤立的。”这两个系统被隔离墙隔离开来，直到按照法律的设想被热力学操作拆除。热力学操作是外部强加的，不受支配系统组成部分的可逆微观动力学定律的制约。它是不可逆转性的原因。本文中的法律陈述符合薛定谔的建议。因果关系在逻辑上优先于第二定律，而不是从第二定律派生出来的。<br />
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===Poincaré recurrence theorem===<br />
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===Poincaré recurrence theorem===<br />
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庞加莱始态复现定理<br />
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{{Main article|Poincaré recurrence theorem}}<br />
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The [[Poincaré recurrence theorem]] considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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The Poincaré recurrence theorem considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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庞加莱始态复现定理认为这是一个孤立的物理系统的理论微观描述。这可以被认为是一个热力学系统的模型后，一个热力学操作已经移除了一个内墙。在足够长的时间之后，系统将返回到一个非常接近初始状态的显微定义状态。庞加莱回归时间是指回归前经过的时间长度。它极其漫长，可能比宇宙的寿命还要长，并且敏感地依赖于被热力学操作拆除的墙体的几何形状。递归定理可能被认为是明显与热力学第二定律相矛盾的。然而，更明显的是，它只不过是一个微观模型，热力学平衡在一个孤立的系统中，通过移除两个系统之间的一道墙而形成。对于一个典型的热力学系统，重现时间是如此之大(比宇宙的生命周期长许多倍) ，以至于在所有的实际目的中，人们都不能观察到重现。然而，人们可能希望想象一下，可以等待庞加莱复发，然后重新插入被热力学操作移除的壁。然后很明显，不可逆性的出现是由于庞加莱循环的完全不可预测性，只要初始状态是热力学平衡，就像宏观热力学的情况一样。即使可以等待，也没有实际可能性选择合适的时间重新插入墙壁。庞加莱始态复现定理为 Loschmidt 悖论提供了一个解决方案。如果一个孤立的热力学系统可以被监测到在越来越多的平均 poincar 递归时间的倍数上，系统的热力学行为将在时间反转下变得不变。<br />
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[[File:James-clerk-maxwell3.jpg|thumb|upright|James Clerk Maxwell]]<br />
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James Clerk Maxwell<br />
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詹姆斯·克拉克·麦克斯韦<br />
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===Maxwell's demon===<br />
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===Maxwell's demon===<br />
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麦克斯韦的魔鬼<br />
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{{unreferenced|section|date=August 2018}}<br />
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{{main article|Maxwell's demon}}<br />
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[[James Clerk Maxwell]] imagined one container divided into two parts, ''A'' and ''B''. Both parts are filled with the same [[gas]] at equal temperatures and placed next to each other, separated by a wall. Observing the [[molecule]]s on both sides, an imaginary [[demon]] guards a microscopic trapdoor in the wall. When a faster-than-average molecule from ''A'' flies towards the trapdoor, the demon opens it, and the molecule will fly from ''A'' to ''B''. The average [[speed]] of the molecules in ''B'' will have increased while in ''A'' they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in ''A'' and increases in ''B'', contrary to the second law of thermodynamics.<br />
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James Clerk Maxwell imagined one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other, separated by a wall. Observing the molecules on both sides, an imaginary demon guards a microscopic trapdoor in the wall. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics.<br />
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詹姆斯·克拉克·麦克斯韦想象一个容器分成两部分，a 和 b。两个部分在相同的温度下充入相同的气体，并被一堵墙隔开，挨着放置。通过观察两边的分子，一个想象中的恶魔守卫着墙上的一个微型暗门。当一个来自 a 的比平均速度快的分子飞向活板门时，魔鬼打开了它，分子就会从 a 飞到 b。B 中分子的平均速度会增加，而 a 中分子的平均速度会减慢。由于平均分子速度与温度相对应，因此 a 中的温度降低，b 中的温度升高，这与热力学第二定律相反。<br />
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One response to this question was suggested in 1929 by [[Leó Szilárd]] and later by [[Léon Brillouin]]. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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One response to this question was suggested in 1929 by Leó Szilárd and later by Léon Brillouin. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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1929年，Leó Szilárd 和后来的 l on brilouin 对这个问题提出了一个答案。Szil rd 指出，现实生活中的麦克斯韦恶魔需要一些测量分子速度的方法，而获取信息需要消耗大量能量。<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between ''A'' and ''B''. It is therefore performing [[thermodynamic operation]]s on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between A and B. It is therefore performing thermodynamic operations on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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麦克斯韦的“魔鬼”不断地改变 a 和 b 之间墙体的渗透性。因此，它是在微观尺度上执行热力学操作，而不仅仅是观察普通的自发或自然的宏观热力学过程。<br />
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==Quotations==<br />
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==Quotations==<br />
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语录<br />
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{{Wikiquote}}<br />
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{{quote|The law that entropy always increases holds, I think, the supreme position among the [[Laws of science|laws of Nature]]. If someone points out to you that your pet theory of the [[universe]] is in disagreement with [[Maxwell's equations]] – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.|Sir [[Arthur Stanley Eddington]], ''The Nature of the Physical World'' (1927)}}<br />
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{{quote|There have been nearly as many formulations of the second law as there have been discussions of it.|Philosopher / Physicist [[Percy Williams Bridgman|P.W. Bridgman]], (1941)}}<br />
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{{quote|Clausius is the author of the sibyllic utterance, "The energy of the universe is constant; the entropy of the universe tends to a maximum." The objectives of continuum thermomechanics stop far short of explaining the "universe", but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size.|[[Clifford Truesdell|Truesdell, C.]], Muncaster, R. G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 17.}}<br />
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==See also==<br />
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==See also==<br />
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参见<br />
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*[[Clausius–Duhem inequality]]<br />
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*[[Fluctuation theorem]]<br />
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*[[Heat death of the universe]]<br />
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*[[History of thermodynamics]]<br />
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*[[Jarzynski equality]]<br />
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*[[Laws of thermodynamics]]<br />
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*[[Maximum entropy thermodynamics]]<br />
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*[[Quantum thermodynamics]]<br />
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*[[Reflections on the Motive Power of Fire]]<br />
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*[[Relativistic heat conduction]]<br />
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*[[Thermal diode]]<br />
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*[[Thermodynamic equilibrium]]<br />
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==References==<br />
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==References==<br />
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参考资料<br />
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===Sources===<br />
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===Sources===<br />
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资料来源<br />
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{{refbegin|50em}}<br />
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*Adkins, C.J. (1968/1983). ''Equilibrium Thermodynamics'', (1st edition 1968), third edition 1983, Cambridge University Press, Cambridge UK, {{ISBN|0-521-25445-0}}.<br />
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*[[Peter Atkins|Atkins, P.W.]], de Paula, J. (2006). ''Atkins' Physical Chemistry'', eighth edition, W.H. Freeman, New York, {{ISBN|978-0-7167-8759-4}}.<br />
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*Attard, P. (2012). ''Non-equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications'', Oxford University Press, Oxford UK, {{ISBN|978-0-19-966276-0}}.<br />
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*Baierlein, R. (1999). ''Thermal Physics'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-59082-5}}.<br />
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*Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics, New York, {{ISBN|0-88318-797-3}}.<br />
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*{{Cite book|title=Concepts in thermal physics|last=Blundell|first=Stephen J.|last2=Blundell|authorlink1=Stephen Blundell|first2=Katherine M.|authorlink2=Katherine Blundell|publisher=[[Oxford University Press]]|year=2010|isbn=9780199562107|edition=2nd|location=Oxford|pages=|oclc=607907330|doi=10.1093/acprof:oso/9780199562091.001.0001 |url=http://cds.cern.ch/record/1235139}}<br />
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*[[Ludwig Boltzmann|Boltzmann, L.]] (1896/1964). ''Lectures on Gas Theory'', translated by S.G. Brush, University of California Press, Berkeley.<br />
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*Borgnakke, C., Sonntag., R.E. (2009). ''Fundamentals of Thermodynamics'', seventh edition, Wiley, {{ISBN|978-0-470-04192-5}}.<br />
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*Buchdahl, H.A. (1966). ''The Concepts of Classical Thermodynamics'', Cambridge University Press, Cambridge UK.<br />
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*[[Percy Williams Bridgman|Bridgman, P.W.]] (1943). ''The Nature of Thermodynamics'', Harvard University Press, Cambridge MA.<br />
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*[[Herbert Callen|Callen, H.B.]] (1960/1985). ''Thermodynamics and an Introduction to Thermostatistics'', (1st edition 1960) 2nd edition 1985, Wiley, New York, {{ISBN|0-471-86256-8}}.<br />
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*{{cite journal|author=C. Carathéodory|author1-link=Constantin Carathéodory<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
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|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
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2010年3月15日 | 第355-386页 | http://gdz.sub.uni-goettingen.de/index.php?id=11&ppn=ppn235181684_0067&dmdid=dmdlog_0033&l=1<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). ''The Second Law of Thermodynamics'', Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können.(p. 363) | doi 10.1007 / bf01450409}.译文可以在这里找到[ http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf ]。同时，在 Kestin，j。(1976).热力学第二定律，Dowden，Hutchinson & Ross，Stroudsburg PA. 。<br />
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*[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986). [http://www.worldcat.org/title/reflections-on-the-motive-power-of-fire-a-critical-edition-with-the-surviving-scientific-manuscripts-translated-and-edited-by-fox-robert/oclc/812944517&referer=brief_results ''Reflections on the motive power of fire''], Manchester University Press, Manchester UK, {{ISBN|0-7190-1741-6}}. [https://archive.org/stream/reflectionsonmot00carnrich#page/n7/mode/2up Also here.]<br />
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*[[Sydney Chapman (mathematician)|Chapman, S.]], [[Thomas George Cowling|Cowling, T.G.]] (1939/1970). ''The Mathematical Theory of Non-uniform gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases'', third edition 1970, Cambridge University Press, London.<br />
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*{{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Ueber Die Bewegende Kraft Der Wärme Und Die Gesetze, Welche Sich Daraus Für Die Wärmelehre Selbst Ableiten Lassen|journal=Annalen der Physik|year=1850|volume=79|issue=4|pages=368–397, 500–524|url=http://gallica.bnf.fr/ark:/12148/bpt6k15164w/f518.image|accessdate=26 June 2012|doi=10.1002/andp.18501550403|bibcode = 1850AnP...155..500C |hdl=2027/uc1.$b242250|hdl-access=free}} Translated into English: {{cite journal|last=Clausius|first=R.|title=On the Moving Force of Heat, and the Laws regarding the Nature of Heat itself which are deducible therefrom|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1851| volume=2|series=4th|issue=VIII|pages=1–21; 102–119 |url=https://archive.org/stream/londonedinburghd02lond#page/1/mode/1up|accessdate=26 June 2012<br />
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10.1080 / 14786445108646819}<br />
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* {{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie|year=1854|journal=Annalen der Physik|volume=xciii|issue=12|pages=481–506|url=http://zfbb.thulb.uni-jena.de/servlets/MCRFileNodeServlet/jportal_derivate_00140956/18541691202_ftp.pdf|accessdate=24 March 2014|doi=10.1002/andp.18541691202<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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| bibcode 1854AnP... 169. 1856年7月 | 第2卷 | 第4页 | 第86页 | https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=242014年3月<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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热力学理论及其在蒸汽机和物体物理性质中的应用 | 1867年 | 出版商 John van Voorst | 位置: 伦敦 | https://archive.org/details/mechanicaltheor04claugoog|quote=editions:pwr_sbkwa8ic。 2012年6月19日<br />
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*Denbigh, K. (1954/1981). ''The Principles of Chemical Equilibrium. With Applications in Chemistry and Chemical Engineering'', fourth edition, Cambridge University Press, Cambridge UK, {{ISBN|0-521-23682-7}}.<br />
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*[[Josiah Willard Gibbs|Gibbs, J.W.]] (1876/1878). On the equilibrium of heterogeneous substances, ''Trans. Conn. Acad.'', '''3''': 108-248, 343-524, reprinted in ''The Collected Works of J. Willard Gibbs, Ph.D, LL. D.'', edited by W.R. Longley, R.G. Van Name, Longmans, Green & Co., New York, 1928, volume 1, pp.&nbsp;55–353.<br />
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*Glansdorff, P., Prigogine, I. (1971). ''Thermodynamic Theory of Structure, Stability, and Fluctuations'', Wiley-Interscience, London, 1971, {{ISBN|0-471-30280-5}}.<br />
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*Gyarmati, I. (1967/1970) ''Non-equilibrium Thermodynamics. Field Theory and Variational Principles'', translated by E. Gyarmati and W.F. Heinz, Springer, New York.<br />
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*[[Charles Kittel|Kittel, C.]], [[Herbert Kroemer|Kroemer, H.]] (1969/1980). ''Thermal Physics'', second edition, Freeman, San Francisco CA, {{ISBN|0-7167-1088-9}}.<br />
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*Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998). ''Modern Thermodynamics: From Heat Engines to Dissipative Structures'', John Wiley & Sons, Chichester, {{ISBN|0-471-97393-9}}.<br />
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*Lebon, G., Jou, D., Casas-Vázquez, J. (2008). ''Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers'', Springer-Verlag, Berlin, {{ISBN|978-3-540-74252-4}}.<br />
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*{{cite journal|ref=harv|title=The Physics and Mathematics of the Second Law of Thermodynamics|last1=Lieb |first1=E. H. |last2=Yngvason |first2=J.|journal=Physics Reports|volume=310|issue=1|pages=1–96 |year=1999|doi=10.1016/S0370-1573(98)00082-9|arxiv = cond-mat/9708200 |bibcode = 1999PhR...310....1L}}<br />
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*Lieb, E.H., Yngvason, J. (2003). The Entropy of Classical Thermodynamics, pp.&nbsp;147–195, Chapter 8 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*{{cite book|ref=harv|first=F.|last=Mandl|title=Statistical physics|edition=second|year=1988|publisher=[[Wiley & Sons]]|isbn=978-0-471-91533-1}}<br />
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*{{cite journal | last1 = Maxwell | first1 = J.C. | authorlink = James Clerk Maxwell | year = 1867 | title = On the dynamical theory of gases | url = | journal = Phil. Trans. R. Soc. Lond. | volume = 157 | issue = | pages = 49–88 | doi = 10.1098/rstl.1867.0004 }}<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985). ''Thermodynamics'', Pitman, London, {{ISBN|0-273-08577-8}}.<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003). Entropy in Nonequilibrium, pp.&nbsp;79–109, Chapter 5 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*Münster, A. (1970), ''Classical Thermodynamics'', translated by E.S. Halberstadt, Wiley–Interscience, London, {{ISBN|0-471-62430-6}}.<br />
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*[[Brian Pippard|Pippard, A.B.]] (1957/1966). ''Elements of Classical Thermodynamics for Advanced Students of Physics'', original publication 1957, reprint 1966, Cambridge University Press, Cambridge UK.<br />
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*[[Max Planck|Planck, M.]] (1897/1903). [https://archive.org/stream/treatiseonthermo00planrich#page/100/mode/2up ''Treatise on Thermodynamics'', translated by A. Ogg, Longmans Green, London, p. 100.]<br />
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*[[Max Planck|Planck. M.]] (1914). [https://archive.org/details/theoryofheatradi00planrich ''The Theory of Heat Radiation''], a translation by Masius, M. of the second German edition, P. Blakiston's Son & Co., Philadelphia.<br />
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*[[Max Planck|Planck, M.]] (1926). Über die Begründung des zweiten Hauptsatzes der Thermodynamik, ''Sitzungsberichte der Preussischen Akademie der Wissenschaften: Physikalisch-mathematische Klasse'': 453–463.<br />
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* Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769-781.<br />
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* {{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}<br />
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*Quinn, T.J. (1983). ''Temperature'', Academic Press, London, {{ISBN|0-12-569680-9}}.<br />
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*{{cite book|ref=harv|first=Y.V.C.|last=Rao|title=An Introduction to thermodynamics|url=https://books.google.com/books?id=iYWiCXziWsEC&pg=PA213|date=2004|publisher=Universities Press|isbn=978-81-7371-461-0|page=213}}<br />
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*Roberts, J.K., Miller, A.R. (1928/1960). ''Heat and Thermodynamics'', (first edition 1928), fifth edition, Blackie & Son Limited, Glasgow.<br />
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*[[Erwin Schrödinger|Schrödinger, E.]] (1950). Irreversibility, ''Proc. R. Ir. Acad.'', '''A53''': 189–195.<br />
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*[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966). ''Elements of Thermodynamics'', Addison-Wesley Publishing, Reading MA.<br />
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*{{cite journal|ref=harv|last=Thomson|first=W.|author-link=William Thomson, 1st Baron Kelvin|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Transactions of the Royal Society of Edinburgh|year=1851|volume=XX|issue=part II|pages=261–268; 289–298|url=https://www.biodiversitylibrary.org/item/126047#page/295/mode/1up}} Also published in {{cite journal|last=Thomson|first=W.|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Philos. Mag. |date=December 1852 |volume=IV |series=4 |issue=22 |page=13|url=https://archive.org/stream/londonedinburghp04maga#page/12/mode/2up |accessdate=25 June 2012}}<br />
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*[[William Thomson, 1st Baron Kelvin|Thomson, W.]] (1852). ''On the universal tendency in nature to the dissipation of mechanical energy'' Philosophical Magazine, Ser. 4, p.&nbsp;304.<br />
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*[[László Tisza|Tisza, L.]] (1966). ''Generalized Thermodynamics'', M.I.T Press, Cambridge MA.<br />
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*[[Clifford Truesdell|Truesdell, C.]] (1980). ''The Tragicomical History of Thermodynamics 1822–1854'', Springer, New York, {{ISBN|0-387-90403-4}}.<br />
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*Uffink, J. (2001). Bluff your way in the second law of thermodynamics, ''Stud. Hist. Phil. Mod. Phys.'', '''32'''(3): 305–394.<br />
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*Uffink, J. (2003). Irreversibility and the Second Law of Thermodynamics, Chapter 7 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003), Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963). ''Lectures in Statistical Mechanics'', American Mathematical Society, Providence RI.<br />
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*[[Mark Zemansky|Zemansky, M.W.]] (1968). ''Heat and Thermodynamics. An Intermediate Textbook'', fifth edition, McGraw-Hill Book Company, New York.<br />
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==Further reading==<br />
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==Further reading==<br />
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进一步阅读<br />
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*Goldstein, Martin, and Inge F., 1993. ''The Refrigerator and the Universe''. Harvard Univ. Press. Chpts. 4–9 contain an introduction to the Second Law, one a bit less technical than this entry. {{ISBN|978-0-674-75324-2}}<br />
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*Leff, Harvey S., and Rex, Andrew F. (eds.) 2003. ''Maxwell's Demon 2 : Entropy, classical and quantum information, computing''. Bristol UK; Philadelphia PA: [[Institute of Physics]]. {{ISBN|978-0-585-49237-7}}<br />
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*{{Cite book | first = J.J. | last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1}}(technical).<br />
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*{{cite book |title=Reflections on the Motive Power of Heat and on Machines Fitted to Develop That Power |last=Carnot |first=Sadi |editor=[[Robert Henry Thurston|Thurston, Robert Henry]] |year=1890 |publisher=J. Wiley & Sons |location=New York |isbn= |pages= }} ([https://books.google.com/books?id=tgdJAAAAIAAJ full text of 1897 ed.]) ([http://www.history.rochester.edu/steam/carnot/1943/ html])<br />
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*Stephen Jay Kline (1999). ''The Low-Down on Entropy and Interpretive Thermodynamics'', La Cañada, CA: DCW Industries. {{ISBN|1-928729-01-0}}.<br />
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*{{cite book | last1 = Kostic | first1 = M | year = 2011 | title = Revisiting The Second Law of Energy Degradation and Entropy Generation: From Sadi Carnot's Ingenious Reasoning to Holistic Generalization | journal = AIP Conf. Proc. | volume = 1411 | issue = 1| pages = 327–350 | doi = 10.1063/1.3665247 | isbn = 978-0-7354-0985-9 | bibcode = 2011AIPC.1411..327K | series = AIP Conference Proceedings | citeseerx = 10.1.1.405.1945 }} also at [https://web.archive.org/web/20130420222450/http://www.kostic.niu.edu/2ndLaw/Revisiting%20The%20Second%20Law%20of%20Energy%20Degradation%20and%20Entropy%20Generation%20-%20From%20Carnot%20to%20Holistic%20Generalization-4.pdf].<br />
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==External links==<br />
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==External links==<br />
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外部链接<br />
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*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics]" – by Lawrence Sklar.<br />
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*[http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node30.html ''Second law of thermodynamics''] in the MIT Course [http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/notes.html ''Unified Thermodynamics and Propulsion''] from Prof. Z. S. Spakovszky<br />
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*[[E.T. Jaynes]], 1988, "[http://bayes.wustl.edu/etj/articles/ccarnot.pdf The evolution of Carnot's principle]," in G. J. Erickson and C. R. Smith (eds.)''Maximum-Entropy and Bayesian Methods in Science and Engineering, Vol'' 1: p.&nbsp;267.<br />
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*[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf Caratheodory, C., "Examination of the foundations of thermodynamics," trans. by D. H. Delphenich]<br />
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* [https://www.bbc.co.uk/programmes/p004y2bm The Second Law of Thermodynamics], BBC Radio 4 discussion with John Gribbin, Peter Atkins & Monica Grady (''In Our Time'', Dec. 16, 2004)<br />
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* [http://mdpi.org/entropy/papers/e6010001.pdf [[Entropy (journal)]], 2004]<br />
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* [https://www.journals.uchicago.edu/doi/abs/10.1086/663835 The Journal of the International Society for the History of Philosophy of Science, 2012]<br />
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{{DEFAULTSORT:Second Law Of Thermodynamics}}<br />
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[[Category:Concepts in physics]]<br />
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Category:Concepts in physics<br />
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分类: 物理概念<br />
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[[Category:Laws of thermodynamics|2]]<br />
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[[Category:Non-equilibrium thermodynamics]]<br />
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Category:Non-equilibrium thermodynamics<br />
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类别: 非平衡态热力学<br />
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[[Category:Philosophy of thermal and statistical physics]]<br />
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Category:Philosophy of thermal and statistical physics<br />
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类别: 热力学和统计物理学哲学<br />
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<small>This page was moved from [[wikipedia:en:Second law of thermodynamics]]. Its edit history can be viewed at [[热力学第二定律/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E7%83%AD%E5%8A%9B%E5%AD%A6%E7%AC%AC%E4%BA%8C%E5%AE%9A%E5%BE%8B&diff=10354热力学第二定律2020-07-16T09:28:54Z<p>Dorr：/* Planck's principle 普朗克原理 */</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Law of physics}}<br />
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{{Thermodynamics|laws}}<br />
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The '''second law of thermodynamics''' states that the total [[entropy]] of an [[isolated system]] can never decrease over time, and is constant if and only if all processes are reversible.<ref>http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node38.html#SECTION05224000000000000000</ref> Isolated systems spontaneously evolve towards [[thermodynamic equilibrium]], the state with maximum entropy. <br />
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The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, and is constant if and only if all processes are reversible. Isolated systems spontaneously evolve towards thermodynamic equilibrium, the state with maximum entropy. <br />
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热力学第二定律指出，孤立系统的总熵永远不会随着时间而减少，而且只有当且仅当所有过程都是可逆的时，总熵才是恒定的。孤立的系统自发地进化到热力学平衡状态，即熵最大的状态。<br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in [[thermodynamic equilibrium]], or is undergoing a (fictive) [[Reversible process (thermodynamics)|reversible process]]. In all processes that occur, including [[spontaneous process]]es,<ref>Atkins and de Paula, p.78</ref> the total entropy of the system and its surroundings increases and the process is [[Irreversible process|irreversible in the thermodynamic sense]]. The increase in entropy accounts for the irreversibility of natural processes, and the [[Arrow of time|asymmetry between future and past]].<ref>{{cite book|last=Zohuri|first=Bahman|title=Dimensional Analysis Beyond the Pi Theorem|url=https://books.google.com/books?id=pRVuDQAAQBAJ|year=2016|publisher=Springer|isbn=978-3-319-45726-0|page=[https://books.google.com.ph/books?id=pRVuDQAAQBAJ&pg=PA111&dq=%22increase+in+entropy+accounts+for+the+irreversibility+of+natural+processes+and+the+asymmetry+between+future+and+past.%22&hl=en&sa=X&ved=0ahUKEwj6spb61tbaAhUFS7wKHfftDtIQ6AEIKDAA#v=onepage&q=%22increase%20in%20entropy%20accounts%20for%20the%20irreversibility%20of%20natural%20processes%20and%20the%20asymmetry%20between%20future%20and%20past.%22 111]}}</ref><br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in thermodynamic equilibrium, or is undergoing a (fictive) reversible process. In all processes that occur, including spontaneous processes, the total entropy of the system and its surroundings increases and the process is irreversible in the thermodynamic sense. The increase in entropy accounts for the irreversibility of natural processes, and the asymmetry between future and past.<br />
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系统及其周围环境的总熵在理想情况下可以保持不变，在这种情况下，系统处于热力学平衡期，或者正在经历一个(虚拟的)可逆过程期。在所有发生的过程中，包括自发过程，系统及其周围环境的总熵增加，这一过程在热力学意义上是不可逆的。熵的增加解释了自然过程的不可逆性，以及未来和过去之间的不对称性。<br />
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Historically, the second law was an [[Empirical evidence|empirical finding]] that was accepted as an axiom of [[thermodynamics|thermodynamic theory]]. [[Statistical mechanics]], classical or [[quantum thermodynamics|quantum]], explains the microscopic origin of the law.<br />
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Historically, the second law was an empirical finding that was accepted as an axiom of thermodynamic theory. Statistical mechanics, classical or quantum, explains the microscopic origin of the law.<br />
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从历史上看，第二定律是一个经验性的发现，被公认为热力学理论的公理。统计力学，无论是经典的还是量子的，都可以解释这个定律的微观起源。<br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist [[Nicolas Léonard Sadi Carnot|Sadi Carnot]], who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as [[Carnot's theorem (thermodynamics)|Carnot's rule]] or limit.<ref>Jaffe, R.L., Taylor, W. (2018). ''The Physics of Energy'', Cambridge University Press, Cambridge UK, pages 150, 151, 259, 772, 743.</ref><br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist Sadi Carnot, who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as Carnot's rule or limit.<br />
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第二定律已经用许多方式表达过了。法国科学家萨迪 · 卡诺在1824年证明了热能在热机中转化为功的效率是有上限的。这方面的第二定律也被称为卡诺的规则或限制。<br />
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==Introduction==<br />
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==Introduction==<br />
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引言<br />
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[[File:Heat flow hot to cold.png|thumb|upright|Heat flow from hot water to cold water.]]<br />
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Heat flow from hot water to cold water.<br />
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热量从热水流向冷水。<br />
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The [[first law of thermodynamics]] provides the definition of the [[internal energy]] of a [[thermodynamic system]], and expresses the law of [[conservation of energy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 40–41.</ref><ref>Munster A. (1970), pp. 8–9, 50–51.</ref> The second law is concerned with the direction of natural processes.<ref>{{harvnb|Mandl|1988}}</ref> It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such [[Phenomenon|phenomena]] are accounted for in terms of [[entropy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 79–107.</ref><ref>Bailyn, M. (1994), Section 71, pp. 113–154.</ref> If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, {{math|''S''}}, increases.<br />
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The first law of thermodynamics provides the definition of the internal energy of a thermodynamic system, and expresses the law of conservation of energy. The second law is concerned with the direction of natural processes. It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such phenomena are accounted for in terms of entropy. If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, , increases.<br />
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能量守恒定律提供了热力学系统内能的定义，并表达了能量守恒定律。第二定律与自然过程的方向有关。它断言自然过程只在一种意义上运行，而且是不可逆的。例如，当有了传导和辐射的路径时，热总是自发地从一个较热的物体流向一个较冷的物体。这种现象可以用熵来解释。如果一个孤立的系统最初是通过内部分隔不可渗透的墙来保持在内部热力学平衡，然后一些操作使墙更具渗透性，那么系统自发地进化到最终达到一个新的内部热力学平衡，其总熵增加。<br />
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In a fictive reversible process, an infinitesimal increment in the entropy ({{math|d''S''}}) of a system is defined to result from an infinitesimal transfer of heat ({{math|δ''Q''}}) to a [[closed system]] (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature ({{math|''T''}}) of the system in equilibrium and the surroundings which supply the heat:<ref>Bailyn, M. (1994), p. 120.</ref><br />
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In a fictive reversible process, an infinitesimal increment in the entropy () of a system is defined to result from an infinitesimal transfer of heat () to a closed system (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature () of the system in equilibrium and the surroundings which supply the heat:<br />
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在虚构的可逆过程中，系统熵的无穷小增量被定义为由无穷小的热量传递()到一个封闭系统(允许能量进入或出去，但不允许物质传递)除以平衡系统和提供热量的环境的共同温度() :<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, idealized fictive reversible process)}.</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, idealized fictive reversible process)}.</math><br />
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Different notations are used for infinitesimal amounts of heat ({{mvar|δ}}) and infinitesimal amounts of entropy ({{mvar|d}}) because entropy is a [[function of state]], while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the [[Clausius Theorem|inequality]] <ref name="MortimerBook"> Mortimer, R. G. ''Physical Chemistry''. 3rd ed., p. 120, Academic Press, 2008. </ref><ref name="FermiBook"> Fermi, E. ''Thermodynamics'', footnote on p. 48, Dover Publications,1956 (still in print). </ref> <br />
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Different notations are used for infinitesimal amounts of heat () and infinitesimal amounts of entropy () because entropy is a function of state, while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the inequality <br />
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用不同的符号表示无穷小量的热量()和无穷小量的熵() ，因为熵是状态的函数，而热量和功一样不是。第二定律要求系统熵的增量满足不等式，对于实际上可能存在的不与环境发生质量交换的无穷小过程，系统熵增量满足不等式<br />
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: <math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process).}</math><br />
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<math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process).}</math><br />
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(实际上可能是封闭系统，不可逆性)。数学<br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature ({{math|''T''}}) and the temperature of the surroundings ({{math|''T''}}<sub>''surr''</sub>).<ref name=":0">Adkins, C.J. (1968/1983), p. 75.</ref><ref name="Munster 45"/> Note that the equality still applies for pure heat flow,<ref name="Schmidt-Rohr 14"> Schmidt-Rohr, K. (2014). "Expansion Work without the External Pressure, and Thermodynamics in Terms of Quasistatic Irreversible Processes" ''J. Chem. Educ.'' '''91''': 402-409. https://dx.doi.org/10.1021/ed3008704 </ref><br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature () and the temperature of the surroundings (<sub>surr</sub>).<br />
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这是因为这种情况下的一般过程可能包括周围环境对系统所做的功，这可能在系统内部产生摩擦或粘滞效应，因为一个化学反应可能正在进行，或者因为热传递实际上只是不可逆地发生，由系统温度()和周围环境温度(sub surr / sub)之间的差分驱动。<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change).}</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change).}</math><br />
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(实际上可能是不改变成分的准静态不可逆性)。数学<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry.<ref name="Oxtoby8th">Oxtoby, D. W; Gillis, H.P., [[Laurie Butler|Butler, L. J.]] (2015).''Principles of Modern Chemistry'', Brooks Cole. p. 617. {{ISBN|978-1305079113}}</ref> <ref name="MortimerBook"></ref> Introducing a set of internal variables <math>\xi</math> to describe the deviation of a thermodynamic system in physical equilibrium (with the required well-defined uniform pressure ''P'' and temperature ''T'')<ref name="Schmidt-Rohr 14"></ref> from the chemical equilibrium state, one can record the equality<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry. from the chemical equilibrium state, one can record the equality<br />
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这是根据测得的热容曲线和相变时的熵变精确确定纯物质的绝对熵的基础。用量热法。从化学平衡来看，人们可以记录平等<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process).}</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process).}</math><br />
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- frac { t }-frac {1}{ j } ， Xi { j } ， delta Xi j， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ,.数学<br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of ''the second principle of thermodynamics'' – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<ref> Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769–781.</ref><ref>{{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}</ref><br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of the second principle of thermodynamics – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<br />
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第二项代表内部变量的工作，这些内部变量可以受到外部影响的干扰，但是系统不能通过内部变量执行任何正功。这种说法介绍了热力学系统在时间上不可能逆转演化的事实，并且可以被认为是热力学第二原理的表述---- 这个表述，当然，相当于熵原理的表述。<br />
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The [[zeroth law of thermodynamics]] in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body.<ref name=dugdale>{{cite book|author=J. S. Dugdale|title=Entropy and its Physical Meaning|publisher=Taylor & Francis|year=1996|isbn=978-0-7484-0569-5|page=13|quote=This law is the basis of temperature.}}</ref> For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows{{how?|date=August 2018}} a distinguished temperature scale, which defines an absolute, [[thermodynamic temperature]], independent of the properties of any particular reference thermometric body.<ref>[[Mark Zemansky|Zemansky, M.W.]] (1968), pp. 207–209.</ref><ref>Quinn, T.J. (1983), p. 8.</ref><br />
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The zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body. For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows a distinguished temperature scale, which defines an absolute, thermodynamic temperature, independent of the properties of any particular reference thermometric body.<br />
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美国热力学第零定律协会在其通常的简短声明中允许承认，相对于热平衡的两个物体具有相同的温度，特别是测试物体具有与参考温度测量物体相同的温度。对于一个物体和另一个物体之间的热平衡，存在着无限多的经验温度标度，通常分别取决于特定参考温度标度的性质。第二定律允许区分温度标度，它定义了一个绝对的热力学温度，与任何特定的参考温度体的性质无关。<br />
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==Various statements of the law==<br />
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==Various statements of the law==<br />
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各种各样的法律陈述<br />
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The second law of thermodynamics may be expressed in many specific ways,<ref name=MIT>{{cite web|title=Concept and Statements of the Second Law|url=http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node37.html|accessdate=2010-10-07 |publisher=web.mit.edu}}</ref> the most prominent classical statements{{sfnp|Lieb|Yngvason|1999}} being the statement by [[Rudolf Clausius]] (1854), the statement by [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] (1851), and the statement in axiomatic thermodynamics by [[Constantin Carathéodory]] (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.{{sfnp|Rao|2004|p=213}}<br />
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The second law of thermodynamics may be expressed in many specific ways, the most prominent classical statements being the statement by Rudolf Clausius (1854), the statement by Lord Kelvin (1851), and the statement in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.<br />
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热力学第二定律可以用许多特定的方式来表达，最突出的经典陈述是 Rudolf Clausius (1854)的陈述，Kelvin 勋爵(1851)的陈述，以及康斯坦丁·卡拉西奥多里(1909)在公理化热力学中的陈述。这些陈述用一般的物理术语描述法律，引用某些过程的不可能性。克劳修斯和开尔文陈述被证明是等价的。<br />
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===Carnot's principle===<br />
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===Carnot's principle===<br />
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卡诺原理<br />
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The historical origin<ref>[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986).</ref> of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a [[Carnot heat engine]], fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the [[caloric theory]] of heat was seriously considered, before the recognition of the [[first law of thermodynamics]], and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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The historical origin of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a Carnot heat engine, fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the caloric theory of heat was seriously considered, before the recognition of the first law of thermodynamics, and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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热力学第二定律的历史起源源于卡诺的原则。它指的是一个卡诺热机周期，虚拟地以被称为准静态的极慢极限模式运行，因此热量和功量在子系统之间传递，这些子系统始终处于它们自己的内部热力学平衡状态。卡诺发动机是热机效率工程师特别感兴趣的理想设备。卡诺认识到卡诺原理的时候，正是热量热理论被认真考虑的时候，正是能量守恒定律理论被认识之前，正是熵概念的数学表达之前。根据第一定律的解释，它在物理上等同于热力学第二定律，并且在今天仍然有效。卡诺最初的论点是从卡路里理论的观点提出的，直到能量守恒定律的发现。以下是他书中的一些例子:<br />
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::...''wherever there exists a difference of temperature, motive power can be produced.''<ref>Carnot, S. (1824/1986), p. 51.</ref><br />
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...wherever there exists a difference of temperature, motive power can be produced.<br />
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只要有温差，就能产生动力。<br />
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::The production of motive power is then due in steam engines not to an actual consumption of caloric, but ''to its transportation from a warm body to a cold body ...''<ref>Carnot, S. (1824/1986), p. 46.</ref><br />
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The production of motive power is then due in steam engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body ...<br />
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动力的产生不是由于蒸汽机实际消耗的热量，而是由于它从一个温暖的物体运输到一个寒冷的物体..。<br />
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::''The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.''<ref>Carnot, S. (1824/1986), p. 68.</ref><br />
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The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.<br />
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热的动力与实现热的媒介无关，热的数量完全取决于两个物体之间的温度，最后是热量的传递。<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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用现代术语来说，卡诺的原则可能更为准确:<br />
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::The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<ref>[[Clifford Truesdell|Truesdell, C.]] (1980), Chapter 5.</ref><ref>Adkins, C.J. (1968/1983), pp. 56–58.</ref><ref>Münster, A. (1970), p. 11.</ref><ref>Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998), pp.67–75.</ref><ref>Lebon, G., Jou, D., Casas-Vázquez, J. (2008), p. 10.</ref><ref>Eu, B.C. (2002), pp. 32–35.</ref><br />
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The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<br />
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准静态或可逆卡诺循环的效率只取决于两个热源的温度，而且无论工作物质是什么，效率是相同的。以这种方式运行的卡诺发动机是使用这两种温度的最有效的热机。<br />
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===Clausius statement===<br />
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===Clausius statement===<br />
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克劳修斯声明<br />
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The German scientist [[Rudolf Clausius]] laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work.{{sfnp|Clausius|1850}} His formulation of the second law, which was published in German in 1854, is known as the ''Clausius statement'':<br />
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The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work. His formulation of the second law, which was published in German in 1854, is known as the Clausius statement:<br />
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1850年，德国科学家 Rudolf Clausius 通过研究热传递和功之间的关系，为热力学第二定律实验室奠定了基础。他在1854年用德语发表的第二定律的提法被称为克劳修斯声明:<br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.{{sfnp|Clausius|1854|p=86}}</blockquote><br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.</blockquote><br />
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热量不可能从一个较冷的物体传递到一个较热的物体，除非同时发生一些其他的变化。 / blockquote<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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克劳修斯的陈述使用了“热量通道”的概念。正如通常在热力学的讨论，这意味着“能量作为热量的净转移” ，并没有提到促成转移的一种方式和其他。<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of [[refrigeration]], for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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如果不对系统进行外部功，热就不能自发地从冷区流向热区，这一点从制冷的普通经验中可以看出。在冰箱中，热量从冷到热，但只有在外部介质——制冷系统的强制作用下才会发生变化。<br />
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===Kelvin statements===<br />
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===Kelvin statements===<br />
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开尔文陈述<br />
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[[William Thomson, 1st Baron Kelvin|Lord Kelvin]] expressed the second law in several wordings.<br />
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Lord Kelvin expressed the second law in several wordings.<br />
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开尔文勋爵用几个字表达了第二定律。<br />
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::It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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没有任何外部机构的帮助，自动机器是不可能把热量从一个物体传递到另一个物体的较高温度。<br />
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::It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.{{sfnp|Thomson|1851}}<br />
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It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.<br />
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通过无生命物质的作用，把物质冷却到周围最冷的物体的温度以下，从物质的任何部分获得机械效应是不可能的。<br />
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===Equivalence of the Clausius and the Kelvin statements===<br />
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===Equivalence of the Clausius and the Kelvin statements===<br />
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克劳修斯和开尔文陈述的等价性<br />
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[[Image:Deriving Kelvin Statement from Clausius Statement.svg|thumb|Derive Kelvin Statement from Clausius Statement]]<br />
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Derive Kelvin Statement from Clausius Statement<br />
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从克劳修斯陈述推导出开尔文陈述<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed [[Carnot engine]] as shown by the figure. The [[Heat engine#Efficiency|efficiency]] of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the [[first law of thermodynamics]], as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the figure. The efficiency of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the first law of thermodynamics, as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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假设有一个引擎违反了开尔文定理: 也就是说，这个引擎以循环的方式吸收热量并将其转化为功，没有任何其他结果。现在配对与反向卡诺发动机如图所示。普通热机的效率为，反向热机的效率为1 / 。这对组合发动机的净效应和唯一效应是将热量从较冷的贮存器转移到较热的贮存器，这违反了克劳修斯的说法。(这是能量守恒定律的结果，因为系统的总能量保持不变，数学文本{输入} + 文本{输出}0暗示 q-frac { eta }-qc / math，所以数学 qc 向左(frac { eta-1 right) / math)。因此，违反开尔文陈述意味着违反克劳修斯陈述，即。克劳修斯的说法暗示了凯尔文的说法。我们可以用类似的方式证明开尔文陈述暗示了克劳修斯陈述，因此两者是等价的。<br />
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===Planck's proposition===<br />
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===Planck's proposition===<br />
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普朗克的命题<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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普朗克提出了直接来自经验的如下命题。这有时被认为是他对第二定律的陈述，但他认为这是第二定律推导的起点。<br />
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::It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 86.</ref><ref>Roberts, J.K., Miller, A.R. (1928/1960), p. 319.</ref><br />
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It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<br />
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建造一台发动机是不可能的，这台发动机将在一个完整的循环中工作，并且除了提高重量和冷却热储以外不会产生任何效果。<br />
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===Relation between Kelvin's statement and Planck's proposition 开尔文表述与普朗克命题的关系===<br />
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It is almost customary in textbooks to speak of the "[[Kelvin-Planck statement]]" of the law, as for example in the text by [[Dirk ter Haar|ter Haar]] and [[Harald Wergeland|Wergeland]].<ref>[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966), p. 17.</ref><br />
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It is almost customary in textbooks to speak of the "Kelvin-Planck statement" of the law, as for example in the text by ter Haar and Wergeland.<br />
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教科书中几乎惯常提及该定律的“'''开尔文-普朗克表述 Kelvin-Planck Statement'''” ，例如'''德克·特哈尔 Diek ter Haar''' 和'''哈拉尔德·沃格兰 Harald Wergeland''' 在文中就是这样表述的。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）Diek ter Haar 中间是小写的<br />
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The '''Kelvin–Planck statement''' (or the ''heat engine statement'') of the second law of thermodynamics states that <br />
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The Kelvin–Planck statement (or the heat engine statement) of the second law of thermodynamics states that <br />
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热力学第二定律的开尔文-普朗克表述（或称“'''热机表述 Heat Engine Statement'''”）指出<br />
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::It is impossible to devise a [[thermodynamic cycle|cyclically]] operating device, the sole effect of which is to absorb energy in the form of heat from a single [[heat reservoir|thermal reservoir]] and to deliver an equivalent amount of [[Work (physics)|work]].<ref name="Rao">{{cite book|last=Rao|first=Y. V. C.|title=Chemical Engineering Thermodynamics|publisher=Universities Press|isbn=978-81-7371-048-3|page=158|year=1997}}</ref><br />
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It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.<br />
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设计一种唯一效果是从单一热源吸收热量并提供等量的功的循环运行装置是不可能的。<br />
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===Planck's statement 普朗克表述===<br />
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Planck stated the second law as follows.<br />
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Planck stated the second law as follows.<br />
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普朗克表述第二定律如下。<br />
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::Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<ref name="Planck 100">[[Max Planck|Planck, M.]] (1897/1903), p. 100.</ref><ref name="Planck 463">[[Max Planck|Planck, M.]] (1926), p. 463, translation by Uffink, J. (2003), p. 131.</ref><ref name="Roberts & Miller 382">Roberts, J.K., Miller, A.R. (1928/1960), p. 382. This source is partly verbatim from Planck's statement, but does not cite Planck. This source calls the statement the principle of the increase of entropy.</ref><br />
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Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<br />
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自然界中发生的任一过程都是沿参与其中的所有物体的熵之和增加的方向进行的。在某些限制下——即对于可逆过程，熵的总和保持不变。<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for ''irreversible phenomena''.<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for irreversible phenomena.<br />
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与普朗克表述非常相似的是'''乌伦贝克 Uhlenbeck'''和'''福特 Ford'''关于不可逆现象的表述。<br />
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::... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<ref>[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963), p. 16.</ref><br />
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... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<br />
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在从一个平衡态到另一个平衡态的不可逆或自发的变化中（例如，当两个物体 A 和 B 接触时的温度平衡过程），熵总是增加。<br />
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===Principle of Carathéodory 卡拉西奥多里原理===<br />
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<!-- [[Caratheodory's principle]] redirects here --><br />
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—— caratheodory 的原理在这里重新定向——<br />
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[[Constantin Carathéodory]] formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<ref>[[Constantin Carathéodory|Carathéodory, C.]] (1909).</ref><br />
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Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<br />
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'''康斯坦丁·卡拉西奥多里 Constantin Carathéodory'''在纯数学公理的基础上进行了热力学公理化。他对第二定律的陈述被称为'''卡拉西奥多里原理 Principle of Carathéodory'''，可以这样表述:<br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.<ref>Buchdahl, H.A. (1966), p. 68.</ref></blockquote><br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.</blockquote><br />
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在绝热封闭系统的任意状态 S 附近，总有从 S 出发不可达的状态。<br />
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With this formulation, he described the concept of [[adiabatic accessibility]] for the first time and provided the foundation for a new subfield of classical thermodynamics, often called [[Ruppeiner geometry|geometrical thermodynamics]]. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic [[process function]], in other words, <math>\delta Q=TdS</math>.<ref name="Sychev1991">{{cite book |last=Sychev |first=V. V. |title=The Differential Equations of Thermodynamics |year=1991 |publisher=Taylor & Francis |isbn=978-1-56032-121-7}}</ref> {{clarify|date=February 2014}}<br />
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With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, <math>\delta Q=TdS</math>. <br />
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公理化处理后，他第一次描述了'''绝热可达性 Adiabatic Accessibility'''的概念，并为经典热力学的一个新的子领域，即通常所说的'''几何热力学 Geometrical Thermodynamics'''奠定了基础。由卡拉西奥多里原理可以推出，准静态转移的热量值是一个可积过程函数，即<math>\delta Q=TdS</math>。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）准静态转移的热量值是一个可积过程函数 存疑<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.<ref name="Munster 45">Münster, A. (1970), p. 45.</ref>{{sfnp|Lieb|Yngvason|1999|p=49}}<ref name="Planck 1926">[[Max Planck|Planck, M.]] (1926).</ref><ref>Buchdahl, H.A. (1966), p. 69.</ref> {{clarify|date=February 2014}}<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium. <br />
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尽管在教科书几乎惯称卡拉西奥多里原理表述了第二定律，并认为其与克劳修斯表述或开尔文-普朗克表述等价，但事实并非如此。为了得到第二定律的所有内容，需要对卡拉西奥多里原理补充普朗克表述，即等量功总是增加一个最初处于自身内部热力学平衡的封闭系统的内部能量。<br />
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===Planck's principle 普朗克原理===<br />
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In 1926, [[Max Planck]] wrote an important paper on the basics of thermodynamics.<ref name="Planck 1926"/><ref>Uffink, J. (2003), pp. 129–132.</ref> He indicated the principle<br />
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In 1926, Max Planck wrote an important paper on the basics of thermodynamics. He indicated the principle<br />
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1926年，'''马克斯·普朗克 Max Planck'''写了一篇关于热力学基础的重要论文。他指出了以下原理<br />
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::The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<ref name="Munster 45"/>{{sfnp |Lieb|Yngvason|1999|p=49}}<br />
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The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<br />
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一个封闭系统的内部能量因绝热过程增加，在整个过程中，系统的体积保持不变。<br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work."<ref>[[Clifford Truesdell|Truesdell, C.]], Muncaster, R.G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 15.</ref> Planck wrote: "The production of heat by friction is irreversible."<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 81.</ref><ref>[[Max Planck|Planck, M.]] (1926), p. 457, Wikipedia editor's translation.</ref><br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work." Planck wrote: "The production of heat by friction is irreversible."<br />
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这个公式没有提到热，没有提到温度，甚至没有提到熵，也不一定隐含地依赖于这些概念，但它暗示了第二定律的内容。一个密切相关的表述为，“摩擦力从来不做正功。”普朗克写道: “摩擦生热是不可逆的。”<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above.<ref>Lieb, E.H., Yngvason, J. (2003), p. 149.</ref> It is relevant that for a system at constant volume and [[Mole (unit)|mole numbers]], the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above. It is relevant that for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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这个普朗克定理没有提到熵，是用物理术语来表述的。它与上面给出的开尔文表述密切相关。相关的是，对于恒定体积和摩尔数的系统，熵是内能的单调函数。然而，普朗克的这个原理实际上并不是普朗克对第二定律的首选表述（见前面小节），而是依赖于熵的概念。<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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Borgnakke 和 Sonntag 提出的以下表述在某种意义上是对普朗克原理的补充。他们没有将其作为第二定律的完整表述:<br />
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::... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<ref>Borgnakke, C., Sonntag., R.E. (2009), p. 304.</ref><br />
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... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<br />
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只有一种方法可以减少[闭合]系统的熵——从系统中转移热量。<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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与普朗克之前提出的原理不同，这一原理明确地用熵的变化来表示。从系统中去除物质也可以减少系统的熵。<br />
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===Statement for a system that has a known expression of its internal energy as a function of its extensive state variables 一个其内能有已知表达式（其扩展状态变量的函数）的系统的表述===<br />
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The second law has been shown to be equivalent to the [[internal energy]] ''U'' being a weakly [[convex function]], when written as a function of extensive properties (mass, volume, entropy, ...).<ref>{{cite book |last1=van Gool |first1=W. |last2=Bruggink |first2=J.J.C. (Eds) |url= |title=Energy and time in the economic and physical sciences |publisher=North-Holland |year=1985 |pages=41–56 |quote= |isbn=978-0-444-87748-2}}</ref><ref>{{Cite journal | last1 = Grubbström | first1 = Robert W. | doi = 10.1016/j.apenergy.2007.01.003 | title = An Attempt to Introduce Dynamics Into Generalised Exergy Considerations| journal = Applied Energy| volume = 84| issue = 7–8 | pages = 701–718 | year = 2007}}</ref> {{clarify|date=February 2014}}<br />
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The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...). <br />
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第二定律被证明等价于弱凸函数内能 U （写成广泛性质（质量，体积，熵，...）的函数时）。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）存疑<br />
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==Corollaries 推论==<br />
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===Perpetual motion of the second kind 第二类永动机===<br />
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{{main article|Perpetual motion}}<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of [[first law of thermodynamics]] by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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在第二定律建立之前，许多志在发明永动机的人试图通过提取环境的巨大内能作为机器动力来突破'''热力学第一定律 First Law of Thermodynamics''' 的限制。这种机器被称为“'''第二类永动机 Perpetual Motion Machine of the Second Kind'''”。第二定律表明这种机器是不可能的。<br />
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===Carnot theorem 卡诺定理===<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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卡诺定理（1824）是一条限制任何可能的发动机的最大效率的原理。效率完全取决于热库和冷库之间的温差。卡诺定理指出:<br />
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*All irreversible heat engines between two heat reservoirs are less efficient than a [[Carnot engine]] operating between the same reservoirs.<br />
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*All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs.<br />
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* 所有不可逆热机的效率低于在两相同热源之间工作的'''卡诺机 Carnot engine'''。<br />
* 所有可逆热机的效率与在两相同热源之间工作的卡诺机相等。<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as [[thermodynamic reversibility]]. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the [[Carnot cycle]]'s reversibility and was condemned to be less efficient.<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the Carnot cycle's reversibility and was condemned to be less efficient.<br />
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在他的理想模型中，热转换成功可以通过逆转循环的运动而恢复，这个概念后来被称为'''热力学可逆性 Thermodynamic Reversibility'''。然而，卡诺进一步假定，一些热量损失了，并没有转化为机械功。因此，没有一个真实的热机能够实现'''卡诺循环 Carnot cycle'''的可逆性，并且被认为效率较低。<br />
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Though formulated in terms of caloric (see the obsolete [[caloric theory]]), rather than [[entropy]], this was an early insight into the second law.<br />
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Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law.<br />
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该理论尽管是用热量表述的（见被取代的'''热质说 Caloric Theory'''），而不是熵，但是它是对第二定律的早期认识。<br />
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===Clausius inequality 克劳修斯不等式===<br />
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The [[Clausius theorem]] (1854) states that in a cyclic process<br />
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The Clausius theorem (1854) states that in a cyclic process<br />
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克劳修斯定理（1854）指出，在一个循环的过程中<br />
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: <math>\oint \frac{\delta Q}{T} \leq 0.</math><br />
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The equality holds in the reversible case<ref>[http://scienceworld.wolfram.com/physics/ClausiusTheorem.html ''Clausius theorem''] at [[Wolfram Research]]</ref> and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function [[entropy]]. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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The equality holds in the reversible case and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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等号在可逆情况下成立，严格不等号在不可逆情况下成立。可逆情况下引入状态函数熵。这是因为在循环过程中，状态函数的变化为零。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）末句存疑<br />
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===Thermodynamic temperature 热力学温度===<br />
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{{main article|Thermodynamic temperature}}<br />
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For an arbitrary heat engine, the efficiency is:<br />
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For an arbitrary heat engine, the efficiency is:<br />
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对于任意热机，效率为:<br />
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: <math>\eta = \frac {W_n}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H} \qquad (1)</math><br />
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where ''W''<sub>n</sub> is for the net work done per cycle. Thus the efficiency depends only on q<sub>C</sub>/q<sub>H</sub>.<br />
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其中 ''W''<sub>n</sub> 表示每个循环所做的净功。因此效率只取决于 q<sub>C</sub>/q<sub>H</sub>。<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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卡诺定理指出，所有在相同热源间运行的可逆发动机的效率是相同的。<br />
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Thus, any reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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Thus, any reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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因此，任何在温度''T''<sub>1</sub>和''T''<sub>2</sub>之间运行的可逆热机必须具有相同的效率，也就是说，效率只是温度的函数:<br />
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<math>\frac{q_C}{q_H} = f(T_H,T_C)\qquad (2).</math><br />
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In addition, a reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between ''T''<sub>1</sub> and another (intermediate) temperature ''T''<sub>2</sub>, and the second between ''T''<sub>2</sub> and''T''<sub>3</sub>. This can only be the case if<br />
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In addition, a reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between T<sub>1</sub> and another (intermediate) temperature T<sub>2</sub>, and the second between T<sub>2</sub> andT<sub>3</sub>. This can only be the case if<br />
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另外，在温度 ''T''<sub>1</sub> 和 ''T''<sub>3</sub> 之间工作的可逆热机必须具有与由分别在温度''T''<sub>1</sub> 和（中间）温度 ''T''<sub>2</sub>之间、在 ''T''<sub>2</sub> 和 ''T''<sub>3</sub> 之间的两个循环组成的系统效率相同。只有下式成立才会出现这种情况:<br />
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: <math>f(T_1,T_3) = \frac{q_3}{q_1} = \frac{q_2 q_3} {q_1 q_2} = f(T_1,T_2)f(T_2,T_3).</math><br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the [[triple point]] of water. Then for any ''T''<sub>2</sub> and ''T''<sub>3</sub>,<br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the triple point of water. Then for any T<sub>2</sub> and T<sub>3</sub>,<br />
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现在考虑如下情形，<math>T_1</math> 是一个固定的参考温度: 水的三相点的温度。则对于任意 T<sub>2</sub> 和 T<sub>3</sub>,<br />
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: <math>f(T_2,T_3) = \frac{f(T_1,T_3)}{f(T_1,T_2)} = \frac{273.16 \cdot f(T_1,T_3)}{273.16 \cdot f(T_1,T_2)}.</math><br />
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Therefore, if thermodynamic temperature is defined by<br />
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Therefore, if thermodynamic temperature is defined by<br />
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因此，如果热力学温度被定义为<br />
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: <math>T = 273.16 \cdot f(T_1,T) \,</math><br />
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then the function ''f'', viewed as a function of thermodynamic temperature, is simply<br />
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then the function f, viewed as a function of thermodynamic temperature, is simply<br />
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那么函数 ''f'' 作为热力学温度的函数，为<br />
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: <math>f(T_2,T_3) = \frac{T_3}{T_2},</math><br />
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and the reference temperature ''T''<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be used{{snd}}the choice here corresponds to the [[Kelvin]] scale.)<br />
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and the reference temperature T<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be usedthe choice here corresponds to the Kelvin scale.)<br />
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参考温度 ''T''<sub>1</sub> 的值为273.16。（任何参考温度和任何正值均可用——此处的选择对应开尔文标度。）<br />
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===Entropy 熵===<br />
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{{main article|Entropy (classical thermodynamics)}}<br />
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According to the [[Clausius theorem|Clausius equality]], for a reversible process<br />
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According to the Clausius equality, for a reversible process<br />
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根据克劳修斯定理，对可逆过程有<br />
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: <math>\oint \frac{\delta Q}{T}=0</math><br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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这意味着线积分 <math>\int_L \frac{\delta Q}{T}</math> 对于可逆过程是路径无关的。<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer<ref name="Schmidt-Rohr 14" /> satisfies<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer satisfies<br />
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所以我们可以定义一个叫做熵的状态函数 S，对于可逆过程或者纯热传递满足<br />
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: <math>dS = \frac{\delta Q}{T} \!</math><br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the [[third law of thermodynamics]], which states that ''S'' = 0 at [[absolute zero]] for perfect crystals.<br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the third law of thermodynamics, which states that S = 0 at absolute zero for perfect crystals.<br />
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据此，只有对上述公式进行积分，才能得到熵的差值。为了获得绝对值，我们需要'''热力学第三定律 Third Law of Thermodynamics'''，它指出绝对零度下完美晶体的 ''S'' = 0。<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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对于任意不可逆过程，由于熵是一个状态函数，我们总是可以将初始状态和最终状态与一个虚拟的可逆过程联系起来，并在这条路径上积分以计算熵的差值。<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the [[Clausius inequality]] on this loop,<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the Clausius inequality on this loop,<br />
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现在把可逆过程逆过来，将其与上述不可逆过程结合。把克劳修斯不等式应用到这个循环,<br />
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: <math>-\Delta S+\int\frac{\delta Q}{T}=\oint\frac{\delta Q}{T}< 0</math><br />
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Thus,<br />
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Thus,<br />
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故,<br />
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: <math>\Delta S \ge \int \frac{\delta Q}{T} \,\!</math><br />
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where the equality holds if the transformation is reversible.<br />
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where the equality holds if the transformation is reversible.<br />
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如果变换可逆，等号成立。<br />
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Notice that if the process is an [[adiabatic process]], then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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Notice that if the process is an adiabatic process, then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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注意，若该过程是一个'''绝热过程 Adiabatic Process'''，则<math>\delta Q=0</math>，故<math>\Delta S\ge 0</math>。<br />
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===Energy, available useful work===<br />
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===Energy, available useful work===<br />
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能量，有用的工作<br />
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{{See also|Exergy}}<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an ''unlimited'' heat reservoir at temperature ''T<sub>R</sub>'' and pressure ''P<sub>R</sub>'' {{snd}}so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain ''T<sub>R</sub>''; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain ''P<sub>R</sub>''.<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an unlimited heat reservoir at temperature T<sub>R</sub> and pressure P<sub>R</sub> so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain T<sub>R</sub>; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain P<sub>R</sub>.<br />
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考虑将第二定律应用于孤立系统(称为整体系统或宇宙)的情形，是一个重要的、具有启发性的理想情形。孤立系统由两部分组成: 感兴趣的子系统和子系统的环境。这些环境被想象为如此之大，以至于它们可以被认为是一个温度为 t 子 r / sub 和压力为 p 子 r / sub 的无限热源，因此无论有多少热量被转移到(或来自)子系统，周围环境的温度将保持 t 子 r / sub; 无论子系统的体积扩大(或收缩)多少，周围环境的压力将保持 p 子 r / sub。<br />
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Whatever changes to ''dS'' and ''dS<sub>R</sub>'' occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy ''S<sub>tot</sub>'' of the isolated total system must not decrease:<br />
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Whatever changes to dS and dS<sub>R</sub> occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy S<sub>tot</sub> of the isolated total system must not decrease:<br />
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根据第二定律，孤立总体系统的熵 s 子系统 / 子系统的熵不能减小，无论子系统和周围环境的熵对 dS 和 dS 子系统 r / 子系统发生什么变化:<br />
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: <math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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<math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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数学 dS + dS r ge 0 / math<br />
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According to the [[first law of thermodynamics]], the change ''dU'' in the internal energy of the sub-system is the sum of the heat ''δq'' added to the sub-system, ''less'' any work ''δw'' done ''by'' the sub-system, ''plus'' any net chemical energy entering the sub-system ''d ∑μ<sub>iR</sub>N<sub>i</sub>'', so that:<br />
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According to the first law of thermodynamics, the change dU in the internal energy of the sub-system is the sum of the heat δq added to the sub-system, less any work δw done by the sub-system, plus any net chemical energy entering the sub-system d ∑μ<sub>iR</sub>N<sub>i</sub>, so that:<br />
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根据能量守恒定律，子系统内部能量的变化 dU 是子系统内部能量 q 的总和，减去子系统所做的任何功，再加上进入子系统 d ∑ sub iR / sub n sub i / sub 的任何净化学能，因此:<br />
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: <math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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<math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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数学 dU delta q- delta w + d ( sum mu iR } n i) ，/ math<br />
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where μ<sub>iR</sub> are the [[chemical potential]]s of chemical species in the external surroundings.<br />
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where μ<sub>iR</sub> are the chemical potentials of chemical species in the external surroundings.<br />
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其中亚 iR / 亚是外部环境中化学物种的化学势。<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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现在热量离开储存器进入子系统是<br />
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: <math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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<math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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数学△ qtr (- dS r) le tr dS / math<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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其中我们首先使用了经典热力学中熵的定义(在统计热力学中，熵变、温度和吸收热量之间的关系可以推导出来) ，然后从上面推导出第二定律的不等式。<br />
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It therefore follows that any net work ''δw'' done by the sub-system must obey<br />
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It therefore follows that any net work δw done by the sub-system must obey<br />
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因此，子系统所做的任何净功都必须服从<br />
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: <math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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<math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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数学增量 w le-dU + tds + sum mu { iR } dN i，/ math<br />
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It is useful to separate the work ''δw'' done by the subsystem into the ''useful'' work ''δw<sub>u</sub>'' that can be done ''by'' the sub-system, over and beyond the work ''p<sub>R</sub> dV'' done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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It is useful to separate the work δw done by the subsystem into the useful work δw<sub>u</sub> that can be done by the sub-system, over and beyond the work p<sub>R</sub> dV done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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将子系统所做的功与子系统所能做的功分离为子系统所能做的有用功，超过了子系统对外界压力所做的功，给出了子系统所能做的有用功(火用)的下列关系式:<br />
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: <math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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<math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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D (u-t r s + p r v-sum { iR } n i) ，/ math<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the ''availability'' or ''[[exergy]]'' ''E'' of the subsystem,<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the availability or exergy E of the subsystem,<br />
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可以很方便地将子系统的右侧定义为热动力位能的精确衍生物，称为子系统的可用性或火用 e,<br />
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: <math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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<math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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数学 e u-t r s + p r v- sum { iR } n i / math<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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因此，第二定律意味着，对于任何可以简单地被视为分为一个子系统和一个与之接触的无限温度和压力储存器的过程,<br />
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: <math> dE + \delta w_u \le 0 \, </math><br />
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<math> dE + \delta w_u \le 0 \, </math><br />
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数学德 delta u le 0，/ math<br />
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i.e. the change in the subsystem's exergy plus the useful work done ''by'' the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done ''on'' the system) must be less than or equal to zero.<br />
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i.e. the change in the subsystem's exergy plus the useful work done by the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done on the system) must be less than or equal to zero.<br />
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也就是。子系统(火用)的变化加上子系统所做的有用功(或者，子系统(火用)的变化不包括任何功，除了压力贮存器所做的功之外，在系统中所做的功)必须小于或等于零。<br />
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In sum, if a proper ''infinite-reservoir-like'' reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in ''E'' for an irreversible process and no change for a reversible process.<br />
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In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in E for an irreversible process and no change for a reversible process.<br />
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总之，如果选择一个合适的类似于无限库的参考状态作为现实世界中的系统环境，那么第二定律预测一个不可逆性的 e 值会减少，而一个可逆过程的 e 值不会变化。<br />
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: <math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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<math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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数学 d = 0 = 0 = 0 = 0 = 0<br />
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This expression together with the associated reference state permits a [[design engineer]] working at the macroscopic scale (above the [[thermodynamic limit]]) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (''Also, see [[process engineer]]''). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (''See [[Exergy efficiency|second law efficiency]]''.)<br />
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This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (Also, see process engineer). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (See second law efficiency.)<br />
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这个表达式连同相关的参考状态允许在宏观的设计工程师利用第二定律，而不需要直接测量或考虑整个孤立系统的熵变。(另外，请参阅流程工程师)。这些变化已经被考虑，假设被考虑的系统可以达到平衡的参考状态而不改变参考状态。还可以找到一个过程或过程集合的效率，将其与可逆的理想状态相比较<br />
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This approach to the Second Law is widely utilized in [[engineering]] practice, [[environmental accounting]], [[systems ecology]], and other disciplines.<br />
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This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines.<br />
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第二定律的这种方法被广泛应用于工程实践、环境会计、系统生态学和其他学科。<br />
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===The second law in chemical thermodynamics===<br />
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===The second law in chemical thermodynamics===<br />
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化学热力学的第二定律<br />
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For a [[spontaneous process|spontaneous chemical process]] in a closed system at constant temperature and pressure without non-''PV'' work, the Clausius inequality Δ''S > Q/T<sub>surr</sub>'' transforms into a condition for the change in [[Gibbs free energy]] <br />
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For a spontaneous chemical process in a closed system at constant temperature and pressure without non-PV work, the Clausius inequality ΔS > Q/T<sub>surr</sub> transforms into a condition for the change in Gibbs free energy <br />
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对于一个封闭系统中的自发化学过程，在没有非 pv 功的情况下，在恒定温度和压力下，将克劳修斯不等式 s q / t sub surr / sub 转化为吉布斯自由能变化的条件<br />
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: <math>\Delta G < 0 </math><br />
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<math>\Delta G < 0 </math><br />
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数学 Delta g 0 / 数学<br />
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or d''G'' < 0. For a similar process at constant temperature and volume, the change in [[Helmholtz free energy]] must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products.<ref name="Oxtoby8th"/> <ref name="MortimerBook"></ref> The chemical equilibrium condition at constant ''T'' and ''p'' without electrical work is d''G'' = 0.<br />
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or dG < 0. For a similar process at constant temperature and volume, the change in Helmholtz free energy must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products. The chemical equilibrium condition at constant T and p without electrical work is dG = 0.<br />
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或者 dG 0。对于一个相似的过程，在恒定的温度和体积下，亥姆霍兹自由能的变化一定是负的，这是 Delta a 0 / math。因此，自由能(g 或 a)变化的负值是过程自发的必要条件。这是热力学第二定律在化学中最有用的形式，其中自由能的变化可以通过列表的生成焓和反应物和产物的标准摩尔熵来计算。在 t 和 p 不变的情况下，化学平衡条件是 dG 0。<br />
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==History==<br />
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==History==<br />
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历史<br />
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{{See also|History of entropy}}<br />
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[[File:Sadi Carnot.jpeg|thumb|upright|Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique]].]]<br />
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Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique.]]<br />
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尼古拉·莱昂纳尔·萨迪·卡诺穿着传统的巴黎综合理工学院制服<br />
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The first theory of the conversion of heat into mechanical work is due to [[Nicolas Léonard Sadi Carnot]] in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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The first theory of the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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第一个把热转化为机械功的理论是在1824年由尼古拉·莱昂纳尔·萨迪·卡诺提出的。他是第一个正确认识到这种转换的效率取决于发动机和环境之间的温差。<br />
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Recognizing the significance of [[James Prescott Joule]]'s work on the conservation of energy, [[Rudolf Clausius]] was the first to formulate the second law during 1850, in this form: heat does not flow ''spontaneously'' from cold to hot bodies. While common knowledge now, this was contrary to the [[caloric theory]] of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law during 1850, in this form: heat does not flow spontaneously from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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认识到詹姆斯·普雷斯科特·焦耳在能量守恒方面工作的重要性，Rudolf Clausius 在1850年第一个提出了第二定律: 热不会自发地从冷物体流向热物体。虽然现在的常识是，这与当时流行的热量理论相反，热量理论认为热量是一种流体。从那里他能够推断出萨迪卡诺定律和熵的定义(1865年)。<br />
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Established during the 19th century, the [[Kelvin-Planck statement|Kelvin-Planck statement of the Second Law]] says, "It is impossible for any device that operates on a [[cyclic process|cycle]] to receive heat from a single [[heat reservoir|reservoir]] and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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Established during the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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建立于19世纪的开尔文-普朗克第二定律(Kelvin-Planck)说，“任何循环运行的设备都不可能从单个蓄热体接收热量，并产生净功。”这被证明相当于克劳修斯的陈述。<br />
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The [[ergodic hypothesis]] is also important for the [[Boltzmann]] approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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遍历假设对玻尔兹曼方法也很重要。它说，在很长一段时间内，在具有相同能量的微观态的相空间的某些区域所花费的时间与这个区域的体积成正比，即。在很长一段时间内，所有可访问的微观状态的可能性都是一样的。同样的，它表明时间平均值和系综平均值是相同的。<br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a [[Macroscopic bodies|macroscopic system]]. This doctrine is obsolescent.<ref>Denbigh, K.G., Denbigh, J.S. (1985). ''Entropy in Relation to Incomplete Knowledge'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-25677-1}}, pp. 43–44.</ref><ref>Grandy, W.T., Jr (2008). ''Entropy and the Time Evolution of Macroscopic Systems'', Oxford University Press, Oxford, {{ISBN|978-0-19-954617-6}}, pp. 55–58.</ref><ref name=Lambert>[http://entropysite.oxy.edu Entropy Sites — A Guide] Content selected by [[Frank L. Lambert]]</ref><br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a macroscopic system. This doctrine is obsolescent.<br />
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有一种传统的学说，从克劳修斯开始，认为熵可以被理解为宏观系统中的分子“无序”。这种学说已经过时了。<br />
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===Account given by Clausius===<br />
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===Account given by Clausius===<br />
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作者: 克劳修斯<br />
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[[File:Clausius-1.jpg|thumb|upright|Rudolf Clausius]]<br />
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Rudolf Clausius<br />
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鲁道夫 · 克劳修斯<br />
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In 1856, the German physicist [[Rudolf Clausius]] stated what he called the "second fundamental theorem in the [[mechanical theory of heat]]" in the following form:{{sfnp|Clausius|1867}}<br />
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In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:<br />
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1856年，德国物理学家鲁道夫 · 克劳修斯阐述了他所谓的“热力学理论中的第二个基本定理” ，其形式如下:<br />
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: <math>\int \frac{\delta Q}{T} = -N</math><br />
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<math>\int \frac{\delta Q}{T} = -N</math><br />
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- n / math<br />
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where ''Q'' is heat, ''T'' is temperature and ''N'' is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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其中 q 是热，t 是温度，n 是一个循环过程中所有未补偿的相变的“等价值”。后来，在1865年，克劳修斯将“等价值”定义为熵。紧接着这个定义，也就是在同一年，第二定律最著名的版本在4月24日苏黎世哲学学会的一次演讲中被宣读，在演讲的最后，克劳修斯总结道:<br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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宇宙的熵趋于最大<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. [[universe]], as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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这句话是第二定律中最广为人知的措辞。由于其语言的松散性，例如:。这个问题的答案是肯定的，因为这个问题的答案是肯定的。开放，封闭，或孤立，许多人认为这个简单的陈述意味着热力学第二定律适用于几乎所有可以想象的主题。这是不正确的; 这个陈述只是一个更加扩展和精确描述的简化版本。<br />
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In terms of time variation, the mathematical statement of the second law for an [[isolated system]] undergoing an arbitrary transformation is:<br />
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In terms of time variation, the mathematical statement of the second law for an isolated system undergoing an arbitrary transformation is:<br />
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就时间变化而言，经历任意变换的孤立系统第二定律的数学表述是:<br />
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: <math>\frac{dS}{dt} \ge 0</math><br />
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<math>\frac{dS}{dt} \ge 0</math><br />
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0 / math<br />
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where<br />
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where<br />
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: ''S'' is the entropy of the system and<br />
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S is the entropy of the system and<br />
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S 是系统的熵<br />
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: ''t'' is [[time]].<br />
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t is time.<br />
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时间到了。<br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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!-- 可逆过程需要与周围环境保持平衡。这对于一个孤立的系统是不可能的。因此，关于可逆过程的讨论已经转移到对封闭系统的分析----<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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平衡后应用等式符号。建立孤立系统第二定律的另一种方法是:<br />
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: <math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math<br />
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with <math> \dot S_{i}</math> the sum of the rate of [[entropy production]] by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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with <math> \dot S_{i}</math> the sum of the rate of entropy production by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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用数学 s { i } / 数学表示系统内所有进程的产生熵之和。这种配方的优点是它显示了产生熵的效果。产生熵是一个非常重要的概念，因为它决定(限制)热机器的效率。与环境温度数学 t { a } / math 相乘，得到所谓的耗散能量数学 p { diss } t { a } dot s { i } / math。<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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封闭系统的第二定律(允许热量交换和移动边界，但不允许物质交换)的表达式是:<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math 来计算数学点 q } + 点 s { i } / math<br />
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Here<br />
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给你<br />
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: <math>\dot Q</math> is the heat flow into the system<br />
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<math>\dot Q</math> is the heat flow into the system<br />
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数学 q / math 是进入系统的热流<br />
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: <math>T</math> is the temperature at the point where the heat enters the system.<br />
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<math>T</math> is the temperature at the point where the heat enters the system.<br />
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Math 是热量进入系统时的温度。<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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等式符号在只有可逆过程在系统内发生的情况下成立。如果发生不可逆过程(在实际操作系统中就是这种情况) ,-符号保持不变。如果系统有多处供热，我们必须求相应项的代数和。<br />
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For open systems (also allowing exchange of matter):<br />
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For open systems (also allowing exchange of matter):<br />
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对于开放系统(也允许物质交换) :<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math 表示数学点 q } + 点 s + 点 s { i } / math<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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这里的数学 s / math 是熵流进入系统，与物质流进入系统有关。它不应该与熵的时间导数混淆。如果物质在几个地方提供，我们必须取这些贡献的代数和。<br />
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==Statistical mechanics==<br />
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==Statistical mechanics==<br />
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统计力学<br />
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[[Statistical mechanics]] gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a [[microstate (statistical mechanics)|microstate]] of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/{{radic|''N''}} where ''N'' is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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Statistical mechanics gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a microstate of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/ where N is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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通过假设物质是由不断运动的原子和分子组成的，统计力学对第二定律给出了解释。系统中每个粒子的一组特定的位置和速度称为系统的微观状态，由于系统的不断运动，系统不断地改变其微观状态。统计力学假设，在平衡状态下，系统可能处于的每个微观状态发生的可能性是相等的，当这个假设被提出时，它直接导致了第二定律必须在统计意义上成立的结论。也就是说，第二定律平均成立，其中 n 是系统中粒子数的1 / 数量级的统计变化。在日常(宏观)情况下，违反第二定律的概率几乎为零。然而，对于粒子数量很少的系统，热力学参数，包括熵，可能显示出与第二定律预测的显著的统计偏差。经典热力学理论没有处理这些统计变量。<br />
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（by Jie XIANG) 在通过假定一种物质由不断运动的原子和分子组成的前提下，统计力学对热力学第二定律进行了解释。系统中每个粒子其特定的位置和速度组成为系统的微观状态，由于（原子和分子的）不断运动，系统也持续改变其微观状态。统计力学假设，在系统力平衡的情况下，其微观状态其实是等可能发生的，进而能够得出结论：热力学第二定律必须在统计学意义上成立。也就是说，第二定律将保持均值不变， 其统计变化为1 /√N，N为系统中的粒子数。在日常(宏观)情况下，违反热力学第二定律的概率几乎为零，但是，对于粒子数量非常少的系统，其热力学参数（包括熵）可能会显示出与第二定律所预测的明显统计偏差。然而经典热力学理论并不处理这些统计变量。<br />
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==Derivation from statistical mechanics==<br />
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==Derivation from statistical mechanics==<br />
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从统计力学导出<br />
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{{Further information|H-theorem}}<br />
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The first mechanical argument of the [[Kinetic theory of gases]] that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to [[James Clerk Maxwell]] in 1860;<ref>{{Cite journal | last1 = Gyenis | first1 = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G }}</ref> [[Ludwig Boltzmann]] with his [[H-theorem]] of 1872 also argued that due to collisions gases should over time tend toward the [[Maxwell–Boltzmann distribution]].<br />
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The first mechanical argument of the Kinetic theory of gases that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to James Clerk Maxwell in 1860; Ludwig Boltzmann with his H-theorem of 1872 also argued that due to collisions gases should over time tend toward the Maxwell–Boltzmann distribution.<br />
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气体动力学理论的第一个力学论证由詹姆斯·克拉克·麦克斯韦在1860年给出，指出分子碰撞引起温度均衡，因此整体趋向于平衡； 路德维希·玻尔兹曼在1872年提出的 H 定理也认为，气体由于碰撞应该随着时间的推移趋向于麦克斯韦-波兹曼分布。<br />
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Due to [[Loschmidt's paradox]], derivations of the Second Law have to make an assumption regarding the past, namely that the system is [[Correlation and dependence|uncorrelated]] at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a [[boundary condition]], and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the [[Big Bang]]), though [[Boltzmann brain|other scenarios]] have also been suggested.<ref name="Hawking AOT">{{cite journal|last=Hawking|first=SW|title=Arrow of time in cosmology|journal=Phys. Rev. D|year=1985|volume=32|issue=10|pages=2489–2495|doi=10.1103/PhysRevD.32.2489|pmid=9956019|bibcode = 1985PhRvD..32.2489H }}</ref><ref>{{cite book | last = Greene | first = Brian | authorlink = Brian Greene | title = The Fabric of the Cosmos | url = https://archive.org/details/fabricofcosmossp00gree | url-access = registration | publisher = Alfred A. Knopf | year = 2004 | page = [https://archive.org/details/fabricofcosmossp00gree/page/171 171] | isbn = 978-0-375-41288-2}}</ref><ref name=Lebowitz>{{cite journal|last=Lebowitz|first=Joel L.|title= Boltzmann's Entropy and Time's Arrow|journal=Physics Today|date=September 1993|volume=46|issue=9|pages=32–38|url=http://users.df.uba.ar/ariel/materias/FT3_2008_1C/papers_pdf/lebowitz_370.pdf|accessdate=2013-02-22|doi=10.1063/1.881363|bibcode = 1993PhT....46i..32L }}</ref><br />
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Due to Loschmidt's paradox, derivations of the Second Law have to make an assumption regarding the past, namely that the system is uncorrelated at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a boundary condition, and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the Big Bang), though other scenarios have also been suggested.<br />
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由于洛施密特悖论，第二定律的导出必须对过去做出一个假设，即系统在过去的某个时刻是不相关的；这样的假设允许进行简单的概率处理。这个假设通常被认为是一个边界条件，因此热力学第二定律最终是过去某个地方的初始条件的结果，可能是在宇宙的开始(大爆炸) 。也有人提出了其他假设。<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the [[Statistical mechanics#Fundamental postulate|fundamental postulate]], also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy.{{citation needed|date=August 2012}} The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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考虑到这些假设，在统计力学中，第二定律不是一个假设，而是统计力学基本假设的一个结果，也被称为等先验概率假设。这个基本假设表明，只要一个人清楚地知道，简单的概率论证只适用于未来，而对于过去，有辅助的信息来源告诉我们，它是低熵的。热力学第二定律的第一部分指出，热孤立系统的熵只能增加。如果我们把熵的概念限制在热平衡系统中，那么热力学第二定律的第一部分是等先验概率假设的一个显然结果。处于热平衡状态的孤立系统包含能量<math>E</math>的熵表示为：<br />
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: <math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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<math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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其中<math>\Omega\left(E\right)</math> 是处于 <math>E</math>和<math>E +\delta E</math这个小区间内的量子态数目。这里的 <math>\delta E</math> 是一个宏观上很小的固定能量区间。严格地说，这意味着熵取决于对<math>\delta E</math>的选择。然而在热力学极限下（例如无穷大系统的极限），狭义的熵(单位体积或单位质量的熵)不依赖于 <math>\delta E</math>。<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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假设我们有一个孤立系统，其宏观状态由许多变量描述。这些宏观变量可以是总体积、活塞在系统中的位置等。从而<math>\Omega</math>将取决于这些变量的值。如果某个变量不是固定的（我们不会在某个位置夹住活塞) ，那么因为在平衡状态下所有可到达状态的等几率到达的，平衡状态下的自由变量会使 <math>\Omega</math> 最大，因为这是平衡状态下最可能的情况。<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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如果该变量最初固定到某个值然后释放，当达到新的平衡时，变量将自我调整使得 <math>\Omega</math> 最大，这意味着熵将增加或保持不变(如果变量初始固定的值恰好是平衡值)。<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's [[H-theorem]], however, proves that the quantity {{math|''H''}} increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the quantity increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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假设我们初始位于一个平衡状态，突然移除了对一个变量的约束。我们做完这件事的时候，可达到的微观状态的数为<math>\Omega</math>，但是系统还没有达到平衡，所以系统处于某些可达到的状态的实际概率还不等于先验概率 <math>1/\Omega</math>。我们已经知道，最终的平衡状态相对于之前的平衡状态，熵会增加或者保持不变。然而，玻耳兹曼的H定理证明系统在不处于平衡态的期间，那个量作为时间的函数单调增加。<br />
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===Derivation of the entropy change for reversible processes===<br />
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===Derivation of the entropy change for reversible processes===<br />
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可逆过程熵变的推导<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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第二定律的第二部分指出，经历可逆过程的系统的熵变是由以下因素给出的:<br />
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: <math>dS =\frac{\delta Q}{T}</math><br />
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<math>dS =\frac{\delta Q}{T}</math><br />
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数学 dS frac delta q }{ t } / math<br />
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where the temperature is defined as:<br />
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where the temperature is defined as:<br />
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其中温度定义为:<br />
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:<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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[数学][[欧米茄][左(右)][右][数学][数学]<br />
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See [[Microcanonical ensemble|here]] for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the [[adiabatic theorem]] of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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请参阅此处对此定义的正当理由。假设系统有一些可以改变的外部参数 x。一般来说，系统的能量本征态将依赖于 x。根据量子力学的绝热定理，在系统哈密顿量无限缓慢变化的极限下，系统将保持在相同的定态，因此系统的能量会随着其所在定态的能量变化而变化。<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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广义力，x，对应于外部变量 x 的定义是，如果 x 增加一个量 dx，那么 math x dx / math 就是系统所做的功。例如，如果 x 是体积，那么 x 是压强。一个已知的系统的广义力在定态数学 e { r } / math 中给出:<br />
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: <math>X = -\frac{dE_{r}}{dx}</math><br />
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<math>X = -\frac{dE_{r}}{dx}</math><br />
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数学 x- frac { dE { r }{ dx } / math<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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由于系统可以在一个数学△ e / math 区间内的任意定态内，我们将系统的广义力定义为上述表达式的期望值:<br />
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: <math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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<math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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数学 x- 左 langle frac { dE { dx }右 rangle，/ math<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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为了计算平均值，我们通过计算在数学 y / math 和数学 y + delta y / math 之间的范围内，其中有多少数学 y / math 能量本征态有一个值来划分 ω 左(e 右) / math 能量本征态。把这个数字命名为 math Omega { y } left (e right) / math，我们有:<br />
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: <math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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<math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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Math Omega left (e right) sum { y } Omega { y } left (e right) ，/ math<br />
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The average defining the generalized force can now be written:<br />
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The average defining the generalized force can now be written:<br />
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定义广义力的平均值现在可以写成:<br />
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: <math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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<math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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数学 x frac {1} Omega 左(e 右) sum { y Omega { y }左(e 右) ，/ math<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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我们可以把它和熵对 x 在恒定能量 e 下的导数联系起来，如下所示。假设我们把 x 改成 x + dx。然后 math Omega left (e right) / math 将会改变，因为能量本征态依赖于 x，导致能量本征态进入或超出数学 e / math 和数学 e + delta e / math 之间的范围。让我们再次关注数学 y / math 和数学 y + delta y / math 之间的能量本征态。由于这些能量本征态的能量增加了 y dx，所有这些能量本征态在 e-y dx 到 e 之间的区间内从 e 以下移动到 e 以上。 有<br />
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: <math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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<math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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数学 n { y }左(e 右) frac { y }左(e 右)}{ delta e } y dx ，/ math<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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这样的能量本征态。如果数学 y dx leq delta e / math，所有这些能量本征态将移动到数学 e / math 和数学 e + delta e / math 之间的范围内，有助于数学 ω / math 的增加。从数学 e + △ e / math 下移到数学 e + △ e / math 上的能量本征态的个数是由数学 n { y }左(e + △ e 右) / math 给出的。区别在于<br />
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: <math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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<math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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数学 n { y }左(e 右)-n { y }左(e + delta e 右) ，/ math<br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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因此是数学增长的净贡献欧米茄 / 数学。请注意，如果 y dx 大于 math delta e / math，那么将会有能量本征态从 e 下面移动到 e + delta e / math 上面。它们在数学 n { y } left (e right) / math 和数学 n { y } left (e + delta e right) / math 中都有计算，因此上述表达式在这种情况下也是有效的。<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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将上面的表达式表示为对 e 的导数，对 y 的求和得到表达式:<br />
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: <math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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数学左(部分欧米茄左(部分欧米茄右){ e }-和 y 左(部分欧米茄右){ x 左(部分欧米茄左(部分欧米茄右)}{ x 右) ，/ 数学<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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关于 x 的欧米茄 / 数学对数导数由以下方式给出:<br />
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: <math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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Math 左( frac { partial ln 左( Omega 右)}{部分 x 右){ e } beta x + 左( frac { partial x 右){ x } ，/ math<br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
<br />
The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
<br />
第一个术语是密集型的，即。它不能根据系统大小进行缩放。相比之下，最后一项的规模与逆系统的规模一样，因此将在热力学极限中消失。因此，我们发现:<br />
<br />
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: <math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
<br />
<math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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数学左(部分 s }右){ e } frac { x }{ t } ，/ math<br />
<br />
<br />
<br />
<br />
<br />
Combining this with<br />
<br />
Combining this with<br />
<br />
结合这一点<br />
<br />
<br />
<br />
<br />
<br />
: <math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
<br />
<math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
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数学左(部分 s }右){ x } frac {1}{ t } ，/ math<br />
<br />
<br />
<br />
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Gives:<br />
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Gives:<br />
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给予:<br />
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: <math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
<br />
<math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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数学 dS 左(部分 s }部分 e }右){ x } dE + 左(部分 s }右){ e } dx frac { t } + frac { t }{ delta q }{ t } ，/ math<br />
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===Derivation for systems described by the canonical ensemble===<br />
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===Derivation for systems described by the canonical ensemble===<br />
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正则系系统的推导<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the [[canonical ensemble]]:<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:<br />
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如果一个系统与某个温度为T的热浴热接触，那么在平衡状态下，关于能量本征值的概率分布值由正则系综给出:<br />
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: <math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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<math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the [[Partition function (statistical mechanics)|partition function]]. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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这里Z是一个使所有概率之和归一化的因子，这个函数被称为配分函数。现在我们考虑对温度和能级所依赖的外部参数的无限小的可逆改变。它遵循熵的一般公式:<br />
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: <math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
<br />
<math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
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<br />
<br />
<br />
<br />
<br />
that<br />
<br />
that<br />
<br />
以及<br />
<br />
<br />
<br />
<br />
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: <math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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<math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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<br />
<br />
<br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
<br />
Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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把正则系综的数学形式 <math>P_{j}</math> 插入，可以得到:<br />
<br />
<br />
<br />
<br />
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: <math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
<br />
<math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
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==Living organisms==<br />
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==Living organisms==<br />
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生命体<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as [[What is Life?|Erwin Schrödinger]], [[Léon Brillouin]]<ref>Léon Brillouin [https://books.google.com/books?hl=en&lr=&id=tPXVbiw_1P0C ''Science and Information Theory''] (Academic Press, 1962) (Dover, 2004)</ref> and [[Life and Energy|Isaac Asimov]]. It is also the topic of current{{When|date=July 2019}} research.{{Citation needed|date=July 2019}}<br />
<br />
There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as Erwin Schrödinger, Léon Brillouin and Isaac Asimov. It is also the topic of current research.<br />
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表述热力学主要有两种方式：(a)描述从一种热力学平衡状态到另一种状态的路径，(b)通过使系统保持不变，同时增加周围环境的总熵的循环过程。这两种方法有助于理解生命过程。这个话题大大超出了本文的范围，但是已经有一些研究者考虑过这个问题，比如埃尔温·薛定谔、莱昂·布里渊和艾萨克.阿西莫夫。这也是当前研究的主题。<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of [[metabolism]], give out breakdown products and heat. Plants [[Photosynthesis|take in radiative energy]] from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
<br />
To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of metabolism, give out breakdown products and heat. Plants take in radiative energy from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
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从近似的角度来看，生命体可以被认为是(b)的一个例子。近似地，一只动物的身体状态每天循环，使得它几乎没有什么变化。动物吸收食物、水和氧气，经过新陈代谢，输出分解的产物和热量。植物吸收来自太阳的辐射能量，这可以认为是热量，以及二氧化碳和水，然后它们释放氧气。它们就是这样生长的，最终会死亡，尸体腐烂，大部分重新变成二氧化碳和水。这可以看作是一个循环过程。总的来说，阳光来自一个高温的源——太阳，它的能量被传递到一个较低的温度汇，例如向太空辐射。这个过程使得植物周围环境的熵增加。因此从循环过程的角度来看，动物和植物服从热力学第二定律。简单的热机效率概念很难适用于这个问题，因为它们假定系统是封闭的。<br />
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From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to [[Non-equilibrium thermodynamics|unsettled current debate or research]]. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
<br />
From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to unsettled current debate or research. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
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从(a)的热力学观点考虑 ，即考虑从一个平衡状态到另一个平衡状态的路径，只会产生一个粗略近似的图像，因为生命体从来不会处于热力学平衡状态。生物体经常必须被认为是开放的系统，因为它们吸收营养物质并排出废弃物。开放系统热力学目前通常从一个热力学平衡状态到另一个状态的路径的角度来考虑，或者是考虑局部热力学平衡近似下的流。生命体的问题可以通过假定一个流不变的稳态来进一步简化。对于这样的近似，熵产生的一般原理在目前有争论和研究的不确定性。尽管如此，从热力学第二定律的这个角度出发衍生出来的想法对生物是有启发意义的。<br />
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==Gravitational systems==<br />
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==Gravitational systems==<br />
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重力系统<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive [[heat capacity]], meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it.<ref name="Grandy 151">Grandy, W.T. (Jr) (2008), p. 151.</ref> Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
<br />
In systems that do not require for their descriptions the general theory of relativity, bodies always have positive heat capacity, meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it. Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
<br />
在不需要描述广义相对论的系统中，物体总是具有正的热容，这意味着温度随着能量的增加而升高。因此，当能量从高温物体流向低温物体时，源温度降低，而汇温度升高; 因此温差随时间而减小。这种情况并不总是发生在那些需要引力和广义相对论的系统中。这样的系统可以自发地朝着质量和能量不均匀分布的方向变化。这在大尺度上适用于宇宙，因此可能很难或不可能将第二定律应用于宇宙。除此之外，广义相对论所描述的系统热力学超出了本文的范围。<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
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{{main article|Non-equilibrium thermodynamics}}<br />
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The theory of classical or [[thermodynamic equilibrium|equilibrium thermodynamics]] is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<ref name="Grandy 151"/><ref>Callen, H.B. (1960/1985), p. 15.</ref><br />
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The theory of classical or equilibrium thermodynamics is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<br />
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经典热力学或平衡热力学的理想化。一个主要的假设或假设，往往甚至没有明确说明，是系统的存在，在他们自己的内部状态的热力学平衡。一般来说，在给定时间内包含一个物理系统的空间区域---- 可能在自然界中找到---- 不是在热力学平衡中，用最严格的术语来解释。从更宽松的角度来说，整个宇宙中没有任何东西是或者曾经是完全精确的热力学平衡。<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of [[thermodynamic equilibrium]]. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable [[thermal fluctuations|fluctuations]]. There is an exception, the case of [[critical point (thermodynamics)|critical states]], which exhibit to the naked eye the phenomenon of [[critical opalescence]]. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of thermodynamic equilibrium. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable fluctuations. There is an exception, the case of critical states, which exhibit to the naked eye the phenomenon of critical opalescence. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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为了物理分析的目的，假设热力学平衡是非常方便的。这种假设可能依赖于试错来证明其正当性。如果这个假设是正确的，它通常是非常有价值和有用的，因为它提供了热力学理论。平衡假设的要素是，一个系统被观察到在无限长的时间内是不变的，而且一个系统中有如此多的粒子，以至于它的微粒性质可以被完全忽略。在这样的平衡假设下，一般来说，没有宏观上可以探测到的波动。有一个例外，即临界状态的例子，它们用肉眼就可以看到临界乳光现象。对于临界状态的实验室研究，需要非常长的观察时间。<br />
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In all cases, the assumption of [[thermodynamic equilibrium]], once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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In all cases, the assumption of thermodynamic equilibrium, once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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在所有情况下，热力学平衡的假设，一旦做出，就意味着没有假定的候选者“涨落”改变系统的熵。<br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<ref>Lieb, E.H., Yngvason, J. (2003), p. 190.</ref><br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<br />
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一个物理系统很容易发生内部的宏观变化，其速度之快足以使熵的恒定性假设失效。或者一个物理系统的粒子太少，以至于微粒的本质在可观测的波动中是显而易见的。那么热力学平衡的假设就会被放弃。对于非平衡态，熵没有绝对的一般定义。<br />
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There are intermediate cases, in which the assumption of local [[thermodynamic equilibrium]] is a very good approximation,<ref>Gyarmati, I. (1967/1970), pp. 4-14.</ref><ref>Glansdorff, P., Prigogine, I. (1971).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003).</ref> but strictly speaking it is still an approximation, not theoretically ideal.<br />
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There are intermediate cases, in which the assumption of local thermodynamic equilibrium is a very good approximation, but strictly speaking it is still an approximation, not theoretically ideal.<br />
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还有一些中间情况，其中局部热力学平衡的假设是一个非常好的近似值，但严格地说它仍然是一个近似值，不是理论上的理想值。<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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对于一般的非平衡状态，考虑其他量的统计力学定义可能是有用的，这些量可以方便地称为‘熵’ ，但是它们不应该与第二定律正确定义的熵混淆或混淆。这些其他的量确实属于统计力学，而不是热力学---- 第二定律的主要领域。<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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宏观观测波动的物理学超出了本文的范围。<br />
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==Arrow of time==<br />
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==Arrow of time==<br />
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时间之箭<br />
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{{Further information|Entropy (arrow of time)}}Thermodynamic Asymmetry in Time<br />
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Thermodynamic Asymmetry in Time<br />
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时间中的热力学不对称性<br />
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{{See also|Arrow of time}}<br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely [[CPT symmetry]], since the second law applies statistically, it is hypothesized, on time-asymmetric ''boundary conditions''.<ref>{{cite web|first=Craig|last=Callender|url=https://plato.stanford.edu/archives/fall2011/entries/time-thermo/|title=Thermodynamic Asymmetry in Time|encyclopedia=Stanford Encyclopedia of Philosophy|date=29 July 2011}}</ref><br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely CPT symmetry, since the second law applies statistically, it is hypothesized, on time-asymmetric boundary conditions.<br />
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热力学第二定律是一个物理定律，它不对称于时间方向的反转。这与已经观察到的物理学基本定律，即 CPT 对称性的概念并不冲突，因为第二定律在统计上适用，它是假设的，是关于时间非对称的边界条件。<br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect ([[Arrow of time#The causal arrow of time|the causal arrow of time]]).<ref>{{cite book | first = J.J.| last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1|display-authors=etal}} chapter 6</ref><br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect (the causal arrow of time).<br />
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第二定律被提出来部分解释时间向前和向后移动的区别，例如为什么原因先于结果(因果关系的时间箭头)。<br />
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==Irreversibility==<br />
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==Irreversibility==<br />
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不可逆性<br />
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Irreversibility in [[thermodynamic process]]es is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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Irreversibility in thermodynamic processes is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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热力学过程的不可逆性是热力学作用不对称性的结果，而不是物体内部不可逆的微观性质的结果。热力学操作是施加在参与机构上的宏观外部干预，而不是从它们的内部性质中得到的。由于没有认识到这一点，产生了被称为“悖论”的说法。<br />
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===Loschmidt's paradox===<br />
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===Loschmidt's paradox===<br />
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洛斯密特悖论<br />
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{{main article|Loschmidt's paradox}}<br />
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[[Loschmidt's paradox]], also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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洛斯密特悖论，也被称为可逆性悖论，其目的在于不可能从描述宏观系统微观演化的时间对称动力学推导出不可逆性。<br />
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In the opinion of [[Erwin Schrödinger|Schrödinger]], "It is now quite obvious in what manner you have to reformulate the law of entropy{{snd}}or for that matter, all other irreversible statements{{snd}}so that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other."<ref>[[Erwin Schrödinger|Schrödinger, E.]] (1950), p.&nbsp;192.</ref> The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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In the opinion of Schrödinger, "It is now quite obvious in what manner you have to reformulate the law of entropyor for that matter, all other irreversible statementsso that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other." The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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根据薛定谔的观点，“现在很明显，你必须用什么方式来重新定义该物质的熵定律，所有其他不可逆的状态，这样它们才能从可逆模型中导出。你不能只谈论一个孤立的系统，而应该至少谈论两个，你可以暂时认为它们与世界其它地方是孤立的，但并不总是相互孤立的。”这两个系统被隔离墙隔离开来，直到按照法律的设想被热力学操作拆除。热力学操作是外部强加的，不受支配系统组成部分的可逆微观动力学定律的制约。它是不可逆转性的原因。本文中的法律陈述符合薛定谔的建议。因果关系在逻辑上优先于第二定律，而不是从第二定律派生出来的。<br />
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===Poincaré recurrence theorem===<br />
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===Poincaré recurrence theorem===<br />
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庞加莱始态复现定理<br />
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{{Main article|Poincaré recurrence theorem}}<br />
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The [[Poincaré recurrence theorem]] considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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The Poincaré recurrence theorem considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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庞加莱始态复现定理认为这是一个孤立的物理系统的理论微观描述。这可以被认为是一个热力学系统的模型后，一个热力学操作已经移除了一个内墙。在足够长的时间之后，系统将返回到一个非常接近初始状态的显微定义状态。庞加莱回归时间是指回归前经过的时间长度。它极其漫长，可能比宇宙的寿命还要长，并且敏感地依赖于被热力学操作拆除的墙体的几何形状。递归定理可能被认为是明显与热力学第二定律相矛盾的。然而，更明显的是，它只不过是一个微观模型，热力学平衡在一个孤立的系统中，通过移除两个系统之间的一道墙而形成。对于一个典型的热力学系统，重现时间是如此之大(比宇宙的生命周期长许多倍) ，以至于在所有的实际目的中，人们都不能观察到重现。然而，人们可能希望想象一下，可以等待庞加莱复发，然后重新插入被热力学操作移除的壁。然后很明显，不可逆性的出现是由于庞加莱循环的完全不可预测性，只要初始状态是热力学平衡，就像宏观热力学的情况一样。即使可以等待，也没有实际可能性选择合适的时间重新插入墙壁。庞加莱始态复现定理为 Loschmidt 悖论提供了一个解决方案。如果一个孤立的热力学系统可以被监测到在越来越多的平均 poincar 递归时间的倍数上，系统的热力学行为将在时间反转下变得不变。<br />
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[[File:James-clerk-maxwell3.jpg|thumb|upright|James Clerk Maxwell]]<br />
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James Clerk Maxwell<br />
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詹姆斯·克拉克·麦克斯韦<br />
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===Maxwell's demon===<br />
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===Maxwell's demon===<br />
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麦克斯韦的魔鬼<br />
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{{unreferenced|section|date=August 2018}}<br />
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{{main article|Maxwell's demon}}<br />
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[[James Clerk Maxwell]] imagined one container divided into two parts, ''A'' and ''B''. Both parts are filled with the same [[gas]] at equal temperatures and placed next to each other, separated by a wall. Observing the [[molecule]]s on both sides, an imaginary [[demon]] guards a microscopic trapdoor in the wall. When a faster-than-average molecule from ''A'' flies towards the trapdoor, the demon opens it, and the molecule will fly from ''A'' to ''B''. The average [[speed]] of the molecules in ''B'' will have increased while in ''A'' they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in ''A'' and increases in ''B'', contrary to the second law of thermodynamics.<br />
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James Clerk Maxwell imagined one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other, separated by a wall. Observing the molecules on both sides, an imaginary demon guards a microscopic trapdoor in the wall. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics.<br />
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詹姆斯·克拉克·麦克斯韦想象一个容器分成两部分，a 和 b。两个部分在相同的温度下充入相同的气体，并被一堵墙隔开，挨着放置。通过观察两边的分子，一个想象中的恶魔守卫着墙上的一个微型暗门。当一个来自 a 的比平均速度快的分子飞向活板门时，魔鬼打开了它，分子就会从 a 飞到 b。B 中分子的平均速度会增加，而 a 中分子的平均速度会减慢。由于平均分子速度与温度相对应，因此 a 中的温度降低，b 中的温度升高，这与热力学第二定律相反。<br />
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One response to this question was suggested in 1929 by [[Leó Szilárd]] and later by [[Léon Brillouin]]. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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One response to this question was suggested in 1929 by Leó Szilárd and later by Léon Brillouin. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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1929年，Leó Szilárd 和后来的 l on brilouin 对这个问题提出了一个答案。Szil rd 指出，现实生活中的麦克斯韦恶魔需要一些测量分子速度的方法，而获取信息需要消耗大量能量。<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between ''A'' and ''B''. It is therefore performing [[thermodynamic operation]]s on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between A and B. It is therefore performing thermodynamic operations on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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麦克斯韦的“魔鬼”不断地改变 a 和 b 之间墙体的渗透性。因此，它是在微观尺度上执行热力学操作，而不仅仅是观察普通的自发或自然的宏观热力学过程。<br />
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==Quotations==<br />
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==Quotations==<br />
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语录<br />
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{{Wikiquote}}<br />
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{{quote|The law that entropy always increases holds, I think, the supreme position among the [[Laws of science|laws of Nature]]. If someone points out to you that your pet theory of the [[universe]] is in disagreement with [[Maxwell's equations]] – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.|Sir [[Arthur Stanley Eddington]], ''The Nature of the Physical World'' (1927)}}<br />
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{{quote|There have been nearly as many formulations of the second law as there have been discussions of it.|Philosopher / Physicist [[Percy Williams Bridgman|P.W. Bridgman]], (1941)}}<br />
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{{quote|Clausius is the author of the sibyllic utterance, "The energy of the universe is constant; the entropy of the universe tends to a maximum." The objectives of continuum thermomechanics stop far short of explaining the "universe", but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size.|[[Clifford Truesdell|Truesdell, C.]], Muncaster, R. G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 17.}}<br />
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==See also==<br />
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参见<br />
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*[[Clausius–Duhem inequality]]<br />
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*[[Fluctuation theorem]]<br />
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*[[Heat death of the universe]]<br />
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*[[History of thermodynamics]]<br />
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*[[Jarzynski equality]]<br />
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*[[Laws of thermodynamics]]<br />
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*[[Maximum entropy thermodynamics]]<br />
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*[[Quantum thermodynamics]]<br />
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*[[Reflections on the Motive Power of Fire]]<br />
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*[[Relativistic heat conduction]]<br />
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*[[Thermal diode]]<br />
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*[[Thermodynamic equilibrium]]<br />
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{{colend}}<br />
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==References==<br />
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==References==<br />
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参考资料<br />
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===Sources===<br />
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资料来源<br />
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{{refbegin|50em}}<br />
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*Adkins, C.J. (1968/1983). ''Equilibrium Thermodynamics'', (1st edition 1968), third edition 1983, Cambridge University Press, Cambridge UK, {{ISBN|0-521-25445-0}}.<br />
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*[[Peter Atkins|Atkins, P.W.]], de Paula, J. (2006). ''Atkins' Physical Chemistry'', eighth edition, W.H. Freeman, New York, {{ISBN|978-0-7167-8759-4}}.<br />
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*Attard, P. (2012). ''Non-equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications'', Oxford University Press, Oxford UK, {{ISBN|978-0-19-966276-0}}.<br />
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*Baierlein, R. (1999). ''Thermal Physics'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-59082-5}}.<br />
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*Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics, New York, {{ISBN|0-88318-797-3}}.<br />
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*{{Cite book|title=Concepts in thermal physics|last=Blundell|first=Stephen J.|last2=Blundell|authorlink1=Stephen Blundell|first2=Katherine M.|authorlink2=Katherine Blundell|publisher=[[Oxford University Press]]|year=2010|isbn=9780199562107|edition=2nd|location=Oxford|pages=|oclc=607907330|doi=10.1093/acprof:oso/9780199562091.001.0001 |url=http://cds.cern.ch/record/1235139}}<br />
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*[[Ludwig Boltzmann|Boltzmann, L.]] (1896/1964). ''Lectures on Gas Theory'', translated by S.G. Brush, University of California Press, Berkeley.<br />
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*Borgnakke, C., Sonntag., R.E. (2009). ''Fundamentals of Thermodynamics'', seventh edition, Wiley, {{ISBN|978-0-470-04192-5}}.<br />
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*Buchdahl, H.A. (1966). ''The Concepts of Classical Thermodynamics'', Cambridge University Press, Cambridge UK.<br />
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*[[Percy Williams Bridgman|Bridgman, P.W.]] (1943). ''The Nature of Thermodynamics'', Harvard University Press, Cambridge MA.<br />
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*[[Herbert Callen|Callen, H.B.]] (1960/1985). ''Thermodynamics and an Introduction to Thermostatistics'', (1st edition 1960) 2nd edition 1985, Wiley, New York, {{ISBN|0-471-86256-8}}.<br />
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*{{cite journal|author=C. Carathéodory|author1-link=Constantin Carathéodory<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
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2010年3月15日 | 第355-386页 | http://gdz.sub.uni-goettingen.de/index.php?id=11&ppn=ppn235181684_0067&dmdid=dmdlog_0033&l=1<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). ''The Second Law of Thermodynamics'', Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können.(p. 363) | doi 10.1007 / bf01450409}.译文可以在这里找到[ http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf ]。同时，在 Kestin，j。(1976).热力学第二定律，Dowden，Hutchinson & Ross，Stroudsburg PA. 。<br />
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*[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986). [http://www.worldcat.org/title/reflections-on-the-motive-power-of-fire-a-critical-edition-with-the-surviving-scientific-manuscripts-translated-and-edited-by-fox-robert/oclc/812944517&referer=brief_results ''Reflections on the motive power of fire''], Manchester University Press, Manchester UK, {{ISBN|0-7190-1741-6}}. [https://archive.org/stream/reflectionsonmot00carnrich#page/n7/mode/2up Also here.]<br />
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*[[Sydney Chapman (mathematician)|Chapman, S.]], [[Thomas George Cowling|Cowling, T.G.]] (1939/1970). ''The Mathematical Theory of Non-uniform gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases'', third edition 1970, Cambridge University Press, London.<br />
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*{{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Ueber Die Bewegende Kraft Der Wärme Und Die Gesetze, Welche Sich Daraus Für Die Wärmelehre Selbst Ableiten Lassen|journal=Annalen der Physik|year=1850|volume=79|issue=4|pages=368–397, 500–524|url=http://gallica.bnf.fr/ark:/12148/bpt6k15164w/f518.image|accessdate=26 June 2012|doi=10.1002/andp.18501550403|bibcode = 1850AnP...155..500C |hdl=2027/uc1.$b242250|hdl-access=free}} Translated into English: {{cite journal|last=Clausius|first=R.|title=On the Moving Force of Heat, and the Laws regarding the Nature of Heat itself which are deducible therefrom|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1851| volume=2|series=4th|issue=VIII|pages=1–21; 102–119 |url=https://archive.org/stream/londonedinburghd02lond#page/1/mode/1up|accessdate=26 June 2012<br />
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|doi=10.1080/14786445108646819}}<br />
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10.1080 / 14786445108646819}<br />
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* {{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie|year=1854|journal=Annalen der Physik|volume=xciii|issue=12|pages=481–506|url=http://zfbb.thulb.uni-jena.de/servlets/MCRFileNodeServlet/jportal_derivate_00140956/18541691202_ftp.pdf|accessdate=24 March 2014|doi=10.1002/andp.18541691202<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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| bibcode 1854AnP... 169. 1856年7月 | 第2卷 | 第4页 | 第86页 | https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=242014年3月<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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热力学理论及其在蒸汽机和物体物理性质中的应用 | 1867年 | 出版商 John van Voorst | 位置: 伦敦 | https://archive.org/details/mechanicaltheor04claugoog|quote=editions:pwr_sbkwa8ic。 2012年6月19日<br />
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*Denbigh, K. (1954/1981). ''The Principles of Chemical Equilibrium. With Applications in Chemistry and Chemical Engineering'', fourth edition, Cambridge University Press, Cambridge UK, {{ISBN|0-521-23682-7}}.<br />
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*Eu, B.C. (2002). ''Generalized Thermodynamics. The Thermodynamics of Irreversible Processes and Generalized Hydrodynamics'', Kluwer Academic Publishers, Dordrecht, {{ISBN|1-4020-0788-4}}.<br />
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*[[Josiah Willard Gibbs|Gibbs, J.W.]] (1876/1878). On the equilibrium of heterogeneous substances, ''Trans. Conn. Acad.'', '''3''': 108-248, 343-524, reprinted in ''The Collected Works of J. Willard Gibbs, Ph.D, LL. D.'', edited by W.R. Longley, R.G. Van Name, Longmans, Green & Co., New York, 1928, volume 1, pp.&nbsp;55–353.<br />
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*Griem, H.R. (2005). ''Principles of Plasma Spectroscopy (Cambridge Monographs on Plasma Physics)'', Cambridge University Press, New York {{ISBN|0-521-61941-6}}.<br />
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*Glansdorff, P., Prigogine, I. (1971). ''Thermodynamic Theory of Structure, Stability, and Fluctuations'', Wiley-Interscience, London, 1971, {{ISBN|0-471-30280-5}}.<br />
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*Grandy, W.T., Jr (2008). [http://global.oup.com/academic/product/entropy-and-the-time-evolution-of-macroscopic-systems-9780199546176?cc=au&lang=en& ''Entropy and the Time Evolution of Macroscopic Systems'']. Oxford University Press. {{ISBN|978-0-19-954617-6}}.<br />
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*Greven, A., Keller, G., Warnecke (editors) (2003). ''Entropy'', Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[Edward A. Guggenheim|Guggenheim, E.A.]] (1949). 'Statistical basis of thermodynamics', ''Research'', '''2''': 450–454.<br />
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*[[Edward A. Guggenheim|Guggenheim, E.A.]] (1967). ''Thermodynamics. An Advanced Treatment for Chemists and Physicists'', fifth revised edition, North Holland, Amsterdam.<br />
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*Gyarmati, I. (1967/1970) ''Non-equilibrium Thermodynamics. Field Theory and Variational Principles'', translated by E. Gyarmati and W.F. Heinz, Springer, New York.<br />
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*[[Charles Kittel|Kittel, C.]], [[Herbert Kroemer|Kroemer, H.]] (1969/1980). ''Thermal Physics'', second edition, Freeman, San Francisco CA, {{ISBN|0-7167-1088-9}}.<br />
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*Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998). ''Modern Thermodynamics: From Heat Engines to Dissipative Structures'', John Wiley & Sons, Chichester, {{ISBN|0-471-97393-9}}.<br />
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*Lebon, G., Jou, D., Casas-Vázquez, J. (2008). ''Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers'', Springer-Verlag, Berlin, {{ISBN|978-3-540-74252-4}}.<br />
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*{{cite journal|ref=harv|title=The Physics and Mathematics of the Second Law of Thermodynamics|last1=Lieb |first1=E. H. |last2=Yngvason |first2=J.|journal=Physics Reports|volume=310|issue=1|pages=1–96 |year=1999|doi=10.1016/S0370-1573(98)00082-9|arxiv = cond-mat/9708200 |bibcode = 1999PhR...310....1L}}<br />
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*Lieb, E.H., Yngvason, J. (2003). The Entropy of Classical Thermodynamics, pp.&nbsp;147–195, Chapter 8 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*{{cite book|ref=harv|first=F.|last=Mandl|title=Statistical physics|edition=second|year=1988|publisher=[[Wiley & Sons]]|isbn=978-0-471-91533-1}}<br />
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*{{cite journal | last1 = Maxwell | first1 = J.C. | authorlink = James Clerk Maxwell | year = 1867 | title = On the dynamical theory of gases | url = | journal = Phil. Trans. R. Soc. Lond. | volume = 157 | issue = | pages = 49–88 | doi = 10.1098/rstl.1867.0004 }}<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985). ''Thermodynamics'', Pitman, London, {{ISBN|0-273-08577-8}}.<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003). Entropy in Nonequilibrium, pp.&nbsp;79–109, Chapter 5 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*Münster, A. (1970), ''Classical Thermodynamics'', translated by E.S. Halberstadt, Wiley–Interscience, London, {{ISBN|0-471-62430-6}}.<br />
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*[[Brian Pippard|Pippard, A.B.]] (1957/1966). ''Elements of Classical Thermodynamics for Advanced Students of Physics'', original publication 1957, reprint 1966, Cambridge University Press, Cambridge UK.<br />
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*[[Max Planck|Planck, M.]] (1897/1903). [https://archive.org/stream/treatiseonthermo00planrich#page/100/mode/2up ''Treatise on Thermodynamics'', translated by A. Ogg, Longmans Green, London, p. 100.]<br />
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*[[Max Planck|Planck. M.]] (1914). [https://archive.org/details/theoryofheatradi00planrich ''The Theory of Heat Radiation''], a translation by Masius, M. of the second German edition, P. Blakiston's Son & Co., Philadelphia.<br />
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*[[Max Planck|Planck, M.]] (1926). Über die Begründung des zweiten Hauptsatzes der Thermodynamik, ''Sitzungsberichte der Preussischen Akademie der Wissenschaften: Physikalisch-mathematische Klasse'': 453–463.<br />
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* Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769-781.<br />
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* {{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}<br />
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*Quinn, T.J. (1983). ''Temperature'', Academic Press, London, {{ISBN|0-12-569680-9}}.<br />
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*{{cite book|ref=harv|first=Y.V.C.|last=Rao|title=An Introduction to thermodynamics|url=https://books.google.com/books?id=iYWiCXziWsEC&pg=PA213|date=2004|publisher=Universities Press|isbn=978-81-7371-461-0|page=213}}<br />
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*Roberts, J.K., Miller, A.R. (1928/1960). ''Heat and Thermodynamics'', (first edition 1928), fifth edition, Blackie & Son Limited, Glasgow.<br />
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*[[Erwin Schrödinger|Schrödinger, E.]] (1950). Irreversibility, ''Proc. R. Ir. Acad.'', '''A53''': 189–195.<br />
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*[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966). ''Elements of Thermodynamics'', Addison-Wesley Publishing, Reading MA.<br />
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*{{cite journal|ref=harv|last=Thomson|first=W.|author-link=William Thomson, 1st Baron Kelvin|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Transactions of the Royal Society of Edinburgh|year=1851|volume=XX|issue=part II|pages=261–268; 289–298|url=https://www.biodiversitylibrary.org/item/126047#page/295/mode/1up}} Also published in {{cite journal|last=Thomson|first=W.|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Philos. Mag. |date=December 1852 |volume=IV |series=4 |issue=22 |page=13|url=https://archive.org/stream/londonedinburghp04maga#page/12/mode/2up |accessdate=25 June 2012}}<br />
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*[[William Thomson, 1st Baron Kelvin|Thomson, W.]] (1852). ''On the universal tendency in nature to the dissipation of mechanical energy'' Philosophical Magazine, Ser. 4, p.&nbsp;304.<br />
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*[[László Tisza|Tisza, L.]] (1966). ''Generalized Thermodynamics'', M.I.T Press, Cambridge MA.<br />
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*[[Clifford Truesdell|Truesdell, C.]] (1980). ''The Tragicomical History of Thermodynamics 1822–1854'', Springer, New York, {{ISBN|0-387-90403-4}}.<br />
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*Uffink, J. (2001). Bluff your way in the second law of thermodynamics, ''Stud. Hist. Phil. Mod. Phys.'', '''32'''(3): 305–394.<br />
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*Uffink, J. (2003). Irreversibility and the Second Law of Thermodynamics, Chapter 7 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003), Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963). ''Lectures in Statistical Mechanics'', American Mathematical Society, Providence RI.<br />
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*[[Mark Zemansky|Zemansky, M.W.]] (1968). ''Heat and Thermodynamics. An Intermediate Textbook'', fifth edition, McGraw-Hill Book Company, New York.<br />
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==Further reading==<br />
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==Further reading==<br />
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进一步阅读<br />
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*Goldstein, Martin, and Inge F., 1993. ''The Refrigerator and the Universe''. Harvard Univ. Press. Chpts. 4–9 contain an introduction to the Second Law, one a bit less technical than this entry. {{ISBN|978-0-674-75324-2}}<br />
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*Leff, Harvey S., and Rex, Andrew F. (eds.) 2003. ''Maxwell's Demon 2 : Entropy, classical and quantum information, computing''. Bristol UK; Philadelphia PA: [[Institute of Physics]]. {{ISBN|978-0-585-49237-7}}<br />
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*{{Cite book | first = J.J. | last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1}}(technical).<br />
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*{{cite book |title=Reflections on the Motive Power of Heat and on Machines Fitted to Develop That Power |last=Carnot |first=Sadi |editor=[[Robert Henry Thurston|Thurston, Robert Henry]] |year=1890 |publisher=J. Wiley & Sons |location=New York |isbn= |pages= }} ([https://books.google.com/books?id=tgdJAAAAIAAJ full text of 1897 ed.]) ([http://www.history.rochester.edu/steam/carnot/1943/ html])<br />
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*Stephen Jay Kline (1999). ''The Low-Down on Entropy and Interpretive Thermodynamics'', La Cañada, CA: DCW Industries. {{ISBN|1-928729-01-0}}.<br />
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*{{cite book | last1 = Kostic | first1 = M | year = 2011 | title = Revisiting The Second Law of Energy Degradation and Entropy Generation: From Sadi Carnot's Ingenious Reasoning to Holistic Generalization | journal = AIP Conf. Proc. | volume = 1411 | issue = 1| pages = 327–350 | doi = 10.1063/1.3665247 | isbn = 978-0-7354-0985-9 | bibcode = 2011AIPC.1411..327K | series = AIP Conference Proceedings | citeseerx = 10.1.1.405.1945 }} also at [https://web.archive.org/web/20130420222450/http://www.kostic.niu.edu/2ndLaw/Revisiting%20The%20Second%20Law%20of%20Energy%20Degradation%20and%20Entropy%20Generation%20-%20From%20Carnot%20to%20Holistic%20Generalization-4.pdf].<br />
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==External links==<br />
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==External links==<br />
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外部链接<br />
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*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics]" – by Lawrence Sklar.<br />
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*[http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node30.html ''Second law of thermodynamics''] in the MIT Course [http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/notes.html ''Unified Thermodynamics and Propulsion''] from Prof. Z. S. Spakovszky<br />
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*[[E.T. Jaynes]], 1988, "[http://bayes.wustl.edu/etj/articles/ccarnot.pdf The evolution of Carnot's principle]," in G. J. Erickson and C. R. Smith (eds.)''Maximum-Entropy and Bayesian Methods in Science and Engineering, Vol'' 1: p.&nbsp;267.<br />
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*[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf Caratheodory, C., "Examination of the foundations of thermodynamics," trans. by D. H. Delphenich]<br />
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* [https://www.bbc.co.uk/programmes/p004y2bm The Second Law of Thermodynamics], BBC Radio 4 discussion with John Gribbin, Peter Atkins & Monica Grady (''In Our Time'', Dec. 16, 2004)<br />
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* [http://mdpi.org/entropy/papers/e6010001.pdf [[Entropy (journal)]], 2004]<br />
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* [https://www.journals.uchicago.edu/doi/abs/10.1086/663835 The Journal of the International Society for the History of Philosophy of Science, 2012]<br />
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{{DEFAULTSORT:Second Law Of Thermodynamics}}<br />
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[[Category:Concepts in physics]]<br />
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Category:Concepts in physics<br />
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分类: 物理概念<br />
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[[Category:Laws of thermodynamics|2]]<br />
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2<br />
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[[Category:Non-equilibrium thermodynamics]]<br />
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Category:Non-equilibrium thermodynamics<br />
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类别: 非平衡态热力学<br />
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[[Category:Philosophy of thermal and statistical physics]]<br />
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Category:Philosophy of thermal and statistical physics<br />
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类别: 热力学和统计物理学哲学<br />
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<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Second law of thermodynamics]]. Its edit history can be viewed at [[热力学第二定律/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E7%83%AD%E5%8A%9B%E5%AD%A6%E7%AC%AC%E4%BA%8C%E5%AE%9A%E5%BE%8B&diff=10353热力学第二定律2020-07-16T09:25:46Z<p>Dorr：/* Relation between Kelvin's statement and Planck's proposition 开尔文表述与普朗克命题的关系 */</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Law of physics}}<br />
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{{Thermodynamics|laws}}<br />
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The '''second law of thermodynamics''' states that the total [[entropy]] of an [[isolated system]] can never decrease over time, and is constant if and only if all processes are reversible.<ref>http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node38.html#SECTION05224000000000000000</ref> Isolated systems spontaneously evolve towards [[thermodynamic equilibrium]], the state with maximum entropy. <br />
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The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, and is constant if and only if all processes are reversible. Isolated systems spontaneously evolve towards thermodynamic equilibrium, the state with maximum entropy. <br />
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热力学第二定律指出，孤立系统的总熵永远不会随着时间而减少，而且只有当且仅当所有过程都是可逆的时，总熵才是恒定的。孤立的系统自发地进化到热力学平衡状态，即熵最大的状态。<br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in [[thermodynamic equilibrium]], or is undergoing a (fictive) [[Reversible process (thermodynamics)|reversible process]]. In all processes that occur, including [[spontaneous process]]es,<ref>Atkins and de Paula, p.78</ref> the total entropy of the system and its surroundings increases and the process is [[Irreversible process|irreversible in the thermodynamic sense]]. The increase in entropy accounts for the irreversibility of natural processes, and the [[Arrow of time|asymmetry between future and past]].<ref>{{cite book|last=Zohuri|first=Bahman|title=Dimensional Analysis Beyond the Pi Theorem|url=https://books.google.com/books?id=pRVuDQAAQBAJ|year=2016|publisher=Springer|isbn=978-3-319-45726-0|page=[https://books.google.com.ph/books?id=pRVuDQAAQBAJ&pg=PA111&dq=%22increase+in+entropy+accounts+for+the+irreversibility+of+natural+processes+and+the+asymmetry+between+future+and+past.%22&hl=en&sa=X&ved=0ahUKEwj6spb61tbaAhUFS7wKHfftDtIQ6AEIKDAA#v=onepage&q=%22increase%20in%20entropy%20accounts%20for%20the%20irreversibility%20of%20natural%20processes%20and%20the%20asymmetry%20between%20future%20and%20past.%22 111]}}</ref><br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in thermodynamic equilibrium, or is undergoing a (fictive) reversible process. In all processes that occur, including spontaneous processes, the total entropy of the system and its surroundings increases and the process is irreversible in the thermodynamic sense. The increase in entropy accounts for the irreversibility of natural processes, and the asymmetry between future and past.<br />
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系统及其周围环境的总熵在理想情况下可以保持不变，在这种情况下，系统处于热力学平衡期，或者正在经历一个(虚拟的)可逆过程期。在所有发生的过程中，包括自发过程，系统及其周围环境的总熵增加，这一过程在热力学意义上是不可逆的。熵的增加解释了自然过程的不可逆性，以及未来和过去之间的不对称性。<br />
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Historically, the second law was an [[Empirical evidence|empirical finding]] that was accepted as an axiom of [[thermodynamics|thermodynamic theory]]. [[Statistical mechanics]], classical or [[quantum thermodynamics|quantum]], explains the microscopic origin of the law.<br />
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Historically, the second law was an empirical finding that was accepted as an axiom of thermodynamic theory. Statistical mechanics, classical or quantum, explains the microscopic origin of the law.<br />
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从历史上看，第二定律是一个经验性的发现，被公认为热力学理论的公理。统计力学，无论是经典的还是量子的，都可以解释这个定律的微观起源。<br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist [[Nicolas Léonard Sadi Carnot|Sadi Carnot]], who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as [[Carnot's theorem (thermodynamics)|Carnot's rule]] or limit.<ref>Jaffe, R.L., Taylor, W. (2018). ''The Physics of Energy'', Cambridge University Press, Cambridge UK, pages 150, 151, 259, 772, 743.</ref><br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist Sadi Carnot, who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as Carnot's rule or limit.<br />
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第二定律已经用许多方式表达过了。法国科学家萨迪 · 卡诺在1824年证明了热能在热机中转化为功的效率是有上限的。这方面的第二定律也被称为卡诺的规则或限制。<br />
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==Introduction==<br />
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==Introduction==<br />
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引言<br />
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[[File:Heat flow hot to cold.png|thumb|upright|Heat flow from hot water to cold water.]]<br />
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Heat flow from hot water to cold water.<br />
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热量从热水流向冷水。<br />
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The [[first law of thermodynamics]] provides the definition of the [[internal energy]] of a [[thermodynamic system]], and expresses the law of [[conservation of energy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 40–41.</ref><ref>Munster A. (1970), pp. 8–9, 50–51.</ref> The second law is concerned with the direction of natural processes.<ref>{{harvnb|Mandl|1988}}</ref> It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such [[Phenomenon|phenomena]] are accounted for in terms of [[entropy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 79–107.</ref><ref>Bailyn, M. (1994), Section 71, pp. 113–154.</ref> If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, {{math|''S''}}, increases.<br />
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The first law of thermodynamics provides the definition of the internal energy of a thermodynamic system, and expresses the law of conservation of energy. The second law is concerned with the direction of natural processes. It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such phenomena are accounted for in terms of entropy. If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, , increases.<br />
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能量守恒定律提供了热力学系统内能的定义，并表达了能量守恒定律。第二定律与自然过程的方向有关。它断言自然过程只在一种意义上运行，而且是不可逆的。例如，当有了传导和辐射的路径时，热总是自发地从一个较热的物体流向一个较冷的物体。这种现象可以用熵来解释。如果一个孤立的系统最初是通过内部分隔不可渗透的墙来保持在内部热力学平衡，然后一些操作使墙更具渗透性，那么系统自发地进化到最终达到一个新的内部热力学平衡，其总熵增加。<br />
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In a fictive reversible process, an infinitesimal increment in the entropy ({{math|d''S''}}) of a system is defined to result from an infinitesimal transfer of heat ({{math|δ''Q''}}) to a [[closed system]] (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature ({{math|''T''}}) of the system in equilibrium and the surroundings which supply the heat:<ref>Bailyn, M. (1994), p. 120.</ref><br />
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In a fictive reversible process, an infinitesimal increment in the entropy () of a system is defined to result from an infinitesimal transfer of heat () to a closed system (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature () of the system in equilibrium and the surroundings which supply the heat:<br />
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在虚构的可逆过程中，系统熵的无穷小增量被定义为由无穷小的热量传递()到一个封闭系统(允许能量进入或出去，但不允许物质传递)除以平衡系统和提供热量的环境的共同温度() :<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, idealized fictive reversible process)}.</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, idealized fictive reversible process)}.</math><br />
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(闭合系统，理想化的可逆过程) / math<br />
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Different notations are used for infinitesimal amounts of heat ({{mvar|δ}}) and infinitesimal amounts of entropy ({{mvar|d}}) because entropy is a [[function of state]], while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the [[Clausius Theorem|inequality]] <ref name="MortimerBook"> Mortimer, R. G. ''Physical Chemistry''. 3rd ed., p. 120, Academic Press, 2008. </ref><ref name="FermiBook"> Fermi, E. ''Thermodynamics'', footnote on p. 48, Dover Publications,1956 (still in print). </ref> <br />
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Different notations are used for infinitesimal amounts of heat () and infinitesimal amounts of entropy () because entropy is a function of state, while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the inequality <br />
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用不同的符号表示无穷小量的热量()和无穷小量的熵() ，因为熵是状态的函数，而热量和功一样不是。第二定律要求系统熵的增量满足不等式，对于实际上可能存在的不与环境发生质量交换的无穷小过程，系统熵增量满足不等式<br />
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: <math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process).}</math><br />
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<math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process).}</math><br />
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(实际上可能是封闭系统，不可逆性)。数学<br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature ({{math|''T''}}) and the temperature of the surroundings ({{math|''T''}}<sub>''surr''</sub>).<ref name=":0">Adkins, C.J. (1968/1983), p. 75.</ref><ref name="Munster 45"/> Note that the equality still applies for pure heat flow,<ref name="Schmidt-Rohr 14"> Schmidt-Rohr, K. (2014). "Expansion Work without the External Pressure, and Thermodynamics in Terms of Quasistatic Irreversible Processes" ''J. Chem. Educ.'' '''91''': 402-409. https://dx.doi.org/10.1021/ed3008704 </ref><br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature () and the temperature of the surroundings (<sub>surr</sub>).<br />
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这是因为这种情况下的一般过程可能包括周围环境对系统所做的功，这可能在系统内部产生摩擦或粘滞效应，因为一个化学反应可能正在进行，或者因为热传递实际上只是不可逆地发生，由系统温度()和周围环境温度(sub surr / sub)之间的差分驱动。<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change).}</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change).}</math><br />
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(实际上可能是不改变成分的准静态不可逆性)。数学<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry.<ref name="Oxtoby8th">Oxtoby, D. W; Gillis, H.P., [[Laurie Butler|Butler, L. J.]] (2015).''Principles of Modern Chemistry'', Brooks Cole. p. 617. {{ISBN|978-1305079113}}</ref> <ref name="MortimerBook"></ref> Introducing a set of internal variables <math>\xi</math> to describe the deviation of a thermodynamic system in physical equilibrium (with the required well-defined uniform pressure ''P'' and temperature ''T'')<ref name="Schmidt-Rohr 14"></ref> from the chemical equilibrium state, one can record the equality<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry. from the chemical equilibrium state, one can record the equality<br />
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这是根据测得的热容曲线和相变时的熵变精确确定纯物质的绝对熵的基础。用量热法。从化学平衡来看，人们可以记录平等<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process).}</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process).}</math><br />
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- frac { t }-frac {1}{ j } ， Xi { j } ， delta Xi j， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ,.数学<br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of ''the second principle of thermodynamics'' – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<ref> Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769–781.</ref><ref>{{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}</ref><br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of the second principle of thermodynamics – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<br />
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第二项代表内部变量的工作，这些内部变量可以受到外部影响的干扰，但是系统不能通过内部变量执行任何正功。这种说法介绍了热力学系统在时间上不可能逆转演化的事实，并且可以被认为是热力学第二原理的表述---- 这个表述，当然，相当于熵原理的表述。<br />
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The [[zeroth law of thermodynamics]] in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body.<ref name=dugdale>{{cite book|author=J. S. Dugdale|title=Entropy and its Physical Meaning|publisher=Taylor & Francis|year=1996|isbn=978-0-7484-0569-5|page=13|quote=This law is the basis of temperature.}}</ref> For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows{{how?|date=August 2018}} a distinguished temperature scale, which defines an absolute, [[thermodynamic temperature]], independent of the properties of any particular reference thermometric body.<ref>[[Mark Zemansky|Zemansky, M.W.]] (1968), pp. 207–209.</ref><ref>Quinn, T.J. (1983), p. 8.</ref><br />
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The zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body. For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows a distinguished temperature scale, which defines an absolute, thermodynamic temperature, independent of the properties of any particular reference thermometric body.<br />
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美国热力学第零定律协会在其通常的简短声明中允许承认，相对于热平衡的两个物体具有相同的温度，特别是测试物体具有与参考温度测量物体相同的温度。对于一个物体和另一个物体之间的热平衡，存在着无限多的经验温度标度，通常分别取决于特定参考温度标度的性质。第二定律允许区分温度标度，它定义了一个绝对的热力学温度，与任何特定的参考温度体的性质无关。<br />
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==Various statements of the law==<br />
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==Various statements of the law==<br />
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各种各样的法律陈述<br />
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The second law of thermodynamics may be expressed in many specific ways,<ref name=MIT>{{cite web|title=Concept and Statements of the Second Law|url=http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node37.html|accessdate=2010-10-07 |publisher=web.mit.edu}}</ref> the most prominent classical statements{{sfnp|Lieb|Yngvason|1999}} being the statement by [[Rudolf Clausius]] (1854), the statement by [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] (1851), and the statement in axiomatic thermodynamics by [[Constantin Carathéodory]] (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.{{sfnp|Rao|2004|p=213}}<br />
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The second law of thermodynamics may be expressed in many specific ways, the most prominent classical statements being the statement by Rudolf Clausius (1854), the statement by Lord Kelvin (1851), and the statement in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.<br />
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热力学第二定律可以用许多特定的方式来表达，最突出的经典陈述是 Rudolf Clausius (1854)的陈述，Kelvin 勋爵(1851)的陈述，以及康斯坦丁·卡拉西奥多里(1909)在公理化热力学中的陈述。这些陈述用一般的物理术语描述法律，引用某些过程的不可能性。克劳修斯和开尔文陈述被证明是等价的。<br />
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===Carnot's principle===<br />
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===Carnot's principle===<br />
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卡诺原理<br />
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The historical origin<ref>[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986).</ref> of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a [[Carnot heat engine]], fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the [[caloric theory]] of heat was seriously considered, before the recognition of the [[first law of thermodynamics]], and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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The historical origin of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a Carnot heat engine, fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the caloric theory of heat was seriously considered, before the recognition of the first law of thermodynamics, and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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热力学第二定律的历史起源源于卡诺的原则。它指的是一个卡诺热机周期，虚拟地以被称为准静态的极慢极限模式运行，因此热量和功量在子系统之间传递，这些子系统始终处于它们自己的内部热力学平衡状态。卡诺发动机是热机效率工程师特别感兴趣的理想设备。卡诺认识到卡诺原理的时候，正是热量热理论被认真考虑的时候，正是能量守恒定律理论被认识之前，正是熵概念的数学表达之前。根据第一定律的解释，它在物理上等同于热力学第二定律，并且在今天仍然有效。卡诺最初的论点是从卡路里理论的观点提出的，直到能量守恒定律的发现。以下是他书中的一些例子:<br />
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::...''wherever there exists a difference of temperature, motive power can be produced.''<ref>Carnot, S. (1824/1986), p. 51.</ref><br />
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...wherever there exists a difference of temperature, motive power can be produced.<br />
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只要有温差，就能产生动力。<br />
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::The production of motive power is then due in steam engines not to an actual consumption of caloric, but ''to its transportation from a warm body to a cold body ...''<ref>Carnot, S. (1824/1986), p. 46.</ref><br />
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The production of motive power is then due in steam engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body ...<br />
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动力的产生不是由于蒸汽机实际消耗的热量，而是由于它从一个温暖的物体运输到一个寒冷的物体..。<br />
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::''The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.''<ref>Carnot, S. (1824/1986), p. 68.</ref><br />
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The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.<br />
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热的动力与实现热的媒介无关，热的数量完全取决于两个物体之间的温度，最后是热量的传递。<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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用现代术语来说，卡诺的原则可能更为准确:<br />
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::The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<ref>[[Clifford Truesdell|Truesdell, C.]] (1980), Chapter 5.</ref><ref>Adkins, C.J. (1968/1983), pp. 56–58.</ref><ref>Münster, A. (1970), p. 11.</ref><ref>Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998), pp.67–75.</ref><ref>Lebon, G., Jou, D., Casas-Vázquez, J. (2008), p. 10.</ref><ref>Eu, B.C. (2002), pp. 32–35.</ref><br />
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The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<br />
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准静态或可逆卡诺循环的效率只取决于两个热源的温度，而且无论工作物质是什么，效率是相同的。以这种方式运行的卡诺发动机是使用这两种温度的最有效的热机。<br />
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===Clausius statement===<br />
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===Clausius statement===<br />
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克劳修斯声明<br />
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The German scientist [[Rudolf Clausius]] laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work.{{sfnp|Clausius|1850}} His formulation of the second law, which was published in German in 1854, is known as the ''Clausius statement'':<br />
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The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work. His formulation of the second law, which was published in German in 1854, is known as the Clausius statement:<br />
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1850年，德国科学家 Rudolf Clausius 通过研究热传递和功之间的关系，为热力学第二定律实验室奠定了基础。他在1854年用德语发表的第二定律的提法被称为克劳修斯声明:<br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.{{sfnp|Clausius|1854|p=86}}</blockquote><br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.</blockquote><br />
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热量不可能从一个较冷的物体传递到一个较热的物体，除非同时发生一些其他的变化。 / blockquote<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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克劳修斯的陈述使用了“热量通道”的概念。正如通常在热力学的讨论，这意味着“能量作为热量的净转移” ，并没有提到促成转移的一种方式和其他。<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of [[refrigeration]], for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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如果不对系统进行外部功，热就不能自发地从冷区流向热区，这一点从制冷的普通经验中可以看出。在冰箱中，热量从冷到热，但只有在外部介质——制冷系统的强制作用下才会发生变化。<br />
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===Kelvin statements===<br />
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===Kelvin statements===<br />
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开尔文陈述<br />
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[[William Thomson, 1st Baron Kelvin|Lord Kelvin]] expressed the second law in several wordings.<br />
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Lord Kelvin expressed the second law in several wordings.<br />
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开尔文勋爵用几个字表达了第二定律。<br />
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::It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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没有任何外部机构的帮助，自动机器是不可能把热量从一个物体传递到另一个物体的较高温度。<br />
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::It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.{{sfnp|Thomson|1851}}<br />
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It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.<br />
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通过无生命物质的作用，把物质冷却到周围最冷的物体的温度以下，从物质的任何部分获得机械效应是不可能的。<br />
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===Equivalence of the Clausius and the Kelvin statements===<br />
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===Equivalence of the Clausius and the Kelvin statements===<br />
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克劳修斯和开尔文陈述的等价性<br />
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[[Image:Deriving Kelvin Statement from Clausius Statement.svg|thumb|Derive Kelvin Statement from Clausius Statement]]<br />
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Derive Kelvin Statement from Clausius Statement<br />
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从克劳修斯陈述推导出开尔文陈述<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed [[Carnot engine]] as shown by the figure. The [[Heat engine#Efficiency|efficiency]] of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the [[first law of thermodynamics]], as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the figure. The efficiency of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the first law of thermodynamics, as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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假设有一个引擎违反了开尔文定理: 也就是说，这个引擎以循环的方式吸收热量并将其转化为功，没有任何其他结果。现在配对与反向卡诺发动机如图所示。普通热机的效率为，反向热机的效率为1 / 。这对组合发动机的净效应和唯一效应是将热量从较冷的贮存器转移到较热的贮存器，这违反了克劳修斯的说法。(这是能量守恒定律的结果，因为系统的总能量保持不变，数学文本{输入} + 文本{输出}0暗示 q-frac { eta }-qc / math，所以数学 qc 向左(frac { eta-1 right) / math)。因此，违反开尔文陈述意味着违反克劳修斯陈述，即。克劳修斯的说法暗示了凯尔文的说法。我们可以用类似的方式证明开尔文陈述暗示了克劳修斯陈述，因此两者是等价的。<br />
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===Planck's proposition===<br />
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===Planck's proposition===<br />
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普朗克的命题<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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普朗克提出了直接来自经验的如下命题。这有时被认为是他对第二定律的陈述，但他认为这是第二定律推导的起点。<br />
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::It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 86.</ref><ref>Roberts, J.K., Miller, A.R. (1928/1960), p. 319.</ref><br />
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It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<br />
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建造一台发动机是不可能的，这台发动机将在一个完整的循环中工作，并且除了提高重量和冷却热储以外不会产生任何效果。<br />
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===Relation between Kelvin's statement and Planck's proposition 开尔文表述与普朗克命题的关系===<br />
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It is almost customary in textbooks to speak of the "[[Kelvin-Planck statement]]" of the law, as for example in the text by [[Dirk ter Haar|ter Haar]] and [[Harald Wergeland|Wergeland]].<ref>[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966), p. 17.</ref><br />
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It is almost customary in textbooks to speak of the "Kelvin-Planck statement" of the law, as for example in the text by ter Haar and Wergeland.<br />
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教科书中几乎惯常提及该定律的“'''开尔文-普朗克表述 Kelvin-Planck Statement'''” ，例如'''德克·特哈尔 Diek ter Haar''' 和'''哈拉尔德·沃格兰 Harald Wergeland''' 在文中就是这样表述的。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）Diek ter Haar 中间是小写的<br />
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The '''Kelvin–Planck statement''' (or the ''heat engine statement'') of the second law of thermodynamics states that <br />
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The Kelvin–Planck statement (or the heat engine statement) of the second law of thermodynamics states that <br />
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热力学第二定律的开尔文-普朗克表述（或称“'''热机表述 Heat Engine Statement'''”）指出<br />
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::It is impossible to devise a [[thermodynamic cycle|cyclically]] operating device, the sole effect of which is to absorb energy in the form of heat from a single [[heat reservoir|thermal reservoir]] and to deliver an equivalent amount of [[Work (physics)|work]].<ref name="Rao">{{cite book|last=Rao|first=Y. V. C.|title=Chemical Engineering Thermodynamics|publisher=Universities Press|isbn=978-81-7371-048-3|page=158|year=1997}}</ref><br />
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It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.<br />
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设计一种唯一效果是从单一热源吸收热量并提供等量的功的循环运行装置是不可能的。<br />
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===Planck's statement 普朗克表述===<br />
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Planck stated the second law as follows.<br />
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Planck stated the second law as follows.<br />
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普朗克表述第二定律如下。<br />
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::Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<ref name="Planck 100">[[Max Planck|Planck, M.]] (1897/1903), p. 100.</ref><ref name="Planck 463">[[Max Planck|Planck, M.]] (1926), p. 463, translation by Uffink, J. (2003), p. 131.</ref><ref name="Roberts & Miller 382">Roberts, J.K., Miller, A.R. (1928/1960), p. 382. This source is partly verbatim from Planck's statement, but does not cite Planck. This source calls the statement the principle of the increase of entropy.</ref><br />
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Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<br />
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自然界中发生的任一过程都是沿参与其中的所有物体的熵之和增加的方向进行的。在某些限制下——即对于可逆过程，熵的总和保持不变。<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for ''irreversible phenomena''.<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for irreversible phenomena.<br />
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与普朗克表述非常相似的是'''乌伦贝克 Uhlenbeck'''和'''福特 Ford'''关于不可逆现象的表述。<br />
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::... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<ref>[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963), p. 16.</ref><br />
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... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<br />
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在从一个平衡态到另一个平衡态的不可逆或自发的变化中（例如，当两个物体 A 和 B 接触时的温度平衡过程），熵总是增加。<br />
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===Principle of Carathéodory 卡拉西奥多里原理===<br />
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<!-- [[Caratheodory's principle]] redirects here --><br />
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—— caratheodory 的原理在这里重新定向——<br />
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[[Constantin Carathéodory]] formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<ref>[[Constantin Carathéodory|Carathéodory, C.]] (1909).</ref><br />
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Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<br />
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'''康斯坦丁·卡拉西奥多里 Constantin Carathéodory'''在纯数学公理的基础上进行了热力学公理化。他对第二定律的陈述被称为'''卡拉西奥多里原理 Principle of Carathéodory'''，可以这样表述:<br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.<ref>Buchdahl, H.A. (1966), p. 68.</ref></blockquote><br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.</blockquote><br />
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在绝热封闭系统的任意状态 S 附近，总有从 S 出发不可达的状态。<br />
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With this formulation, he described the concept of [[adiabatic accessibility]] for the first time and provided the foundation for a new subfield of classical thermodynamics, often called [[Ruppeiner geometry|geometrical thermodynamics]]. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic [[process function]], in other words, <math>\delta Q=TdS</math>.<ref name="Sychev1991">{{cite book |last=Sychev |first=V. V. |title=The Differential Equations of Thermodynamics |year=1991 |publisher=Taylor & Francis |isbn=978-1-56032-121-7}}</ref> {{clarify|date=February 2014}}<br />
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With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, <math>\delta Q=TdS</math>. <br />
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公理化处理后，他第一次描述了'''绝热可达性 Adiabatic Accessibility'''的概念，并为经典热力学的一个新的子领域，即通常所说的'''几何热力学 Geometrical Thermodynamics'''奠定了基础。由卡拉西奥多里原理可以推出，准静态转移的热量值是一个可积过程函数，即<math>\delta Q=TdS</math>。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）准静态转移的热量值是一个可积过程函数 存疑<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.<ref name="Munster 45">Münster, A. (1970), p. 45.</ref>{{sfnp|Lieb|Yngvason|1999|p=49}}<ref name="Planck 1926">[[Max Planck|Planck, M.]] (1926).</ref><ref>Buchdahl, H.A. (1966), p. 69.</ref> {{clarify|date=February 2014}}<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium. <br />
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尽管在教科书几乎惯称卡拉西奥多里原理表述了第二定律，并认为其与克劳修斯表述或开尔文-普朗克表述等价，但事实并非如此。为了得到第二定律的所有内容，需要对卡拉西奥多里原理补充普朗克表述，即等量功总是增加一个最初处于自身内部热力学平衡的封闭系统的内部能量。<br />
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===Planck's principle 普朗克原理===<br />
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In 1926, [[Max Planck]] wrote an important paper on the basics of thermodynamics.<ref name="Planck 1926"/><ref>Uffink, J. (2003), pp. 129–132.</ref> He indicated the principle<br />
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In 1926, Max Planck wrote an important paper on the basics of thermodynamics. He indicated the principle<br />
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1926年，'''马克斯·普朗克 Max Planck'''写了一篇关于热力学基础的重要论文。他指出了以下原理<br />
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::The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<ref name="Munster 45"/>{{sfnp |Lieb|Yngvason|1999|p=49}}<br />
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The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<br />
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一个封闭系统的内部能量因绝热过程增加，在整个过程中，系统的体积保持不变。<br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work."<ref>[[Clifford Truesdell|Truesdell, C.]], Muncaster, R.G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 15.</ref> Planck wrote: "The production of heat by friction is irreversible."<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 81.</ref><ref>[[Max Planck|Planck, M.]] (1926), p. 457, Wikipedia editor's translation.</ref><br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work." Planck wrote: "The production of heat by friction is irreversible."<br />
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这个公式没有提到热，没有提到温度，甚至没有提到熵，也不一定隐含地依赖于这些概念，但它暗示了第二定律的内容。一个密切相关的表述为，“摩擦力从来不做正功。”普朗克写道: “摩擦生热是不可逆的。”<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above.<ref>Lieb, E.H., Yngvason, J. (2003), p. 149.</ref> It is relevant that for a system at constant volume and [[Mole (unit)|mole numbers]], the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above. It is relevant that for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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更不用说熵了，这个普朗克定理是用物理术语来表述的。它与上面给出的开尔文表述密切相关。相关的是，对于恒定体积和摩尔数的系统，熵是内能的单调函数。然而，普朗克的这个原理实际上并不是普朗克对第二定律的首选表述（见前面小节），而是依赖于熵的概念。<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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Borgnakke 和 Sonntag 提出的以下表述在某种意义上是对普朗克原理的补充。他们没有将其作为第二定律的完整表述:<br />
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::... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<ref>Borgnakke, C., Sonntag., R.E. (2009), p. 304.</ref><br />
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... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<br />
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只有一种方法可以减少[闭合]系统的熵——从系统中转移热量。<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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与普朗克之前提出的原理不同，这一原理明确地用熵的变化来表示。从系统中去除物质也可以减少系统的熵。<br />
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===Statement for a system that has a known expression of its internal energy as a function of its extensive state variables 一个其内能有已知表达式（其扩展状态变量的函数）的系统的表述===<br />
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The second law has been shown to be equivalent to the [[internal energy]] ''U'' being a weakly [[convex function]], when written as a function of extensive properties (mass, volume, entropy, ...).<ref>{{cite book |last1=van Gool |first1=W. |last2=Bruggink |first2=J.J.C. (Eds) |url= |title=Energy and time in the economic and physical sciences |publisher=North-Holland |year=1985 |pages=41–56 |quote= |isbn=978-0-444-87748-2}}</ref><ref>{{Cite journal | last1 = Grubbström | first1 = Robert W. | doi = 10.1016/j.apenergy.2007.01.003 | title = An Attempt to Introduce Dynamics Into Generalised Exergy Considerations| journal = Applied Energy| volume = 84| issue = 7–8 | pages = 701–718 | year = 2007}}</ref> {{clarify|date=February 2014}}<br />
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The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...). <br />
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第二定律被证明等价于弱凸函数内能 U （写成广泛性质（质量，体积，熵，...）的函数时）。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）存疑<br />
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==Corollaries 推论==<br />
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===Perpetual motion of the second kind 第二类永动机===<br />
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{{main article|Perpetual motion}}<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of [[first law of thermodynamics]] by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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在第二定律建立之前，许多志在发明永动机的人试图通过提取环境的巨大内能作为机器动力来突破'''热力学第一定律 First Law of Thermodynamics''' 的限制。这种机器被称为“'''第二类永动机 Perpetual Motion Machine of the Second Kind'''”。第二定律表明这种机器是不可能的。<br />
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===Carnot theorem 卡诺定理===<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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卡诺定理（1824）是一条限制任何可能的发动机的最大效率的原理。效率完全取决于热库和冷库之间的温差。卡诺定理指出:<br />
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*All irreversible heat engines between two heat reservoirs are less efficient than a [[Carnot engine]] operating between the same reservoirs.<br />
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*All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs.<br />
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* 所有不可逆热机的效率低于在两相同热源之间工作的'''卡诺机 Carnot engine'''。<br />
* 所有可逆热机的效率与在两相同热源之间工作的卡诺机相等。<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as [[thermodynamic reversibility]]. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the [[Carnot cycle]]'s reversibility and was condemned to be less efficient.<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the Carnot cycle's reversibility and was condemned to be less efficient.<br />
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在他的理想模型中，热转换成功可以通过逆转循环的运动而恢复，这个概念后来被称为'''热力学可逆性 Thermodynamic Reversibility'''。然而，卡诺进一步假定，一些热量损失了，并没有转化为机械功。因此，没有一个真实的热机能够实现'''卡诺循环 Carnot cycle'''的可逆性，并且被认为效率较低。<br />
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Though formulated in terms of caloric (see the obsolete [[caloric theory]]), rather than [[entropy]], this was an early insight into the second law.<br />
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Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law.<br />
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该理论尽管是用热量表述的（见被取代的'''热质说 Caloric Theory'''），而不是熵，但是它是对第二定律的早期认识。<br />
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===Clausius inequality 克劳修斯不等式===<br />
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The [[Clausius theorem]] (1854) states that in a cyclic process<br />
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The Clausius theorem (1854) states that in a cyclic process<br />
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克劳修斯定理（1854）指出，在一个循环的过程中<br />
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: <math>\oint \frac{\delta Q}{T} \leq 0.</math><br />
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The equality holds in the reversible case<ref>[http://scienceworld.wolfram.com/physics/ClausiusTheorem.html ''Clausius theorem''] at [[Wolfram Research]]</ref> and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function [[entropy]]. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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The equality holds in the reversible case and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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等号在可逆情况下成立，严格不等号在不可逆情况下成立。可逆情况下引入状态函数熵。这是因为在循环过程中，状态函数的变化为零。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）末句存疑<br />
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===Thermodynamic temperature 热力学温度===<br />
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{{main article|Thermodynamic temperature}}<br />
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For an arbitrary heat engine, the efficiency is:<br />
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For an arbitrary heat engine, the efficiency is:<br />
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对于任意热机，效率为:<br />
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: <math>\eta = \frac {W_n}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H} \qquad (1)</math><br />
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where ''W''<sub>n</sub> is for the net work done per cycle. Thus the efficiency depends only on q<sub>C</sub>/q<sub>H</sub>.<br />
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其中 ''W''<sub>n</sub> 表示每个循环所做的净功。因此效率只取决于 q<sub>C</sub>/q<sub>H</sub>。<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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卡诺定理指出，所有在相同热源间运行的可逆发动机的效率是相同的。<br />
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Thus, any reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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Thus, any reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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因此，任何在温度''T''<sub>1</sub>和''T''<sub>2</sub>之间运行的可逆热机必须具有相同的效率，也就是说，效率只是温度的函数:<br />
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<math>\frac{q_C}{q_H} = f(T_H,T_C)\qquad (2).</math><br />
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In addition, a reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between ''T''<sub>1</sub> and another (intermediate) temperature ''T''<sub>2</sub>, and the second between ''T''<sub>2</sub> and''T''<sub>3</sub>. This can only be the case if<br />
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In addition, a reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between T<sub>1</sub> and another (intermediate) temperature T<sub>2</sub>, and the second between T<sub>2</sub> andT<sub>3</sub>. This can only be the case if<br />
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另外，在温度 ''T''<sub>1</sub> 和 ''T''<sub>3</sub> 之间工作的可逆热机必须具有与由分别在温度''T''<sub>1</sub> 和（中间）温度 ''T''<sub>2</sub>之间、在 ''T''<sub>2</sub> 和 ''T''<sub>3</sub> 之间的两个循环组成的系统效率相同。只有下式成立才会出现这种情况:<br />
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: <math>f(T_1,T_3) = \frac{q_3}{q_1} = \frac{q_2 q_3} {q_1 q_2} = f(T_1,T_2)f(T_2,T_3).</math><br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the [[triple point]] of water. Then for any ''T''<sub>2</sub> and ''T''<sub>3</sub>,<br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the triple point of water. Then for any T<sub>2</sub> and T<sub>3</sub>,<br />
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现在考虑如下情形，<math>T_1</math> 是一个固定的参考温度: 水的三相点的温度。则对于任意 T<sub>2</sub> 和 T<sub>3</sub>,<br />
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: <math>f(T_2,T_3) = \frac{f(T_1,T_3)}{f(T_1,T_2)} = \frac{273.16 \cdot f(T_1,T_3)}{273.16 \cdot f(T_1,T_2)}.</math><br />
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Therefore, if thermodynamic temperature is defined by<br />
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Therefore, if thermodynamic temperature is defined by<br />
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因此，如果热力学温度被定义为<br />
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: <math>T = 273.16 \cdot f(T_1,T) \,</math><br />
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then the function ''f'', viewed as a function of thermodynamic temperature, is simply<br />
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then the function f, viewed as a function of thermodynamic temperature, is simply<br />
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那么函数 ''f'' 作为热力学温度的函数，为<br />
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: <math>f(T_2,T_3) = \frac{T_3}{T_2},</math><br />
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and the reference temperature ''T''<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be used{{snd}}the choice here corresponds to the [[Kelvin]] scale.)<br />
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and the reference temperature T<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be usedthe choice here corresponds to the Kelvin scale.)<br />
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参考温度 ''T''<sub>1</sub> 的值为273.16。（任何参考温度和任何正值均可用——此处的选择对应开尔文标度。）<br />
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===Entropy 熵===<br />
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{{main article|Entropy (classical thermodynamics)}}<br />
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According to the [[Clausius theorem|Clausius equality]], for a reversible process<br />
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According to the Clausius equality, for a reversible process<br />
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根据克劳修斯定理，对可逆过程有<br />
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: <math>\oint \frac{\delta Q}{T}=0</math><br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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这意味着线积分 <math>\int_L \frac{\delta Q}{T}</math> 对于可逆过程是路径无关的。<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer<ref name="Schmidt-Rohr 14" /> satisfies<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer satisfies<br />
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所以我们可以定义一个叫做熵的状态函数 S，对于可逆过程或者纯热传递满足<br />
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: <math>dS = \frac{\delta Q}{T} \!</math><br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the [[third law of thermodynamics]], which states that ''S'' = 0 at [[absolute zero]] for perfect crystals.<br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the third law of thermodynamics, which states that S = 0 at absolute zero for perfect crystals.<br />
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据此，只有对上述公式进行积分，才能得到熵的差值。为了获得绝对值，我们需要'''热力学第三定律 Third Law of Thermodynamics'''，它指出绝对零度下完美晶体的 ''S'' = 0。<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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对于任意不可逆过程，由于熵是一个状态函数，我们总是可以将初始状态和最终状态与一个虚拟的可逆过程联系起来，并在这条路径上积分以计算熵的差值。<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the [[Clausius inequality]] on this loop,<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the Clausius inequality on this loop,<br />
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现在把可逆过程逆过来，将其与上述不可逆过程结合。把克劳修斯不等式应用到这个循环,<br />
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: <math>-\Delta S+\int\frac{\delta Q}{T}=\oint\frac{\delta Q}{T}< 0</math><br />
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Thus,<br />
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Thus,<br />
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故,<br />
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: <math>\Delta S \ge \int \frac{\delta Q}{T} \,\!</math><br />
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where the equality holds if the transformation is reversible.<br />
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where the equality holds if the transformation is reversible.<br />
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如果变换可逆，等号成立。<br />
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Notice that if the process is an [[adiabatic process]], then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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Notice that if the process is an adiabatic process, then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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注意，若该过程是一个'''绝热过程 Adiabatic Process'''，则<math>\delta Q=0</math>，故<math>\Delta S\ge 0</math>。<br />
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===Energy, available useful work===<br />
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===Energy, available useful work===<br />
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能量，有用的工作<br />
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{{See also|Exergy}}<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an ''unlimited'' heat reservoir at temperature ''T<sub>R</sub>'' and pressure ''P<sub>R</sub>'' {{snd}}so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain ''T<sub>R</sub>''; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain ''P<sub>R</sub>''.<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an unlimited heat reservoir at temperature T<sub>R</sub> and pressure P<sub>R</sub> so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain T<sub>R</sub>; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain P<sub>R</sub>.<br />
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考虑将第二定律应用于孤立系统(称为整体系统或宇宙)的情形，是一个重要的、具有启发性的理想情形。孤立系统由两部分组成: 感兴趣的子系统和子系统的环境。这些环境被想象为如此之大，以至于它们可以被认为是一个温度为 t 子 r / sub 和压力为 p 子 r / sub 的无限热源，因此无论有多少热量被转移到(或来自)子系统，周围环境的温度将保持 t 子 r / sub; 无论子系统的体积扩大(或收缩)多少，周围环境的压力将保持 p 子 r / sub。<br />
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Whatever changes to ''dS'' and ''dS<sub>R</sub>'' occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy ''S<sub>tot</sub>'' of the isolated total system must not decrease:<br />
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Whatever changes to dS and dS<sub>R</sub> occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy S<sub>tot</sub> of the isolated total system must not decrease:<br />
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根据第二定律，孤立总体系统的熵 s 子系统 / 子系统的熵不能减小，无论子系统和周围环境的熵对 dS 和 dS 子系统 r / 子系统发生什么变化:<br />
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: <math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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<math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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数学 dS + dS r ge 0 / math<br />
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According to the [[first law of thermodynamics]], the change ''dU'' in the internal energy of the sub-system is the sum of the heat ''δq'' added to the sub-system, ''less'' any work ''δw'' done ''by'' the sub-system, ''plus'' any net chemical energy entering the sub-system ''d ∑μ<sub>iR</sub>N<sub>i</sub>'', so that:<br />
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According to the first law of thermodynamics, the change dU in the internal energy of the sub-system is the sum of the heat δq added to the sub-system, less any work δw done by the sub-system, plus any net chemical energy entering the sub-system d ∑μ<sub>iR</sub>N<sub>i</sub>, so that:<br />
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根据能量守恒定律，子系统内部能量的变化 dU 是子系统内部能量 q 的总和，减去子系统所做的任何功，再加上进入子系统 d ∑ sub iR / sub n sub i / sub 的任何净化学能，因此:<br />
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: <math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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<math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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数学 dU delta q- delta w + d ( sum mu iR } n i) ，/ math<br />
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where μ<sub>iR</sub> are the [[chemical potential]]s of chemical species in the external surroundings.<br />
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where μ<sub>iR</sub> are the chemical potentials of chemical species in the external surroundings.<br />
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其中亚 iR / 亚是外部环境中化学物种的化学势。<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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现在热量离开储存器进入子系统是<br />
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: <math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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<math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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其中我们首先使用了经典热力学中熵的定义(在统计热力学中，熵变、温度和吸收热量之间的关系可以推导出来) ，然后从上面推导出第二定律的不等式。<br />
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It therefore follows that any net work ''δw'' done by the sub-system must obey<br />
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It therefore follows that any net work δw done by the sub-system must obey<br />
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因此，子系统所做的任何净功都必须服从<br />
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: <math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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<math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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数学增量 w le-dU + tds + sum mu { iR } dN i，/ math<br />
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It is useful to separate the work ''δw'' done by the subsystem into the ''useful'' work ''δw<sub>u</sub>'' that can be done ''by'' the sub-system, over and beyond the work ''p<sub>R</sub> dV'' done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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It is useful to separate the work δw done by the subsystem into the useful work δw<sub>u</sub> that can be done by the sub-system, over and beyond the work p<sub>R</sub> dV done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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将子系统所做的功与子系统所能做的功分离为子系统所能做的有用功，超过了子系统对外界压力所做的功，给出了子系统所能做的有用功(火用)的下列关系式:<br />
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: <math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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<math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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D (u-t r s + p r v-sum { iR } n i) ，/ math<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the ''availability'' or ''[[exergy]]'' ''E'' of the subsystem,<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the availability or exergy E of the subsystem,<br />
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可以很方便地将子系统的右侧定义为热动力位能的精确衍生物，称为子系统的可用性或火用 e,<br />
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: <math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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<math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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数学 e u-t r s + p r v- sum { iR } n i / math<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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因此，第二定律意味着，对于任何可以简单地被视为分为一个子系统和一个与之接触的无限温度和压力储存器的过程,<br />
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: <math> dE + \delta w_u \le 0 \, </math><br />
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<math> dE + \delta w_u \le 0 \, </math><br />
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数学德 delta u le 0，/ math<br />
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i.e. the change in the subsystem's exergy plus the useful work done ''by'' the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done ''on'' the system) must be less than or equal to zero.<br />
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i.e. the change in the subsystem's exergy plus the useful work done by the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done on the system) must be less than or equal to zero.<br />
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也就是。子系统(火用)的变化加上子系统所做的有用功(或者，子系统(火用)的变化不包括任何功，除了压力贮存器所做的功之外，在系统中所做的功)必须小于或等于零。<br />
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In sum, if a proper ''infinite-reservoir-like'' reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in ''E'' for an irreversible process and no change for a reversible process.<br />
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In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in E for an irreversible process and no change for a reversible process.<br />
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总之，如果选择一个合适的类似于无限库的参考状态作为现实世界中的系统环境，那么第二定律预测一个不可逆性的 e 值会减少，而一个可逆过程的 e 值不会变化。<br />
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: <math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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<math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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数学 d = 0 = 0 = 0 = 0 = 0<br />
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This expression together with the associated reference state permits a [[design engineer]] working at the macroscopic scale (above the [[thermodynamic limit]]) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (''Also, see [[process engineer]]''). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (''See [[Exergy efficiency|second law efficiency]]''.)<br />
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This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (Also, see process engineer). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (See second law efficiency.)<br />
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这个表达式连同相关的参考状态允许在宏观的设计工程师利用第二定律，而不需要直接测量或考虑整个孤立系统的熵变。(另外，请参阅流程工程师)。这些变化已经被考虑，假设被考虑的系统可以达到平衡的参考状态而不改变参考状态。还可以找到一个过程或过程集合的效率，将其与可逆的理想状态相比较<br />
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This approach to the Second Law is widely utilized in [[engineering]] practice, [[environmental accounting]], [[systems ecology]], and other disciplines.<br />
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This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines.<br />
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第二定律的这种方法被广泛应用于工程实践、环境会计、系统生态学和其他学科。<br />
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===The second law in chemical thermodynamics===<br />
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===The second law in chemical thermodynamics===<br />
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化学热力学的第二定律<br />
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For a [[spontaneous process|spontaneous chemical process]] in a closed system at constant temperature and pressure without non-''PV'' work, the Clausius inequality Δ''S > Q/T<sub>surr</sub>'' transforms into a condition for the change in [[Gibbs free energy]] <br />
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For a spontaneous chemical process in a closed system at constant temperature and pressure without non-PV work, the Clausius inequality ΔS > Q/T<sub>surr</sub> transforms into a condition for the change in Gibbs free energy <br />
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对于一个封闭系统中的自发化学过程，在没有非 pv 功的情况下，在恒定温度和压力下，将克劳修斯不等式 s q / t sub surr / sub 转化为吉布斯自由能变化的条件<br />
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: <math>\Delta G < 0 </math><br />
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数学 Delta g 0 / 数学<br />
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or d''G'' < 0. For a similar process at constant temperature and volume, the change in [[Helmholtz free energy]] must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products.<ref name="Oxtoby8th"/> <ref name="MortimerBook"></ref> The chemical equilibrium condition at constant ''T'' and ''p'' without electrical work is d''G'' = 0.<br />
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or dG < 0. For a similar process at constant temperature and volume, the change in Helmholtz free energy must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products. The chemical equilibrium condition at constant T and p without electrical work is dG = 0.<br />
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或者 dG 0。对于一个相似的过程，在恒定的温度和体积下，亥姆霍兹自由能的变化一定是负的，这是 Delta a 0 / math。因此，自由能(g 或 a)变化的负值是过程自发的必要条件。这是热力学第二定律在化学中最有用的形式，其中自由能的变化可以通过列表的生成焓和反应物和产物的标准摩尔熵来计算。在 t 和 p 不变的情况下，化学平衡条件是 dG 0。<br />
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==History==<br />
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==History==<br />
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历史<br />
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{{See also|History of entropy}}<br />
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[[File:Sadi Carnot.jpeg|thumb|upright|Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique]].]]<br />
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Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique.]]<br />
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尼古拉·莱昂纳尔·萨迪·卡诺穿着传统的巴黎综合理工学院制服<br />
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The first theory of the conversion of heat into mechanical work is due to [[Nicolas Léonard Sadi Carnot]] in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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The first theory of the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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第一个把热转化为机械功的理论是在1824年由尼古拉·莱昂纳尔·萨迪·卡诺提出的。他是第一个正确认识到这种转换的效率取决于发动机和环境之间的温差。<br />
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Recognizing the significance of [[James Prescott Joule]]'s work on the conservation of energy, [[Rudolf Clausius]] was the first to formulate the second law during 1850, in this form: heat does not flow ''spontaneously'' from cold to hot bodies. While common knowledge now, this was contrary to the [[caloric theory]] of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law during 1850, in this form: heat does not flow spontaneously from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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认识到詹姆斯·普雷斯科特·焦耳在能量守恒方面工作的重要性，Rudolf Clausius 在1850年第一个提出了第二定律: 热不会自发地从冷物体流向热物体。虽然现在的常识是，这与当时流行的热量理论相反，热量理论认为热量是一种流体。从那里他能够推断出萨迪卡诺定律和熵的定义(1865年)。<br />
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Established during the 19th century, the [[Kelvin-Planck statement|Kelvin-Planck statement of the Second Law]] says, "It is impossible for any device that operates on a [[cyclic process|cycle]] to receive heat from a single [[heat reservoir|reservoir]] and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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Established during the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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建立于19世纪的开尔文-普朗克第二定律(Kelvin-Planck)说，“任何循环运行的设备都不可能从单个蓄热体接收热量，并产生净功。”这被证明相当于克劳修斯的陈述。<br />
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The [[ergodic hypothesis]] is also important for the [[Boltzmann]] approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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遍历假设对玻尔兹曼方法也很重要。它说，在很长一段时间内，在具有相同能量的微观态的相空间的某些区域所花费的时间与这个区域的体积成正比，即。在很长一段时间内，所有可访问的微观状态的可能性都是一样的。同样的，它表明时间平均值和系综平均值是相同的。<br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a [[Macroscopic bodies|macroscopic system]]. This doctrine is obsolescent.<ref>Denbigh, K.G., Denbigh, J.S. (1985). ''Entropy in Relation to Incomplete Knowledge'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-25677-1}}, pp. 43–44.</ref><ref>Grandy, W.T., Jr (2008). ''Entropy and the Time Evolution of Macroscopic Systems'', Oxford University Press, Oxford, {{ISBN|978-0-19-954617-6}}, pp. 55–58.</ref><ref name=Lambert>[http://entropysite.oxy.edu Entropy Sites — A Guide] Content selected by [[Frank L. Lambert]]</ref><br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a macroscopic system. This doctrine is obsolescent.<br />
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有一种传统的学说，从克劳修斯开始，认为熵可以被理解为宏观系统中的分子“无序”。这种学说已经过时了。<br />
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===Account given by Clausius===<br />
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===Account given by Clausius===<br />
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作者: 克劳修斯<br />
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[[File:Clausius-1.jpg|thumb|upright|Rudolf Clausius]]<br />
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Rudolf Clausius<br />
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鲁道夫 · 克劳修斯<br />
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In 1856, the German physicist [[Rudolf Clausius]] stated what he called the "second fundamental theorem in the [[mechanical theory of heat]]" in the following form:{{sfnp|Clausius|1867}}<br />
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In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:<br />
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1856年，德国物理学家鲁道夫 · 克劳修斯阐述了他所谓的“热力学理论中的第二个基本定理” ，其形式如下:<br />
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: <math>\int \frac{\delta Q}{T} = -N</math><br />
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<math>\int \frac{\delta Q}{T} = -N</math><br />
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- n / math<br />
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where ''Q'' is heat, ''T'' is temperature and ''N'' is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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其中 q 是热，t 是温度，n 是一个循环过程中所有未补偿的相变的“等价值”。后来，在1865年，克劳修斯将“等价值”定义为熵。紧接着这个定义，也就是在同一年，第二定律最著名的版本在4月24日苏黎世哲学学会的一次演讲中被宣读，在演讲的最后，克劳修斯总结道:<br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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宇宙的熵趋于最大<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. [[universe]], as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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这句话是第二定律中最广为人知的措辞。由于其语言的松散性，例如:。这个问题的答案是肯定的，因为这个问题的答案是肯定的。开放，封闭，或孤立，许多人认为这个简单的陈述意味着热力学第二定律适用于几乎所有可以想象的主题。这是不正确的; 这个陈述只是一个更加扩展和精确描述的简化版本。<br />
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In terms of time variation, the mathematical statement of the second law for an [[isolated system]] undergoing an arbitrary transformation is:<br />
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In terms of time variation, the mathematical statement of the second law for an isolated system undergoing an arbitrary transformation is:<br />
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就时间变化而言，经历任意变换的孤立系统第二定律的数学表述是:<br />
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: <math>\frac{dS}{dt} \ge 0</math><br />
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<math>\frac{dS}{dt} \ge 0</math><br />
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0 / math<br />
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where<br />
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where<br />
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在哪里<br />
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: ''S'' is the entropy of the system and<br />
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S is the entropy of the system and<br />
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S 是系统的熵<br />
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: ''t'' is [[time]].<br />
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t is time.<br />
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时间到了。<br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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!-- 可逆过程需要与周围环境保持平衡。这对于一个孤立的系统是不可能的。因此，关于可逆过程的讨论已经转移到对封闭系统的分析----<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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平衡后应用等式符号。建立孤立系统第二定律的另一种方法是:<br />
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: <math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math<br />
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with <math> \dot S_{i}</math> the sum of the rate of [[entropy production]] by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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with <math> \dot S_{i}</math> the sum of the rate of entropy production by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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用数学 s { i } / 数学表示系统内所有进程的产生熵之和。这种配方的优点是它显示了产生熵的效果。产生熵是一个非常重要的概念，因为它决定(限制)热机器的效率。与环境温度数学 t { a } / math 相乘，得到所谓的耗散能量数学 p { diss } t { a } dot s { i } / math。<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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封闭系统的第二定律(允许热量交换和移动边界，但不允许物质交换)的表达式是:<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math 来计算数学点 q } + 点 s { i } / math<br />
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Here<br />
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Here<br />
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给你<br />
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: <math>\dot Q</math> is the heat flow into the system<br />
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<math>\dot Q</math> is the heat flow into the system<br />
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数学 q / math 是进入系统的热流<br />
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: <math>T</math> is the temperature at the point where the heat enters the system.<br />
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<math>T</math> is the temperature at the point where the heat enters the system.<br />
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Math 是热量进入系统时的温度。<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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等式符号在只有可逆过程在系统内发生的情况下成立。如果发生不可逆过程(在实际操作系统中就是这种情况) ,-符号保持不变。如果系统有多处供热，我们必须求相应项的代数和。<br />
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For open systems (also allowing exchange of matter):<br />
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For open systems (also allowing exchange of matter):<br />
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对于开放系统(也允许物质交换) :<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math 表示数学点 q } + 点 s + 点 s { i } / math<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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这里的数学 s / math 是熵流进入系统，与物质流进入系统有关。它不应该与熵的时间导数混淆。如果物质在几个地方提供，我们必须取这些贡献的代数和。<br />
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==Statistical mechanics==<br />
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==Statistical mechanics==<br />
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统计力学<br />
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[[Statistical mechanics]] gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a [[microstate (statistical mechanics)|microstate]] of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/{{radic|''N''}} where ''N'' is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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Statistical mechanics gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a microstate of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/ where N is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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通过假设物质是由不断运动的原子和分子组成的，统计力学对第二定律给出了解释。系统中每个粒子的一组特定的位置和速度称为系统的微观状态，由于系统的不断运动，系统不断地改变其微观状态。统计力学假设，在平衡状态下，系统可能处于的每个微观状态发生的可能性是相等的，当这个假设被提出时，它直接导致了第二定律必须在统计意义上成立的结论。也就是说，第二定律平均成立，其中 n 是系统中粒子数的1 / 数量级的统计变化。在日常(宏观)情况下，违反第二定律的概率几乎为零。然而，对于粒子数量很少的系统，热力学参数，包括熵，可能显示出与第二定律预测的显著的统计偏差。经典热力学理论没有处理这些统计变量。<br />
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（by Jie XIANG) 在通过假定一种物质由不断运动的原子和分子组成的前提下，统计力学对热力学第二定律进行了解释。系统中每个粒子其特定的位置和速度组成为系统的微观状态，由于（原子和分子的）不断运动，系统也持续改变其微观状态。统计力学假设，在系统力平衡的情况下，其微观状态其实是等可能发生的，进而能够得出结论：热力学第二定律必须在统计学意义上成立。也就是说，第二定律将保持均值不变， 其统计变化为1 /√N，N为系统中的粒子数。在日常(宏观)情况下，违反热力学第二定律的概率几乎为零，但是，对于粒子数量非常少的系统，其热力学参数（包括熵）可能会显示出与第二定律所预测的明显统计偏差。然而经典热力学理论并不处理这些统计变量。<br />
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==Derivation from statistical mechanics==<br />
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==Derivation from statistical mechanics==<br />
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从统计力学导出<br />
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{{Further information|H-theorem}}<br />
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The first mechanical argument of the [[Kinetic theory of gases]] that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to [[James Clerk Maxwell]] in 1860;<ref>{{Cite journal | last1 = Gyenis | first1 = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G }}</ref> [[Ludwig Boltzmann]] with his [[H-theorem]] of 1872 also argued that due to collisions gases should over time tend toward the [[Maxwell–Boltzmann distribution]].<br />
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The first mechanical argument of the Kinetic theory of gases that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to James Clerk Maxwell in 1860; Ludwig Boltzmann with his H-theorem of 1872 also argued that due to collisions gases should over time tend toward the Maxwell–Boltzmann distribution.<br />
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气体动力学理论的第一个力学论证由詹姆斯·克拉克·麦克斯韦在1860年给出，指出分子碰撞引起温度均衡，因此整体趋向于平衡； 路德维希·玻尔兹曼在1872年提出的 H 定理也认为，气体由于碰撞应该随着时间的推移趋向于麦克斯韦-波兹曼分布。<br />
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Due to [[Loschmidt's paradox]], derivations of the Second Law have to make an assumption regarding the past, namely that the system is [[Correlation and dependence|uncorrelated]] at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a [[boundary condition]], and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the [[Big Bang]]), though [[Boltzmann brain|other scenarios]] have also been suggested.<ref name="Hawking AOT">{{cite journal|last=Hawking|first=SW|title=Arrow of time in cosmology|journal=Phys. Rev. D|year=1985|volume=32|issue=10|pages=2489–2495|doi=10.1103/PhysRevD.32.2489|pmid=9956019|bibcode = 1985PhRvD..32.2489H }}</ref><ref>{{cite book | last = Greene | first = Brian | authorlink = Brian Greene | title = The Fabric of the Cosmos | url = https://archive.org/details/fabricofcosmossp00gree | url-access = registration | publisher = Alfred A. Knopf | year = 2004 | page = [https://archive.org/details/fabricofcosmossp00gree/page/171 171] | isbn = 978-0-375-41288-2}}</ref><ref name=Lebowitz>{{cite journal|last=Lebowitz|first=Joel L.|title= Boltzmann's Entropy and Time's Arrow|journal=Physics Today|date=September 1993|volume=46|issue=9|pages=32–38|url=http://users.df.uba.ar/ariel/materias/FT3_2008_1C/papers_pdf/lebowitz_370.pdf|accessdate=2013-02-22|doi=10.1063/1.881363|bibcode = 1993PhT....46i..32L }}</ref><br />
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Due to Loschmidt's paradox, derivations of the Second Law have to make an assumption regarding the past, namely that the system is uncorrelated at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a boundary condition, and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the Big Bang), though other scenarios have also been suggested.<br />
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由于洛施密特悖论，第二定律的导出必须对过去做出一个假设，即系统在过去的某个时刻是不相关的；这样的假设允许进行简单的概率处理。这个假设通常被认为是一个边界条件，因此热力学第二定律最终是过去某个地方的初始条件的结果，可能是在宇宙的开始(大爆炸) 。也有人提出了其他假设。<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the [[Statistical mechanics#Fundamental postulate|fundamental postulate]], also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy.{{citation needed|date=August 2012}} The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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考虑到这些假设，在统计力学中，第二定律不是一个假设，而是统计力学基本假设的一个结果，也被称为等先验概率假设。这个基本假设表明，只要一个人清楚地知道，简单的概率论证只适用于未来，而对于过去，有辅助的信息来源告诉我们，它是低熵的。热力学第二定律的第一部分指出，热孤立系统的熵只能增加。如果我们把熵的概念限制在热平衡系统中，那么热力学第二定律的第一部分是等先验概率假设的一个显然结果。处于热平衡状态的孤立系统包含能量<math>E</math>的熵表示为：<br />
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: <math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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<math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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其中<math>\Omega\left(E\right)</math> 是处于 <math>E</math>和<math>E +\delta E</math这个小区间内的量子态数目。这里的 <math>\delta E</math> 是一个宏观上很小的固定能量区间。严格地说，这意味着熵取决于对<math>\delta E</math>的选择。然而在热力学极限下（例如无穷大系统的极限），狭义的熵(单位体积或单位质量的熵)不依赖于 <math>\delta E</math>。<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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假设我们有一个孤立系统，其宏观状态由许多变量描述。这些宏观变量可以是总体积、活塞在系统中的位置等。从而<math>\Omega</math>将取决于这些变量的值。如果某个变量不是固定的（我们不会在某个位置夹住活塞) ，那么因为在平衡状态下所有可到达状态的等几率到达的，平衡状态下的自由变量会使 <math>\Omega</math> 最大，因为这是平衡状态下最可能的情况。<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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如果该变量最初固定到某个值然后释放，当达到新的平衡时，变量将自我调整使得 <math>\Omega</math> 最大，这意味着熵将增加或保持不变(如果变量初始固定的值恰好是平衡值)。<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's [[H-theorem]], however, proves that the quantity {{math|''H''}} increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the quantity increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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假设我们初始位于一个平衡状态，突然移除了对一个变量的约束。我们做完这件事的时候，可达到的微观状态的数为<math>\Omega</math>，但是系统还没有达到平衡，所以系统处于某些可达到的状态的实际概率还不等于先验概率 <math>1/\Omega</math>。我们已经知道，最终的平衡状态相对于之前的平衡状态，熵会增加或者保持不变。然而，玻耳兹曼的H定理证明系统在不处于平衡态的期间，那个量作为时间的函数单调增加。<br />
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===Derivation of the entropy change for reversible processes===<br />
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===Derivation of the entropy change for reversible processes===<br />
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可逆过程熵变的推导<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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第二定律的第二部分指出，经历可逆过程的系统的熵变是由以下因素给出的:<br />
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: <math>dS =\frac{\delta Q}{T}</math><br />
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<math>dS =\frac{\delta Q}{T}</math><br />
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数学 dS frac delta q }{ t } / math<br />
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where the temperature is defined as:<br />
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where the temperature is defined as:<br />
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其中温度定义为:<br />
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:<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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[数学][[欧米茄][左(右)][右][数学][数学]<br />
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See [[Microcanonical ensemble|here]] for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the [[adiabatic theorem]] of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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请参阅此处对此定义的正当理由。假设系统有一些可以改变的外部参数 x。一般来说，系统的能量本征态将依赖于 x。根据量子力学的绝热定理，在系统哈密顿量无限缓慢变化的极限下，系统将保持在相同的定态，因此系统的能量会随着其所在定态的能量变化而变化。<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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广义力，x，对应于外部变量 x 的定义是，如果 x 增加一个量 dx，那么 math x dx / math 就是系统所做的功。例如，如果 x 是体积，那么 x 是压强。一个已知的系统的广义力在定态数学 e { r } / math 中给出:<br />
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: <math>X = -\frac{dE_{r}}{dx}</math><br />
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<math>X = -\frac{dE_{r}}{dx}</math><br />
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数学 x- frac { dE { r }{ dx } / math<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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由于系统可以在一个数学△ e / math 区间内的任意定态内，我们将系统的广义力定义为上述表达式的期望值:<br />
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: <math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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<math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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数学 x- 左 langle frac { dE { dx }右 rangle，/ math<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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为了计算平均值，我们通过计算在数学 y / math 和数学 y + delta y / math 之间的范围内，其中有多少数学 y / math 能量本征态有一个值来划分 ω 左(e 右) / math 能量本征态。把这个数字命名为 math Omega { y } left (e right) / math，我们有:<br />
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: <math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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<math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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Math Omega left (e right) sum { y } Omega { y } left (e right) ，/ math<br />
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The average defining the generalized force can now be written:<br />
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The average defining the generalized force can now be written:<br />
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定义广义力的平均值现在可以写成:<br />
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: <math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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<math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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数学 x frac {1} Omega 左(e 右) sum { y Omega { y }左(e 右) ，/ math<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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我们可以把它和熵对 x 在恒定能量 e 下的导数联系起来，如下所示。假设我们把 x 改成 x + dx。然后 math Omega left (e right) / math 将会改变，因为能量本征态依赖于 x，导致能量本征态进入或超出数学 e / math 和数学 e + delta e / math 之间的范围。让我们再次关注数学 y / math 和数学 y + delta y / math 之间的能量本征态。由于这些能量本征态的能量增加了 y dx，所有这些能量本征态在 e-y dx 到 e 之间的区间内从 e 以下移动到 e 以上。 有<br />
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: <math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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<math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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数学 n { y }左(e 右) frac { y }左(e 右)}{ delta e } y dx ，/ math<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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这样的能量本征态。如果数学 y dx leq delta e / math，所有这些能量本征态将移动到数学 e / math 和数学 e + delta e / math 之间的范围内，有助于数学 ω / math 的增加。从数学 e + △ e / math 下移到数学 e + △ e / math 上的能量本征态的个数是由数学 n { y }左(e + △ e 右) / math 给出的。区别在于<br />
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: <math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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<math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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数学 n { y }左(e 右)-n { y }左(e + delta e 右) ，/ math<br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
<br />
is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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因此是数学增长的净贡献欧米茄 / 数学。请注意，如果 y dx 大于 math delta e / math，那么将会有能量本征态从 e 下面移动到 e + delta e / math 上面。它们在数学 n { y } left (e right) / math 和数学 n { y } left (e + delta e right) / math 中都有计算，因此上述表达式在这种情况下也是有效的。<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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将上面的表达式表示为对 e 的导数，对 y 的求和得到表达式:<br />
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: <math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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数学左(部分欧米茄左(部分欧米茄右){ e }-和 y 左(部分欧米茄右){ x 左(部分欧米茄左(部分欧米茄右)}{ x 右) ，/ 数学<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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关于 x 的欧米茄 / 数学对数导数由以下方式给出:<br />
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: <math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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Math 左( frac { partial ln 左( Omega 右)}{部分 x 右){ e } beta x + 左( frac { partial x 右){ x } ，/ math<br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
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第一个术语是密集型的，即。它不能根据系统大小进行缩放。相比之下，最后一项的规模与逆系统的规模一样，因此将在热力学极限中消失。因此，我们发现:<br />
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: <math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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<math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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数学左(部分 s }右){ e } frac { x }{ t } ，/ math<br />
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Combining this with<br />
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Combining this with<br />
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结合这一点<br />
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: <math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
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<math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
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数学左(部分 s }右){ x } frac {1}{ t } ，/ math<br />
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Gives:<br />
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Gives:<br />
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给予:<br />
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: <math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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<math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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数学 dS 左(部分 s }部分 e }右){ x } dE + 左(部分 s }右){ e } dx frac { t } + frac { t }{ delta q }{ t } ，/ math<br />
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===Derivation for systems described by the canonical ensemble===<br />
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===Derivation for systems described by the canonical ensemble===<br />
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正则系系统的推导<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the [[canonical ensemble]]:<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:<br />
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如果一个系统与某个温度为T的热浴热接触，那么在平衡状态下，关于能量本征值的概率分布值由正则系综给出:<br />
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: <math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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<math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the [[Partition function (statistical mechanics)|partition function]]. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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这里Z是一个使所有概率之和归一化的因子，这个函数被称为配分函数。现在我们考虑对温度和能级所依赖的外部参数的无限小的可逆改变。它遵循熵的一般公式:<br />
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: <math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
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<math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
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that<br />
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that<br />
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以及<br />
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: <math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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<math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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把正则系综的数学形式 <math>P_{j}</math> 插入，可以得到:<br />
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: <math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
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<math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
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==Living organisms==<br />
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==Living organisms==<br />
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生命体<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as [[What is Life?|Erwin Schrödinger]], [[Léon Brillouin]]<ref>Léon Brillouin [https://books.google.com/books?hl=en&lr=&id=tPXVbiw_1P0C ''Science and Information Theory''] (Academic Press, 1962) (Dover, 2004)</ref> and [[Life and Energy|Isaac Asimov]]. It is also the topic of current{{When|date=July 2019}} research.{{Citation needed|date=July 2019}}<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as Erwin Schrödinger, Léon Brillouin and Isaac Asimov. It is also the topic of current research.<br />
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表述热力学主要有两种方式：(a)描述从一种热力学平衡状态到另一种状态的路径，(b)通过使系统保持不变，同时增加周围环境的总熵的循环过程。这两种方法有助于理解生命过程。这个话题大大超出了本文的范围，但是已经有一些研究者考虑过这个问题，比如埃尔温·薛定谔、莱昂·布里渊和艾萨克.阿西莫夫。这也是当前研究的主题。<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of [[metabolism]], give out breakdown products and heat. Plants [[Photosynthesis|take in radiative energy]] from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of metabolism, give out breakdown products and heat. Plants take in radiative energy from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
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从近似的角度来看，生命体可以被认为是(b)的一个例子。近似地，一只动物的身体状态每天循环，使得它几乎没有什么变化。动物吸收食物、水和氧气，经过新陈代谢，输出分解的产物和热量。植物吸收来自太阳的辐射能量，这可以认为是热量，以及二氧化碳和水，然后它们释放氧气。它们就是这样生长的，最终会死亡，尸体腐烂，大部分重新变成二氧化碳和水。这可以看作是一个循环过程。总的来说，阳光来自一个高温的源——太阳，它的能量被传递到一个较低的温度汇，例如向太空辐射。这个过程使得植物周围环境的熵增加。因此从循环过程的角度来看，动物和植物服从热力学第二定律。简单的热机效率概念很难适用于这个问题，因为它们假定系统是封闭的。<br />
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From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to [[Non-equilibrium thermodynamics|unsettled current debate or research]]. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
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From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to unsettled current debate or research. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
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从(a)的热力学观点考虑 ，即考虑从一个平衡状态到另一个平衡状态的路径，只会产生一个粗略近似的图像，因为生命体从来不会处于热力学平衡状态。生物体经常必须被认为是开放的系统，因为它们吸收营养物质并排出废弃物。开放系统热力学目前通常从一个热力学平衡状态到另一个状态的路径的角度来考虑，或者是考虑局部热力学平衡近似下的流。生命体的问题可以通过假定一个流不变的稳态来进一步简化。对于这样的近似，熵产生的一般原理在目前有争论和研究的不确定性。尽管如此，从热力学第二定律的这个角度出发衍生出来的想法对生物是有启发意义的。<br />
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==Gravitational systems==<br />
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==Gravitational systems==<br />
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重力系统<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive [[heat capacity]], meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it.<ref name="Grandy 151">Grandy, W.T. (Jr) (2008), p. 151.</ref> Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive heat capacity, meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it. Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
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在不需要描述广义相对论的系统中，物体总是具有正的热容，这意味着温度随着能量的增加而升高。因此，当能量从高温物体流向低温物体时，源温度降低，而汇温度升高; 因此温差随时间而减小。这种情况并不总是发生在那些需要引力和广义相对论的系统中。这样的系统可以自发地朝着质量和能量不均匀分布的方向变化。这在大尺度上适用于宇宙，因此可能很难或不可能将第二定律应用于宇宙。除此之外，广义相对论所描述的系统热力学超出了本文的范围。<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
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{{main article|Non-equilibrium thermodynamics}}<br />
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The theory of classical or [[thermodynamic equilibrium|equilibrium thermodynamics]] is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<ref name="Grandy 151"/><ref>Callen, H.B. (1960/1985), p. 15.</ref><br />
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The theory of classical or equilibrium thermodynamics is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<br />
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经典热力学或平衡热力学的理想化。一个主要的假设或假设，往往甚至没有明确说明，是系统的存在，在他们自己的内部状态的热力学平衡。一般来说，在给定时间内包含一个物理系统的空间区域---- 可能在自然界中找到---- 不是在热力学平衡中，用最严格的术语来解释。从更宽松的角度来说，整个宇宙中没有任何东西是或者曾经是完全精确的热力学平衡。<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of [[thermodynamic equilibrium]]. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable [[thermal fluctuations|fluctuations]]. There is an exception, the case of [[critical point (thermodynamics)|critical states]], which exhibit to the naked eye the phenomenon of [[critical opalescence]]. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of thermodynamic equilibrium. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable fluctuations. There is an exception, the case of critical states, which exhibit to the naked eye the phenomenon of critical opalescence. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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为了物理分析的目的，假设热力学平衡是非常方便的。这种假设可能依赖于试错来证明其正当性。如果这个假设是正确的，它通常是非常有价值和有用的，因为它提供了热力学理论。平衡假设的要素是，一个系统被观察到在无限长的时间内是不变的，而且一个系统中有如此多的粒子，以至于它的微粒性质可以被完全忽略。在这样的平衡假设下，一般来说，没有宏观上可以探测到的波动。有一个例外，即临界状态的例子，它们用肉眼就可以看到临界乳光现象。对于临界状态的实验室研究，需要非常长的观察时间。<br />
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In all cases, the assumption of [[thermodynamic equilibrium]], once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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In all cases, the assumption of thermodynamic equilibrium, once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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在所有情况下，热力学平衡的假设，一旦做出，就意味着没有假定的候选者“涨落”改变系统的熵。<br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<ref>Lieb, E.H., Yngvason, J. (2003), p. 190.</ref><br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<br />
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一个物理系统很容易发生内部的宏观变化，其速度之快足以使熵的恒定性假设失效。或者一个物理系统的粒子太少，以至于微粒的本质在可观测的波动中是显而易见的。那么热力学平衡的假设就会被放弃。对于非平衡态，熵没有绝对的一般定义。<br />
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There are intermediate cases, in which the assumption of local [[thermodynamic equilibrium]] is a very good approximation,<ref>Gyarmati, I. (1967/1970), pp. 4-14.</ref><ref>Glansdorff, P., Prigogine, I. (1971).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003).</ref> but strictly speaking it is still an approximation, not theoretically ideal.<br />
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There are intermediate cases, in which the assumption of local thermodynamic equilibrium is a very good approximation, but strictly speaking it is still an approximation, not theoretically ideal.<br />
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还有一些中间情况，其中局部热力学平衡的假设是一个非常好的近似值，但严格地说它仍然是一个近似值，不是理论上的理想值。<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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对于一般的非平衡状态，考虑其他量的统计力学定义可能是有用的，这些量可以方便地称为‘熵’ ，但是它们不应该与第二定律正确定义的熵混淆或混淆。这些其他的量确实属于统计力学，而不是热力学---- 第二定律的主要领域。<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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宏观观测波动的物理学超出了本文的范围。<br />
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==Arrow of time==<br />
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==Arrow of time==<br />
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时间之箭<br />
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{{Further information|Entropy (arrow of time)}}Thermodynamic Asymmetry in Time<br />
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Thermodynamic Asymmetry in Time<br />
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时间中的热力学不对称性<br />
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{{See also|Arrow of time}}<br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely [[CPT symmetry]], since the second law applies statistically, it is hypothesized, on time-asymmetric ''boundary conditions''.<ref>{{cite web|first=Craig|last=Callender|url=https://plato.stanford.edu/archives/fall2011/entries/time-thermo/|title=Thermodynamic Asymmetry in Time|encyclopedia=Stanford Encyclopedia of Philosophy|date=29 July 2011}}</ref><br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely CPT symmetry, since the second law applies statistically, it is hypothesized, on time-asymmetric boundary conditions.<br />
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热力学第二定律是一个物理定律，它不对称于时间方向的反转。这与已经观察到的物理学基本定律，即 CPT 对称性的概念并不冲突，因为第二定律在统计上适用，它是假设的，是关于时间非对称的边界条件。<br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect ([[Arrow of time#The causal arrow of time|the causal arrow of time]]).<ref>{{cite book | first = J.J.| last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1|display-authors=etal}} chapter 6</ref><br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect (the causal arrow of time).<br />
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第二定律被提出来部分解释时间向前和向后移动的区别，例如为什么原因先于结果(因果关系的时间箭头)。<br />
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==Irreversibility==<br />
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==Irreversibility==<br />
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不可逆性<br />
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Irreversibility in [[thermodynamic process]]es is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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Irreversibility in thermodynamic processes is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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热力学过程的不可逆性是热力学作用不对称性的结果，而不是物体内部不可逆的微观性质的结果。热力学操作是施加在参与机构上的宏观外部干预，而不是从它们的内部性质中得到的。由于没有认识到这一点，产生了被称为“悖论”的说法。<br />
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===Loschmidt's paradox===<br />
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===Loschmidt's paradox===<br />
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洛斯密特悖论<br />
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{{main article|Loschmidt's paradox}}<br />
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[[Loschmidt's paradox]], also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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洛斯密特悖论，也被称为可逆性悖论，其目的在于不可能从描述宏观系统微观演化的时间对称动力学推导出不可逆性。<br />
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In the opinion of [[Erwin Schrödinger|Schrödinger]], "It is now quite obvious in what manner you have to reformulate the law of entropy{{snd}}or for that matter, all other irreversible statements{{snd}}so that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other."<ref>[[Erwin Schrödinger|Schrödinger, E.]] (1950), p.&nbsp;192.</ref> The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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In the opinion of Schrödinger, "It is now quite obvious in what manner you have to reformulate the law of entropyor for that matter, all other irreversible statementsso that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other." The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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根据薛定谔的观点，“现在很明显，你必须用什么方式来重新定义该物质的熵定律，所有其他不可逆的状态，这样它们才能从可逆模型中导出。你不能只谈论一个孤立的系统，而应该至少谈论两个，你可以暂时认为它们与世界其它地方是孤立的，但并不总是相互孤立的。”这两个系统被隔离墙隔离开来，直到按照法律的设想被热力学操作拆除。热力学操作是外部强加的，不受支配系统组成部分的可逆微观动力学定律的制约。它是不可逆转性的原因。本文中的法律陈述符合薛定谔的建议。因果关系在逻辑上优先于第二定律，而不是从第二定律派生出来的。<br />
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===Poincaré recurrence theorem===<br />
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===Poincaré recurrence theorem===<br />
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庞加莱始态复现定理<br />
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{{Main article|Poincaré recurrence theorem}}<br />
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The [[Poincaré recurrence theorem]] considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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The Poincaré recurrence theorem considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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庞加莱始态复现定理认为这是一个孤立的物理系统的理论微观描述。这可以被认为是一个热力学系统的模型后，一个热力学操作已经移除了一个内墙。在足够长的时间之后，系统将返回到一个非常接近初始状态的显微定义状态。庞加莱回归时间是指回归前经过的时间长度。它极其漫长，可能比宇宙的寿命还要长，并且敏感地依赖于被热力学操作拆除的墙体的几何形状。递归定理可能被认为是明显与热力学第二定律相矛盾的。然而，更明显的是，它只不过是一个微观模型，热力学平衡在一个孤立的系统中，通过移除两个系统之间的一道墙而形成。对于一个典型的热力学系统，重现时间是如此之大(比宇宙的生命周期长许多倍) ，以至于在所有的实际目的中，人们都不能观察到重现。然而，人们可能希望想象一下，可以等待庞加莱复发，然后重新插入被热力学操作移除的壁。然后很明显，不可逆性的出现是由于庞加莱循环的完全不可预测性，只要初始状态是热力学平衡，就像宏观热力学的情况一样。即使可以等待，也没有实际可能性选择合适的时间重新插入墙壁。庞加莱始态复现定理为 Loschmidt 悖论提供了一个解决方案。如果一个孤立的热力学系统可以被监测到在越来越多的平均 poincar 递归时间的倍数上，系统的热力学行为将在时间反转下变得不变。<br />
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[[File:James-clerk-maxwell3.jpg|thumb|upright|James Clerk Maxwell]]<br />
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James Clerk Maxwell<br />
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詹姆斯·克拉克·麦克斯韦<br />
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===Maxwell's demon===<br />
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===Maxwell's demon===<br />
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麦克斯韦的魔鬼<br />
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{{unreferenced|section|date=August 2018}}<br />
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{{main article|Maxwell's demon}}<br />
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[[James Clerk Maxwell]] imagined one container divided into two parts, ''A'' and ''B''. Both parts are filled with the same [[gas]] at equal temperatures and placed next to each other, separated by a wall. Observing the [[molecule]]s on both sides, an imaginary [[demon]] guards a microscopic trapdoor in the wall. When a faster-than-average molecule from ''A'' flies towards the trapdoor, the demon opens it, and the molecule will fly from ''A'' to ''B''. The average [[speed]] of the molecules in ''B'' will have increased while in ''A'' they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in ''A'' and increases in ''B'', contrary to the second law of thermodynamics.<br />
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James Clerk Maxwell imagined one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other, separated by a wall. Observing the molecules on both sides, an imaginary demon guards a microscopic trapdoor in the wall. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics.<br />
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詹姆斯·克拉克·麦克斯韦想象一个容器分成两部分，a 和 b。两个部分在相同的温度下充入相同的气体，并被一堵墙隔开，挨着放置。通过观察两边的分子，一个想象中的恶魔守卫着墙上的一个微型暗门。当一个来自 a 的比平均速度快的分子飞向活板门时，魔鬼打开了它，分子就会从 a 飞到 b。B 中分子的平均速度会增加，而 a 中分子的平均速度会减慢。由于平均分子速度与温度相对应，因此 a 中的温度降低，b 中的温度升高，这与热力学第二定律相反。<br />
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One response to this question was suggested in 1929 by [[Leó Szilárd]] and later by [[Léon Brillouin]]. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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One response to this question was suggested in 1929 by Leó Szilárd and later by Léon Brillouin. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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1929年，Leó Szilárd 和后来的 l on brilouin 对这个问题提出了一个答案。Szil rd 指出，现实生活中的麦克斯韦恶魔需要一些测量分子速度的方法，而获取信息需要消耗大量能量。<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between ''A'' and ''B''. It is therefore performing [[thermodynamic operation]]s on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between A and B. It is therefore performing thermodynamic operations on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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麦克斯韦的“魔鬼”不断地改变 a 和 b 之间墙体的渗透性。因此，它是在微观尺度上执行热力学操作，而不仅仅是观察普通的自发或自然的宏观热力学过程。<br />
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==Quotations==<br />
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==Quotations==<br />
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语录<br />
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{{Wikiquote}}<br />
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{{quote|The law that entropy always increases holds, I think, the supreme position among the [[Laws of science|laws of Nature]]. If someone points out to you that your pet theory of the [[universe]] is in disagreement with [[Maxwell's equations]] – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.|Sir [[Arthur Stanley Eddington]], ''The Nature of the Physical World'' (1927)}}<br />
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{{quote|There have been nearly as many formulations of the second law as there have been discussions of it.|Philosopher / Physicist [[Percy Williams Bridgman|P.W. Bridgman]], (1941)}}<br />
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{{quote|Clausius is the author of the sibyllic utterance, "The energy of the universe is constant; the entropy of the universe tends to a maximum." The objectives of continuum thermomechanics stop far short of explaining the "universe", but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size.|[[Clifford Truesdell|Truesdell, C.]], Muncaster, R. G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 17.}}<br />
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==See also==<br />
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==See also==<br />
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参见<br />
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*[[Clausius–Duhem inequality]]<br />
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*[[Fluctuation theorem]]<br />
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*[[Heat death of the universe]]<br />
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*[[History of thermodynamics]]<br />
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*[[Jarzynski equality]]<br />
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*[[Laws of thermodynamics]]<br />
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*[[Maximum entropy thermodynamics]]<br />
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*[[Quantum thermodynamics]]<br />
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*[[Reflections on the Motive Power of Fire]]<br />
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*[[Relativistic heat conduction]]<br />
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*[[Thermal diode]]<br />
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*[[Thermodynamic equilibrium]]<br />
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{{colend}}<br />
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==References==<br />
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==References==<br />
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参考资料<br />
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===Sources===<br />
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资料来源<br />
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{{refbegin|50em}}<br />
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*Adkins, C.J. (1968/1983). ''Equilibrium Thermodynamics'', (1st edition 1968), third edition 1983, Cambridge University Press, Cambridge UK, {{ISBN|0-521-25445-0}}.<br />
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*[[Peter Atkins|Atkins, P.W.]], de Paula, J. (2006). ''Atkins' Physical Chemistry'', eighth edition, W.H. Freeman, New York, {{ISBN|978-0-7167-8759-4}}.<br />
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*Attard, P. (2012). ''Non-equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications'', Oxford University Press, Oxford UK, {{ISBN|978-0-19-966276-0}}.<br />
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*Baierlein, R. (1999). ''Thermal Physics'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-59082-5}}.<br />
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*Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics, New York, {{ISBN|0-88318-797-3}}.<br />
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*{{Cite book|title=Concepts in thermal physics|last=Blundell|first=Stephen J.|last2=Blundell|authorlink1=Stephen Blundell|first2=Katherine M.|authorlink2=Katherine Blundell|publisher=[[Oxford University Press]]|year=2010|isbn=9780199562107|edition=2nd|location=Oxford|pages=|oclc=607907330|doi=10.1093/acprof:oso/9780199562091.001.0001 |url=http://cds.cern.ch/record/1235139}}<br />
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*[[Ludwig Boltzmann|Boltzmann, L.]] (1896/1964). ''Lectures on Gas Theory'', translated by S.G. Brush, University of California Press, Berkeley.<br />
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*Borgnakke, C., Sonntag., R.E. (2009). ''Fundamentals of Thermodynamics'', seventh edition, Wiley, {{ISBN|978-0-470-04192-5}}.<br />
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*Buchdahl, H.A. (1966). ''The Concepts of Classical Thermodynamics'', Cambridge University Press, Cambridge UK.<br />
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*[[Percy Williams Bridgman|Bridgman, P.W.]] (1943). ''The Nature of Thermodynamics'', Harvard University Press, Cambridge MA.<br />
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*[[Herbert Callen|Callen, H.B.]] (1960/1985). ''Thermodynamics and an Introduction to Thermostatistics'', (1st edition 1960) 2nd edition 1985, Wiley, New York, {{ISBN|0-471-86256-8}}.<br />
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*{{cite journal|author=C. Carathéodory|author1-link=Constantin Carathéodory<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
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|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
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2010年3月15日 | 第355-386页 | http://gdz.sub.uni-goettingen.de/index.php?id=11&ppn=ppn235181684_0067&dmdid=dmdlog_0033&l=1<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). ''The Second Law of Thermodynamics'', Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können.(p. 363) | doi 10.1007 / bf01450409}.译文可以在这里找到[ http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf ]。同时，在 Kestin，j。(1976).热力学第二定律，Dowden，Hutchinson & Ross，Stroudsburg PA. 。<br />
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*[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986). [http://www.worldcat.org/title/reflections-on-the-motive-power-of-fire-a-critical-edition-with-the-surviving-scientific-manuscripts-translated-and-edited-by-fox-robert/oclc/812944517&referer=brief_results ''Reflections on the motive power of fire''], Manchester University Press, Manchester UK, {{ISBN|0-7190-1741-6}}. [https://archive.org/stream/reflectionsonmot00carnrich#page/n7/mode/2up Also here.]<br />
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*[[Sydney Chapman (mathematician)|Chapman, S.]], [[Thomas George Cowling|Cowling, T.G.]] (1939/1970). ''The Mathematical Theory of Non-uniform gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases'', third edition 1970, Cambridge University Press, London.<br />
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*{{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Ueber Die Bewegende Kraft Der Wärme Und Die Gesetze, Welche Sich Daraus Für Die Wärmelehre Selbst Ableiten Lassen|journal=Annalen der Physik|year=1850|volume=79|issue=4|pages=368–397, 500–524|url=http://gallica.bnf.fr/ark:/12148/bpt6k15164w/f518.image|accessdate=26 June 2012|doi=10.1002/andp.18501550403|bibcode = 1850AnP...155..500C |hdl=2027/uc1.$b242250|hdl-access=free}} Translated into English: {{cite journal|last=Clausius|first=R.|title=On the Moving Force of Heat, and the Laws regarding the Nature of Heat itself which are deducible therefrom|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1851| volume=2|series=4th|issue=VIII|pages=1–21; 102–119 |url=https://archive.org/stream/londonedinburghd02lond#page/1/mode/1up|accessdate=26 June 2012<br />
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|doi=10.1080/14786445108646819}}<br />
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|doi=10.1080/14786445108646819}}<br />
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10.1080 / 14786445108646819}<br />
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* {{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie|year=1854|journal=Annalen der Physik|volume=xciii|issue=12|pages=481–506|url=http://zfbb.thulb.uni-jena.de/servlets/MCRFileNodeServlet/jportal_derivate_00140956/18541691202_ftp.pdf|accessdate=24 March 2014|doi=10.1002/andp.18541691202<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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| bibcode 1854AnP... 169. 1856年7月 | 第2卷 | 第4页 | 第86页 | https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=242014年3月<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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热力学理论及其在蒸汽机和物体物理性质中的应用 | 1867年 | 出版商 John van Voorst | 位置: 伦敦 | https://archive.org/details/mechanicaltheor04claugoog|quote=editions:pwr_sbkwa8ic。 2012年6月19日<br />
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*Denbigh, K. (1954/1981). ''The Principles of Chemical Equilibrium. With Applications in Chemistry and Chemical Engineering'', fourth edition, Cambridge University Press, Cambridge UK, {{ISBN|0-521-23682-7}}.<br />
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*Eu, B.C. (2002). ''Generalized Thermodynamics. The Thermodynamics of Irreversible Processes and Generalized Hydrodynamics'', Kluwer Academic Publishers, Dordrecht, {{ISBN|1-4020-0788-4}}.<br />
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*[[Josiah Willard Gibbs|Gibbs, J.W.]] (1876/1878). On the equilibrium of heterogeneous substances, ''Trans. Conn. Acad.'', '''3''': 108-248, 343-524, reprinted in ''The Collected Works of J. Willard Gibbs, Ph.D, LL. D.'', edited by W.R. Longley, R.G. Van Name, Longmans, Green & Co., New York, 1928, volume 1, pp.&nbsp;55–353.<br />
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*Griem, H.R. (2005). ''Principles of Plasma Spectroscopy (Cambridge Monographs on Plasma Physics)'', Cambridge University Press, New York {{ISBN|0-521-61941-6}}.<br />
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*Glansdorff, P., Prigogine, I. (1971). ''Thermodynamic Theory of Structure, Stability, and Fluctuations'', Wiley-Interscience, London, 1971, {{ISBN|0-471-30280-5}}.<br />
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*Grandy, W.T., Jr (2008). [http://global.oup.com/academic/product/entropy-and-the-time-evolution-of-macroscopic-systems-9780199546176?cc=au&lang=en& ''Entropy and the Time Evolution of Macroscopic Systems'']. Oxford University Press. {{ISBN|978-0-19-954617-6}}.<br />
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*Greven, A., Keller, G., Warnecke (editors) (2003). ''Entropy'', Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[Edward A. Guggenheim|Guggenheim, E.A.]] (1949). 'Statistical basis of thermodynamics', ''Research'', '''2''': 450–454.<br />
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*[[Edward A. Guggenheim|Guggenheim, E.A.]] (1967). ''Thermodynamics. An Advanced Treatment for Chemists and Physicists'', fifth revised edition, North Holland, Amsterdam.<br />
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*Gyarmati, I. (1967/1970) ''Non-equilibrium Thermodynamics. Field Theory and Variational Principles'', translated by E. Gyarmati and W.F. Heinz, Springer, New York.<br />
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*[[Charles Kittel|Kittel, C.]], [[Herbert Kroemer|Kroemer, H.]] (1969/1980). ''Thermal Physics'', second edition, Freeman, San Francisco CA, {{ISBN|0-7167-1088-9}}.<br />
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*Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998). ''Modern Thermodynamics: From Heat Engines to Dissipative Structures'', John Wiley & Sons, Chichester, {{ISBN|0-471-97393-9}}.<br />
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*Lebon, G., Jou, D., Casas-Vázquez, J. (2008). ''Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers'', Springer-Verlag, Berlin, {{ISBN|978-3-540-74252-4}}.<br />
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*{{cite journal|ref=harv|title=The Physics and Mathematics of the Second Law of Thermodynamics|last1=Lieb |first1=E. H. |last2=Yngvason |first2=J.|journal=Physics Reports|volume=310|issue=1|pages=1–96 |year=1999|doi=10.1016/S0370-1573(98)00082-9|arxiv = cond-mat/9708200 |bibcode = 1999PhR...310....1L}}<br />
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*Lieb, E.H., Yngvason, J. (2003). The Entropy of Classical Thermodynamics, pp.&nbsp;147–195, Chapter 8 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*{{cite book|ref=harv|first=F.|last=Mandl|title=Statistical physics|edition=second|year=1988|publisher=[[Wiley & Sons]]|isbn=978-0-471-91533-1}}<br />
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*{{cite journal | last1 = Maxwell | first1 = J.C. | authorlink = James Clerk Maxwell | year = 1867 | title = On the dynamical theory of gases | url = | journal = Phil. Trans. R. Soc. Lond. | volume = 157 | issue = | pages = 49–88 | doi = 10.1098/rstl.1867.0004 }}<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985). ''Thermodynamics'', Pitman, London, {{ISBN|0-273-08577-8}}.<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003). Entropy in Nonequilibrium, pp.&nbsp;79–109, Chapter 5 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*Münster, A. (1970), ''Classical Thermodynamics'', translated by E.S. Halberstadt, Wiley–Interscience, London, {{ISBN|0-471-62430-6}}.<br />
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*[[Brian Pippard|Pippard, A.B.]] (1957/1966). ''Elements of Classical Thermodynamics for Advanced Students of Physics'', original publication 1957, reprint 1966, Cambridge University Press, Cambridge UK.<br />
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*[[Max Planck|Planck, M.]] (1897/1903). [https://archive.org/stream/treatiseonthermo00planrich#page/100/mode/2up ''Treatise on Thermodynamics'', translated by A. Ogg, Longmans Green, London, p. 100.]<br />
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*[[Max Planck|Planck. M.]] (1914). [https://archive.org/details/theoryofheatradi00planrich ''The Theory of Heat Radiation''], a translation by Masius, M. of the second German edition, P. Blakiston's Son & Co., Philadelphia.<br />
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*[[Max Planck|Planck, M.]] (1926). Über die Begründung des zweiten Hauptsatzes der Thermodynamik, ''Sitzungsberichte der Preussischen Akademie der Wissenschaften: Physikalisch-mathematische Klasse'': 453–463.<br />
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* Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769-781.<br />
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* {{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}<br />
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*Quinn, T.J. (1983). ''Temperature'', Academic Press, London, {{ISBN|0-12-569680-9}}.<br />
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*{{cite book|ref=harv|first=Y.V.C.|last=Rao|title=An Introduction to thermodynamics|url=https://books.google.com/books?id=iYWiCXziWsEC&pg=PA213|date=2004|publisher=Universities Press|isbn=978-81-7371-461-0|page=213}}<br />
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*Roberts, J.K., Miller, A.R. (1928/1960). ''Heat and Thermodynamics'', (first edition 1928), fifth edition, Blackie & Son Limited, Glasgow.<br />
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*[[Erwin Schrödinger|Schrödinger, E.]] (1950). Irreversibility, ''Proc. R. Ir. Acad.'', '''A53''': 189–195.<br />
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*[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966). ''Elements of Thermodynamics'', Addison-Wesley Publishing, Reading MA.<br />
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*{{cite journal|ref=harv|last=Thomson|first=W.|author-link=William Thomson, 1st Baron Kelvin|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Transactions of the Royal Society of Edinburgh|year=1851|volume=XX|issue=part II|pages=261–268; 289–298|url=https://www.biodiversitylibrary.org/item/126047#page/295/mode/1up}} Also published in {{cite journal|last=Thomson|first=W.|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Philos. Mag. |date=December 1852 |volume=IV |series=4 |issue=22 |page=13|url=https://archive.org/stream/londonedinburghp04maga#page/12/mode/2up |accessdate=25 June 2012}}<br />
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*[[William Thomson, 1st Baron Kelvin|Thomson, W.]] (1852). ''On the universal tendency in nature to the dissipation of mechanical energy'' Philosophical Magazine, Ser. 4, p.&nbsp;304.<br />
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*[[László Tisza|Tisza, L.]] (1966). ''Generalized Thermodynamics'', M.I.T Press, Cambridge MA.<br />
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*[[Clifford Truesdell|Truesdell, C.]] (1980). ''The Tragicomical History of Thermodynamics 1822–1854'', Springer, New York, {{ISBN|0-387-90403-4}}.<br />
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*Uffink, J. (2001). Bluff your way in the second law of thermodynamics, ''Stud. Hist. Phil. Mod. Phys.'', '''32'''(3): 305–394.<br />
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*Uffink, J. (2003). Irreversibility and the Second Law of Thermodynamics, Chapter 7 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003), Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963). ''Lectures in Statistical Mechanics'', American Mathematical Society, Providence RI.<br />
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*[[Mark Zemansky|Zemansky, M.W.]] (1968). ''Heat and Thermodynamics. An Intermediate Textbook'', fifth edition, McGraw-Hill Book Company, New York.<br />
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==Further reading==<br />
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==Further reading==<br />
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进一步阅读<br />
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*Goldstein, Martin, and Inge F., 1993. ''The Refrigerator and the Universe''. Harvard Univ. Press. Chpts. 4–9 contain an introduction to the Second Law, one a bit less technical than this entry. {{ISBN|978-0-674-75324-2}}<br />
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*Leff, Harvey S., and Rex, Andrew F. (eds.) 2003. ''Maxwell's Demon 2 : Entropy, classical and quantum information, computing''. Bristol UK; Philadelphia PA: [[Institute of Physics]]. {{ISBN|978-0-585-49237-7}}<br />
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*{{Cite book | first = J.J. | last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1}}(technical).<br />
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*{{cite book |title=Reflections on the Motive Power of Heat and on Machines Fitted to Develop That Power |last=Carnot |first=Sadi |editor=[[Robert Henry Thurston|Thurston, Robert Henry]] |year=1890 |publisher=J. Wiley & Sons |location=New York |isbn= |pages= }} ([https://books.google.com/books?id=tgdJAAAAIAAJ full text of 1897 ed.]) ([http://www.history.rochester.edu/steam/carnot/1943/ html])<br />
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*Stephen Jay Kline (1999). ''The Low-Down on Entropy and Interpretive Thermodynamics'', La Cañada, CA: DCW Industries. {{ISBN|1-928729-01-0}}.<br />
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*{{cite book | last1 = Kostic | first1 = M | year = 2011 | title = Revisiting The Second Law of Energy Degradation and Entropy Generation: From Sadi Carnot's Ingenious Reasoning to Holistic Generalization | journal = AIP Conf. Proc. | volume = 1411 | issue = 1| pages = 327–350 | doi = 10.1063/1.3665247 | isbn = 978-0-7354-0985-9 | bibcode = 2011AIPC.1411..327K | series = AIP Conference Proceedings | citeseerx = 10.1.1.405.1945 }} also at [https://web.archive.org/web/20130420222450/http://www.kostic.niu.edu/2ndLaw/Revisiting%20The%20Second%20Law%20of%20Energy%20Degradation%20and%20Entropy%20Generation%20-%20From%20Carnot%20to%20Holistic%20Generalization-4.pdf].<br />
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==External links==<br />
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==External links==<br />
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外部链接<br />
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*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics]" – by Lawrence Sklar.<br />
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*[http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node30.html ''Second law of thermodynamics''] in the MIT Course [http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/notes.html ''Unified Thermodynamics and Propulsion''] from Prof. Z. S. Spakovszky<br />
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*[[E.T. Jaynes]], 1988, "[http://bayes.wustl.edu/etj/articles/ccarnot.pdf The evolution of Carnot's principle]," in G. J. Erickson and C. R. Smith (eds.)''Maximum-Entropy and Bayesian Methods in Science and Engineering, Vol'' 1: p.&nbsp;267.<br />
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*[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf Caratheodory, C., "Examination of the foundations of thermodynamics," trans. by D. H. Delphenich]<br />
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* [https://www.bbc.co.uk/programmes/p004y2bm The Second Law of Thermodynamics], BBC Radio 4 discussion with John Gribbin, Peter Atkins & Monica Grady (''In Our Time'', Dec. 16, 2004)<br />
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* [http://mdpi.org/entropy/papers/e6010001.pdf [[Entropy (journal)]], 2004]<br />
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* [https://www.journals.uchicago.edu/doi/abs/10.1086/663835 The Journal of the International Society for the History of Philosophy of Science, 2012]<br />
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Category:Concepts in physics<br />
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分类: 物理概念<br />
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[[Category:Laws of thermodynamics|2]]<br />
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[[Category:Non-equilibrium thermodynamics]]<br />
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Category:Non-equilibrium thermodynamics<br />
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类别: 非平衡态热力学<br />
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[[Category:Philosophy of thermal and statistical physics]]<br />
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Category:Philosophy of thermal and statistical physics<br />
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类别: 热力学和统计物理学哲学<br />
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<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Second law of thermodynamics]]. Its edit history can be viewed at [[热力学第二定律/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E7%83%AD%E5%8A%9B%E5%AD%A6%E7%AC%AC%E4%BA%8C%E5%AE%9A%E5%BE%8B&diff=10352热力学第二定律2020-07-16T09:24:10Z<p>Dorr：/* Perpetual motion of the second kind 第二类永动机 */</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Law of physics}}<br />
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{{Thermodynamics|laws}}<br />
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The '''second law of thermodynamics''' states that the total [[entropy]] of an [[isolated system]] can never decrease over time, and is constant if and only if all processes are reversible.<ref>http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node38.html#SECTION05224000000000000000</ref> Isolated systems spontaneously evolve towards [[thermodynamic equilibrium]], the state with maximum entropy. <br />
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The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, and is constant if and only if all processes are reversible. Isolated systems spontaneously evolve towards thermodynamic equilibrium, the state with maximum entropy. <br />
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热力学第二定律指出，孤立系统的总熵永远不会随着时间而减少，而且只有当且仅当所有过程都是可逆的时，总熵才是恒定的。孤立的系统自发地进化到热力学平衡状态，即熵最大的状态。<br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in [[thermodynamic equilibrium]], or is undergoing a (fictive) [[Reversible process (thermodynamics)|reversible process]]. In all processes that occur, including [[spontaneous process]]es,<ref>Atkins and de Paula, p.78</ref> the total entropy of the system and its surroundings increases and the process is [[Irreversible process|irreversible in the thermodynamic sense]]. The increase in entropy accounts for the irreversibility of natural processes, and the [[Arrow of time|asymmetry between future and past]].<ref>{{cite book|last=Zohuri|first=Bahman|title=Dimensional Analysis Beyond the Pi Theorem|url=https://books.google.com/books?id=pRVuDQAAQBAJ|year=2016|publisher=Springer|isbn=978-3-319-45726-0|page=[https://books.google.com.ph/books?id=pRVuDQAAQBAJ&pg=PA111&dq=%22increase+in+entropy+accounts+for+the+irreversibility+of+natural+processes+and+the+asymmetry+between+future+and+past.%22&hl=en&sa=X&ved=0ahUKEwj6spb61tbaAhUFS7wKHfftDtIQ6AEIKDAA#v=onepage&q=%22increase%20in%20entropy%20accounts%20for%20the%20irreversibility%20of%20natural%20processes%20and%20the%20asymmetry%20between%20future%20and%20past.%22 111]}}</ref><br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in thermodynamic equilibrium, or is undergoing a (fictive) reversible process. In all processes that occur, including spontaneous processes, the total entropy of the system and its surroundings increases and the process is irreversible in the thermodynamic sense. The increase in entropy accounts for the irreversibility of natural processes, and the asymmetry between future and past.<br />
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系统及其周围环境的总熵在理想情况下可以保持不变，在这种情况下，系统处于热力学平衡期，或者正在经历一个(虚拟的)可逆过程期。在所有发生的过程中，包括自发过程，系统及其周围环境的总熵增加，这一过程在热力学意义上是不可逆的。熵的增加解释了自然过程的不可逆性，以及未来和过去之间的不对称性。<br />
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Historically, the second law was an [[Empirical evidence|empirical finding]] that was accepted as an axiom of [[thermodynamics|thermodynamic theory]]. [[Statistical mechanics]], classical or [[quantum thermodynamics|quantum]], explains the microscopic origin of the law.<br />
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Historically, the second law was an empirical finding that was accepted as an axiom of thermodynamic theory. Statistical mechanics, classical or quantum, explains the microscopic origin of the law.<br />
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从历史上看，第二定律是一个经验性的发现，被公认为热力学理论的公理。统计力学，无论是经典的还是量子的，都可以解释这个定律的微观起源。<br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist [[Nicolas Léonard Sadi Carnot|Sadi Carnot]], who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as [[Carnot's theorem (thermodynamics)|Carnot's rule]] or limit.<ref>Jaffe, R.L., Taylor, W. (2018). ''The Physics of Energy'', Cambridge University Press, Cambridge UK, pages 150, 151, 259, 772, 743.</ref><br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist Sadi Carnot, who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as Carnot's rule or limit.<br />
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第二定律已经用许多方式表达过了。法国科学家萨迪 · 卡诺在1824年证明了热能在热机中转化为功的效率是有上限的。这方面的第二定律也被称为卡诺的规则或限制。<br />
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==Introduction==<br />
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==Introduction==<br />
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引言<br />
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[[File:Heat flow hot to cold.png|thumb|upright|Heat flow from hot water to cold water.]]<br />
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Heat flow from hot water to cold water.<br />
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热量从热水流向冷水。<br />
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The [[first law of thermodynamics]] provides the definition of the [[internal energy]] of a [[thermodynamic system]], and expresses the law of [[conservation of energy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 40–41.</ref><ref>Munster A. (1970), pp. 8–9, 50–51.</ref> The second law is concerned with the direction of natural processes.<ref>{{harvnb|Mandl|1988}}</ref> It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such [[Phenomenon|phenomena]] are accounted for in terms of [[entropy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 79–107.</ref><ref>Bailyn, M. (1994), Section 71, pp. 113–154.</ref> If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, {{math|''S''}}, increases.<br />
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The first law of thermodynamics provides the definition of the internal energy of a thermodynamic system, and expresses the law of conservation of energy. The second law is concerned with the direction of natural processes. It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such phenomena are accounted for in terms of entropy. If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, , increases.<br />
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能量守恒定律提供了热力学系统内能的定义，并表达了能量守恒定律。第二定律与自然过程的方向有关。它断言自然过程只在一种意义上运行，而且是不可逆的。例如，当有了传导和辐射的路径时，热总是自发地从一个较热的物体流向一个较冷的物体。这种现象可以用熵来解释。如果一个孤立的系统最初是通过内部分隔不可渗透的墙来保持在内部热力学平衡，然后一些操作使墙更具渗透性，那么系统自发地进化到最终达到一个新的内部热力学平衡，其总熵增加。<br />
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In a fictive reversible process, an infinitesimal increment in the entropy ({{math|d''S''}}) of a system is defined to result from an infinitesimal transfer of heat ({{math|δ''Q''}}) to a [[closed system]] (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature ({{math|''T''}}) of the system in equilibrium and the surroundings which supply the heat:<ref>Bailyn, M. (1994), p. 120.</ref><br />
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In a fictive reversible process, an infinitesimal increment in the entropy () of a system is defined to result from an infinitesimal transfer of heat () to a closed system (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature () of the system in equilibrium and the surroundings which supply the heat:<br />
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在虚构的可逆过程中，系统熵的无穷小增量被定义为由无穷小的热量传递()到一个封闭系统(允许能量进入或出去，但不允许物质传递)除以平衡系统和提供热量的环境的共同温度() :<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, idealized fictive reversible process)}.</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, idealized fictive reversible process)}.</math><br />
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(闭合系统，理想化的可逆过程) / math<br />
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Different notations are used for infinitesimal amounts of heat ({{mvar|δ}}) and infinitesimal amounts of entropy ({{mvar|d}}) because entropy is a [[function of state]], while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the [[Clausius Theorem|inequality]] <ref name="MortimerBook"> Mortimer, R. G. ''Physical Chemistry''. 3rd ed., p. 120, Academic Press, 2008. </ref><ref name="FermiBook"> Fermi, E. ''Thermodynamics'', footnote on p. 48, Dover Publications,1956 (still in print). </ref> <br />
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Different notations are used for infinitesimal amounts of heat () and infinitesimal amounts of entropy () because entropy is a function of state, while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the inequality <br />
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用不同的符号表示无穷小量的热量()和无穷小量的熵() ，因为熵是状态的函数，而热量和功一样不是。第二定律要求系统熵的增量满足不等式，对于实际上可能存在的不与环境发生质量交换的无穷小过程，系统熵增量满足不等式<br />
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: <math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process).}</math><br />
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<math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process).}</math><br />
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(实际上可能是封闭系统，不可逆性)。数学<br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature ({{math|''T''}}) and the temperature of the surroundings ({{math|''T''}}<sub>''surr''</sub>).<ref name=":0">Adkins, C.J. (1968/1983), p. 75.</ref><ref name="Munster 45"/> Note that the equality still applies for pure heat flow,<ref name="Schmidt-Rohr 14"> Schmidt-Rohr, K. (2014). "Expansion Work without the External Pressure, and Thermodynamics in Terms of Quasistatic Irreversible Processes" ''J. Chem. Educ.'' '''91''': 402-409. https://dx.doi.org/10.1021/ed3008704 </ref><br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature () and the temperature of the surroundings (<sub>surr</sub>).<br />
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这是因为这种情况下的一般过程可能包括周围环境对系统所做的功，这可能在系统内部产生摩擦或粘滞效应，因为一个化学反应可能正在进行，或者因为热传递实际上只是不可逆地发生，由系统温度()和周围环境温度(sub surr / sub)之间的差分驱动。<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change).}</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change).}</math><br />
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(实际上可能是不改变成分的准静态不可逆性)。数学<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry.<ref name="Oxtoby8th">Oxtoby, D. W; Gillis, H.P., [[Laurie Butler|Butler, L. J.]] (2015).''Principles of Modern Chemistry'', Brooks Cole. p. 617. {{ISBN|978-1305079113}}</ref> <ref name="MortimerBook"></ref> Introducing a set of internal variables <math>\xi</math> to describe the deviation of a thermodynamic system in physical equilibrium (with the required well-defined uniform pressure ''P'' and temperature ''T'')<ref name="Schmidt-Rohr 14"></ref> from the chemical equilibrium state, one can record the equality<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry. from the chemical equilibrium state, one can record the equality<br />
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这是根据测得的热容曲线和相变时的熵变精确确定纯物质的绝对熵的基础。用量热法。从化学平衡来看，人们可以记录平等<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process).}</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process).}</math><br />
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- frac { t }-frac {1}{ j } ， Xi { j } ， delta Xi j， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ,.数学<br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of ''the second principle of thermodynamics'' – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<ref> Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769–781.</ref><ref>{{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}</ref><br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of the second principle of thermodynamics – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<br />
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第二项代表内部变量的工作，这些内部变量可以受到外部影响的干扰，但是系统不能通过内部变量执行任何正功。这种说法介绍了热力学系统在时间上不可能逆转演化的事实，并且可以被认为是热力学第二原理的表述---- 这个表述，当然，相当于熵原理的表述。<br />
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The [[zeroth law of thermodynamics]] in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body.<ref name=dugdale>{{cite book|author=J. S. Dugdale|title=Entropy and its Physical Meaning|publisher=Taylor & Francis|year=1996|isbn=978-0-7484-0569-5|page=13|quote=This law is the basis of temperature.}}</ref> For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows{{how?|date=August 2018}} a distinguished temperature scale, which defines an absolute, [[thermodynamic temperature]], independent of the properties of any particular reference thermometric body.<ref>[[Mark Zemansky|Zemansky, M.W.]] (1968), pp. 207–209.</ref><ref>Quinn, T.J. (1983), p. 8.</ref><br />
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The zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body. For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows a distinguished temperature scale, which defines an absolute, thermodynamic temperature, independent of the properties of any particular reference thermometric body.<br />
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美国热力学第零定律协会在其通常的简短声明中允许承认，相对于热平衡的两个物体具有相同的温度，特别是测试物体具有与参考温度测量物体相同的温度。对于一个物体和另一个物体之间的热平衡，存在着无限多的经验温度标度，通常分别取决于特定参考温度标度的性质。第二定律允许区分温度标度，它定义了一个绝对的热力学温度，与任何特定的参考温度体的性质无关。<br />
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==Various statements of the law==<br />
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==Various statements of the law==<br />
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各种各样的法律陈述<br />
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The second law of thermodynamics may be expressed in many specific ways,<ref name=MIT>{{cite web|title=Concept and Statements of the Second Law|url=http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node37.html|accessdate=2010-10-07 |publisher=web.mit.edu}}</ref> the most prominent classical statements{{sfnp|Lieb|Yngvason|1999}} being the statement by [[Rudolf Clausius]] (1854), the statement by [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] (1851), and the statement in axiomatic thermodynamics by [[Constantin Carathéodory]] (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.{{sfnp|Rao|2004|p=213}}<br />
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The second law of thermodynamics may be expressed in many specific ways, the most prominent classical statements being the statement by Rudolf Clausius (1854), the statement by Lord Kelvin (1851), and the statement in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.<br />
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热力学第二定律可以用许多特定的方式来表达，最突出的经典陈述是 Rudolf Clausius (1854)的陈述，Kelvin 勋爵(1851)的陈述，以及康斯坦丁·卡拉西奥多里(1909)在公理化热力学中的陈述。这些陈述用一般的物理术语描述法律，引用某些过程的不可能性。克劳修斯和开尔文陈述被证明是等价的。<br />
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===Carnot's principle===<br />
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===Carnot's principle===<br />
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卡诺原理<br />
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The historical origin<ref>[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986).</ref> of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a [[Carnot heat engine]], fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the [[caloric theory]] of heat was seriously considered, before the recognition of the [[first law of thermodynamics]], and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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The historical origin of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a Carnot heat engine, fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the caloric theory of heat was seriously considered, before the recognition of the first law of thermodynamics, and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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热力学第二定律的历史起源源于卡诺的原则。它指的是一个卡诺热机周期，虚拟地以被称为准静态的极慢极限模式运行，因此热量和功量在子系统之间传递，这些子系统始终处于它们自己的内部热力学平衡状态。卡诺发动机是热机效率工程师特别感兴趣的理想设备。卡诺认识到卡诺原理的时候，正是热量热理论被认真考虑的时候，正是能量守恒定律理论被认识之前，正是熵概念的数学表达之前。根据第一定律的解释，它在物理上等同于热力学第二定律，并且在今天仍然有效。卡诺最初的论点是从卡路里理论的观点提出的，直到能量守恒定律的发现。以下是他书中的一些例子:<br />
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::...''wherever there exists a difference of temperature, motive power can be produced.''<ref>Carnot, S. (1824/1986), p. 51.</ref><br />
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...wherever there exists a difference of temperature, motive power can be produced.<br />
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只要有温差，就能产生动力。<br />
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::The production of motive power is then due in steam engines not to an actual consumption of caloric, but ''to its transportation from a warm body to a cold body ...''<ref>Carnot, S. (1824/1986), p. 46.</ref><br />
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The production of motive power is then due in steam engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body ...<br />
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动力的产生不是由于蒸汽机实际消耗的热量，而是由于它从一个温暖的物体运输到一个寒冷的物体..。<br />
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::''The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.''<ref>Carnot, S. (1824/1986), p. 68.</ref><br />
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The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.<br />
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热的动力与实现热的媒介无关，热的数量完全取决于两个物体之间的温度，最后是热量的传递。<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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用现代术语来说，卡诺的原则可能更为准确:<br />
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::The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<ref>[[Clifford Truesdell|Truesdell, C.]] (1980), Chapter 5.</ref><ref>Adkins, C.J. (1968/1983), pp. 56–58.</ref><ref>Münster, A. (1970), p. 11.</ref><ref>Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998), pp.67–75.</ref><ref>Lebon, G., Jou, D., Casas-Vázquez, J. (2008), p. 10.</ref><ref>Eu, B.C. (2002), pp. 32–35.</ref><br />
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The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<br />
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准静态或可逆卡诺循环的效率只取决于两个热源的温度，而且无论工作物质是什么，效率是相同的。以这种方式运行的卡诺发动机是使用这两种温度的最有效的热机。<br />
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===Clausius statement===<br />
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===Clausius statement===<br />
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克劳修斯声明<br />
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The German scientist [[Rudolf Clausius]] laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work.{{sfnp|Clausius|1850}} His formulation of the second law, which was published in German in 1854, is known as the ''Clausius statement'':<br />
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The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work. His formulation of the second law, which was published in German in 1854, is known as the Clausius statement:<br />
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1850年，德国科学家 Rudolf Clausius 通过研究热传递和功之间的关系，为热力学第二定律实验室奠定了基础。他在1854年用德语发表的第二定律的提法被称为克劳修斯声明:<br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.{{sfnp|Clausius|1854|p=86}}</blockquote><br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.</blockquote><br />
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热量不可能从一个较冷的物体传递到一个较热的物体，除非同时发生一些其他的变化。 / blockquote<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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克劳修斯的陈述使用了“热量通道”的概念。正如通常在热力学的讨论，这意味着“能量作为热量的净转移” ，并没有提到促成转移的一种方式和其他。<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of [[refrigeration]], for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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如果不对系统进行外部功，热就不能自发地从冷区流向热区，这一点从制冷的普通经验中可以看出。在冰箱中，热量从冷到热，但只有在外部介质——制冷系统的强制作用下才会发生变化。<br />
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===Kelvin statements===<br />
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===Kelvin statements===<br />
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开尔文陈述<br />
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[[William Thomson, 1st Baron Kelvin|Lord Kelvin]] expressed the second law in several wordings.<br />
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Lord Kelvin expressed the second law in several wordings.<br />
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开尔文勋爵用几个字表达了第二定律。<br />
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::It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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没有任何外部机构的帮助，自动机器是不可能把热量从一个物体传递到另一个物体的较高温度。<br />
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::It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.{{sfnp|Thomson|1851}}<br />
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It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.<br />
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通过无生命物质的作用，把物质冷却到周围最冷的物体的温度以下，从物质的任何部分获得机械效应是不可能的。<br />
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===Equivalence of the Clausius and the Kelvin statements===<br />
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===Equivalence of the Clausius and the Kelvin statements===<br />
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克劳修斯和开尔文陈述的等价性<br />
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[[Image:Deriving Kelvin Statement from Clausius Statement.svg|thumb|Derive Kelvin Statement from Clausius Statement]]<br />
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Derive Kelvin Statement from Clausius Statement<br />
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从克劳修斯陈述推导出开尔文陈述<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed [[Carnot engine]] as shown by the figure. The [[Heat engine#Efficiency|efficiency]] of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the [[first law of thermodynamics]], as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the figure. The efficiency of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the first law of thermodynamics, as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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假设有一个引擎违反了开尔文定理: 也就是说，这个引擎以循环的方式吸收热量并将其转化为功，没有任何其他结果。现在配对与反向卡诺发动机如图所示。普通热机的效率为，反向热机的效率为1 / 。这对组合发动机的净效应和唯一效应是将热量从较冷的贮存器转移到较热的贮存器，这违反了克劳修斯的说法。(这是能量守恒定律的结果，因为系统的总能量保持不变，数学文本{输入} + 文本{输出}0暗示 q-frac { eta }-qc / math，所以数学 qc 向左(frac { eta-1 right) / math)。因此，违反开尔文陈述意味着违反克劳修斯陈述，即。克劳修斯的说法暗示了凯尔文的说法。我们可以用类似的方式证明开尔文陈述暗示了克劳修斯陈述，因此两者是等价的。<br />
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===Planck's proposition===<br />
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===Planck's proposition===<br />
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普朗克的命题<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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普朗克提出了直接来自经验的如下命题。这有时被认为是他对第二定律的陈述，但他认为这是第二定律推导的起点。<br />
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::It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 86.</ref><ref>Roberts, J.K., Miller, A.R. (1928/1960), p. 319.</ref><br />
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It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<br />
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建造一台发动机是不可能的，这台发动机将在一个完整的循环中工作，并且除了提高重量和冷却热储以外不会产生任何效果。<br />
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===Relation between Kelvin's statement and Planck's proposition 开尔文表述与普朗克命题的关系===<br />
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It is almost customary in textbooks to speak of the "[[Kelvin-Planck statement]]" of the law, as for example in the text by [[Dirk ter Haar|ter Haar]] and [[Harald Wergeland|Wergeland]].<ref>[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966), p. 17.</ref><br />
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It is almost customary in textbooks to speak of the "Kelvin-Planck statement" of the law, as for example in the text by ter Haar and Wergeland.<br />
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教科书中几乎惯常提及该定律的“'''开尔文-普朗克表述 Kelvin-Planck Statement'''” ，例如'''德克·特哈尔 Diek ter Haar''' 和'''哈拉尔德·沃格兰 Harald Wergeland''' 在文中就是这样表述的。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）Diek ter Haar 中间是小写的<br />
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The '''Kelvin–Planck statement''' (or the ''heat engine statement'') of the second law of thermodynamics states that <br />
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The Kelvin–Planck statement (or the heat engine statement) of the second law of thermodynamics states that <br />
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热力学第二定律的开尔文-普朗克表述（或称“热机表述 Heat Engine Statement”）指出<br />
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::It is impossible to devise a [[thermodynamic cycle|cyclically]] operating device, the sole effect of which is to absorb energy in the form of heat from a single [[heat reservoir|thermal reservoir]] and to deliver an equivalent amount of [[Work (physics)|work]].<ref name="Rao">{{cite book|last=Rao|first=Y. V. C.|title=Chemical Engineering Thermodynamics|publisher=Universities Press|isbn=978-81-7371-048-3|page=158|year=1997}}</ref><br />
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It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.<br />
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设计一种唯一效果是从单一热源吸收热量并提供等量的功的循环运行装置是不可能的。<br />
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===Planck's statement 普朗克表述===<br />
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Planck stated the second law as follows.<br />
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Planck stated the second law as follows.<br />
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普朗克表述第二定律如下。<br />
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::Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<ref name="Planck 100">[[Max Planck|Planck, M.]] (1897/1903), p. 100.</ref><ref name="Planck 463">[[Max Planck|Planck, M.]] (1926), p. 463, translation by Uffink, J. (2003), p. 131.</ref><ref name="Roberts & Miller 382">Roberts, J.K., Miller, A.R. (1928/1960), p. 382. This source is partly verbatim from Planck's statement, but does not cite Planck. This source calls the statement the principle of the increase of entropy.</ref><br />
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Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<br />
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自然界中发生的任一过程都是沿参与其中的所有物体的熵之和增加的方向进行的。在某些限制下——即对于可逆过程，熵的总和保持不变。<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for ''irreversible phenomena''.<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for irreversible phenomena.<br />
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与普朗克表述非常相似的是'''乌伦贝克 Uhlenbeck'''和'''福特 Ford'''关于不可逆现象的表述。<br />
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::... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<ref>[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963), p. 16.</ref><br />
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... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<br />
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在从一个平衡态到另一个平衡态的不可逆或自发的变化中（例如，当两个物体 A 和 B 接触时的温度平衡过程），熵总是增加。<br />
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===Principle of Carathéodory 卡拉西奥多里原理===<br />
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<!-- [[Caratheodory's principle]] redirects here --><br />
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—— caratheodory 的原理在这里重新定向——<br />
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[[Constantin Carathéodory]] formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<ref>[[Constantin Carathéodory|Carathéodory, C.]] (1909).</ref><br />
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Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<br />
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'''康斯坦丁·卡拉西奥多里 Constantin Carathéodory'''在纯数学公理的基础上进行了热力学公理化。他对第二定律的陈述被称为'''卡拉西奥多里原理 Principle of Carathéodory'''，可以这样表述:<br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.<ref>Buchdahl, H.A. (1966), p. 68.</ref></blockquote><br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.</blockquote><br />
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在绝热封闭系统的任意状态 S 附近，总有从 S 出发不可达的状态。<br />
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With this formulation, he described the concept of [[adiabatic accessibility]] for the first time and provided the foundation for a new subfield of classical thermodynamics, often called [[Ruppeiner geometry|geometrical thermodynamics]]. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic [[process function]], in other words, <math>\delta Q=TdS</math>.<ref name="Sychev1991">{{cite book |last=Sychev |first=V. V. |title=The Differential Equations of Thermodynamics |year=1991 |publisher=Taylor & Francis |isbn=978-1-56032-121-7}}</ref> {{clarify|date=February 2014}}<br />
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With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, <math>\delta Q=TdS</math>. <br />
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公理化处理后，他第一次描述了'''绝热可达性 Adiabatic Accessibility'''的概念，并为经典热力学的一个新的子领域，即通常所说的'''几何热力学 Geometrical Thermodynamics'''奠定了基础。由卡拉西奥多里原理可以推出，准静态转移的热量值是一个可积过程函数，即<math>\delta Q=TdS</math>。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）准静态转移的热量值是一个可积过程函数 存疑<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.<ref name="Munster 45">Münster, A. (1970), p. 45.</ref>{{sfnp|Lieb|Yngvason|1999|p=49}}<ref name="Planck 1926">[[Max Planck|Planck, M.]] (1926).</ref><ref>Buchdahl, H.A. (1966), p. 69.</ref> {{clarify|date=February 2014}}<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium. <br />
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尽管在教科书几乎惯称卡拉西奥多里原理表述了第二定律，并认为其与克劳修斯表述或开尔文-普朗克表述等价，但事实并非如此。为了得到第二定律的所有内容，需要对卡拉西奥多里原理补充普朗克表述，即等量功总是增加一个最初处于自身内部热力学平衡的封闭系统的内部能量。<br />
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===Planck's principle 普朗克原理===<br />
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In 1926, [[Max Planck]] wrote an important paper on the basics of thermodynamics.<ref name="Planck 1926"/><ref>Uffink, J. (2003), pp. 129–132.</ref> He indicated the principle<br />
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In 1926, Max Planck wrote an important paper on the basics of thermodynamics. He indicated the principle<br />
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1926年，'''马克斯·普朗克 Max Planck'''写了一篇关于热力学基础的重要论文。他指出了以下原理<br />
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::The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<ref name="Munster 45"/>{{sfnp |Lieb|Yngvason|1999|p=49}}<br />
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The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<br />
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一个封闭系统的内部能量因绝热过程增加，在整个过程中，系统的体积保持不变。<br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work."<ref>[[Clifford Truesdell|Truesdell, C.]], Muncaster, R.G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 15.</ref> Planck wrote: "The production of heat by friction is irreversible."<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 81.</ref><ref>[[Max Planck|Planck, M.]] (1926), p. 457, Wikipedia editor's translation.</ref><br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work." Planck wrote: "The production of heat by friction is irreversible."<br />
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这个公式没有提到热，没有提到温度，甚至没有提到熵，也不一定隐含地依赖于这些概念，但它暗示了第二定律的内容。一个密切相关的表述为，“摩擦力从来不做正功。”普朗克写道: “摩擦生热是不可逆的。”<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above.<ref>Lieb, E.H., Yngvason, J. (2003), p. 149.</ref> It is relevant that for a system at constant volume and [[Mole (unit)|mole numbers]], the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above. It is relevant that for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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更不用说熵了，这个普朗克定理是用物理术语来表述的。它与上面给出的开尔文表述密切相关。相关的是，对于恒定体积和摩尔数的系统，熵是内能的单调函数。然而，普朗克的这个原理实际上并不是普朗克对第二定律的首选表述（见前面小节），而是依赖于熵的概念。<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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Borgnakke 和 Sonntag 提出的以下表述在某种意义上是对普朗克原理的补充。他们没有将其作为第二定律的完整表述:<br />
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::... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<ref>Borgnakke, C., Sonntag., R.E. (2009), p. 304.</ref><br />
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... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<br />
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只有一种方法可以减少[闭合]系统的熵——从系统中转移热量。<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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与普朗克之前提出的原理不同，这一原理明确地用熵的变化来表示。从系统中去除物质也可以减少系统的熵。<br />
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===Statement for a system that has a known expression of its internal energy as a function of its extensive state variables 一个其内能有已知表达式（其扩展状态变量的函数）的系统的表述===<br />
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The second law has been shown to be equivalent to the [[internal energy]] ''U'' being a weakly [[convex function]], when written as a function of extensive properties (mass, volume, entropy, ...).<ref>{{cite book |last1=van Gool |first1=W. |last2=Bruggink |first2=J.J.C. (Eds) |url= |title=Energy and time in the economic and physical sciences |publisher=North-Holland |year=1985 |pages=41–56 |quote= |isbn=978-0-444-87748-2}}</ref><ref>{{Cite journal | last1 = Grubbström | first1 = Robert W. | doi = 10.1016/j.apenergy.2007.01.003 | title = An Attempt to Introduce Dynamics Into Generalised Exergy Considerations| journal = Applied Energy| volume = 84| issue = 7–8 | pages = 701–718 | year = 2007}}</ref> {{clarify|date=February 2014}}<br />
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The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...). <br />
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第二定律被证明等价于弱凸函数内能 U （写成广泛性质（质量，体积，熵，...）的函数时）。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）存疑<br />
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==Corollaries 推论==<br />
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===Perpetual motion of the second kind 第二类永动机===<br />
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{{main article|Perpetual motion}}<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of [[first law of thermodynamics]] by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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在第二定律建立之前，许多志在发明永动机的人试图通过提取环境的巨大内能作为机器动力来突破'''热力学第一定律 First Law of Thermodynamics''' 的限制。这种机器被称为“'''第二类永动机 Perpetual Motion Machine of the Second Kind'''”。第二定律表明这种机器是不可能的。<br />
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===Carnot theorem 卡诺定理===<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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卡诺定理（1824）是一条限制任何可能的发动机的最大效率的原理。效率完全取决于热库和冷库之间的温差。卡诺定理指出:<br />
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*All irreversible heat engines between two heat reservoirs are less efficient than a [[Carnot engine]] operating between the same reservoirs.<br />
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*All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs.<br />
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* 所有不可逆热机的效率低于在两相同热源之间工作的'''卡诺机 Carnot engine'''。<br />
* 所有可逆热机的效率与在两相同热源之间工作的卡诺机相等。<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as [[thermodynamic reversibility]]. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the [[Carnot cycle]]'s reversibility and was condemned to be less efficient.<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the Carnot cycle's reversibility and was condemned to be less efficient.<br />
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在他的理想模型中，热转换成功可以通过逆转循环的运动而恢复，这个概念后来被称为'''热力学可逆性 Thermodynamic Reversibility'''。然而，卡诺进一步假定，一些热量损失了，并没有转化为机械功。因此，没有一个真实的热机能够实现'''卡诺循环 Carnot cycle'''的可逆性，并且被认为效率较低。<br />
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Though formulated in terms of caloric (see the obsolete [[caloric theory]]), rather than [[entropy]], this was an early insight into the second law.<br />
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Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law.<br />
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该理论尽管是用热量表述的（见被取代的'''热质说 Caloric Theory'''），而不是熵，但是它是对第二定律的早期认识。<br />
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===Clausius inequality 克劳修斯不等式===<br />
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The [[Clausius theorem]] (1854) states that in a cyclic process<br />
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The Clausius theorem (1854) states that in a cyclic process<br />
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克劳修斯定理（1854）指出，在一个循环的过程中<br />
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: <math>\oint \frac{\delta Q}{T} \leq 0.</math><br />
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The equality holds in the reversible case<ref>[http://scienceworld.wolfram.com/physics/ClausiusTheorem.html ''Clausius theorem''] at [[Wolfram Research]]</ref> and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function [[entropy]]. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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The equality holds in the reversible case and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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等号在可逆情况下成立，严格不等号在不可逆情况下成立。可逆情况下引入状态函数熵。这是因为在循环过程中，状态函数的变化为零。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）末句存疑<br />
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===Thermodynamic temperature 热力学温度===<br />
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{{main article|Thermodynamic temperature}}<br />
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For an arbitrary heat engine, the efficiency is:<br />
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For an arbitrary heat engine, the efficiency is:<br />
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对于任意热机，效率为:<br />
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: <math>\eta = \frac {W_n}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H} \qquad (1)</math><br />
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where ''W''<sub>n</sub> is for the net work done per cycle. Thus the efficiency depends only on q<sub>C</sub>/q<sub>H</sub>.<br />
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其中 ''W''<sub>n</sub> 表示每个循环所做的净功。因此效率只取决于 q<sub>C</sub>/q<sub>H</sub>。<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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卡诺定理指出，所有在相同热源间运行的可逆发动机的效率是相同的。<br />
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Thus, any reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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Thus, any reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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因此，任何在温度''T''<sub>1</sub>和''T''<sub>2</sub>之间运行的可逆热机必须具有相同的效率，也就是说，效率只是温度的函数:<br />
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<math>\frac{q_C}{q_H} = f(T_H,T_C)\qquad (2).</math><br />
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In addition, a reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between ''T''<sub>1</sub> and another (intermediate) temperature ''T''<sub>2</sub>, and the second between ''T''<sub>2</sub> and''T''<sub>3</sub>. This can only be the case if<br />
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In addition, a reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between T<sub>1</sub> and another (intermediate) temperature T<sub>2</sub>, and the second between T<sub>2</sub> andT<sub>3</sub>. This can only be the case if<br />
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另外，在温度 ''T''<sub>1</sub> 和 ''T''<sub>3</sub> 之间工作的可逆热机必须具有与由分别在温度''T''<sub>1</sub> 和（中间）温度 ''T''<sub>2</sub>之间、在 ''T''<sub>2</sub> 和 ''T''<sub>3</sub> 之间的两个循环组成的系统效率相同。只有下式成立才会出现这种情况:<br />
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: <math>f(T_1,T_3) = \frac{q_3}{q_1} = \frac{q_2 q_3} {q_1 q_2} = f(T_1,T_2)f(T_2,T_3).</math><br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the [[triple point]] of water. Then for any ''T''<sub>2</sub> and ''T''<sub>3</sub>,<br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the triple point of water. Then for any T<sub>2</sub> and T<sub>3</sub>,<br />
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现在考虑如下情形，<math>T_1</math> 是一个固定的参考温度: 水的三相点的温度。则对于任意 T<sub>2</sub> 和 T<sub>3</sub>,<br />
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: <math>f(T_2,T_3) = \frac{f(T_1,T_3)}{f(T_1,T_2)} = \frac{273.16 \cdot f(T_1,T_3)}{273.16 \cdot f(T_1,T_2)}.</math><br />
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Therefore, if thermodynamic temperature is defined by<br />
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Therefore, if thermodynamic temperature is defined by<br />
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因此，如果热力学温度被定义为<br />
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: <math>T = 273.16 \cdot f(T_1,T) \,</math><br />
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then the function ''f'', viewed as a function of thermodynamic temperature, is simply<br />
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then the function f, viewed as a function of thermodynamic temperature, is simply<br />
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那么函数 ''f'' 作为热力学温度的函数，为<br />
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: <math>f(T_2,T_3) = \frac{T_3}{T_2},</math><br />
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and the reference temperature ''T''<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be used{{snd}}the choice here corresponds to the [[Kelvin]] scale.)<br />
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and the reference temperature T<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be usedthe choice here corresponds to the Kelvin scale.)<br />
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参考温度 ''T''<sub>1</sub> 的值为273.16。（任何参考温度和任何正值均可用——此处的选择对应开尔文标度。）<br />
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===Entropy 熵===<br />
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{{main article|Entropy (classical thermodynamics)}}<br />
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According to the [[Clausius theorem|Clausius equality]], for a reversible process<br />
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According to the Clausius equality, for a reversible process<br />
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根据克劳修斯定理，对可逆过程有<br />
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: <math>\oint \frac{\delta Q}{T}=0</math><br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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这意味着线积分 <math>\int_L \frac{\delta Q}{T}</math> 对于可逆过程是路径无关的。<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer<ref name="Schmidt-Rohr 14" /> satisfies<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer satisfies<br />
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所以我们可以定义一个叫做熵的状态函数 S，对于可逆过程或者纯热传递满足<br />
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: <math>dS = \frac{\delta Q}{T} \!</math><br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the [[third law of thermodynamics]], which states that ''S'' = 0 at [[absolute zero]] for perfect crystals.<br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the third law of thermodynamics, which states that S = 0 at absolute zero for perfect crystals.<br />
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据此，只有对上述公式进行积分，才能得到熵的差值。为了获得绝对值，我们需要'''热力学第三定律 Third Law of Thermodynamics'''，它指出绝对零度下完美晶体的 ''S'' = 0。<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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对于任意不可逆过程，由于熵是一个状态函数，我们总是可以将初始状态和最终状态与一个虚拟的可逆过程联系起来，并在这条路径上积分以计算熵的差值。<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the [[Clausius inequality]] on this loop,<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the Clausius inequality on this loop,<br />
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现在把可逆过程逆过来，将其与上述不可逆过程结合。把克劳修斯不等式应用到这个循环,<br />
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: <math>-\Delta S+\int\frac{\delta Q}{T}=\oint\frac{\delta Q}{T}< 0</math><br />
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Thus,<br />
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Thus,<br />
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故,<br />
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: <math>\Delta S \ge \int \frac{\delta Q}{T} \,\!</math><br />
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where the equality holds if the transformation is reversible.<br />
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where the equality holds if the transformation is reversible.<br />
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如果变换可逆，等号成立。<br />
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Notice that if the process is an [[adiabatic process]], then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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Notice that if the process is an adiabatic process, then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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注意，若该过程是一个'''绝热过程 Adiabatic Process'''，则<math>\delta Q=0</math>，故<math>\Delta S\ge 0</math>。<br />
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===Energy, available useful work===<br />
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===Energy, available useful work===<br />
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能量，有用的工作<br />
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{{See also|Exergy}}<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an ''unlimited'' heat reservoir at temperature ''T<sub>R</sub>'' and pressure ''P<sub>R</sub>'' {{snd}}so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain ''T<sub>R</sub>''; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain ''P<sub>R</sub>''.<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an unlimited heat reservoir at temperature T<sub>R</sub> and pressure P<sub>R</sub> so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain T<sub>R</sub>; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain P<sub>R</sub>.<br />
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考虑将第二定律应用于孤立系统(称为整体系统或宇宙)的情形，是一个重要的、具有启发性的理想情形。孤立系统由两部分组成: 感兴趣的子系统和子系统的环境。这些环境被想象为如此之大，以至于它们可以被认为是一个温度为 t 子 r / sub 和压力为 p 子 r / sub 的无限热源，因此无论有多少热量被转移到(或来自)子系统，周围环境的温度将保持 t 子 r / sub; 无论子系统的体积扩大(或收缩)多少，周围环境的压力将保持 p 子 r / sub。<br />
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Whatever changes to ''dS'' and ''dS<sub>R</sub>'' occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy ''S<sub>tot</sub>'' of the isolated total system must not decrease:<br />
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Whatever changes to dS and dS<sub>R</sub> occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy S<sub>tot</sub> of the isolated total system must not decrease:<br />
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根据第二定律，孤立总体系统的熵 s 子系统 / 子系统的熵不能减小，无论子系统和周围环境的熵对 dS 和 dS 子系统 r / 子系统发生什么变化:<br />
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: <math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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<math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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数学 dS + dS r ge 0 / math<br />
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According to the [[first law of thermodynamics]], the change ''dU'' in the internal energy of the sub-system is the sum of the heat ''δq'' added to the sub-system, ''less'' any work ''δw'' done ''by'' the sub-system, ''plus'' any net chemical energy entering the sub-system ''d ∑μ<sub>iR</sub>N<sub>i</sub>'', so that:<br />
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According to the first law of thermodynamics, the change dU in the internal energy of the sub-system is the sum of the heat δq added to the sub-system, less any work δw done by the sub-system, plus any net chemical energy entering the sub-system d ∑μ<sub>iR</sub>N<sub>i</sub>, so that:<br />
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根据能量守恒定律，子系统内部能量的变化 dU 是子系统内部能量 q 的总和，减去子系统所做的任何功，再加上进入子系统 d ∑ sub iR / sub n sub i / sub 的任何净化学能，因此:<br />
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: <math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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<math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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数学 dU delta q- delta w + d ( sum mu iR } n i) ，/ math<br />
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where μ<sub>iR</sub> are the [[chemical potential]]s of chemical species in the external surroundings.<br />
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where μ<sub>iR</sub> are the chemical potentials of chemical species in the external surroundings.<br />
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其中亚 iR / 亚是外部环境中化学物种的化学势。<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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现在热量离开储存器进入子系统是<br />
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: <math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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<math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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数学△ qtr (- dS r) le tr dS / math<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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其中我们首先使用了经典热力学中熵的定义(在统计热力学中，熵变、温度和吸收热量之间的关系可以推导出来) ，然后从上面推导出第二定律的不等式。<br />
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It therefore follows that any net work ''δw'' done by the sub-system must obey<br />
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It therefore follows that any net work δw done by the sub-system must obey<br />
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因此，子系统所做的任何净功都必须服从<br />
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: <math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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<math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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数学增量 w le-dU + tds + sum mu { iR } dN i，/ math<br />
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It is useful to separate the work ''δw'' done by the subsystem into the ''useful'' work ''δw<sub>u</sub>'' that can be done ''by'' the sub-system, over and beyond the work ''p<sub>R</sub> dV'' done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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It is useful to separate the work δw done by the subsystem into the useful work δw<sub>u</sub> that can be done by the sub-system, over and beyond the work p<sub>R</sub> dV done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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将子系统所做的功与子系统所能做的功分离为子系统所能做的有用功，超过了子系统对外界压力所做的功，给出了子系统所能做的有用功(火用)的下列关系式:<br />
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: <math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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<math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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D (u-t r s + p r v-sum { iR } n i) ，/ math<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the ''availability'' or ''[[exergy]]'' ''E'' of the subsystem,<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the availability or exergy E of the subsystem,<br />
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可以很方便地将子系统的右侧定义为热动力位能的精确衍生物，称为子系统的可用性或火用 e,<br />
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: <math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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<math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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数学 e u-t r s + p r v- sum { iR } n i / math<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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因此，第二定律意味着，对于任何可以简单地被视为分为一个子系统和一个与之接触的无限温度和压力储存器的过程,<br />
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: <math> dE + \delta w_u \le 0 \, </math><br />
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<math> dE + \delta w_u \le 0 \, </math><br />
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数学德 delta u le 0，/ math<br />
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i.e. the change in the subsystem's exergy plus the useful work done ''by'' the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done ''on'' the system) must be less than or equal to zero.<br />
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i.e. the change in the subsystem's exergy plus the useful work done by the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done on the system) must be less than or equal to zero.<br />
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也就是。子系统(火用)的变化加上子系统所做的有用功(或者，子系统(火用)的变化不包括任何功，除了压力贮存器所做的功之外，在系统中所做的功)必须小于或等于零。<br />
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In sum, if a proper ''infinite-reservoir-like'' reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in ''E'' for an irreversible process and no change for a reversible process.<br />
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In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in E for an irreversible process and no change for a reversible process.<br />
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总之，如果选择一个合适的类似于无限库的参考状态作为现实世界中的系统环境，那么第二定律预测一个不可逆性的 e 值会减少，而一个可逆过程的 e 值不会变化。<br />
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: <math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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<math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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数学 d = 0 = 0 = 0 = 0 = 0<br />
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This expression together with the associated reference state permits a [[design engineer]] working at the macroscopic scale (above the [[thermodynamic limit]]) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (''Also, see [[process engineer]]''). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (''See [[Exergy efficiency|second law efficiency]]''.)<br />
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This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (Also, see process engineer). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (See second law efficiency.)<br />
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这个表达式连同相关的参考状态允许在宏观的设计工程师利用第二定律，而不需要直接测量或考虑整个孤立系统的熵变。(另外，请参阅流程工程师)。这些变化已经被考虑，假设被考虑的系统可以达到平衡的参考状态而不改变参考状态。还可以找到一个过程或过程集合的效率，将其与可逆的理想状态相比较<br />
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This approach to the Second Law is widely utilized in [[engineering]] practice, [[environmental accounting]], [[systems ecology]], and other disciplines.<br />
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This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines.<br />
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第二定律的这种方法被广泛应用于工程实践、环境会计、系统生态学和其他学科。<br />
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===The second law in chemical thermodynamics===<br />
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===The second law in chemical thermodynamics===<br />
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化学热力学的第二定律<br />
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For a [[spontaneous process|spontaneous chemical process]] in a closed system at constant temperature and pressure without non-''PV'' work, the Clausius inequality Δ''S > Q/T<sub>surr</sub>'' transforms into a condition for the change in [[Gibbs free energy]] <br />
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For a spontaneous chemical process in a closed system at constant temperature and pressure without non-PV work, the Clausius inequality ΔS > Q/T<sub>surr</sub> transforms into a condition for the change in Gibbs free energy <br />
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对于一个封闭系统中的自发化学过程，在没有非 pv 功的情况下，在恒定温度和压力下，将克劳修斯不等式 s q / t sub surr / sub 转化为吉布斯自由能变化的条件<br />
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: <math>\Delta G < 0 </math><br />
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数学 Delta g 0 / 数学<br />
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or d''G'' < 0. For a similar process at constant temperature and volume, the change in [[Helmholtz free energy]] must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products.<ref name="Oxtoby8th"/> <ref name="MortimerBook"></ref> The chemical equilibrium condition at constant ''T'' and ''p'' without electrical work is d''G'' = 0.<br />
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or dG < 0. For a similar process at constant temperature and volume, the change in Helmholtz free energy must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products. The chemical equilibrium condition at constant T and p without electrical work is dG = 0.<br />
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或者 dG 0。对于一个相似的过程，在恒定的温度和体积下，亥姆霍兹自由能的变化一定是负的，这是 Delta a 0 / math。因此，自由能(g 或 a)变化的负值是过程自发的必要条件。这是热力学第二定律在化学中最有用的形式，其中自由能的变化可以通过列表的生成焓和反应物和产物的标准摩尔熵来计算。在 t 和 p 不变的情况下，化学平衡条件是 dG 0。<br />
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==History==<br />
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==History==<br />
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历史<br />
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{{See also|History of entropy}}<br />
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[[File:Sadi Carnot.jpeg|thumb|upright|Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique]].]]<br />
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Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique.]]<br />
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尼古拉·莱昂纳尔·萨迪·卡诺穿着传统的巴黎综合理工学院制服<br />
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The first theory of the conversion of heat into mechanical work is due to [[Nicolas Léonard Sadi Carnot]] in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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The first theory of the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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第一个把热转化为机械功的理论是在1824年由尼古拉·莱昂纳尔·萨迪·卡诺提出的。他是第一个正确认识到这种转换的效率取决于发动机和环境之间的温差。<br />
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Recognizing the significance of [[James Prescott Joule]]'s work on the conservation of energy, [[Rudolf Clausius]] was the first to formulate the second law during 1850, in this form: heat does not flow ''spontaneously'' from cold to hot bodies. While common knowledge now, this was contrary to the [[caloric theory]] of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law during 1850, in this form: heat does not flow spontaneously from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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认识到詹姆斯·普雷斯科特·焦耳在能量守恒方面工作的重要性，Rudolf Clausius 在1850年第一个提出了第二定律: 热不会自发地从冷物体流向热物体。虽然现在的常识是，这与当时流行的热量理论相反，热量理论认为热量是一种流体。从那里他能够推断出萨迪卡诺定律和熵的定义(1865年)。<br />
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Established during the 19th century, the [[Kelvin-Planck statement|Kelvin-Planck statement of the Second Law]] says, "It is impossible for any device that operates on a [[cyclic process|cycle]] to receive heat from a single [[heat reservoir|reservoir]] and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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Established during the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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建立于19世纪的开尔文-普朗克第二定律(Kelvin-Planck)说，“任何循环运行的设备都不可能从单个蓄热体接收热量，并产生净功。”这被证明相当于克劳修斯的陈述。<br />
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The [[ergodic hypothesis]] is also important for the [[Boltzmann]] approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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遍历假设对玻尔兹曼方法也很重要。它说，在很长一段时间内，在具有相同能量的微观态的相空间的某些区域所花费的时间与这个区域的体积成正比，即。在很长一段时间内，所有可访问的微观状态的可能性都是一样的。同样的，它表明时间平均值和系综平均值是相同的。<br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a [[Macroscopic bodies|macroscopic system]]. This doctrine is obsolescent.<ref>Denbigh, K.G., Denbigh, J.S. (1985). ''Entropy in Relation to Incomplete Knowledge'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-25677-1}}, pp. 43–44.</ref><ref>Grandy, W.T., Jr (2008). ''Entropy and the Time Evolution of Macroscopic Systems'', Oxford University Press, Oxford, {{ISBN|978-0-19-954617-6}}, pp. 55–58.</ref><ref name=Lambert>[http://entropysite.oxy.edu Entropy Sites — A Guide] Content selected by [[Frank L. Lambert]]</ref><br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a macroscopic system. This doctrine is obsolescent.<br />
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有一种传统的学说，从克劳修斯开始，认为熵可以被理解为宏观系统中的分子“无序”。这种学说已经过时了。<br />
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===Account given by Clausius===<br />
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===Account given by Clausius===<br />
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作者: 克劳修斯<br />
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[[File:Clausius-1.jpg|thumb|upright|Rudolf Clausius]]<br />
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Rudolf Clausius<br />
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鲁道夫 · 克劳修斯<br />
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In 1856, the German physicist [[Rudolf Clausius]] stated what he called the "second fundamental theorem in the [[mechanical theory of heat]]" in the following form:{{sfnp|Clausius|1867}}<br />
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In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:<br />
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1856年，德国物理学家鲁道夫 · 克劳修斯阐述了他所谓的“热力学理论中的第二个基本定理” ，其形式如下:<br />
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: <math>\int \frac{\delta Q}{T} = -N</math><br />
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<math>\int \frac{\delta Q}{T} = -N</math><br />
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- n / math<br />
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where ''Q'' is heat, ''T'' is temperature and ''N'' is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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其中 q 是热，t 是温度，n 是一个循环过程中所有未补偿的相变的“等价值”。后来，在1865年，克劳修斯将“等价值”定义为熵。紧接着这个定义，也就是在同一年，第二定律最著名的版本在4月24日苏黎世哲学学会的一次演讲中被宣读，在演讲的最后，克劳修斯总结道:<br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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宇宙的熵趋于最大<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. [[universe]], as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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这句话是第二定律中最广为人知的措辞。由于其语言的松散性，例如:。这个问题的答案是肯定的，因为这个问题的答案是肯定的。开放，封闭，或孤立，许多人认为这个简单的陈述意味着热力学第二定律适用于几乎所有可以想象的主题。这是不正确的; 这个陈述只是一个更加扩展和精确描述的简化版本。<br />
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In terms of time variation, the mathematical statement of the second law for an [[isolated system]] undergoing an arbitrary transformation is:<br />
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In terms of time variation, the mathematical statement of the second law for an isolated system undergoing an arbitrary transformation is:<br />
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就时间变化而言，经历任意变换的孤立系统第二定律的数学表述是:<br />
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: <math>\frac{dS}{dt} \ge 0</math><br />
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<math>\frac{dS}{dt} \ge 0</math><br />
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0 / math<br />
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where<br />
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where<br />
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在哪里<br />
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: ''S'' is the entropy of the system and<br />
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S is the entropy of the system and<br />
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S 是系统的熵<br />
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: ''t'' is [[time]].<br />
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t is time.<br />
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时间到了。<br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> <br />
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!-- 可逆过程需要与周围环境保持平衡。这对于一个孤立的系统是不可能的。因此，关于可逆过程的讨论已经转移到对封闭系统的分析----<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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平衡后应用等式符号。建立孤立系统第二定律的另一种方法是:<br />
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: <math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math<br />
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with <math> \dot S_{i}</math> the sum of the rate of [[entropy production]] by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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with <math> \dot S_{i}</math> the sum of the rate of entropy production by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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用数学 s { i } / 数学表示系统内所有进程的产生熵之和。这种配方的优点是它显示了产生熵的效果。产生熵是一个非常重要的概念，因为它决定(限制)热机器的效率。与环境温度数学 t { a } / math 相乘，得到所谓的耗散能量数学 p { diss } t { a } dot s { i } / math。<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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封闭系统的第二定律(允许热量交换和移动边界，但不允许物质交换)的表达式是:<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math 来计算数学点 q } + 点 s { i } / math<br />
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Here<br />
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给你<br />
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: <math>\dot Q</math> is the heat flow into the system<br />
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<math>\dot Q</math> is the heat flow into the system<br />
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数学 q / math 是进入系统的热流<br />
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: <math>T</math> is the temperature at the point where the heat enters the system.<br />
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<math>T</math> is the temperature at the point where the heat enters the system.<br />
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Math 是热量进入系统时的温度。<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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等式符号在只有可逆过程在系统内发生的情况下成立。如果发生不可逆过程(在实际操作系统中就是这种情况) ,-符号保持不变。如果系统有多处供热，我们必须求相应项的代数和。<br />
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For open systems (also allowing exchange of matter):<br />
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For open systems (also allowing exchange of matter):<br />
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对于开放系统(也允许物质交换) :<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math 表示数学点 q } + 点 s + 点 s { i } / math<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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这里的数学 s / math 是熵流进入系统，与物质流进入系统有关。它不应该与熵的时间导数混淆。如果物质在几个地方提供，我们必须取这些贡献的代数和。<br />
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==Statistical mechanics==<br />
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==Statistical mechanics==<br />
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统计力学<br />
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[[Statistical mechanics]] gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a [[microstate (statistical mechanics)|microstate]] of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/{{radic|''N''}} where ''N'' is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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Statistical mechanics gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a microstate of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/ where N is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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通过假设物质是由不断运动的原子和分子组成的，统计力学对第二定律给出了解释。系统中每个粒子的一组特定的位置和速度称为系统的微观状态，由于系统的不断运动，系统不断地改变其微观状态。统计力学假设，在平衡状态下，系统可能处于的每个微观状态发生的可能性是相等的，当这个假设被提出时，它直接导致了第二定律必须在统计意义上成立的结论。也就是说，第二定律平均成立，其中 n 是系统中粒子数的1 / 数量级的统计变化。在日常(宏观)情况下，违反第二定律的概率几乎为零。然而，对于粒子数量很少的系统，热力学参数，包括熵，可能显示出与第二定律预测的显著的统计偏差。经典热力学理论没有处理这些统计变量。<br />
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（by Jie XIANG) 在通过假定一种物质由不断运动的原子和分子组成的前提下，统计力学对热力学第二定律进行了解释。系统中每个粒子其特定的位置和速度组成为系统的微观状态，由于（原子和分子的）不断运动，系统也持续改变其微观状态。统计力学假设，在系统力平衡的情况下，其微观状态其实是等可能发生的，进而能够得出结论：热力学第二定律必须在统计学意义上成立。也就是说，第二定律将保持均值不变， 其统计变化为1 /√N，N为系统中的粒子数。在日常(宏观)情况下，违反热力学第二定律的概率几乎为零，但是，对于粒子数量非常少的系统，其热力学参数（包括熵）可能会显示出与第二定律所预测的明显统计偏差。然而经典热力学理论并不处理这些统计变量。<br />
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==Derivation from statistical mechanics==<br />
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==Derivation from statistical mechanics==<br />
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从统计力学导出<br />
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{{Further information|H-theorem}}<br />
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The first mechanical argument of the [[Kinetic theory of gases]] that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to [[James Clerk Maxwell]] in 1860;<ref>{{Cite journal | last1 = Gyenis | first1 = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G }}</ref> [[Ludwig Boltzmann]] with his [[H-theorem]] of 1872 also argued that due to collisions gases should over time tend toward the [[Maxwell–Boltzmann distribution]].<br />
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The first mechanical argument of the Kinetic theory of gases that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to James Clerk Maxwell in 1860; Ludwig Boltzmann with his H-theorem of 1872 also argued that due to collisions gases should over time tend toward the Maxwell–Boltzmann distribution.<br />
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气体动力学理论的第一个力学论证由詹姆斯·克拉克·麦克斯韦在1860年给出，指出分子碰撞引起温度均衡，因此整体趋向于平衡； 路德维希·玻尔兹曼在1872年提出的 H 定理也认为，气体由于碰撞应该随着时间的推移趋向于麦克斯韦-波兹曼分布。<br />
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Due to [[Loschmidt's paradox]], derivations of the Second Law have to make an assumption regarding the past, namely that the system is [[Correlation and dependence|uncorrelated]] at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a [[boundary condition]], and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the [[Big Bang]]), though [[Boltzmann brain|other scenarios]] have also been suggested.<ref name="Hawking AOT">{{cite journal|last=Hawking|first=SW|title=Arrow of time in cosmology|journal=Phys. Rev. D|year=1985|volume=32|issue=10|pages=2489–2495|doi=10.1103/PhysRevD.32.2489|pmid=9956019|bibcode = 1985PhRvD..32.2489H }}</ref><ref>{{cite book | last = Greene | first = Brian | authorlink = Brian Greene | title = The Fabric of the Cosmos | url = https://archive.org/details/fabricofcosmossp00gree | url-access = registration | publisher = Alfred A. Knopf | year = 2004 | page = [https://archive.org/details/fabricofcosmossp00gree/page/171 171] | isbn = 978-0-375-41288-2}}</ref><ref name=Lebowitz>{{cite journal|last=Lebowitz|first=Joel L.|title= Boltzmann's Entropy and Time's Arrow|journal=Physics Today|date=September 1993|volume=46|issue=9|pages=32–38|url=http://users.df.uba.ar/ariel/materias/FT3_2008_1C/papers_pdf/lebowitz_370.pdf|accessdate=2013-02-22|doi=10.1063/1.881363|bibcode = 1993PhT....46i..32L }}</ref><br />
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Due to Loschmidt's paradox, derivations of the Second Law have to make an assumption regarding the past, namely that the system is uncorrelated at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a boundary condition, and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the Big Bang), though other scenarios have also been suggested.<br />
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由于洛施密特悖论，第二定律的导出必须对过去做出一个假设，即系统在过去的某个时刻是不相关的；这样的假设允许进行简单的概率处理。这个假设通常被认为是一个边界条件，因此热力学第二定律最终是过去某个地方的初始条件的结果，可能是在宇宙的开始(大爆炸) 。也有人提出了其他假设。<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the [[Statistical mechanics#Fundamental postulate|fundamental postulate]], also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy.{{citation needed|date=August 2012}} The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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考虑到这些假设，在统计力学中，第二定律不是一个假设，而是统计力学基本假设的一个结果，也被称为等先验概率假设。这个基本假设表明，只要一个人清楚地知道，简单的概率论证只适用于未来，而对于过去，有辅助的信息来源告诉我们，它是低熵的。热力学第二定律的第一部分指出，热孤立系统的熵只能增加。如果我们把熵的概念限制在热平衡系统中，那么热力学第二定律的第一部分是等先验概率假设的一个显然结果。处于热平衡状态的孤立系统包含能量<math>E</math>的熵表示为：<br />
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: <math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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<math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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其中<math>\Omega\left(E\right)</math> 是处于 <math>E</math>和<math>E +\delta E</math这个小区间内的量子态数目。这里的 <math>\delta E</math> 是一个宏观上很小的固定能量区间。严格地说，这意味着熵取决于对<math>\delta E</math>的选择。然而在热力学极限下（例如无穷大系统的极限），狭义的熵(单位体积或单位质量的熵)不依赖于 <math>\delta E</math>。<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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假设我们有一个孤立系统，其宏观状态由许多变量描述。这些宏观变量可以是总体积、活塞在系统中的位置等。从而<math>\Omega</math>将取决于这些变量的值。如果某个变量不是固定的（我们不会在某个位置夹住活塞) ，那么因为在平衡状态下所有可到达状态的等几率到达的，平衡状态下的自由变量会使 <math>\Omega</math> 最大，因为这是平衡状态下最可能的情况。<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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如果该变量最初固定到某个值然后释放，当达到新的平衡时，变量将自我调整使得 <math>\Omega</math> 最大，这意味着熵将增加或保持不变(如果变量初始固定的值恰好是平衡值)。<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's [[H-theorem]], however, proves that the quantity {{math|''H''}} increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the quantity increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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假设我们初始位于一个平衡状态，突然移除了对一个变量的约束。我们做完这件事的时候，可达到的微观状态的数为<math>\Omega</math>，但是系统还没有达到平衡，所以系统处于某些可达到的状态的实际概率还不等于先验概率 <math>1/\Omega</math>。我们已经知道，最终的平衡状态相对于之前的平衡状态，熵会增加或者保持不变。然而，玻耳兹曼的H定理证明系统在不处于平衡态的期间，那个量作为时间的函数单调增加。<br />
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===Derivation of the entropy change for reversible processes===<br />
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===Derivation of the entropy change for reversible processes===<br />
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可逆过程熵变的推导<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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第二定律的第二部分指出，经历可逆过程的系统的熵变是由以下因素给出的:<br />
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: <math>dS =\frac{\delta Q}{T}</math><br />
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<math>dS =\frac{\delta Q}{T}</math><br />
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数学 dS frac delta q }{ t } / math<br />
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where the temperature is defined as:<br />
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where the temperature is defined as:<br />
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其中温度定义为:<br />
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:<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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[数学][[欧米茄][左(右)][右][数学][数学]<br />
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See [[Microcanonical ensemble|here]] for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the [[adiabatic theorem]] of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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请参阅此处对此定义的正当理由。假设系统有一些可以改变的外部参数 x。一般来说，系统的能量本征态将依赖于 x。根据量子力学的绝热定理，在系统哈密顿量无限缓慢变化的极限下，系统将保持在相同的定态，因此系统的能量会随着其所在定态的能量变化而变化。<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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广义力，x，对应于外部变量 x 的定义是，如果 x 增加一个量 dx，那么 math x dx / math 就是系统所做的功。例如，如果 x 是体积，那么 x 是压强。一个已知的系统的广义力在定态数学 e { r } / math 中给出:<br />
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: <math>X = -\frac{dE_{r}}{dx}</math><br />
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<math>X = -\frac{dE_{r}}{dx}</math><br />
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数学 x- frac { dE { r }{ dx } / math<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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由于系统可以在一个数学△ e / math 区间内的任意定态内，我们将系统的广义力定义为上述表达式的期望值:<br />
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: <math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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<math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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数学 x- 左 langle frac { dE { dx }右 rangle，/ math<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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为了计算平均值，我们通过计算在数学 y / math 和数学 y + delta y / math 之间的范围内，其中有多少数学 y / math 能量本征态有一个值来划分 ω 左(e 右) / math 能量本征态。把这个数字命名为 math Omega { y } left (e right) / math，我们有:<br />
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: <math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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<math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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Math Omega left (e right) sum { y } Omega { y } left (e right) ，/ math<br />
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<br />
<br />
<br />
<br />
The average defining the generalized force can now be written:<br />
<br />
The average defining the generalized force can now be written:<br />
<br />
定义广义力的平均值现在可以写成:<br />
<br />
<br />
<br />
<br />
<br />
: <math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
<br />
<math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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数学 x frac {1} Omega 左(e 右) sum { y Omega { y }左(e 右) ，/ math<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
<br />
We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
<br />
我们可以把它和熵对 x 在恒定能量 e 下的导数联系起来，如下所示。假设我们把 x 改成 x + dx。然后 math Omega left (e right) / math 将会改变，因为能量本征态依赖于 x，导致能量本征态进入或超出数学 e / math 和数学 e + delta e / math 之间的范围。让我们再次关注数学 y / math 和数学 y + delta y / math 之间的能量本征态。由于这些能量本征态的能量增加了 y dx，所有这些能量本征态在 e-y dx 到 e 之间的区间内从 e 以下移动到 e 以上。 有<br />
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<br />
<br />
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: <math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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<math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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数学 n { y }左(e 右) frac { y }左(e 右)}{ delta e } y dx ，/ math<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
<br />
such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
<br />
这样的能量本征态。如果数学 y dx leq delta e / math，所有这些能量本征态将移动到数学 e / math 和数学 e + delta e / math 之间的范围内，有助于数学 ω / math 的增加。从数学 e + △ e / math 下移到数学 e + △ e / math 上的能量本征态的个数是由数学 n { y }左(e + △ e 右) / math 给出的。区别在于<br />
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: <math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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<math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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数学 n { y }左(e 右)-n { y }左(e + delta e 右) ，/ math<br />
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<br />
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<br />
is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
<br />
is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
<br />
因此是数学增长的净贡献欧米茄 / 数学。请注意，如果 y dx 大于 math delta e / math，那么将会有能量本征态从 e 下面移动到 e + delta e / math 上面。它们在数学 n { y } left (e right) / math 和数学 n { y } left (e + delta e right) / math 中都有计算，因此上述表达式在这种情况下也是有效的。<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
<br />
Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
<br />
将上面的表达式表示为对 e 的导数，对 y 的求和得到表达式:<br />
<br />
<br />
<br />
<br />
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: <math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
<br />
<math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
<br />
数学左(部分欧米茄左(部分欧米茄右){ e }-和 y 左(部分欧米茄右){ x 左(部分欧米茄左(部分欧米茄右)}{ x 右) ，/ 数学<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
<br />
The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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关于 x 的欧米茄 / 数学对数导数由以下方式给出:<br />
<br />
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<br />
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: <math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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Math 左( frac { partial ln 左( Omega 右)}{部分 x 右){ e } beta x + 左( frac { partial x 右){ x } ，/ math<br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
<br />
The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
<br />
第一个术语是密集型的，即。它不能根据系统大小进行缩放。相比之下，最后一项的规模与逆系统的规模一样，因此将在热力学极限中消失。因此，我们发现:<br />
<br />
<br />
<br />
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: <math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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<math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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数学左(部分 s }右){ e } frac { x }{ t } ，/ math<br />
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<br />
<br />
<br />
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Combining this with<br />
<br />
Combining this with<br />
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结合这一点<br />
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<br />
<br />
<br />
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: <math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
<br />
<math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
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数学左(部分 s }右){ x } frac {1}{ t } ，/ math<br />
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<br />
<br />
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<br />
Gives:<br />
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Gives:<br />
<br />
给予:<br />
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<br />
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: <math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
<br />
<math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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数学 dS 左(部分 s }部分 e }右){ x } dE + 左(部分 s }右){ e } dx frac { t } + frac { t }{ delta q }{ t } ，/ math<br />
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===Derivation for systems described by the canonical ensemble===<br />
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===Derivation for systems described by the canonical ensemble===<br />
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正则系系统的推导<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the [[canonical ensemble]]:<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:<br />
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如果一个系统与某个温度为T的热浴热接触，那么在平衡状态下，关于能量本征值的概率分布值由正则系综给出:<br />
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: <math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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<math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the [[Partition function (statistical mechanics)|partition function]]. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
<br />
这里Z是一个使所有概率之和归一化的因子，这个函数被称为配分函数。现在我们考虑对温度和能级所依赖的外部参数的无限小的可逆改变。它遵循熵的一般公式:<br />
<br />
<br />
<br />
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: <math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
<br />
<math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
<br />
<br />
<br />
<br />
<br />
<br />
that<br />
<br />
that<br />
<br />
以及<br />
<br />
<br />
<br />
<br />
<br />
: <math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
<br />
<math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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<br />
<br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
<br />
Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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把正则系综的数学形式 <math>P_{j}</math> 插入，可以得到:<br />
<br />
<br />
<br />
<br />
<br />
: <math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
<br />
<math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
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==Living organisms==<br />
<br />
==Living organisms==<br />
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生命体<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as [[What is Life?|Erwin Schrödinger]], [[Léon Brillouin]]<ref>Léon Brillouin [https://books.google.com/books?hl=en&lr=&id=tPXVbiw_1P0C ''Science and Information Theory''] (Academic Press, 1962) (Dover, 2004)</ref> and [[Life and Energy|Isaac Asimov]]. It is also the topic of current{{When|date=July 2019}} research.{{Citation needed|date=July 2019}}<br />
<br />
There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as Erwin Schrödinger, Léon Brillouin and Isaac Asimov. It is also the topic of current research.<br />
<br />
表述热力学主要有两种方式：(a)描述从一种热力学平衡状态到另一种状态的路径，(b)通过使系统保持不变，同时增加周围环境的总熵的循环过程。这两种方法有助于理解生命过程。这个话题大大超出了本文的范围，但是已经有一些研究者考虑过这个问题，比如埃尔温·薛定谔、莱昂·布里渊和艾萨克.阿西莫夫。这也是当前研究的主题。<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of [[metabolism]], give out breakdown products and heat. Plants [[Photosynthesis|take in radiative energy]] from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
<br />
To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of metabolism, give out breakdown products and heat. Plants take in radiative energy from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
<br />
从近似的角度来看，生命体可以被认为是(b)的一个例子。近似地，一只动物的身体状态每天循环，使得它几乎没有什么变化。动物吸收食物、水和氧气，经过新陈代谢，输出分解的产物和热量。植物吸收来自太阳的辐射能量，这可以认为是热量，以及二氧化碳和水，然后它们释放氧气。它们就是这样生长的，最终会死亡，尸体腐烂，大部分重新变成二氧化碳和水。这可以看作是一个循环过程。总的来说，阳光来自一个高温的源——太阳，它的能量被传递到一个较低的温度汇，例如向太空辐射。这个过程使得植物周围环境的熵增加。因此从循环过程的角度来看，动物和植物服从热力学第二定律。简单的热机效率概念很难适用于这个问题，因为它们假定系统是封闭的。<br />
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From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to [[Non-equilibrium thermodynamics|unsettled current debate or research]]. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
<br />
From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to unsettled current debate or research. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
<br />
从(a)的热力学观点考虑 ，即考虑从一个平衡状态到另一个平衡状态的路径，只会产生一个粗略近似的图像，因为生命体从来不会处于热力学平衡状态。生物体经常必须被认为是开放的系统，因为它们吸收营养物质并排出废弃物。开放系统热力学目前通常从一个热力学平衡状态到另一个状态的路径的角度来考虑，或者是考虑局部热力学平衡近似下的流。生命体的问题可以通过假定一个流不变的稳态来进一步简化。对于这样的近似，熵产生的一般原理在目前有争论和研究的不确定性。尽管如此，从热力学第二定律的这个角度出发衍生出来的想法对生物是有启发意义的。<br />
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==Gravitational systems==<br />
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==Gravitational systems==<br />
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重力系统<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive [[heat capacity]], meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it.<ref name="Grandy 151">Grandy, W.T. (Jr) (2008), p. 151.</ref> Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
<br />
In systems that do not require for their descriptions the general theory of relativity, bodies always have positive heat capacity, meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it. Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
<br />
在不需要描述广义相对论的系统中，物体总是具有正的热容，这意味着温度随着能量的增加而升高。因此，当能量从高温物体流向低温物体时，源温度降低，而汇温度升高; 因此温差随时间而减小。这种情况并不总是发生在那些需要引力和广义相对论的系统中。这样的系统可以自发地朝着质量和能量不均匀分布的方向变化。这在大尺度上适用于宇宙，因此可能很难或不可能将第二定律应用于宇宙。除此之外，广义相对论所描述的系统热力学超出了本文的范围。<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
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{{main article|Non-equilibrium thermodynamics}}<br />
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The theory of classical or [[thermodynamic equilibrium|equilibrium thermodynamics]] is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<ref name="Grandy 151"/><ref>Callen, H.B. (1960/1985), p. 15.</ref><br />
<br />
The theory of classical or equilibrium thermodynamics is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<br />
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经典热力学或平衡热力学的理想化。一个主要的假设或假设，往往甚至没有明确说明，是系统的存在，在他们自己的内部状态的热力学平衡。一般来说，在给定时间内包含一个物理系统的空间区域---- 可能在自然界中找到---- 不是在热力学平衡中，用最严格的术语来解释。从更宽松的角度来说，整个宇宙中没有任何东西是或者曾经是完全精确的热力学平衡。<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of [[thermodynamic equilibrium]]. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable [[thermal fluctuations|fluctuations]]. There is an exception, the case of [[critical point (thermodynamics)|critical states]], which exhibit to the naked eye the phenomenon of [[critical opalescence]]. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of thermodynamic equilibrium. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable fluctuations. There is an exception, the case of critical states, which exhibit to the naked eye the phenomenon of critical opalescence. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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为了物理分析的目的，假设热力学平衡是非常方便的。这种假设可能依赖于试错来证明其正当性。如果这个假设是正确的，它通常是非常有价值和有用的，因为它提供了热力学理论。平衡假设的要素是，一个系统被观察到在无限长的时间内是不变的，而且一个系统中有如此多的粒子，以至于它的微粒性质可以被完全忽略。在这样的平衡假设下，一般来说，没有宏观上可以探测到的波动。有一个例外，即临界状态的例子，它们用肉眼就可以看到临界乳光现象。对于临界状态的实验室研究，需要非常长的观察时间。<br />
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In all cases, the assumption of [[thermodynamic equilibrium]], once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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In all cases, the assumption of thermodynamic equilibrium, once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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在所有情况下，热力学平衡的假设，一旦做出，就意味着没有假定的候选者“涨落”改变系统的熵。<br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<ref>Lieb, E.H., Yngvason, J. (2003), p. 190.</ref><br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<br />
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一个物理系统很容易发生内部的宏观变化，其速度之快足以使熵的恒定性假设失效。或者一个物理系统的粒子太少，以至于微粒的本质在可观测的波动中是显而易见的。那么热力学平衡的假设就会被放弃。对于非平衡态，熵没有绝对的一般定义。<br />
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There are intermediate cases, in which the assumption of local [[thermodynamic equilibrium]] is a very good approximation,<ref>Gyarmati, I. (1967/1970), pp. 4-14.</ref><ref>Glansdorff, P., Prigogine, I. (1971).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003).</ref> but strictly speaking it is still an approximation, not theoretically ideal.<br />
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There are intermediate cases, in which the assumption of local thermodynamic equilibrium is a very good approximation, but strictly speaking it is still an approximation, not theoretically ideal.<br />
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还有一些中间情况，其中局部热力学平衡的假设是一个非常好的近似值，但严格地说它仍然是一个近似值，不是理论上的理想值。<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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对于一般的非平衡状态，考虑其他量的统计力学定义可能是有用的，这些量可以方便地称为‘熵’ ，但是它们不应该与第二定律正确定义的熵混淆或混淆。这些其他的量确实属于统计力学，而不是热力学---- 第二定律的主要领域。<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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宏观观测波动的物理学超出了本文的范围。<br />
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==Arrow of time==<br />
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==Arrow of time==<br />
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时间之箭<br />
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{{Further information|Entropy (arrow of time)}}Thermodynamic Asymmetry in Time<br />
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Thermodynamic Asymmetry in Time<br />
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时间中的热力学不对称性<br />
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{{See also|Arrow of time}}<br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely [[CPT symmetry]], since the second law applies statistically, it is hypothesized, on time-asymmetric ''boundary conditions''.<ref>{{cite web|first=Craig|last=Callender|url=https://plato.stanford.edu/archives/fall2011/entries/time-thermo/|title=Thermodynamic Asymmetry in Time|encyclopedia=Stanford Encyclopedia of Philosophy|date=29 July 2011}}</ref><br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely CPT symmetry, since the second law applies statistically, it is hypothesized, on time-asymmetric boundary conditions.<br />
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热力学第二定律是一个物理定律，它不对称于时间方向的反转。这与已经观察到的物理学基本定律，即 CPT 对称性的概念并不冲突，因为第二定律在统计上适用，它是假设的，是关于时间非对称的边界条件。<br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect ([[Arrow of time#The causal arrow of time|the causal arrow of time]]).<ref>{{cite book | first = J.J.| last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1|display-authors=etal}} chapter 6</ref><br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect (the causal arrow of time).<br />
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第二定律被提出来部分解释时间向前和向后移动的区别，例如为什么原因先于结果(因果关系的时间箭头)。<br />
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==Irreversibility==<br />
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==Irreversibility==<br />
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不可逆性<br />
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Irreversibility in [[thermodynamic process]]es is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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Irreversibility in thermodynamic processes is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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热力学过程的不可逆性是热力学作用不对称性的结果，而不是物体内部不可逆的微观性质的结果。热力学操作是施加在参与机构上的宏观外部干预，而不是从它们的内部性质中得到的。由于没有认识到这一点，产生了被称为“悖论”的说法。<br />
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===Loschmidt's paradox===<br />
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===Loschmidt's paradox===<br />
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洛斯密特悖论<br />
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{{main article|Loschmidt's paradox}}<br />
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[[Loschmidt's paradox]], also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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洛斯密特悖论，也被称为可逆性悖论，其目的在于不可能从描述宏观系统微观演化的时间对称动力学推导出不可逆性。<br />
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In the opinion of [[Erwin Schrödinger|Schrödinger]], "It is now quite obvious in what manner you have to reformulate the law of entropy{{snd}}or for that matter, all other irreversible statements{{snd}}so that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other."<ref>[[Erwin Schrödinger|Schrödinger, E.]] (1950), p.&nbsp;192.</ref> The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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In the opinion of Schrödinger, "It is now quite obvious in what manner you have to reformulate the law of entropyor for that matter, all other irreversible statementsso that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other." The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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根据薛定谔的观点，“现在很明显，你必须用什么方式来重新定义该物质的熵定律，所有其他不可逆的状态，这样它们才能从可逆模型中导出。你不能只谈论一个孤立的系统，而应该至少谈论两个，你可以暂时认为它们与世界其它地方是孤立的，但并不总是相互孤立的。”这两个系统被隔离墙隔离开来，直到按照法律的设想被热力学操作拆除。热力学操作是外部强加的，不受支配系统组成部分的可逆微观动力学定律的制约。它是不可逆转性的原因。本文中的法律陈述符合薛定谔的建议。因果关系在逻辑上优先于第二定律，而不是从第二定律派生出来的。<br />
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===Poincaré recurrence theorem===<br />
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===Poincaré recurrence theorem===<br />
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庞加莱始态复现定理<br />
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{{Main article|Poincaré recurrence theorem}}<br />
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The [[Poincaré recurrence theorem]] considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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The Poincaré recurrence theorem considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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庞加莱始态复现定理认为这是一个孤立的物理系统的理论微观描述。这可以被认为是一个热力学系统的模型后，一个热力学操作已经移除了一个内墙。在足够长的时间之后，系统将返回到一个非常接近初始状态的显微定义状态。庞加莱回归时间是指回归前经过的时间长度。它极其漫长，可能比宇宙的寿命还要长，并且敏感地依赖于被热力学操作拆除的墙体的几何形状。递归定理可能被认为是明显与热力学第二定律相矛盾的。然而，更明显的是，它只不过是一个微观模型，热力学平衡在一个孤立的系统中，通过移除两个系统之间的一道墙而形成。对于一个典型的热力学系统，重现时间是如此之大(比宇宙的生命周期长许多倍) ，以至于在所有的实际目的中，人们都不能观察到重现。然而，人们可能希望想象一下，可以等待庞加莱复发，然后重新插入被热力学操作移除的壁。然后很明显，不可逆性的出现是由于庞加莱循环的完全不可预测性，只要初始状态是热力学平衡，就像宏观热力学的情况一样。即使可以等待，也没有实际可能性选择合适的时间重新插入墙壁。庞加莱始态复现定理为 Loschmidt 悖论提供了一个解决方案。如果一个孤立的热力学系统可以被监测到在越来越多的平均 poincar 递归时间的倍数上，系统的热力学行为将在时间反转下变得不变。<br />
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[[File:James-clerk-maxwell3.jpg|thumb|upright|James Clerk Maxwell]]<br />
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James Clerk Maxwell<br />
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詹姆斯·克拉克·麦克斯韦<br />
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===Maxwell's demon===<br />
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===Maxwell's demon===<br />
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麦克斯韦的魔鬼<br />
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{{unreferenced|section|date=August 2018}}<br />
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{{main article|Maxwell's demon}}<br />
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[[James Clerk Maxwell]] imagined one container divided into two parts, ''A'' and ''B''. Both parts are filled with the same [[gas]] at equal temperatures and placed next to each other, separated by a wall. Observing the [[molecule]]s on both sides, an imaginary [[demon]] guards a microscopic trapdoor in the wall. When a faster-than-average molecule from ''A'' flies towards the trapdoor, the demon opens it, and the molecule will fly from ''A'' to ''B''. The average [[speed]] of the molecules in ''B'' will have increased while in ''A'' they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in ''A'' and increases in ''B'', contrary to the second law of thermodynamics.<br />
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James Clerk Maxwell imagined one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other, separated by a wall. Observing the molecules on both sides, an imaginary demon guards a microscopic trapdoor in the wall. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics.<br />
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詹姆斯·克拉克·麦克斯韦想象一个容器分成两部分，a 和 b。两个部分在相同的温度下充入相同的气体，并被一堵墙隔开，挨着放置。通过观察两边的分子，一个想象中的恶魔守卫着墙上的一个微型暗门。当一个来自 a 的比平均速度快的分子飞向活板门时，魔鬼打开了它，分子就会从 a 飞到 b。B 中分子的平均速度会增加，而 a 中分子的平均速度会减慢。由于平均分子速度与温度相对应，因此 a 中的温度降低，b 中的温度升高，这与热力学第二定律相反。<br />
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One response to this question was suggested in 1929 by [[Leó Szilárd]] and later by [[Léon Brillouin]]. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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One response to this question was suggested in 1929 by Leó Szilárd and later by Léon Brillouin. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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1929年，Leó Szilárd 和后来的 l on brilouin 对这个问题提出了一个答案。Szil rd 指出，现实生活中的麦克斯韦恶魔需要一些测量分子速度的方法，而获取信息需要消耗大量能量。<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between ''A'' and ''B''. It is therefore performing [[thermodynamic operation]]s on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between A and B. It is therefore performing thermodynamic operations on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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麦克斯韦的“魔鬼”不断地改变 a 和 b 之间墙体的渗透性。因此，它是在微观尺度上执行热力学操作，而不仅仅是观察普通的自发或自然的宏观热力学过程。<br />
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==Quotations==<br />
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==Quotations==<br />
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语录<br />
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{{Wikiquote}}<br />
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{{quote|The law that entropy always increases holds, I think, the supreme position among the [[Laws of science|laws of Nature]]. If someone points out to you that your pet theory of the [[universe]] is in disagreement with [[Maxwell's equations]] – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.|Sir [[Arthur Stanley Eddington]], ''The Nature of the Physical World'' (1927)}}<br />
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{{quote|There have been nearly as many formulations of the second law as there have been discussions of it.|Philosopher / Physicist [[Percy Williams Bridgman|P.W. Bridgman]], (1941)}}<br />
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{{quote|Clausius is the author of the sibyllic utterance, "The energy of the universe is constant; the entropy of the universe tends to a maximum." The objectives of continuum thermomechanics stop far short of explaining the "universe", but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size.|[[Clifford Truesdell|Truesdell, C.]], Muncaster, R. G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 17.}}<br />
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==See also==<br />
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参见<br />
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*[[Clausius–Duhem inequality]]<br />
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*[[Fluctuation theorem]]<br />
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*[[Heat death of the universe]]<br />
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*[[History of thermodynamics]]<br />
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*[[Jarzynski equality]]<br />
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*[[Laws of thermodynamics]]<br />
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*[[Maximum entropy thermodynamics]]<br />
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*[[Quantum thermodynamics]]<br />
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*[[Reflections on the Motive Power of Fire]]<br />
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*[[Relativistic heat conduction]]<br />
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*[[Thermal diode]]<br />
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*[[Thermodynamic equilibrium]]<br />
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{{colend}}<br />
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==References==<br />
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==References==<br />
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参考资料<br />
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{{reflist|20em}}<br />
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===Sources===<br />
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===Sources===<br />
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资料来源<br />
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{{refbegin|50em}}<br />
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*Adkins, C.J. (1968/1983). ''Equilibrium Thermodynamics'', (1st edition 1968), third edition 1983, Cambridge University Press, Cambridge UK, {{ISBN|0-521-25445-0}}.<br />
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*[[Peter Atkins|Atkins, P.W.]], de Paula, J. (2006). ''Atkins' Physical Chemistry'', eighth edition, W.H. Freeman, New York, {{ISBN|978-0-7167-8759-4}}.<br />
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*Attard, P. (2012). ''Non-equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications'', Oxford University Press, Oxford UK, {{ISBN|978-0-19-966276-0}}.<br />
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*Baierlein, R. (1999). ''Thermal Physics'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-59082-5}}.<br />
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*Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics, New York, {{ISBN|0-88318-797-3}}.<br />
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*{{Cite book|title=Concepts in thermal physics|last=Blundell|first=Stephen J.|last2=Blundell|authorlink1=Stephen Blundell|first2=Katherine M.|authorlink2=Katherine Blundell|publisher=[[Oxford University Press]]|year=2010|isbn=9780199562107|edition=2nd|location=Oxford|pages=|oclc=607907330|doi=10.1093/acprof:oso/9780199562091.001.0001 |url=http://cds.cern.ch/record/1235139}}<br />
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*[[Ludwig Boltzmann|Boltzmann, L.]] (1896/1964). ''Lectures on Gas Theory'', translated by S.G. Brush, University of California Press, Berkeley.<br />
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*Borgnakke, C., Sonntag., R.E. (2009). ''Fundamentals of Thermodynamics'', seventh edition, Wiley, {{ISBN|978-0-470-04192-5}}.<br />
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*Buchdahl, H.A. (1966). ''The Concepts of Classical Thermodynamics'', Cambridge University Press, Cambridge UK.<br />
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*[[Percy Williams Bridgman|Bridgman, P.W.]] (1943). ''The Nature of Thermodynamics'', Harvard University Press, Cambridge MA.<br />
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*[[Herbert Callen|Callen, H.B.]] (1960/1985). ''Thermodynamics and an Introduction to Thermostatistics'', (1st edition 1960) 2nd edition 1985, Wiley, New York, {{ISBN|0-471-86256-8}}.<br />
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*{{cite journal|author=C. Carathéodory|author1-link=Constantin Carathéodory<br />
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<br />
|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
<br />
|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
<br />
|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
<br />
2010年3月15日 | 第355-386页 | http://gdz.sub.uni-goettingen.de/index.php?id=11&ppn=ppn235181684_0067&dmdid=dmdlog_0033&l=1<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). ''The Second Law of Thermodynamics'', Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
<br />
|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
<br />
|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können.(p. 363) | doi 10.1007 / bf01450409}.译文可以在这里找到[ http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf ]。同时，在 Kestin，j。(1976).热力学第二定律，Dowden，Hutchinson & Ross，Stroudsburg PA. 。<br />
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*[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986). [http://www.worldcat.org/title/reflections-on-the-motive-power-of-fire-a-critical-edition-with-the-surviving-scientific-manuscripts-translated-and-edited-by-fox-robert/oclc/812944517&referer=brief_results ''Reflections on the motive power of fire''], Manchester University Press, Manchester UK, {{ISBN|0-7190-1741-6}}. [https://archive.org/stream/reflectionsonmot00carnrich#page/n7/mode/2up Also here.]<br />
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*[[Sydney Chapman (mathematician)|Chapman, S.]], [[Thomas George Cowling|Cowling, T.G.]] (1939/1970). ''The Mathematical Theory of Non-uniform gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases'', third edition 1970, Cambridge University Press, London.<br />
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*{{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Ueber Die Bewegende Kraft Der Wärme Und Die Gesetze, Welche Sich Daraus Für Die Wärmelehre Selbst Ableiten Lassen|journal=Annalen der Physik|year=1850|volume=79|issue=4|pages=368–397, 500–524|url=http://gallica.bnf.fr/ark:/12148/bpt6k15164w/f518.image|accessdate=26 June 2012|doi=10.1002/andp.18501550403|bibcode = 1850AnP...155..500C |hdl=2027/uc1.$b242250|hdl-access=free}} Translated into English: {{cite journal|last=Clausius|first=R.|title=On the Moving Force of Heat, and the Laws regarding the Nature of Heat itself which are deducible therefrom|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1851| volume=2|series=4th|issue=VIII|pages=1–21; 102–119 |url=https://archive.org/stream/londonedinburghd02lond#page/1/mode/1up|accessdate=26 June 2012<br />
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|doi=10.1080/14786445108646819}}<br />
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|doi=10.1080/14786445108646819}}<br />
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10.1080 / 14786445108646819}<br />
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* {{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie|year=1854|journal=Annalen der Physik|volume=xciii|issue=12|pages=481–506|url=http://zfbb.thulb.uni-jena.de/servlets/MCRFileNodeServlet/jportal_derivate_00140956/18541691202_ftp.pdf|accessdate=24 March 2014|doi=10.1002/andp.18541691202<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
<br />
| bibcode 1854AnP... 169. 1856年7月 | 第2卷 | 第4页 | 第86页 | https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=242014年3月<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
<br />
热力学理论及其在蒸汽机和物体物理性质中的应用 | 1867年 | 出版商 John van Voorst | 位置: 伦敦 | https://archive.org/details/mechanicaltheor04claugoog|quote=editions:pwr_sbkwa8ic。 2012年6月19日<br />
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}}<br />
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*Denbigh, K. (1954/1981). ''The Principles of Chemical Equilibrium. With Applications in Chemistry and Chemical Engineering'', fourth edition, Cambridge University Press, Cambridge UK, {{ISBN|0-521-23682-7}}.<br />
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*Eu, B.C. (2002). ''Generalized Thermodynamics. The Thermodynamics of Irreversible Processes and Generalized Hydrodynamics'', Kluwer Academic Publishers, Dordrecht, {{ISBN|1-4020-0788-4}}.<br />
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*[[Josiah Willard Gibbs|Gibbs, J.W.]] (1876/1878). On the equilibrium of heterogeneous substances, ''Trans. Conn. Acad.'', '''3''': 108-248, 343-524, reprinted in ''The Collected Works of J. Willard Gibbs, Ph.D, LL. D.'', edited by W.R. Longley, R.G. Van Name, Longmans, Green & Co., New York, 1928, volume 1, pp.&nbsp;55–353.<br />
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*Griem, H.R. (2005). ''Principles of Plasma Spectroscopy (Cambridge Monographs on Plasma Physics)'', Cambridge University Press, New York {{ISBN|0-521-61941-6}}.<br />
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*Glansdorff, P., Prigogine, I. (1971). ''Thermodynamic Theory of Structure, Stability, and Fluctuations'', Wiley-Interscience, London, 1971, {{ISBN|0-471-30280-5}}.<br />
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*Grandy, W.T., Jr (2008). [http://global.oup.com/academic/product/entropy-and-the-time-evolution-of-macroscopic-systems-9780199546176?cc=au&lang=en& ''Entropy and the Time Evolution of Macroscopic Systems'']. Oxford University Press. {{ISBN|978-0-19-954617-6}}.<br />
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*Greven, A., Keller, G., Warnecke (editors) (2003). ''Entropy'', Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[Edward A. Guggenheim|Guggenheim, E.A.]] (1949). 'Statistical basis of thermodynamics', ''Research'', '''2''': 450–454.<br />
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*[[Edward A. Guggenheim|Guggenheim, E.A.]] (1967). ''Thermodynamics. An Advanced Treatment for Chemists and Physicists'', fifth revised edition, North Holland, Amsterdam.<br />
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*Gyarmati, I. (1967/1970) ''Non-equilibrium Thermodynamics. Field Theory and Variational Principles'', translated by E. Gyarmati and W.F. Heinz, Springer, New York.<br />
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*[[Charles Kittel|Kittel, C.]], [[Herbert Kroemer|Kroemer, H.]] (1969/1980). ''Thermal Physics'', second edition, Freeman, San Francisco CA, {{ISBN|0-7167-1088-9}}.<br />
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*Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998). ''Modern Thermodynamics: From Heat Engines to Dissipative Structures'', John Wiley & Sons, Chichester, {{ISBN|0-471-97393-9}}.<br />
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*Lebon, G., Jou, D., Casas-Vázquez, J. (2008). ''Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers'', Springer-Verlag, Berlin, {{ISBN|978-3-540-74252-4}}.<br />
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*{{cite journal|ref=harv|title=The Physics and Mathematics of the Second Law of Thermodynamics|last1=Lieb |first1=E. H. |last2=Yngvason |first2=J.|journal=Physics Reports|volume=310|issue=1|pages=1–96 |year=1999|doi=10.1016/S0370-1573(98)00082-9|arxiv = cond-mat/9708200 |bibcode = 1999PhR...310....1L}}<br />
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*Lieb, E.H., Yngvason, J. (2003). The Entropy of Classical Thermodynamics, pp.&nbsp;147–195, Chapter 8 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*{{cite book|ref=harv|first=F.|last=Mandl|title=Statistical physics|edition=second|year=1988|publisher=[[Wiley & Sons]]|isbn=978-0-471-91533-1}}<br />
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*{{cite journal | last1 = Maxwell | first1 = J.C. | authorlink = James Clerk Maxwell | year = 1867 | title = On the dynamical theory of gases | url = | journal = Phil. Trans. R. Soc. Lond. | volume = 157 | issue = | pages = 49–88 | doi = 10.1098/rstl.1867.0004 }}<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985). ''Thermodynamics'', Pitman, London, {{ISBN|0-273-08577-8}}.<br />
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*[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003). Entropy in Nonequilibrium, pp.&nbsp;79–109, Chapter 5 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003).<br />
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*Münster, A. (1970), ''Classical Thermodynamics'', translated by E.S. Halberstadt, Wiley–Interscience, London, {{ISBN|0-471-62430-6}}.<br />
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*[[Brian Pippard|Pippard, A.B.]] (1957/1966). ''Elements of Classical Thermodynamics for Advanced Students of Physics'', original publication 1957, reprint 1966, Cambridge University Press, Cambridge UK.<br />
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*[[Max Planck|Planck, M.]] (1897/1903). [https://archive.org/stream/treatiseonthermo00planrich#page/100/mode/2up ''Treatise on Thermodynamics'', translated by A. Ogg, Longmans Green, London, p. 100.]<br />
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*[[Max Planck|Planck. M.]] (1914). [https://archive.org/details/theoryofheatradi00planrich ''The Theory of Heat Radiation''], a translation by Masius, M. of the second German edition, P. Blakiston's Son & Co., Philadelphia.<br />
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*[[Max Planck|Planck, M.]] (1926). Über die Begründung des zweiten Hauptsatzes der Thermodynamik, ''Sitzungsberichte der Preussischen Akademie der Wissenschaften: Physikalisch-mathematische Klasse'': 453–463.<br />
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* Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769-781.<br />
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* {{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}<br />
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*Quinn, T.J. (1983). ''Temperature'', Academic Press, London, {{ISBN|0-12-569680-9}}.<br />
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*{{cite book|ref=harv|first=Y.V.C.|last=Rao|title=An Introduction to thermodynamics|url=https://books.google.com/books?id=iYWiCXziWsEC&pg=PA213|date=2004|publisher=Universities Press|isbn=978-81-7371-461-0|page=213}}<br />
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*Roberts, J.K., Miller, A.R. (1928/1960). ''Heat and Thermodynamics'', (first edition 1928), fifth edition, Blackie & Son Limited, Glasgow.<br />
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*[[Erwin Schrödinger|Schrödinger, E.]] (1950). Irreversibility, ''Proc. R. Ir. Acad.'', '''A53''': 189–195.<br />
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*[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966). ''Elements of Thermodynamics'', Addison-Wesley Publishing, Reading MA.<br />
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*{{cite journal|ref=harv|last=Thomson|first=W.|author-link=William Thomson, 1st Baron Kelvin|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Transactions of the Royal Society of Edinburgh|year=1851|volume=XX|issue=part II|pages=261–268; 289–298|url=https://www.biodiversitylibrary.org/item/126047#page/295/mode/1up}} Also published in {{cite journal|last=Thomson|first=W.|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Philos. Mag. |date=December 1852 |volume=IV |series=4 |issue=22 |page=13|url=https://archive.org/stream/londonedinburghp04maga#page/12/mode/2up |accessdate=25 June 2012}}<br />
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*[[William Thomson, 1st Baron Kelvin|Thomson, W.]] (1852). ''On the universal tendency in nature to the dissipation of mechanical energy'' Philosophical Magazine, Ser. 4, p.&nbsp;304.<br />
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*[[László Tisza|Tisza, L.]] (1966). ''Generalized Thermodynamics'', M.I.T Press, Cambridge MA.<br />
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*[[Clifford Truesdell|Truesdell, C.]] (1980). ''The Tragicomical History of Thermodynamics 1822–1854'', Springer, New York, {{ISBN|0-387-90403-4}}.<br />
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*Uffink, J. (2001). Bluff your way in the second law of thermodynamics, ''Stud. Hist. Phil. Mod. Phys.'', '''32'''(3): 305–394.<br />
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*Uffink, J. (2003). Irreversibility and the Second Law of Thermodynamics, Chapter 7 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003), Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963). ''Lectures in Statistical Mechanics'', American Mathematical Society, Providence RI.<br />
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*[[Mark Zemansky|Zemansky, M.W.]] (1968). ''Heat and Thermodynamics. An Intermediate Textbook'', fifth edition, McGraw-Hill Book Company, New York.<br />
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==Further reading==<br />
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==Further reading==<br />
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进一步阅读<br />
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*Goldstein, Martin, and Inge F., 1993. ''The Refrigerator and the Universe''. Harvard Univ. Press. Chpts. 4–9 contain an introduction to the Second Law, one a bit less technical than this entry. {{ISBN|978-0-674-75324-2}}<br />
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*Leff, Harvey S., and Rex, Andrew F. (eds.) 2003. ''Maxwell's Demon 2 : Entropy, classical and quantum information, computing''. Bristol UK; Philadelphia PA: [[Institute of Physics]]. {{ISBN|978-0-585-49237-7}}<br />
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*{{Cite book | first = J.J. | last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1}}(technical).<br />
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*{{cite book |title=Reflections on the Motive Power of Heat and on Machines Fitted to Develop That Power |last=Carnot |first=Sadi |editor=[[Robert Henry Thurston|Thurston, Robert Henry]] |year=1890 |publisher=J. Wiley & Sons |location=New York |isbn= |pages= }} ([https://books.google.com/books?id=tgdJAAAAIAAJ full text of 1897 ed.]) ([http://www.history.rochester.edu/steam/carnot/1943/ html])<br />
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*Stephen Jay Kline (1999). ''The Low-Down on Entropy and Interpretive Thermodynamics'', La Cañada, CA: DCW Industries. {{ISBN|1-928729-01-0}}.<br />
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*{{cite book | last1 = Kostic | first1 = M | year = 2011 | title = Revisiting The Second Law of Energy Degradation and Entropy Generation: From Sadi Carnot's Ingenious Reasoning to Holistic Generalization | journal = AIP Conf. Proc. | volume = 1411 | issue = 1| pages = 327–350 | doi = 10.1063/1.3665247 | isbn = 978-0-7354-0985-9 | bibcode = 2011AIPC.1411..327K | series = AIP Conference Proceedings | citeseerx = 10.1.1.405.1945 }} also at [https://web.archive.org/web/20130420222450/http://www.kostic.niu.edu/2ndLaw/Revisiting%20The%20Second%20Law%20of%20Energy%20Degradation%20and%20Entropy%20Generation%20-%20From%20Carnot%20to%20Holistic%20Generalization-4.pdf].<br />
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==External links==<br />
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==External links==<br />
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外部链接<br />
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*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics]" – by Lawrence Sklar.<br />
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*[http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node30.html ''Second law of thermodynamics''] in the MIT Course [http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/notes.html ''Unified Thermodynamics and Propulsion''] from Prof. Z. S. Spakovszky<br />
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*[[E.T. Jaynes]], 1988, "[http://bayes.wustl.edu/etj/articles/ccarnot.pdf The evolution of Carnot's principle]," in G. J. Erickson and C. R. Smith (eds.)''Maximum-Entropy and Bayesian Methods in Science and Engineering, Vol'' 1: p.&nbsp;267.<br />
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*[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf Caratheodory, C., "Examination of the foundations of thermodynamics," trans. by D. H. Delphenich]<br />
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* [https://www.bbc.co.uk/programmes/p004y2bm The Second Law of Thermodynamics], BBC Radio 4 discussion with John Gribbin, Peter Atkins & Monica Grady (''In Our Time'', Dec. 16, 2004)<br />
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* [http://mdpi.org/entropy/papers/e6010001.pdf [[Entropy (journal)]], 2004]<br />
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* [https://www.journals.uchicago.edu/doi/abs/10.1086/663835 The Journal of the International Society for the History of Philosophy of Science, 2012]<br />
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{{DEFAULTSORT:Second Law Of Thermodynamics}}<br />
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[[Category:Concepts in physics]]<br />
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Category:Concepts in physics<br />
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分类: 物理概念<br />
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[[Category:Laws of thermodynamics|2]]<br />
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[[Category:Non-equilibrium thermodynamics]]<br />
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Category:Non-equilibrium thermodynamics<br />
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类别: 非平衡态热力学<br />
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[[Category:Philosophy of thermal and statistical physics]]<br />
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Category:Philosophy of thermal and statistical physics<br />
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类别: 热力学和统计物理学哲学<br />
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<small>This page was moved from [[wikipedia:en:Second law of thermodynamics]]. Its edit history can be viewed at [[热力学第二定律/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E7%83%AD%E5%8A%9B%E5%AD%A6%E7%AC%AC%E4%BA%8C%E5%AE%9A%E5%BE%8B&diff=10350热力学第二定律2020-07-16T09:18:54Z<p>Dorr：</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Law of physics}}<br />
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{{Thermodynamics|laws}}<br />
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The '''second law of thermodynamics''' states that the total [[entropy]] of an [[isolated system]] can never decrease over time, and is constant if and only if all processes are reversible.<ref>http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node38.html#SECTION05224000000000000000</ref> Isolated systems spontaneously evolve towards [[thermodynamic equilibrium]], the state with maximum entropy. <br />
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The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, and is constant if and only if all processes are reversible. Isolated systems spontaneously evolve towards thermodynamic equilibrium, the state with maximum entropy. <br />
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热力学第二定律指出，孤立系统的总熵永远不会随着时间而减少，而且只有当且仅当所有过程都是可逆的时，总熵才是恒定的。孤立的系统自发地进化到热力学平衡状态，即熵最大的状态。<br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in [[thermodynamic equilibrium]], or is undergoing a (fictive) [[Reversible process (thermodynamics)|reversible process]]. In all processes that occur, including [[spontaneous process]]es,<ref>Atkins and de Paula, p.78</ref> the total entropy of the system and its surroundings increases and the process is [[Irreversible process|irreversible in the thermodynamic sense]]. The increase in entropy accounts for the irreversibility of natural processes, and the [[Arrow of time|asymmetry between future and past]].<ref>{{cite book|last=Zohuri|first=Bahman|title=Dimensional Analysis Beyond the Pi Theorem|url=https://books.google.com/books?id=pRVuDQAAQBAJ|year=2016|publisher=Springer|isbn=978-3-319-45726-0|page=[https://books.google.com.ph/books?id=pRVuDQAAQBAJ&pg=PA111&dq=%22increase+in+entropy+accounts+for+the+irreversibility+of+natural+processes+and+the+asymmetry+between+future+and+past.%22&hl=en&sa=X&ved=0ahUKEwj6spb61tbaAhUFS7wKHfftDtIQ6AEIKDAA#v=onepage&q=%22increase%20in%20entropy%20accounts%20for%20the%20irreversibility%20of%20natural%20processes%20and%20the%20asymmetry%20between%20future%20and%20past.%22 111]}}</ref><br />
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The total entropy of a system and its surroundings can remain constant in ideal cases where the system is in thermodynamic equilibrium, or is undergoing a (fictive) reversible process. In all processes that occur, including spontaneous processes, the total entropy of the system and its surroundings increases and the process is irreversible in the thermodynamic sense. The increase in entropy accounts for the irreversibility of natural processes, and the asymmetry between future and past.<br />
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系统及其周围环境的总熵在理想情况下可以保持不变，在这种情况下，系统处于热力学平衡期，或者正在经历一个(虚拟的)可逆过程期。在所有发生的过程中，包括自发过程，系统及其周围环境的总熵增加，这一过程在热力学意义上是不可逆的。熵的增加解释了自然过程的不可逆性，以及未来和过去之间的不对称性。<br />
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Historically, the second law was an [[Empirical evidence|empirical finding]] that was accepted as an axiom of [[thermodynamics|thermodynamic theory]]. [[Statistical mechanics]], classical or [[quantum thermodynamics|quantum]], explains the microscopic origin of the law.<br />
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Historically, the second law was an empirical finding that was accepted as an axiom of thermodynamic theory. Statistical mechanics, classical or quantum, explains the microscopic origin of the law.<br />
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从历史上看，第二定律是一个经验性的发现，被公认为热力学理论的公理。统计力学，无论是经典的还是量子的，都可以解释这个定律的微观起源。<br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist [[Nicolas Léonard Sadi Carnot|Sadi Carnot]], who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as [[Carnot's theorem (thermodynamics)|Carnot's rule]] or limit.<ref>Jaffe, R.L., Taylor, W. (2018). ''The Physics of Energy'', Cambridge University Press, Cambridge UK, pages 150, 151, 259, 772, 743.</ref><br />
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The second law has been expressed in many ways. Its first formulation is credited to the French scientist Sadi Carnot, who in 1824 showed that there is an upper limit to the efficiency of conversion of heat to work in a heat engine. This aspect of the second law is also known as Carnot's rule or limit.<br />
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第二定律已经用许多方式表达过了。法国科学家萨迪 · 卡诺在1824年证明了热能在热机中转化为功的效率是有上限的。这方面的第二定律也被称为卡诺的规则或限制。<br />
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==Introduction==<br />
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==Introduction==<br />
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引言<br />
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[[File:Heat flow hot to cold.png|thumb|upright|Heat flow from hot water to cold water.]]<br />
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Heat flow from hot water to cold water.<br />
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热量从热水流向冷水。<br />
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The [[first law of thermodynamics]] provides the definition of the [[internal energy]] of a [[thermodynamic system]], and expresses the law of [[conservation of energy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 40–41.</ref><ref>Munster A. (1970), pp. 8–9, 50–51.</ref> The second law is concerned with the direction of natural processes.<ref>{{harvnb|Mandl|1988}}</ref> It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such [[Phenomenon|phenomena]] are accounted for in terms of [[entropy]].<ref>[[Max Planck|Planck, M.]] (1897/1903), pp. 79–107.</ref><ref>Bailyn, M. (1994), Section 71, pp. 113–154.</ref> If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, {{math|''S''}}, increases.<br />
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The first law of thermodynamics provides the definition of the internal energy of a thermodynamic system, and expresses the law of conservation of energy. The second law is concerned with the direction of natural processes. It asserts that a natural process runs only in one sense, and is not reversible. For example, when a path for conduction and radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such phenomena are accounted for in terms of entropy. If an isolated system is held initially in internal thermodynamic equilibrium by internal partitioning impermeable walls, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, , increases.<br />
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能量守恒定律提供了热力学系统内能的定义，并表达了能量守恒定律。第二定律与自然过程的方向有关。它断言自然过程只在一种意义上运行，而且是不可逆的。例如，当有了传导和辐射的路径时，热总是自发地从一个较热的物体流向一个较冷的物体。这种现象可以用熵来解释。如果一个孤立的系统最初是通过内部分隔不可渗透的墙来保持在内部热力学平衡，然后一些操作使墙更具渗透性，那么系统自发地进化到最终达到一个新的内部热力学平衡，其总熵增加。<br />
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In a fictive reversible process, an infinitesimal increment in the entropy ({{math|d''S''}}) of a system is defined to result from an infinitesimal transfer of heat ({{math|δ''Q''}}) to a [[closed system]] (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature ({{math|''T''}}) of the system in equilibrium and the surroundings which supply the heat:<ref>Bailyn, M. (1994), p. 120.</ref><br />
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In a fictive reversible process, an infinitesimal increment in the entropy () of a system is defined to result from an infinitesimal transfer of heat () to a closed system (which allows the entry or exit of energy – but not transfer of matter) divided by the common temperature () of the system in equilibrium and the surroundings which supply the heat:<br />
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在虚构的可逆过程中，系统熵的无穷小增量被定义为由无穷小的热量传递()到一个封闭系统(允许能量进入或出去，但不允许物质传递)除以平衡系统和提供热量的环境的共同温度() :<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, idealized fictive reversible process)}.</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, idealized fictive reversible process)}.</math><br />
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(闭合系统，理想化的可逆过程) / math<br />
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Different notations are used for infinitesimal amounts of heat ({{mvar|δ}}) and infinitesimal amounts of entropy ({{mvar|d}}) because entropy is a [[function of state]], while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the [[Clausius Theorem|inequality]] <ref name="MortimerBook"> Mortimer, R. G. ''Physical Chemistry''. 3rd ed., p. 120, Academic Press, 2008. </ref><ref name="FermiBook"> Fermi, E. ''Thermodynamics'', footnote on p. 48, Dover Publications,1956 (still in print). </ref> <br />
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Different notations are used for infinitesimal amounts of heat () and infinitesimal amounts of entropy () because entropy is a function of state, while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the inequality <br />
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用不同的符号表示无穷小量的热量()和无穷小量的熵() ，因为熵是状态的函数，而热量和功一样不是。第二定律要求系统熵的增量满足不等式，对于实际上可能存在的不与环境发生质量交换的无穷小过程，系统熵增量满足不等式<br />
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: <math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process).}</math><br />
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<math>\mathrm dS > \frac{\delta Q}{T_{surr}} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible, irreversible process).}</math><br />
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(实际上可能是封闭系统，不可逆性)。数学<br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature ({{math|''T''}}) and the temperature of the surroundings ({{math|''T''}}<sub>''surr''</sub>).<ref name=":0">Adkins, C.J. (1968/1983), p. 75.</ref><ref name="Munster 45"/> Note that the equality still applies for pure heat flow,<ref name="Schmidt-Rohr 14"> Schmidt-Rohr, K. (2014). "Expansion Work without the External Pressure, and Thermodynamics in Terms of Quasistatic Irreversible Processes" ''J. Chem. Educ.'' '''91''': 402-409. https://dx.doi.org/10.1021/ed3008704 </ref><br />
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This is because a general process for this case may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature () and the temperature of the surroundings (<sub>surr</sub>).<br />
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这是因为这种情况下的一般过程可能包括周围环境对系统所做的功，这可能在系统内部产生摩擦或粘滞效应，因为一个化学反应可能正在进行，或者因为热传递实际上只是不可逆地发生，由系统温度()和周围环境温度(sub surr / sub)之间的差分驱动。<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change).}</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(actually possible quasistatic irreversible process without composition change).}</math><br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry.<ref name="Oxtoby8th">Oxtoby, D. W; Gillis, H.P., [[Laurie Butler|Butler, L. J.]] (2015).''Principles of Modern Chemistry'', Brooks Cole. p. 617. {{ISBN|978-1305079113}}</ref> <ref name="MortimerBook"></ref> Introducing a set of internal variables <math>\xi</math> to describe the deviation of a thermodynamic system in physical equilibrium (with the required well-defined uniform pressure ''P'' and temperature ''T'')<ref name="Schmidt-Rohr 14"></ref> from the chemical equilibrium state, one can record the equality<br />
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which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry. from the chemical equilibrium state, one can record the equality<br />
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这是根据测得的热容曲线和相变时的熵变精确确定纯物质的绝对熵的基础。用量热法。从化学平衡来看，人们可以记录平等<br />
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: <math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process).}</math><br />
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<math>\mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, actually possible quasistatic irreversible process).}</math><br />
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- frac { t }-frac {1}{ j } ， Xi { j } ， delta Xi j， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ,.数学<br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of ''the second principle of thermodynamics'' – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<ref> Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769–781.</ref><ref>{{Cite journal | doi=10.1155/2013/906136|title = A Derivation of the Main Relations of Nonequilibrium Thermodynamics| journal=ISRN Thermodynamics| volume=2013| pages=1–9|year = 2013|last1 = Pokrovskii|first1 = Vladimir N.|doi-access=free}}</ref><br />
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The second term represents work of internal variables that can be perturbed by external influences, but the system cannot perform any positive work via internal variables. This statement introduces the impossibility of the reversion of evolution of the thermodynamic system in time and can be considered as a formulation of the second principle of thermodynamics – the formulation, which is, of course, equivalent to the formulation of the principle in terms of entropy.<br />
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第二项代表内部变量的工作，这些内部变量可以受到外部影响的干扰，但是系统不能通过内部变量执行任何正功。这种说法介绍了热力学系统在时间上不可能逆转演化的事实，并且可以被认为是热力学第二原理的表述---- 这个表述，当然，相当于熵原理的表述。<br />
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The [[zeroth law of thermodynamics]] in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body.<ref name=dugdale>{{cite book|author=J. S. Dugdale|title=Entropy and its Physical Meaning|publisher=Taylor & Francis|year=1996|isbn=978-0-7484-0569-5|page=13|quote=This law is the basis of temperature.}}</ref> For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows{{how?|date=August 2018}} a distinguished temperature scale, which defines an absolute, [[thermodynamic temperature]], independent of the properties of any particular reference thermometric body.<ref>[[Mark Zemansky|Zemansky, M.W.]] (1968), pp. 207–209.</ref><ref>Quinn, T.J. (1983), p. 8.</ref><br />
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The zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body. For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows a distinguished temperature scale, which defines an absolute, thermodynamic temperature, independent of the properties of any particular reference thermometric body.<br />
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美国热力学第零定律协会在其通常的简短声明中允许承认，相对于热平衡的两个物体具有相同的温度，特别是测试物体具有与参考温度测量物体相同的温度。对于一个物体和另一个物体之间的热平衡，存在着无限多的经验温度标度，通常分别取决于特定参考温度标度的性质。第二定律允许区分温度标度，它定义了一个绝对的热力学温度，与任何特定的参考温度体的性质无关。<br />
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==Various statements of the law==<br />
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==Various statements of the law==<br />
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各种各样的法律陈述<br />
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The second law of thermodynamics may be expressed in many specific ways,<ref name=MIT>{{cite web|title=Concept and Statements of the Second Law|url=http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node37.html|accessdate=2010-10-07 |publisher=web.mit.edu}}</ref> the most prominent classical statements{{sfnp|Lieb|Yngvason|1999}} being the statement by [[Rudolf Clausius]] (1854), the statement by [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] (1851), and the statement in axiomatic thermodynamics by [[Constantin Carathéodory]] (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.{{sfnp|Rao|2004|p=213}}<br />
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The second law of thermodynamics may be expressed in many specific ways, the most prominent classical statements being the statement by Rudolf Clausius (1854), the statement by Lord Kelvin (1851), and the statement in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.<br />
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热力学第二定律可以用许多特定的方式来表达，最突出的经典陈述是 Rudolf Clausius (1854)的陈述，Kelvin 勋爵(1851)的陈述，以及康斯坦丁·卡拉西奥多里(1909)在公理化热力学中的陈述。这些陈述用一般的物理术语描述法律，引用某些过程的不可能性。克劳修斯和开尔文陈述被证明是等价的。<br />
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===Carnot's principle===<br />
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===Carnot's principle===<br />
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卡诺原理<br />
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The historical origin<ref>[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986).</ref> of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a [[Carnot heat engine]], fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the [[caloric theory]] of heat was seriously considered, before the recognition of the [[first law of thermodynamics]], and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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The historical origin of the second law of thermodynamics was in Carnot's principle. It refers to a cycle of a Carnot heat engine, fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. The Carnot engine is an idealized device of special interest to engineers who are concerned with the efficiency of heat engines. Carnot's principle was recognized by Carnot at a time when the caloric theory of heat was seriously considered, before the recognition of the first law of thermodynamics, and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, it is physically equivalent to the second law of thermodynamics, and remains valid today. Carnot's original arguments were made from the viewpoint of the caloric theory, before the discovery of the first law of thermodynamics. Some samples from his book are:<br />
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热力学第二定律的历史起源源于卡诺的原则。它指的是一个卡诺热机周期，虚拟地以被称为准静态的极慢极限模式运行，因此热量和功量在子系统之间传递，这些子系统始终处于它们自己的内部热力学平衡状态。卡诺发动机是热机效率工程师特别感兴趣的理想设备。卡诺认识到卡诺原理的时候，正是热量热理论被认真考虑的时候，正是能量守恒定律理论被认识之前，正是熵概念的数学表达之前。根据第一定律的解释，它在物理上等同于热力学第二定律，并且在今天仍然有效。卡诺最初的论点是从卡路里理论的观点提出的，直到能量守恒定律的发现。以下是他书中的一些例子:<br />
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::...''wherever there exists a difference of temperature, motive power can be produced.''<ref>Carnot, S. (1824/1986), p. 51.</ref><br />
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...wherever there exists a difference of temperature, motive power can be produced.<br />
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只要有温差，就能产生动力。<br />
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::The production of motive power is then due in steam engines not to an actual consumption of caloric, but ''to its transportation from a warm body to a cold body ...''<ref>Carnot, S. (1824/1986), p. 46.</ref><br />
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The production of motive power is then due in steam engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body ...<br />
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动力的产生不是由于蒸汽机实际消耗的热量，而是由于它从一个温暖的物体运输到一个寒冷的物体..。<br />
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::''The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.''<ref>Carnot, S. (1824/1986), p. 68.</ref><br />
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The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.<br />
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热的动力与实现热的媒介无关，热的数量完全取决于两个物体之间的温度，最后是热量的传递。<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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In modern terms, Carnot's principle may be stated more precisely:<br />
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用现代术语来说，卡诺的原则可能更为准确:<br />
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::The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<ref>[[Clifford Truesdell|Truesdell, C.]] (1980), Chapter 5.</ref><ref>Adkins, C.J. (1968/1983), pp. 56–58.</ref><ref>Münster, A. (1970), p. 11.</ref><ref>Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998), pp.67–75.</ref><ref>Lebon, G., Jou, D., Casas-Vázquez, J. (2008), p. 10.</ref><ref>Eu, B.C. (2002), pp. 32–35.</ref><br />
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The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.<br />
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准静态或可逆卡诺循环的效率只取决于两个热源的温度，而且无论工作物质是什么，效率是相同的。以这种方式运行的卡诺发动机是使用这两种温度的最有效的热机。<br />
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===Clausius statement===<br />
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===Clausius statement===<br />
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克劳修斯声明<br />
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The German scientist [[Rudolf Clausius]] laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work.{{sfnp|Clausius|1850}} His formulation of the second law, which was published in German in 1854, is known as the ''Clausius statement'':<br />
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The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work. His formulation of the second law, which was published in German in 1854, is known as the Clausius statement:<br />
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1850年，德国科学家 Rudolf Clausius 通过研究热传递和功之间的关系，为热力学第二定律实验室奠定了基础。他在1854年用德语发表的第二定律的提法被称为克劳修斯声明:<br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.{{sfnp|Clausius|1854|p=86}}</blockquote><br />
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<blockquote>Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.</blockquote><br />
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热量不可能从一个较冷的物体传递到一个较热的物体，除非同时发生一些其他的变化。 / blockquote<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.<br />
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克劳修斯的陈述使用了“热量通道”的概念。正如通常在热力学的讨论，这意味着“能量作为热量的净转移” ，并没有提到促成转移的一种方式和其他。<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of [[refrigeration]], for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, the refrigeration system.<br />
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如果不对系统进行外部功，热就不能自发地从冷区流向热区，这一点从制冷的普通经验中可以看出。在冰箱中，热量从冷到热，但只有在外部介质——制冷系统的强制作用下才会发生变化。<br />
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===Kelvin statements===<br />
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===Kelvin statements===<br />
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开尔文陈述<br />
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[[William Thomson, 1st Baron Kelvin|Lord Kelvin]] expressed the second law in several wordings.<br />
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Lord Kelvin expressed the second law in several wordings.<br />
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开尔文勋爵用几个字表达了第二定律。<br />
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::It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.<br />
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没有任何外部机构的帮助，自动机器是不可能把热量从一个物体传递到另一个物体的较高温度。<br />
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::It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.{{sfnp|Thomson|1851}}<br />
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It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.<br />
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通过无生命物质的作用，把物质冷却到周围最冷的物体的温度以下，从物质的任何部分获得机械效应是不可能的。<br />
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===Equivalence of the Clausius and the Kelvin statements===<br />
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===Equivalence of the Clausius and the Kelvin statements===<br />
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克劳修斯和开尔文陈述的等价性<br />
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[[Image:Deriving Kelvin Statement from Clausius Statement.svg|thumb|Derive Kelvin Statement from Clausius Statement]]<br />
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Derive Kelvin Statement from Clausius Statement<br />
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从克劳修斯陈述推导出开尔文陈述<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed [[Carnot engine]] as shown by the figure. The [[Heat engine#Efficiency|efficiency]] of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the [[first law of thermodynamics]], as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the figure. The efficiency of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat <math>\Delta Q=Q\left(\frac{1}{\eta}-1\right)</math> from the cooler reservoir to the hotter one, which violates the Clausius statement. (This is a consequence of the first law of thermodynamics, as for the total system's energy to remain the same, <math> \text{Input}+\text{Output}=0 \implies Q-\frac{Q}{\eta} = -Q_c </math>, so therefore <math> Q_c=Q\left( \frac{1}{\eta}-1\right) </math> ). Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.<br />
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假设有一个引擎违反了开尔文定理: 也就是说，这个引擎以循环的方式吸收热量并将其转化为功，没有任何其他结果。现在配对与反向卡诺发动机如图所示。普通热机的效率为，反向热机的效率为1 / 。这对组合发动机的净效应和唯一效应是将热量从较冷的贮存器转移到较热的贮存器，这违反了克劳修斯的说法。(这是能量守恒定律的结果，因为系统的总能量保持不变，数学文本{输入} + 文本{输出}0暗示 q-frac { eta }-qc / math，所以数学 qc 向左(frac { eta-1 right) / math)。因此，违反开尔文陈述意味着违反克劳修斯陈述，即。克劳修斯的说法暗示了凯尔文的说法。我们可以用类似的方式证明开尔文陈述暗示了克劳修斯陈述，因此两者是等价的。<br />
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===Planck's proposition===<br />
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===Planck's proposition===<br />
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普朗克的命题<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.<br />
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普朗克提出了直接来自经验的如下命题。这有时被认为是他对第二定律的陈述，但他认为这是第二定律推导的起点。<br />
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::It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 86.</ref><ref>Roberts, J.K., Miller, A.R. (1928/1960), p. 319.</ref><br />
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It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.<br />
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建造一台发动机是不可能的，这台发动机将在一个完整的循环中工作，并且除了提高重量和冷却热储以外不会产生任何效果。<br />
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===Relation between Kelvin's statement and Planck's proposition 开尔文表述与普朗克命题的关系===<br />
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It is almost customary in textbooks to speak of the "[[Kelvin-Planck statement]]" of the law, as for example in the text by [[Dirk ter Haar|ter Haar]] and [[Harald Wergeland|Wergeland]].<ref>[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966), p. 17.</ref><br />
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It is almost customary in textbooks to speak of the "Kelvin-Planck statement" of the law, as for example in the text by ter Haar and Wergeland.<br />
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教科书中几乎惯常提及该定律的“'''开尔文-普朗克表述 Kelvin-Planck Statement'''” ，例如'''德克·特哈尔 Diek ter Haar''' 和'''哈拉尔德·沃格兰 Harald Wergeland''' 在文中就是这样表述的。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）Diek ter Haar 中间是小写的<br />
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The '''Kelvin–Planck statement''' (or the ''heat engine statement'') of the second law of thermodynamics states that <br />
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The Kelvin–Planck statement (or the heat engine statement) of the second law of thermodynamics states that <br />
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热力学第二定律的开尔文-普朗克表述（或称“热机表述 Heat Engine Statement”）指出<br />
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::It is impossible to devise a [[thermodynamic cycle|cyclically]] operating device, the sole effect of which is to absorb energy in the form of heat from a single [[heat reservoir|thermal reservoir]] and to deliver an equivalent amount of [[Work (physics)|work]].<ref name="Rao">{{cite book|last=Rao|first=Y. V. C.|title=Chemical Engineering Thermodynamics|publisher=Universities Press|isbn=978-81-7371-048-3|page=158|year=1997}}</ref><br />
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It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.<br />
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设计一种唯一效果是从单一热源吸收热量并提供等量的功的循环运行装置是不可能的。<br />
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===Planck's statement 普朗克表述===<br />
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Planck stated the second law as follows.<br />
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Planck stated the second law as follows.<br />
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普朗克表述第二定律如下。<br />
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::Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<ref name="Planck 100">[[Max Planck|Planck, M.]] (1897/1903), p. 100.</ref><ref name="Planck 463">[[Max Planck|Planck, M.]] (1926), p. 463, translation by Uffink, J. (2003), p. 131.</ref><ref name="Roberts & Miller 382">Roberts, J.K., Miller, A.R. (1928/1960), p. 382. This source is partly verbatim from Planck's statement, but does not cite Planck. This source calls the statement the principle of the increase of entropy.</ref><br />
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Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.<br />
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自然界中发生的任一过程都是沿参与其中的所有物体的熵之和增加的方向进行的。在某些限制下——即对于可逆过程，熵的总和保持不变。<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for ''irreversible phenomena''.<br />
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Rather like Planck's statement is that of Uhlenbeck and Ford for irreversible phenomena.<br />
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与普朗克表述非常相似的是'''乌伦贝克 Uhlenbeck'''和'''福特 Ford'''关于不可逆现象的表述。<br />
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::... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<ref>[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963), p. 16.</ref><br />
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... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.<br />
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在从一个平衡态到另一个平衡态的不可逆或自发的变化中（例如，当两个物体 A 和 B 接触时的温度平衡过程），熵总是增加。<br />
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===Principle of Carathéodory 卡拉西奥多里原理===<br />
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—— caratheodory 的原理在这里重新定向——<br />
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[[Constantin Carathéodory]] formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<ref>[[Constantin Carathéodory|Carathéodory, C.]] (1909).</ref><br />
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Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:<br />
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'''康斯坦丁·卡拉西奥多里 Constantin Carathéodory'''在纯数学公理的基础上进行了热力学公理化。他对第二定律的陈述被称为'''卡拉西奥多里原理 Principle of Carathéodory'''，可以这样表述:<br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.<ref>Buchdahl, H.A. (1966), p. 68.</ref></blockquote><br />
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<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.</blockquote><br />
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在绝热封闭系统的任意状态 S 附近，总有从 S 出发不可达的状态。<br />
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With this formulation, he described the concept of [[adiabatic accessibility]] for the first time and provided the foundation for a new subfield of classical thermodynamics, often called [[Ruppeiner geometry|geometrical thermodynamics]]. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic [[process function]], in other words, <math>\delta Q=TdS</math>.<ref name="Sychev1991">{{cite book |last=Sychev |first=V. V. |title=The Differential Equations of Thermodynamics |year=1991 |publisher=Taylor & Francis |isbn=978-1-56032-121-7}}</ref> {{clarify|date=February 2014}}<br />
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With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, <math>\delta Q=TdS</math>. <br />
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公理化处理后，他第一次描述了'''绝热可达性 Adiabatic Accessibility'''的概念，并为经典热力学的一个新的子领域，即通常所说的'''几何热力学 Geometrical Thermodynamics'''奠定了基础。由卡拉西奥多里原理可以推出，准静态转移的热量值是一个可积过程函数，即<math>\delta Q=TdS</math>。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）准静态转移的热量值是一个可积过程函数 存疑<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.<ref name="Munster 45">Münster, A. (1970), p. 45.</ref>{{sfnp|Lieb|Yngvason|1999|p=49}}<ref name="Planck 1926">[[Max Planck|Planck, M.]] (1926).</ref><ref>Buchdahl, H.A. (1966), p. 69.</ref> {{clarify|date=February 2014}}<br />
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Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium. <br />
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尽管在教科书几乎惯称卡拉西奥多里原理表述了第二定律，并认为其与克劳修斯表述或开尔文-普朗克表述等价，但事实并非如此。为了得到第二定律的所有内容，需要对卡拉西奥多里原理补充普朗克表述，即等量功总是增加一个最初处于自身内部热力学平衡的封闭系统的内部能量。<br />
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===Planck's principle 普朗克原理===<br />
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In 1926, [[Max Planck]] wrote an important paper on the basics of thermodynamics.<ref name="Planck 1926"/><ref>Uffink, J. (2003), pp. 129–132.</ref> He indicated the principle<br />
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In 1926, Max Planck wrote an important paper on the basics of thermodynamics. He indicated the principle<br />
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1926年，'''马克斯·普朗克 Max Planck'''写了一篇关于热力学基础的重要论文。他指出了以下原理<br />
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::The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<ref name="Munster 45"/>{{sfnp |Lieb|Yngvason|1999|p=49}}<br />
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The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.<br />
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一个封闭系统的内部能量因绝热过程增加，在整个过程中，系统的体积保持不变。<br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work."<ref>[[Clifford Truesdell|Truesdell, C.]], Muncaster, R.G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 15.</ref> Planck wrote: "The production of heat by friction is irreversible."<ref>[[Max Planck|Planck, M.]] (1897/1903), p. 81.</ref><ref>[[Max Planck|Planck, M.]] (1926), p. 457, Wikipedia editor's translation.</ref><br />
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This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work." Planck wrote: "The production of heat by friction is irreversible."<br />
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这个公式没有提到热，没有提到温度，甚至没有提到熵，也不一定隐含地依赖于这些概念，但它暗示了第二定律的内容。一个密切相关的表述为，“摩擦力从来不做正功。”普朗克写道: “摩擦生热是不可逆的。”<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above.<ref>Lieb, E.H., Yngvason, J. (2003), p. 149.</ref> It is relevant that for a system at constant volume and [[Mole (unit)|mole numbers]], the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above. It is relevant that for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.<br />
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更不用说熵了，这个普朗克定理是用物理术语来表述的。它与上面给出的开尔文表述密切相关。相关的是，对于恒定体积和摩尔数的系统，熵是内能的单调函数。然而，普朗克的这个原理实际上并不是普朗克对第二定律的首选表述（见前面小节），而是依赖于熵的概念。<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:<br />
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Borgnakke 和 Sonntag 提出的以下表述在某种意义上是对普朗克原理的补充。他们没有将其作为第二定律的完整表述:<br />
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::... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<ref>Borgnakke, C., Sonntag., R.E. (2009), p. 304.</ref><br />
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... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.<br />
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只有一种方法可以减少[闭合]系统的熵——从系统中转移热量。<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.<br />
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与普朗克之前提出的原理不同，这一原理明确地用熵的变化来表示。从系统中去除物质也可以减少系统的熵。<br />
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===Statement for a system that has a known expression of its internal energy as a function of its extensive state variables 一个其内能有已知表达式（其扩展状态变量的函数）的系统的表述===<br />
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The second law has been shown to be equivalent to the [[internal energy]] ''U'' being a weakly [[convex function]], when written as a function of extensive properties (mass, volume, entropy, ...).<ref>{{cite book |last1=van Gool |first1=W. |last2=Bruggink |first2=J.J.C. (Eds) |url= |title=Energy and time in the economic and physical sciences |publisher=North-Holland |year=1985 |pages=41–56 |quote= |isbn=978-0-444-87748-2}}</ref><ref>{{Cite journal | last1 = Grubbström | first1 = Robert W. | doi = 10.1016/j.apenergy.2007.01.003 | title = An Attempt to Introduce Dynamics Into Generalised Exergy Considerations| journal = Applied Energy| volume = 84| issue = 7–8 | pages = 701–718 | year = 2007}}</ref> {{clarify|date=February 2014}}<br />
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The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...). <br />
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第二定律被证明等价于弱凸函数内能 U （写成广泛性质（质量，体积，熵，...）的函数时）。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）存疑<br />
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==Corollaries 推论==<br />
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===Perpetual motion of the second kind 第二类永动机===<br />
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{{main article|Perpetual motion}}<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of [[first law of thermodynamics]] by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.<br />
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在第二定律建立之前，许多志在发明永动机的人试图通过提取环境的巨大内能作为机器动力来突破'''热力学第一定律 First Law of Thermodynamics''' 的限制。这种机器被称为“第二类永动机 Perpetual Motion Machine of the Second Kind”。第二定律表明这种机器是不可能的。<br />
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===Carnot theorem 卡诺定理===<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:<br />
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卡诺定理（1824）是一条限制任何可能的发动机的最大效率的原理。效率完全取决于热库和冷库之间的温差。卡诺定理指出:<br />
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*All irreversible heat engines between two heat reservoirs are less efficient than a [[Carnot engine]] operating between the same reservoirs.<br />
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*All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs.<br />
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* 所有不可逆热机的效率低于在两相同热源之间工作的'''卡诺机 Carnot engine'''。<br />
* 所有可逆热机的效率与在两相同热源之间工作的卡诺机相等。<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as [[thermodynamic reversibility]]. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the [[Carnot cycle]]'s reversibility and was condemned to be less efficient.<br />
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In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the Carnot cycle's reversibility and was condemned to be less efficient.<br />
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在他的理想模型中，热转换成功可以通过逆转循环的运动而恢复，这个概念后来被称为'''热力学可逆性 Thermodynamic Reversibility'''。然而，卡诺进一步假定，一些热量损失了，并没有转化为机械功。因此，没有一个真实的热机能够实现'''卡诺循环 Carnot cycle'''的可逆性，并且被认为效率较低。<br />
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Though formulated in terms of caloric (see the obsolete [[caloric theory]]), rather than [[entropy]], this was an early insight into the second law.<br />
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Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law.<br />
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该理论尽管是用热量表述的（见被取代的'''热质说 Caloric Theory'''），而不是熵，但是它是对第二定律的早期认识。<br />
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===Clausius inequality 克劳修斯不等式===<br />
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The [[Clausius theorem]] (1854) states that in a cyclic process<br />
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The Clausius theorem (1854) states that in a cyclic process<br />
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克劳修斯定理（1854）指出，在一个循环的过程中<br />
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: <math>\oint \frac{\delta Q}{T} \leq 0.</math><br />
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The equality holds in the reversible case<ref>[http://scienceworld.wolfram.com/physics/ClausiusTheorem.html ''Clausius theorem''] at [[Wolfram Research]]</ref> and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function [[entropy]]. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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The equality holds in the reversible case and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality.<br />
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等号在可逆情况下成立，严格不等号在不可逆情况下成立。可逆情况下引入状态函数熵。这是因为在循环过程中，状态函数的变化为零。<br />
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--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）末句存疑<br />
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===Thermodynamic temperature 热力学温度===<br />
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{{main article|Thermodynamic temperature}}<br />
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For an arbitrary heat engine, the efficiency is:<br />
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For an arbitrary heat engine, the efficiency is:<br />
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对于任意热机，效率为:<br />
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: <math>\eta = \frac {W_n}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H} \qquad (1)</math><br />
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where ''W''<sub>n</sub> is for the net work done per cycle. Thus the efficiency depends only on q<sub>C</sub>/q<sub>H</sub>.<br />
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其中 ''W''<sub>n</sub> 表示每个循环所做的净功。因此效率只取决于 q<sub>C</sub>/q<sub>H</sub>。<br />
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[[Carnot theorem (thermodynamics)|Carnot's theorem]] states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient.<br />
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卡诺定理指出，所有在相同热源间运行的可逆发动机的效率是相同的。<br />
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Thus, any reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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Thus, any reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only: <br />
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因此，任何在温度''T''<sub>1</sub>和''T''<sub>2</sub>之间运行的可逆热机必须具有相同的效率，也就是说，效率只是温度的函数:<br />
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<math>\frac{q_C}{q_H} = f(T_H,T_C)\qquad (2).</math><br />
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In addition, a reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between ''T''<sub>1</sub> and another (intermediate) temperature ''T''<sub>2</sub>, and the second between ''T''<sub>2</sub> and''T''<sub>3</sub>. This can only be the case if<br />
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In addition, a reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between T<sub>1</sub> and another (intermediate) temperature T<sub>2</sub>, and the second between T<sub>2</sub> andT<sub>3</sub>. This can only be the case if<br />
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另外，在温度 ''T''<sub>1</sub> 和 ''T''<sub>3</sub> 之间工作的可逆热机必须具有与由分别在温度''T''<sub>1</sub> 和（中间）温度 ''T''<sub>2</sub>之间、在 ''T''<sub>2</sub> 和 ''T''<sub>3</sub> 之间的两个循环组成的系统效率相同。只有下式成立才会出现这种情况:<br />
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: <math>f(T_1,T_3) = \frac{q_3}{q_1} = \frac{q_2 q_3} {q_1 q_2} = f(T_1,T_2)f(T_2,T_3).</math><br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the [[triple point]] of water. Then for any ''T''<sub>2</sub> and ''T''<sub>3</sub>,<br />
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Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the triple point of water. Then for any T<sub>2</sub> and T<sub>3</sub>,<br />
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现在考虑如下情形，<math>T_1</math> 是一个固定的参考温度: 水的三相点的温度。则对于任意 T<sub>2</sub> 和 T<sub>3</sub>,<br />
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: <math>f(T_2,T_3) = \frac{f(T_1,T_3)}{f(T_1,T_2)} = \frac{273.16 \cdot f(T_1,T_3)}{273.16 \cdot f(T_1,T_2)}.</math><br />
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Therefore, if thermodynamic temperature is defined by<br />
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Therefore, if thermodynamic temperature is defined by<br />
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因此，如果热力学温度被定义为<br />
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: <math>T = 273.16 \cdot f(T_1,T) \,</math><br />
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then the function ''f'', viewed as a function of thermodynamic temperature, is simply<br />
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then the function f, viewed as a function of thermodynamic temperature, is simply<br />
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那么函数 ''f'' 作为热力学温度的函数，为<br />
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: <math>f(T_2,T_3) = \frac{T_3}{T_2},</math><br />
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and the reference temperature ''T''<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be used{{snd}}the choice here corresponds to the [[Kelvin]] scale.)<br />
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and the reference temperature T<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be usedthe choice here corresponds to the Kelvin scale.)<br />
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参考温度 ''T''<sub>1</sub> 的值为273.16。（任何参考温度和任何正值均可用——此处的选择对应开尔文标度。）<br />
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===Entropy 熵===<br />
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{{main article|Entropy (classical thermodynamics)}}<br />
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According to the [[Clausius theorem|Clausius equality]], for a reversible process<br />
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According to the Clausius equality, for a reversible process<br />
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根据克劳修斯定理，对可逆过程有<br />
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: <math>\oint \frac{\delta Q}{T}=0</math><br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.<br />
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这意味着线积分 <math>\int_L \frac{\delta Q}{T}</math> 对于可逆过程是路径无关的。<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer<ref name="Schmidt-Rohr 14" /> satisfies<br />
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So we can define a state function S called entropy, which for a reversible process or for pure heat transfer satisfies<br />
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所以我们可以定义一个叫做熵的状态函数 S，对于可逆过程或者纯热传递满足<br />
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: <math>dS = \frac{\delta Q}{T} \!</math><br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the [[third law of thermodynamics]], which states that ''S'' = 0 at [[absolute zero]] for perfect crystals.<br />
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With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the third law of thermodynamics, which states that S = 0 at absolute zero for perfect crystals.<br />
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据此，只有对上述公式进行积分，才能得到熵的差值。为了获得绝对值，我们需要'''热力学第三定律 Third Law of Thermodynamics'''，它指出绝对零度下完美晶体的 ''S'' = 0。<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.<br />
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对于任意不可逆过程，由于熵是一个状态函数，我们总是可以将初始状态和最终状态与一个虚拟的可逆过程联系起来，并在这条路径上积分以计算熵的差值。<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the [[Clausius inequality]] on this loop,<br />
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Now reverse the reversible process and combine it with the said irreversible process. Applying the Clausius inequality on this loop,<br />
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现在把可逆过程逆过来，将其与上述不可逆过程结合。把克劳修斯不等式应用到这个循环,<br />
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: <math>-\Delta S+\int\frac{\delta Q}{T}=\oint\frac{\delta Q}{T}< 0</math><br />
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Thus,<br />
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Thus,<br />
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故,<br />
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: <math>\Delta S \ge \int \frac{\delta Q}{T} \,\!</math><br />
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where the equality holds if the transformation is reversible.<br />
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where the equality holds if the transformation is reversible.<br />
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如果变换可逆，等号成立。<br />
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Notice that if the process is an [[adiabatic process]], then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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Notice that if the process is an adiabatic process, then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.<br />
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注意，若该过程是一个'''绝热过程 Adiabatic Process'''，则<math>\delta Q=0</math>，故<math>\Delta S\ge 0</math>。<br />
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===Energy, available useful work===<br />
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===Energy, available useful work===<br />
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能量，有用的工作<br />
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{{See also|Exergy}}<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an ''unlimited'' heat reservoir at temperature ''T<sub>R</sub>'' and pressure ''P<sub>R</sub>'' {{snd}}so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain ''T<sub>R</sub>''; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain ''P<sub>R</sub>''.<br />
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An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an unlimited heat reservoir at temperature T<sub>R</sub> and pressure P<sub>R</sub> so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain T<sub>R</sub>; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain P<sub>R</sub>.<br />
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考虑将第二定律应用于孤立系统(称为整体系统或宇宙)的情形，是一个重要的、具有启发性的理想情形。孤立系统由两部分组成: 感兴趣的子系统和子系统的环境。这些环境被想象为如此之大，以至于它们可以被认为是一个温度为 t 子 r / sub 和压力为 p 子 r / sub 的无限热源，因此无论有多少热量被转移到(或来自)子系统，周围环境的温度将保持 t 子 r / sub; 无论子系统的体积扩大(或收缩)多少，周围环境的压力将保持 p 子 r / sub。<br />
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Whatever changes to ''dS'' and ''dS<sub>R</sub>'' occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy ''S<sub>tot</sub>'' of the isolated total system must not decrease:<br />
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Whatever changes to dS and dS<sub>R</sub> occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy S<sub>tot</sub> of the isolated total system must not decrease:<br />
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根据第二定律，孤立总体系统的熵 s 子系统 / 子系统的熵不能减小，无论子系统和周围环境的熵对 dS 和 dS 子系统 r / 子系统发生什么变化:<br />
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: <math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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<math> dS_{\mathrm{tot}}= dS + dS_R \ge 0 </math><br />
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数学 dS + dS r ge 0 / math<br />
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According to the [[first law of thermodynamics]], the change ''dU'' in the internal energy of the sub-system is the sum of the heat ''δq'' added to the sub-system, ''less'' any work ''δw'' done ''by'' the sub-system, ''plus'' any net chemical energy entering the sub-system ''d ∑μ<sub>iR</sub>N<sub>i</sub>'', so that:<br />
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According to the first law of thermodynamics, the change dU in the internal energy of the sub-system is the sum of the heat δq added to the sub-system, less any work δw done by the sub-system, plus any net chemical energy entering the sub-system d ∑μ<sub>iR</sub>N<sub>i</sub>, so that:<br />
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根据能量守恒定律，子系统内部能量的变化 dU 是子系统内部能量 q 的总和，减去子系统所做的任何功，再加上进入子系统 d ∑ sub iR / sub n sub i / sub 的任何净化学能，因此:<br />
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: <math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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<math> dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,</math><br />
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数学 dU delta q- delta w + d ( sum mu iR } n i) ，/ math<br />
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where μ<sub>iR</sub> are the [[chemical potential]]s of chemical species in the external surroundings.<br />
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where μ<sub>iR</sub> are the chemical potentials of chemical species in the external surroundings.<br />
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其中亚 iR / 亚是外部环境中化学物种的化学势。<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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Now the heat leaving the reservoir and entering the sub-system is<br />
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现在热量离开储存器进入子系统是<br />
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: <math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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<math> \delta q = T_R (-dS_R) \le T_R dS </math><br />
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数学△ qtr (- dS r) le tr dS / math<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.<br />
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其中我们首先使用了经典热力学中熵的定义(在统计热力学中，熵变、温度和吸收热量之间的关系可以推导出来) ，然后从上面推导出第二定律的不等式。<br />
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It therefore follows that any net work ''δw'' done by the sub-system must obey<br />
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It therefore follows that any net work δw done by the sub-system must obey<br />
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因此，子系统所做的任何净功都必须服从<br />
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: <math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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<math> \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,</math><br />
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数学增量 w le-dU + tds + sum mu { iR } dN i，/ math<br />
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It is useful to separate the work ''δw'' done by the subsystem into the ''useful'' work ''δw<sub>u</sub>'' that can be done ''by'' the sub-system, over and beyond the work ''p<sub>R</sub> dV'' done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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It is useful to separate the work δw done by the subsystem into the useful work δw<sub>u</sub> that can be done by the sub-system, over and beyond the work p<sub>R</sub> dV done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:<br />
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将子系统所做的功与子系统所能做的功分离为子系统所能做的有用功，超过了子系统对外界压力所做的功，给出了子系统所能做的有用功(火用)的下列关系式:<br />
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: <math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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<math> \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,</math><br />
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D (u-t r s + p r v-sum { iR } n i) ，/ math<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the ''availability'' or ''[[exergy]]'' ''E'' of the subsystem,<br />
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It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the availability or exergy E of the subsystem,<br />
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可以很方便地将子系统的右侧定义为热动力位能的精确衍生物，称为子系统的可用性或火用 e,<br />
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: <math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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<math> E = U - T_R S + p_R V - \sum \mu_{iR} N_i </math><br />
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数学 e u-t r s + p r v- sum { iR } n i / math<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,<br />
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因此，第二定律意味着，对于任何可以简单地被视为分为一个子系统和一个与之接触的无限温度和压力储存器的过程,<br />
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: <math> dE + \delta w_u \le 0 \, </math><br />
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<math> dE + \delta w_u \le 0 \, </math><br />
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数学德 delta u le 0，/ math<br />
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i.e. the change in the subsystem's exergy plus the useful work done ''by'' the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done ''on'' the system) must be less than or equal to zero.<br />
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i.e. the change in the subsystem's exergy plus the useful work done by the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done on the system) must be less than or equal to zero.<br />
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也就是。子系统(火用)的变化加上子系统所做的有用功(或者，子系统(火用)的变化不包括任何功，除了压力贮存器所做的功之外，在系统中所做的功)必须小于或等于零。<br />
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In sum, if a proper ''infinite-reservoir-like'' reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in ''E'' for an irreversible process and no change for a reversible process.<br />
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In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in E for an irreversible process and no change for a reversible process.<br />
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总之，如果选择一个合适的类似于无限库的参考状态作为现实世界中的系统环境，那么第二定律预测一个不可逆性的 e 值会减少，而一个可逆过程的 e 值不会变化。<br />
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: <math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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<math>dS_{tot} \ge 0 </math> Is equivalent to <math> dE + \delta w_u \le 0 </math><br />
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数学 d = 0 = 0 = 0 = 0 = 0<br />
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This expression together with the associated reference state permits a [[design engineer]] working at the macroscopic scale (above the [[thermodynamic limit]]) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (''Also, see [[process engineer]]''). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (''See [[Exergy efficiency|second law efficiency]]''.)<br />
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This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (Also, see process engineer). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (See second law efficiency.)<br />
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这个表达式连同相关的参考状态允许在宏观的设计工程师利用第二定律，而不需要直接测量或考虑整个孤立系统的熵变。(另外，请参阅流程工程师)。这些变化已经被考虑，假设被考虑的系统可以达到平衡的参考状态而不改变参考状态。还可以找到一个过程或过程集合的效率，将其与可逆的理想状态相比较<br />
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This approach to the Second Law is widely utilized in [[engineering]] practice, [[environmental accounting]], [[systems ecology]], and other disciplines.<br />
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This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines.<br />
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第二定律的这种方法被广泛应用于工程实践、环境会计、系统生态学和其他学科。<br />
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===The second law in chemical thermodynamics===<br />
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===The second law in chemical thermodynamics===<br />
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化学热力学的第二定律<br />
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For a [[spontaneous process|spontaneous chemical process]] in a closed system at constant temperature and pressure without non-''PV'' work, the Clausius inequality Δ''S > Q/T<sub>surr</sub>'' transforms into a condition for the change in [[Gibbs free energy]] <br />
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For a spontaneous chemical process in a closed system at constant temperature and pressure without non-PV work, the Clausius inequality ΔS > Q/T<sub>surr</sub> transforms into a condition for the change in Gibbs free energy <br />
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对于一个封闭系统中的自发化学过程，在没有非 pv 功的情况下，在恒定温度和压力下，将克劳修斯不等式 s q / t sub surr / sub 转化为吉布斯自由能变化的条件<br />
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: <math>\Delta G < 0 </math><br />
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<math>\Delta G < 0 </math><br />
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数学 Delta g 0 / 数学<br />
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or d''G'' < 0. For a similar process at constant temperature and volume, the change in [[Helmholtz free energy]] must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products.<ref name="Oxtoby8th"/> <ref name="MortimerBook"></ref> The chemical equilibrium condition at constant ''T'' and ''p'' without electrical work is d''G'' = 0.<br />
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or dG < 0. For a similar process at constant temperature and volume, the change in Helmholtz free energy must be negative, <math>\Delta A < 0 </math>. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products. The chemical equilibrium condition at constant T and p without electrical work is dG = 0.<br />
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或者 dG 0。对于一个相似的过程，在恒定的温度和体积下，亥姆霍兹自由能的变化一定是负的，这是 Delta a 0 / math。因此，自由能(g 或 a)变化的负值是过程自发的必要条件。这是热力学第二定律在化学中最有用的形式，其中自由能的变化可以通过列表的生成焓和反应物和产物的标准摩尔熵来计算。在 t 和 p 不变的情况下，化学平衡条件是 dG 0。<br />
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==History==<br />
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==History==<br />
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历史<br />
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{{See also|History of entropy}}<br />
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[[File:Sadi Carnot.jpeg|thumb|upright|Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique]].]]<br />
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Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique.]]<br />
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尼古拉·莱昂纳尔·萨迪·卡诺穿着传统的巴黎综合理工学院制服<br />
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The first theory of the conversion of heat into mechanical work is due to [[Nicolas Léonard Sadi Carnot]] in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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The first theory of the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.<br />
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第一个把热转化为机械功的理论是在1824年由尼古拉·莱昂纳尔·萨迪·卡诺提出的。他是第一个正确认识到这种转换的效率取决于发动机和环境之间的温差。<br />
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Recognizing the significance of [[James Prescott Joule]]'s work on the conservation of energy, [[Rudolf Clausius]] was the first to formulate the second law during 1850, in this form: heat does not flow ''spontaneously'' from cold to hot bodies. While common knowledge now, this was contrary to the [[caloric theory]] of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law during 1850, in this form: heat does not flow spontaneously from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).<br />
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认识到詹姆斯·普雷斯科特·焦耳在能量守恒方面工作的重要性，Rudolf Clausius 在1850年第一个提出了第二定律: 热不会自发地从冷物体流向热物体。虽然现在的常识是，这与当时流行的热量理论相反，热量理论认为热量是一种流体。从那里他能够推断出萨迪卡诺定律和熵的定义(1865年)。<br />
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Established during the 19th century, the [[Kelvin-Planck statement|Kelvin-Planck statement of the Second Law]] says, "It is impossible for any device that operates on a [[cyclic process|cycle]] to receive heat from a single [[heat reservoir|reservoir]] and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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Established during the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.<br />
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建立于19世纪的开尔文-普朗克第二定律(Kelvin-Planck)说，“任何循环运行的设备都不可能从单个蓄热体接收热量，并产生净功。”这被证明相当于克劳修斯的陈述。<br />
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The [[ergodic hypothesis]] is also important for the [[Boltzmann]] approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.<br />
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遍历假设对玻尔兹曼方法也很重要。它说，在很长一段时间内，在具有相同能量的微观态的相空间的某些区域所花费的时间与这个区域的体积成正比，即。在很长一段时间内，所有可访问的微观状态的可能性都是一样的。同样的，它表明时间平均值和系综平均值是相同的。<br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a [[Macroscopic bodies|macroscopic system]]. This doctrine is obsolescent.<ref>Denbigh, K.G., Denbigh, J.S. (1985). ''Entropy in Relation to Incomplete Knowledge'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-25677-1}}, pp. 43–44.</ref><ref>Grandy, W.T., Jr (2008). ''Entropy and the Time Evolution of Macroscopic Systems'', Oxford University Press, Oxford, {{ISBN|978-0-19-954617-6}}, pp. 55–58.</ref><ref name=Lambert>[http://entropysite.oxy.edu Entropy Sites — A Guide] Content selected by [[Frank L. Lambert]]</ref><br />
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There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a macroscopic system. This doctrine is obsolescent.<br />
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有一种传统的学说，从克劳修斯开始，认为熵可以被理解为宏观系统中的分子“无序”。这种学说已经过时了。<br />
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===Account given by Clausius===<br />
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===Account given by Clausius===<br />
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作者: 克劳修斯<br />
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[[File:Clausius-1.jpg|thumb|upright|Rudolf Clausius]]<br />
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Rudolf Clausius<br />
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鲁道夫 · 克劳修斯<br />
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In 1856, the German physicist [[Rudolf Clausius]] stated what he called the "second fundamental theorem in the [[mechanical theory of heat]]" in the following form:{{sfnp|Clausius|1867}}<br />
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In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:<br />
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1856年，德国物理学家鲁道夫 · 克劳修斯阐述了他所谓的“热力学理论中的第二个基本定理” ，其形式如下:<br />
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: <math>\int \frac{\delta Q}{T} = -N</math><br />
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<math>\int \frac{\delta Q}{T} = -N</math><br />
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- n / math<br />
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where ''Q'' is heat, ''T'' is temperature and ''N'' is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:<br />
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其中 q 是热，t 是温度，n 是一个循环过程中所有未补偿的相变的“等价值”。后来，在1865年，克劳修斯将“等价值”定义为熵。紧接着这个定义，也就是在同一年，第二定律最著名的版本在4月24日苏黎世哲学学会的一次演讲中被宣读，在演讲的最后，克劳修斯总结道:<br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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<blockquote>The entropy of the universe tends to a maximum.</blockquote><br />
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宇宙的熵趋于最大<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. [[universe]], as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.<br />
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这句话是第二定律中最广为人知的措辞。由于其语言的松散性，例如:。这个问题的答案是肯定的，因为这个问题的答案是肯定的。开放，封闭，或孤立，许多人认为这个简单的陈述意味着热力学第二定律适用于几乎所有可以想象的主题。这是不正确的; 这个陈述只是一个更加扩展和精确描述的简化版本。<br />
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In terms of time variation, the mathematical statement of the second law for an [[isolated system]] undergoing an arbitrary transformation is:<br />
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In terms of time variation, the mathematical statement of the second law for an isolated system undergoing an arbitrary transformation is:<br />
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就时间变化而言，经历任意变换的孤立系统第二定律的数学表述是:<br />
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: <math>\frac{dS}{dt} \ge 0</math><br />
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<math>\frac{dS}{dt} \ge 0</math><br />
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0 / math<br />
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where<br />
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where<br />
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: ''S'' is the entropy of the system and<br />
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S is the entropy of the system and<br />
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S 是系统的熵<br />
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: ''t'' is [[time]].<br />
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t is time.<br />
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时间到了。<br />
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!-- 可逆过程需要与周围环境保持平衡。这对于一个孤立的系统是不可能的。因此，关于可逆过程的讨论已经转移到对封闭系统的分析----<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:<br />
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平衡后应用等式符号。建立孤立系统第二定律的另一种方法是:<br />
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: <math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math<br />
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with <math> \dot S_{i}</math> the sum of the rate of [[entropy production]] by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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with <math> \dot S_{i}</math> the sum of the rate of entropy production by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_{a}</math> it gives the so-called dissipated energy <math> P_{diss}=T_{a}\dot S_{i}</math>.<br />
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用数学 s { i } / 数学表示系统内所有进程的产生熵之和。这种配方的优点是它显示了产生熵的效果。产生熵是一个非常重要的概念，因为它决定(限制)热机器的效率。与环境温度数学 t { a } / math 相乘，得到所谓的耗散能量数学 p { diss } t { a } dot s { i } / math。<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:<br />
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封闭系统的第二定律(允许热量交换和移动边界，但不允许物质交换)的表达式是:<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math 来计算数学点 q } + 点 s { i } / math<br />
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: <math>\dot Q</math> is the heat flow into the system<br />
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<math>\dot Q</math> is the heat flow into the system<br />
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数学 q / math 是进入系统的热流<br />
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: <math>T</math> is the temperature at the point where the heat enters the system.<br />
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<math>T</math> is the temperature at the point where the heat enters the system.<br />
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Math 是热量进入系统时的温度。<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.<br />
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等式符号在只有可逆过程在系统内发生的情况下成立。如果发生不可逆过程(在实际操作系统中就是这种情况) ,-符号保持不变。如果系统有多处供热，我们必须求相应项的代数和。<br />
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For open systems (also allowing exchange of matter):<br />
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For open systems (also allowing exchange of matter):<br />
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对于开放系统(也允许物质交换) :<br />
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: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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<math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}</math> with <math> \dot S_{i} \ge 0</math><br />
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用数学点 s { i } ge 0 / math 表示数学点 q } + 点 s + 点 s { i } / math<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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Here <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.<br />
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这里的数学 s / math 是熵流进入系统，与物质流进入系统有关。它不应该与熵的时间导数混淆。如果物质在几个地方提供，我们必须取这些贡献的代数和。<br />
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==Statistical mechanics==<br />
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==Statistical mechanics==<br />
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统计力学<br />
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[[Statistical mechanics]] gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a [[microstate (statistical mechanics)|microstate]] of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/{{radic|''N''}} where ''N'' is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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Statistical mechanics gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a microstate of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/ where N is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.<br />
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通过假设物质是由不断运动的原子和分子组成的，统计力学对第二定律给出了解释。系统中每个粒子的一组特定的位置和速度称为系统的微观状态，由于系统的不断运动，系统不断地改变其微观状态。统计力学假设，在平衡状态下，系统可能处于的每个微观状态发生的可能性是相等的，当这个假设被提出时，它直接导致了第二定律必须在统计意义上成立的结论。也就是说，第二定律平均成立，其中 n 是系统中粒子数的1 / 数量级的统计变化。在日常(宏观)情况下，违反第二定律的概率几乎为零。然而，对于粒子数量很少的系统，热力学参数，包括熵，可能显示出与第二定律预测的显著的统计偏差。经典热力学理论没有处理这些统计变量。<br />
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（by Jie XIANG) 在通过假定一种物质由不断运动的原子和分子组成的前提下，统计力学对热力学第二定律进行了解释。系统中每个粒子其特定的位置和速度组成为系统的微观状态，由于（原子和分子的）不断运动，系统也持续改变其微观状态。统计力学假设，在系统力平衡的情况下，其微观状态其实是等可能发生的，进而能够得出结论：热力学第二定律必须在统计学意义上成立。也就是说，第二定律将保持均值不变， 其统计变化为1 /√N，N为系统中的粒子数。在日常(宏观)情况下，违反热力学第二定律的概率几乎为零，但是，对于粒子数量非常少的系统，其热力学参数（包括熵）可能会显示出与第二定律所预测的明显统计偏差。然而经典热力学理论并不处理这些统计变量。<br />
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==Derivation from statistical mechanics==<br />
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==Derivation from statistical mechanics==<br />
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从统计力学导出<br />
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{{Further information|H-theorem}}<br />
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The first mechanical argument of the [[Kinetic theory of gases]] that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to [[James Clerk Maxwell]] in 1860;<ref>{{Cite journal | last1 = Gyenis | first1 = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G }}</ref> [[Ludwig Boltzmann]] with his [[H-theorem]] of 1872 also argued that due to collisions gases should over time tend toward the [[Maxwell–Boltzmann distribution]].<br />
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The first mechanical argument of the Kinetic theory of gases that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium was due to James Clerk Maxwell in 1860; Ludwig Boltzmann with his H-theorem of 1872 also argued that due to collisions gases should over time tend toward the Maxwell–Boltzmann distribution.<br />
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气体动力学理论的第一个力学论证由詹姆斯·克拉克·麦克斯韦在1860年给出，指出分子碰撞引起温度均衡，因此整体趋向于平衡； 路德维希·玻尔兹曼在1872年提出的 H 定理也认为，气体由于碰撞应该随着时间的推移趋向于麦克斯韦-波兹曼分布。<br />
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Due to [[Loschmidt's paradox]], derivations of the Second Law have to make an assumption regarding the past, namely that the system is [[Correlation and dependence|uncorrelated]] at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a [[boundary condition]], and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the [[Big Bang]]), though [[Boltzmann brain|other scenarios]] have also been suggested.<ref name="Hawking AOT">{{cite journal|last=Hawking|first=SW|title=Arrow of time in cosmology|journal=Phys. Rev. D|year=1985|volume=32|issue=10|pages=2489–2495|doi=10.1103/PhysRevD.32.2489|pmid=9956019|bibcode = 1985PhRvD..32.2489H }}</ref><ref>{{cite book | last = Greene | first = Brian | authorlink = Brian Greene | title = The Fabric of the Cosmos | url = https://archive.org/details/fabricofcosmossp00gree | url-access = registration | publisher = Alfred A. Knopf | year = 2004 | page = [https://archive.org/details/fabricofcosmossp00gree/page/171 171] | isbn = 978-0-375-41288-2}}</ref><ref name=Lebowitz>{{cite journal|last=Lebowitz|first=Joel L.|title= Boltzmann's Entropy and Time's Arrow|journal=Physics Today|date=September 1993|volume=46|issue=9|pages=32–38|url=http://users.df.uba.ar/ariel/materias/FT3_2008_1C/papers_pdf/lebowitz_370.pdf|accessdate=2013-02-22|doi=10.1063/1.881363|bibcode = 1993PhT....46i..32L }}</ref><br />
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Due to Loschmidt's paradox, derivations of the Second Law have to make an assumption regarding the past, namely that the system is uncorrelated at some time in the past; this allows for simple probabilistic treatment. This assumption is usually thought as a boundary condition, and thus the second Law is ultimately a consequence of the initial conditions somewhere in the past, probably at the beginning of the universe (the Big Bang), though other scenarios have also been suggested.<br />
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由于洛施密特悖论，第二定律的导出必须对过去做出一个假设，即系统在过去的某个时刻是不相关的；这样的假设允许进行简单的概率处理。这个假设通常被认为是一个边界条件，因此热力学第二定律最终是过去某个地方的初始条件的结果，可能是在宇宙的开始(大爆炸) 。也有人提出了其他假设。<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the [[Statistical mechanics#Fundamental postulate|fundamental postulate]], also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy.{{citation needed|date=August 2012}} The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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Given these assumptions, in statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermally isolated system can only increase, is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of <math>E</math> is:<br />
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考虑到这些假设，在统计力学中，第二定律不是一个假设，而是统计力学基本假设的一个结果，也被称为等先验概率假设。这个基本假设表明，只要一个人清楚地知道，简单的概率论证只适用于未来，而对于过去，有辅助的信息来源告诉我们，它是低熵的。热力学第二定律的第一部分指出，热孤立系统的熵只能增加。如果我们把熵的概念限制在热平衡系统中，那么热力学第二定律的第一部分是等先验概率假设的一个显然结果。处于热平衡状态的孤立系统包含能量<math>E</math>的熵表示为：<br />
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: <math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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<math>S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\,</math><br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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where <math>\Omega\left(E\right)</math> is the number of quantum states in a small interval between <math>E</math> and <math>E +\delta E</math>. Here <math>\delta E</math> is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of <math>\delta E</math>. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on <math>\delta E</math>.<br />
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其中<math>\Omega\left(E\right)</math> 是处于 <math>E</math>和<math>E +\delta E</math这个小区间内的量子态数目。这里的 <math>\delta E</math> 是一个宏观上很小的固定能量区间。严格地说，这意味着熵取决于对<math>\delta E</math>的选择。然而在热力学极限下（例如无穷大系统的极限），狭义的熵(单位体积或单位质量的熵)不依赖于 <math>\delta E</math>。<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then <math>\Omega</math> will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that <math>\Omega</math> is maximized as that is the most probable situation in equilibrium.<br />
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假设我们有一个孤立系统，其宏观状态由许多变量描述。这些宏观变量可以是总体积、活塞在系统中的位置等。从而<math>\Omega</math>将取决于这些变量的值。如果某个变量不是固定的（我们不会在某个位置夹住活塞) ，那么因为在平衡状态下所有可到达状态的等几率到达的，平衡状态下的自由变量会使 <math>\Omega</math> 最大，因为这是平衡状态下最可能的情况。<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that <math>\Omega</math> is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).<br />
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如果该变量最初固定到某个值然后释放，当达到新的平衡时，变量将自我调整使得 <math>\Omega</math> 最大，这意味着熵将增加或保持不变(如果变量初始固定的值恰好是平衡值)。<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's [[H-theorem]], however, proves that the quantity {{math|''H''}} increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number <math>\Omega</math> of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of <math>1/\Omega</math>. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the quantity increases monotonically as a function of time during the intermediate out of equilibrium state.<br />
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假设我们初始位于一个平衡状态，突然移除了对一个变量的约束。我们做完这件事的时候，可达到的微观状态的数为<math>\Omega</math>，但是系统还没有达到平衡，所以系统处于某些可达到的状态的实际概率还不等于先验概率 <math>1/\Omega</math>。我们已经知道，最终的平衡状态相对于之前的平衡状态，熵会增加或者保持不变。然而，玻耳兹曼的H定理证明系统在不处于平衡态的期间，那个量作为时间的函数单调增加。<br />
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===Derivation of the entropy change for reversible processes===<br />
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===Derivation of the entropy change for reversible processes===<br />
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可逆过程熵变的推导<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:<br />
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第二定律的第二部分指出，经历可逆过程的系统的熵变是由以下因素给出的:<br />
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: <math>dS =\frac{\delta Q}{T}</math><br />
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<math>dS =\frac{\delta Q}{T}</math><br />
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数学 dS frac delta q }{ t } / math<br />
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where the temperature is defined as:<br />
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where the temperature is defined as:<br />
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其中温度定义为:<br />
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:<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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<math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math><br />
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[数学][[欧米茄][左(右)][右][数学][数学]<br />
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See [[Microcanonical ensemble|here]] for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the [[adiabatic theorem]] of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.<br />
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请参阅此处对此定义的正当理由。假设系统有一些可以改变的外部参数 x。一般来说，系统的能量本征态将依赖于 x。根据量子力学的绝热定理，在系统哈密顿量无限缓慢变化的极限下，系统将保持在相同的定态，因此系统的能量会随着其所在定态的能量变化而变化。<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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The generalized force, X, corresponding to the external variable x is defined such that <math>X dx</math> is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:<br />
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广义力，x，对应于外部变量 x 的定义是，如果 x 增加一个量 dx，那么 math x dx / math 就是系统所做的功。例如，如果 x 是体积，那么 x 是压强。一个已知的系统的广义力在定态数学 e { r } / math 中给出:<br />
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: <math>X = -\frac{dE_{r}}{dx}</math><br />
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<math>X = -\frac{dE_{r}}{dx}</math><br />
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数学 x- frac { dE { r }{ dx } / math<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:<br />
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由于系统可以在一个数学△ e / math 区间内的任意定态内，我们将系统的广义力定义为上述表达式的期望值:<br />
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: <math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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<math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math><br />
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数学 x- 左 langle frac { dE { dx }右 rangle，/ math<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:<br />
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为了计算平均值，我们通过计算在数学 y / math 和数学 y + delta y / math 之间的范围内，其中有多少数学 y / math 能量本征态有一个值来划分 ω 左(e 右) / math 能量本征态。把这个数字命名为 math Omega { y } left (e right) / math，我们有:<br />
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: <math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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<math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math><br />
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Math Omega left (e right) sum { y } Omega { y } left (e right) ，/ math<br />
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The average defining the generalized force can now be written:<br />
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The average defining the generalized force can now be written:<br />
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定义广义力的平均值现在可以写成:<br />
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: <math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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<math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math><br />
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数学 x frac {1} Omega 左(e 右) sum { y Omega { y }左(e 右) ，/ math<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math>\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are<br />
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我们可以把它和熵对 x 在恒定能量 e 下的导数联系起来，如下所示。假设我们把 x 改成 x + dx。然后 math Omega left (e right) / math 将会改变，因为能量本征态依赖于 x，导致能量本征态进入或超出数学 e / math 和数学 e + delta e / math 之间的范围。让我们再次关注数学 y / math 和数学 y + delta y / math 之间的能量本征态。由于这些能量本征态的能量增加了 y dx，所有这些能量本征态在 e-y dx 到 e 之间的区间内从 e 以下移动到 e 以上。 有<br />
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: <math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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<math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math><br />
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数学 n { y }左(e 右) frac { y }左(e 右)}{ delta e } y dx ，/ math<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference<br />
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这样的能量本征态。如果数学 y dx leq delta e / math，所有这些能量本征态将移动到数学 e / math 和数学 e + delta e / math 之间的范围内，有助于数学 ω / math 的增加。从数学 e + △ e / math 下移到数学 e + △ e / math 上的能量本征态的个数是由数学 n { y }左(e + △ e 右) / math 给出的。区别在于<br />
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: <math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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<math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math><br />
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数学 n { y }左(e 右)-n { y }左(e + delta e 右) ，/ math<br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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is thus the net contribution to the increase in <math>\Omega</math>. Note that if Y dx is larger than <math>\delta E</math> there will be the energy eigenstates that move from below E to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.<br />
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因此是数学增长的净贡献欧米茄 / 数学。请注意，如果 y dx 大于 math delta e / math，那么将会有能量本征态从 e 下面移动到 e + delta e / math 上面。它们在数学 n { y } left (e right) / math 和数学 n { y } left (e + delta e right) / math 中都有计算，因此上述表达式在这种情况下也是有效的。<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:<br />
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将上面的表达式表示为对 e 的导数，对 y 的求和得到表达式:<br />
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: <math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math><br />
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数学左(部分欧米茄左(部分欧米茄右){ e }-和 y 左(部分欧米茄右){ x 左(部分欧米茄左(部分欧米茄右)}{ x 右) ，/ 数学<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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The logarithmic derivative of <math>\Omega</math> with respect to x is thus given by:<br />
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关于 x 的欧米茄 / 数学对数导数由以下方式给出:<br />
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: <math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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<math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math><br />
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Math 左( frac { partial ln 左( Omega 右)}{部分 x 右){ e } beta x + 左( frac { partial x 右){ x } ，/ math<br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
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The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:<br />
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第一个术语是密集型的，即。它不能根据系统大小进行缩放。相比之下，最后一项的规模与逆系统的规模一样，因此将在热力学极限中消失。因此，我们发现:<br />
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: <math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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<math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math><br />
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数学左(部分 s }右){ e } frac { x }{ t } ，/ math<br />
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Combining this with<br />
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Combining this with<br />
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结合这一点<br />
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: <math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
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<math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math><br />
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数学左(部分 s }右){ x } frac {1}{ t } ，/ math<br />
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给予:<br />
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: <math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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<math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math><br />
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数学 dS 左(部分 s }部分 e }右){ x } dE + 左(部分 s }右){ e } dx frac { t } + frac { t }{ delta q }{ t } ，/ math<br />
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===Derivation for systems described by the canonical ensemble===<br />
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===Derivation for systems described by the canonical ensemble===<br />
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正则系系统的推导<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the [[canonical ensemble]]:<br />
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If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:<br />
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如果一个系统与某个温度为T的热浴热接触，那么在平衡状态下，关于能量本征值的概率分布值由正则系综给出:<br />
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: <math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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<math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math><br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the [[Partition function (statistical mechanics)|partition function]]. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:<br />
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这里Z是一个使所有概率之和归一化的因子，这个函数被称为配分函数。现在我们考虑对温度和能级所依赖的外部参数的无限小的可逆改变。它遵循熵的一般公式:<br />
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: <math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
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<math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math><br />
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that<br />
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that<br />
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以及<br />
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: <math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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<math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math><br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:<br />
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把正则系综的数学形式 <math>P_{j}</math> 插入，可以得到:<br />
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: <math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
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<math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math><br />
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==Living organisms==<br />
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==Living organisms==<br />
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生命体<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as [[What is Life?|Erwin Schrödinger]], [[Léon Brillouin]]<ref>Léon Brillouin [https://books.google.com/books?hl=en&lr=&id=tPXVbiw_1P0C ''Science and Information Theory''] (Academic Press, 1962) (Dover, 2004)</ref> and [[Life and Energy|Isaac Asimov]]. It is also the topic of current{{When|date=July 2019}} research.{{Citation needed|date=July 2019}}<br />
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There are two principal ways of formulating thermodynamics, (a) through passages from one state of thermodynamic equilibrium to another, and (b) through cyclic processes, by which the system is left unchanged, while the total entropy of the surroundings is increased. These two ways help to understand the processes of life. This topic is mostly beyond the scope of this present article, but has been considered by several authors, such as Erwin Schrödinger, Léon Brillouin and Isaac Asimov. It is also the topic of current research.<br />
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表述热力学主要有两种方式：(a)描述从一种热力学平衡状态到另一种状态的路径，(b)通过使系统保持不变，同时增加周围环境的总熵的循环过程。这两种方法有助于理解生命过程。这个话题大大超出了本文的范围，但是已经有一些研究者考虑过这个问题，比如埃尔温·薛定谔、莱昂·布里渊和艾萨克.阿西莫夫。这也是当前研究的主题。<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of [[metabolism]], give out breakdown products and heat. Plants [[Photosynthesis|take in radiative energy]] from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
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To a fair approximation, living organisms may be considered as examples of (b). Approximately, an animal's physical state cycles by the day, leaving the animal nearly unchanged. Animals take in food, water, and oxygen, and, as a result of metabolism, give out breakdown products and heat. Plants take in radiative energy from the sun, which may be regarded as heat, and carbon dioxide and water. They give out oxygen. In this way they grow. Eventually they die, and their remains rot away, turning mostly back into carbon dioxide and water. This can be regarded as a cyclic process. Overall, the sunlight is from a high temperature source, the sun, and its energy is passed to a lower temperature sink, i.e. radiated into space. This is an increase of entropy of the surroundings of the plant. Thus animals and plants obey the second law of thermodynamics, considered in terms of cyclic processes. Simple concepts of efficiency of heat engines are hardly applicable to this problem because they assume closed systems.<br />
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从近似的角度来看，生命体可以被认为是(b)的一个例子。近似地，一只动物的身体状态每天循环，使得它几乎没有什么变化。动物吸收食物、水和氧气，经过新陈代谢，输出分解的产物和热量。植物吸收来自太阳的辐射能量，这可以认为是热量，以及二氧化碳和水，然后它们释放氧气。它们就是这样生长的，最终会死亡，尸体腐烂，大部分重新变成二氧化碳和水。这可以看作是一个循环过程。总的来说，阳光来自一个高温的源——太阳，它的能量被传递到一个较低的温度汇，例如向太空辐射。这个过程使得植物周围环境的熵增加。因此从循环过程的角度来看，动物和植物服从热力学第二定律。简单的热机效率概念很难适用于这个问题，因为它们假定系统是封闭的。<br />
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From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to [[Non-equilibrium thermodynamics|unsettled current debate or research]]. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
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From the thermodynamic viewpoint that considers (a), passages from one equilibrium state to another, only a roughly approximate picture appears, because living organisms are never in states of thermodynamic equilibrium. Living organisms must often be considered as open systems, because they take in nutrients and give out waste products. Thermodynamics of open systems is currently often considered in terms of passages from one state of thermodynamic equilibrium to another, or in terms of flows in the approximation of local thermodynamic equilibrium. The problem for living organisms may be further simplified by the approximation of assuming a steady state with unchanging flows. General principles of entropy production for such approximations are subject to unsettled current debate or research. Nevertheless, ideas derived from this viewpoint on the second law of thermodynamics are enlightening about living creatures.<br />
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从(a)的热力学观点考虑 ，即考虑从一个平衡状态到另一个平衡状态的路径，只会产生一个粗略近似的图像，因为生命体从来不会处于热力学平衡状态。生物体经常必须被认为是开放的系统，因为它们吸收营养物质并排出废弃物。开放系统热力学目前通常从一个热力学平衡状态到另一个状态的路径的角度来考虑，或者是考虑局部热力学平衡近似下的流。生命体的问题可以通过假定一个流不变的稳态来进一步简化。对于这样的近似，熵产生的一般原理在目前有争论和研究的不确定性。尽管如此，从热力学第二定律的这个角度出发衍生出来的想法对生物是有启发意义的。<br />
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==Gravitational systems==<br />
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==Gravitational systems==<br />
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重力系统<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive [[heat capacity]], meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it.<ref name="Grandy 151">Grandy, W.T. (Jr) (2008), p. 151.</ref> Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
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In systems that do not require for their descriptions the general theory of relativity, bodies always have positive heat capacity, meaning that the temperature rises with energy. Therefore, when energy flows from a high-temperature object to a low-temperature object, the source temperature is decreased while the sink temperature is increased; hence temperature differences tend to diminish over time. This is not always the case for systems in which the gravitational force is important and the general theory of relativity is required. Such systems can spontaneously change towards uneven spread of mass and energy. This applies to the universe in large scale, and consequently it may be difficult or impossible to apply the second law to it. Beyond this, the thermodynamics of systems described by the general theory of relativity is beyond the scope of the present article.<br />
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在不需要描述广义相对论的系统中，物体总是具有正的热容，这意味着温度随着能量的增加而升高。因此，当能量从高温物体流向低温物体时，源温度降低，而汇温度升高; 因此温差随时间而减小。这种情况并不总是发生在那些需要引力和广义相对论的系统中。这样的系统可以自发地朝着质量和能量不均匀分布的方向变化。这在大尺度上适用于宇宙，因此可能很难或不可能将第二定律应用于宇宙。除此之外，广义相对论所描述的系统热力学超出了本文的范围。<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
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==Non-equilibrium states==<br />
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{{main article|Non-equilibrium thermodynamics}}<br />
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The theory of classical or [[thermodynamic equilibrium|equilibrium thermodynamics]] is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<ref name="Grandy 151"/><ref>Callen, H.B. (1960/1985), p. 15.</ref><br />
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The theory of classical or equilibrium thermodynamics is idealized. A main postulate or assumption, often not even explicitly stated, is the existence of systems in their own internal states of thermodynamic equilibrium. In general, a region of space containing a physical system at a given time, that may be found in nature, is not in thermodynamic equilibrium, read in the most stringent terms. In looser terms, nothing in the entire universe is or has ever been truly in exact thermodynamic equilibrium.<br />
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经典热力学或平衡热力学的理想化。一个主要的假设或假设，往往甚至没有明确说明，是系统的存在，在他们自己的内部状态的热力学平衡。一般来说，在给定时间内包含一个物理系统的空间区域---- 可能在自然界中找到---- 不是在热力学平衡中，用最严格的术语来解释。从更宽松的角度来说，整个宇宙中没有任何东西是或者曾经是完全精确的热力学平衡。<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of [[thermodynamic equilibrium]]. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable [[thermal fluctuations|fluctuations]]. There is an exception, the case of [[critical point (thermodynamics)|critical states]], which exhibit to the naked eye the phenomenon of [[critical opalescence]]. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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For purposes of physical analysis, it is often enough convenient to make an assumption of thermodynamic equilibrium. Such an assumption may rely on trial and error for its justification. If the assumption is justified, it can often be very valuable and useful because it makes available the theory of thermodynamics. Elements of the equilibrium assumption are that a system is observed to be unchanging over an indefinitely long time, and that there are so many particles in a system, that its particulate nature can be entirely ignored. Under such an equilibrium assumption, in general, there are no macroscopically detectable fluctuations. There is an exception, the case of critical states, which exhibit to the naked eye the phenomenon of critical opalescence. For laboratory studies of critical states, exceptionally long observation times are needed.<br />
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为了物理分析的目的，假设热力学平衡是非常方便的。这种假设可能依赖于试错来证明其正当性。如果这个假设是正确的，它通常是非常有价值和有用的，因为它提供了热力学理论。平衡假设的要素是，一个系统被观察到在无限长的时间内是不变的，而且一个系统中有如此多的粒子，以至于它的微粒性质可以被完全忽略。在这样的平衡假设下，一般来说，没有宏观上可以探测到的波动。有一个例外，即临界状态的例子，它们用肉眼就可以看到临界乳光现象。对于临界状态的实验室研究，需要非常长的观察时间。<br />
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In all cases, the assumption of [[thermodynamic equilibrium]], once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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In all cases, the assumption of thermodynamic equilibrium, once made, implies as a consequence that no putative candidate "fluctuation" alters the entropy of the system.<br />
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在所有情况下，热力学平衡的假设，一旦做出，就意味着没有假定的候选者“涨落”改变系统的熵。<br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<ref>Lieb, E.H., Yngvason, J. (2003), p. 190.</ref><br />
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It can easily happen that a physical system exhibits internal macroscopic changes that are fast enough to invalidate the assumption of the constancy of the entropy. Or that a physical system has so few particles that the particulate nature is manifest in observable fluctuations. Then the assumption of thermodynamic equilibrium is to be abandoned. There is no unqualified general definition of entropy for non-equilibrium states.<br />
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一个物理系统很容易发生内部的宏观变化，其速度之快足以使熵的恒定性假设失效。或者一个物理系统的粒子太少，以至于微粒的本质在可观测的波动中是显而易见的。那么热力学平衡的假设就会被放弃。对于非平衡态，熵没有绝对的一般定义。<br />
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There are intermediate cases, in which the assumption of local [[thermodynamic equilibrium]] is a very good approximation,<ref>Gyarmati, I. (1967/1970), pp. 4-14.</ref><ref>Glansdorff, P., Prigogine, I. (1971).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (1985).</ref><ref>[[Gottfried Wilhelm Leibniz Prize|Müller, I.]] (2003).</ref> but strictly speaking it is still an approximation, not theoretically ideal.<br />
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There are intermediate cases, in which the assumption of local thermodynamic equilibrium is a very good approximation, but strictly speaking it is still an approximation, not theoretically ideal.<br />
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还有一些中间情况，其中局部热力学平衡的假设是一个非常好的近似值，但严格地说它仍然是一个近似值，不是理论上的理想值。<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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For non-equilibrium situations in general, it may be useful to consider statistical mechanical definitions of other quantities that may be conveniently called 'entropy', but they should not be confused or conflated with thermodynamic entropy properly defined for the second law. These other quantities indeed belong to statistical mechanics, not to thermodynamics, the primary realm of the second law.<br />
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对于一般的非平衡状态，考虑其他量的统计力学定义可能是有用的，这些量可以方便地称为‘熵’ ，但是它们不应该与第二定律正确定义的熵混淆或混淆。这些其他的量确实属于统计力学，而不是热力学---- 第二定律的主要领域。<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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The physics of macroscopically observable fluctuations is beyond the scope of this article.<br />
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宏观观测波动的物理学超出了本文的范围。<br />
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==Arrow of time==<br />
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==Arrow of time==<br />
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时间之箭<br />
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{{Further information|Entropy (arrow of time)}}Thermodynamic Asymmetry in Time<br />
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Thermodynamic Asymmetry in Time<br />
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时间中的热力学不对称性<br />
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{{See also|Arrow of time}}<br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely [[CPT symmetry]], since the second law applies statistically, it is hypothesized, on time-asymmetric ''boundary conditions''.<ref>{{cite web|first=Craig|last=Callender|url=https://plato.stanford.edu/archives/fall2011/entries/time-thermo/|title=Thermodynamic Asymmetry in Time|encyclopedia=Stanford Encyclopedia of Philosophy|date=29 July 2011}}</ref><br />
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The second law of thermodynamics is a physical law that is not symmetric to reversal of the time direction. This does not conflict with notions that have been observed of the fundamental laws of physics, namely CPT symmetry, since the second law applies statistically, it is hypothesized, on time-asymmetric boundary conditions.<br />
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热力学第二定律是一个物理定律，它不对称于时间方向的反转。这与已经观察到的物理学基本定律，即 CPT 对称性的概念并不冲突，因为第二定律在统计上适用，它是假设的，是关于时间非对称的边界条件。<br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect ([[Arrow of time#The causal arrow of time|the causal arrow of time]]).<ref>{{cite book | first = J.J.| last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1|display-authors=etal}} chapter 6</ref><br />
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The second law has been proposed to supply a partial explanation of the difference between moving forward and backwards in time, such as why the cause precedes the effect (the causal arrow of time).<br />
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第二定律被提出来部分解释时间向前和向后移动的区别，例如为什么原因先于结果(因果关系的时间箭头)。<br />
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==Irreversibility==<br />
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==Irreversibility==<br />
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不可逆性<br />
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Irreversibility in [[thermodynamic process]]es is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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Irreversibility in thermodynamic processes is a consequence of the asymmetric character of thermodynamic operations, and not of any internally irreversible microscopic properties of the bodies. Thermodynamic operations are macroscopic external interventions imposed on the participating bodies, not derived from their internal properties. There are reputed "paradoxes" that arise from failure to recognize this.<br />
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热力学过程的不可逆性是热力学作用不对称性的结果，而不是物体内部不可逆的微观性质的结果。热力学操作是施加在参与机构上的宏观外部干预，而不是从它们的内部性质中得到的。由于没有认识到这一点，产生了被称为“悖论”的说法。<br />
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===Loschmidt's paradox===<br />
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===Loschmidt's paradox===<br />
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洛斯密特悖论<br />
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{{main article|Loschmidt's paradox}}<br />
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[[Loschmidt's paradox]], also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.<br />
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洛斯密特悖论，也被称为可逆性悖论，其目的在于不可能从描述宏观系统微观演化的时间对称动力学推导出不可逆性。<br />
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In the opinion of [[Erwin Schrödinger|Schrödinger]], "It is now quite obvious in what manner you have to reformulate the law of entropy{{snd}}or for that matter, all other irreversible statements{{snd}}so that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other."<ref>[[Erwin Schrödinger|Schrödinger, E.]] (1950), p.&nbsp;192.</ref> The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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In the opinion of Schrödinger, "It is now quite obvious in what manner you have to reformulate the law of entropyor for that matter, all other irreversible statementsso that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other." The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.<br />
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根据薛定谔的观点，“现在很明显，你必须用什么方式来重新定义该物质的熵定律，所有其他不可逆的状态，这样它们才能从可逆模型中导出。你不能只谈论一个孤立的系统，而应该至少谈论两个，你可以暂时认为它们与世界其它地方是孤立的，但并不总是相互孤立的。”这两个系统被隔离墙隔离开来，直到按照法律的设想被热力学操作拆除。热力学操作是外部强加的，不受支配系统组成部分的可逆微观动力学定律的制约。它是不可逆转性的原因。本文中的法律陈述符合薛定谔的建议。因果关系在逻辑上优先于第二定律，而不是从第二定律派生出来的。<br />
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===Poincaré recurrence theorem===<br />
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===Poincaré recurrence theorem===<br />
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庞加莱始态复现定理<br />
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{{Main article|Poincaré recurrence theorem}}<br />
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The [[Poincaré recurrence theorem]] considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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The Poincaré recurrence theorem considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.<br />
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庞加莱始态复现定理认为这是一个孤立的物理系统的理论微观描述。这可以被认为是一个热力学系统的模型后，一个热力学操作已经移除了一个内墙。在足够长的时间之后，系统将返回到一个非常接近初始状态的显微定义状态。庞加莱回归时间是指回归前经过的时间长度。它极其漫长，可能比宇宙的寿命还要长，并且敏感地依赖于被热力学操作拆除的墙体的几何形状。递归定理可能被认为是明显与热力学第二定律相矛盾的。然而，更明显的是，它只不过是一个微观模型，热力学平衡在一个孤立的系统中，通过移除两个系统之间的一道墙而形成。对于一个典型的热力学系统，重现时间是如此之大(比宇宙的生命周期长许多倍) ，以至于在所有的实际目的中，人们都不能观察到重现。然而，人们可能希望想象一下，可以等待庞加莱复发，然后重新插入被热力学操作移除的壁。然后很明显，不可逆性的出现是由于庞加莱循环的完全不可预测性，只要初始状态是热力学平衡，就像宏观热力学的情况一样。即使可以等待，也没有实际可能性选择合适的时间重新插入墙壁。庞加莱始态复现定理为 Loschmidt 悖论提供了一个解决方案。如果一个孤立的热力学系统可以被监测到在越来越多的平均 poincar 递归时间的倍数上，系统的热力学行为将在时间反转下变得不变。<br />
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[[File:James-clerk-maxwell3.jpg|thumb|upright|James Clerk Maxwell]]<br />
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James Clerk Maxwell<br />
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詹姆斯·克拉克·麦克斯韦<br />
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===Maxwell's demon===<br />
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===Maxwell's demon===<br />
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麦克斯韦的魔鬼<br />
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{{main article|Maxwell's demon}}<br />
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[[James Clerk Maxwell]] imagined one container divided into two parts, ''A'' and ''B''. Both parts are filled with the same [[gas]] at equal temperatures and placed next to each other, separated by a wall. Observing the [[molecule]]s on both sides, an imaginary [[demon]] guards a microscopic trapdoor in the wall. When a faster-than-average molecule from ''A'' flies towards the trapdoor, the demon opens it, and the molecule will fly from ''A'' to ''B''. The average [[speed]] of the molecules in ''B'' will have increased while in ''A'' they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in ''A'' and increases in ''B'', contrary to the second law of thermodynamics.<br />
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James Clerk Maxwell imagined one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other, separated by a wall. Observing the molecules on both sides, an imaginary demon guards a microscopic trapdoor in the wall. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics.<br />
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詹姆斯·克拉克·麦克斯韦想象一个容器分成两部分，a 和 b。两个部分在相同的温度下充入相同的气体，并被一堵墙隔开，挨着放置。通过观察两边的分子，一个想象中的恶魔守卫着墙上的一个微型暗门。当一个来自 a 的比平均速度快的分子飞向活板门时，魔鬼打开了它，分子就会从 a 飞到 b。B 中分子的平均速度会增加，而 a 中分子的平均速度会减慢。由于平均分子速度与温度相对应，因此 a 中的温度降低，b 中的温度升高，这与热力学第二定律相反。<br />
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One response to this question was suggested in 1929 by [[Leó Szilárd]] and later by [[Léon Brillouin]]. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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One response to this question was suggested in 1929 by Leó Szilárd and later by Léon Brillouin. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy.<br />
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1929年，Leó Szilárd 和后来的 l on brilouin 对这个问题提出了一个答案。Szil rd 指出，现实生活中的麦克斯韦恶魔需要一些测量分子速度的方法，而获取信息需要消耗大量能量。<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between ''A'' and ''B''. It is therefore performing [[thermodynamic operation]]s on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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Maxwell's 'demon' repeatedly alters the permeability of the wall between A and B. It is therefore performing thermodynamic operations on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.<br />
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麦克斯韦的“魔鬼”不断地改变 a 和 b 之间墙体的渗透性。因此，它是在微观尺度上执行热力学操作，而不仅仅是观察普通的自发或自然的宏观热力学过程。<br />
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==Quotations==<br />
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语录<br />
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{{Wikiquote}}<br />
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{{quote|The law that entropy always increases holds, I think, the supreme position among the [[Laws of science|laws of Nature]]. If someone points out to you that your pet theory of the [[universe]] is in disagreement with [[Maxwell's equations]] – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.|Sir [[Arthur Stanley Eddington]], ''The Nature of the Physical World'' (1927)}}<br />
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{{quote|There have been nearly as many formulations of the second law as there have been discussions of it.|Philosopher / Physicist [[Percy Williams Bridgman|P.W. Bridgman]], (1941)}}<br />
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{{quote|Clausius is the author of the sibyllic utterance, "The energy of the universe is constant; the entropy of the universe tends to a maximum." The objectives of continuum thermomechanics stop far short of explaining the "universe", but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size.|[[Clifford Truesdell|Truesdell, C.]], Muncaster, R. G. (1980). ''Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics'', Academic Press, New York, {{ISBN|0-12-701350-4}}, p. 17.}}<br />
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==See also==<br />
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==See also==<br />
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参见<br />
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*[[Clausius–Duhem inequality]]<br />
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*[[Fluctuation theorem]]<br />
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*[[Heat death of the universe]]<br />
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*[[History of thermodynamics]]<br />
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*[[Jarzynski equality]]<br />
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*[[Laws of thermodynamics]]<br />
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*[[Maximum entropy thermodynamics]]<br />
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*[[Quantum thermodynamics]]<br />
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*[[Reflections on the Motive Power of Fire]]<br />
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*[[Relativistic heat conduction]]<br />
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*[[Thermal diode]]<br />
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*[[Thermodynamic equilibrium]]<br />
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==References==<br />
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==References==<br />
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参考资料<br />
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===Sources===<br />
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===Sources===<br />
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资料来源<br />
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{{refbegin|50em}}<br />
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*Adkins, C.J. (1968/1983). ''Equilibrium Thermodynamics'', (1st edition 1968), third edition 1983, Cambridge University Press, Cambridge UK, {{ISBN|0-521-25445-0}}.<br />
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*[[Peter Atkins|Atkins, P.W.]], de Paula, J. (2006). ''Atkins' Physical Chemistry'', eighth edition, W.H. Freeman, New York, {{ISBN|978-0-7167-8759-4}}.<br />
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*Attard, P. (2012). ''Non-equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications'', Oxford University Press, Oxford UK, {{ISBN|978-0-19-966276-0}}.<br />
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*Baierlein, R. (1999). ''Thermal Physics'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-59082-5}}.<br />
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*Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics, New York, {{ISBN|0-88318-797-3}}.<br />
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*{{Cite book|title=Concepts in thermal physics|last=Blundell|first=Stephen J.|last2=Blundell|authorlink1=Stephen Blundell|first2=Katherine M.|authorlink2=Katherine Blundell|publisher=[[Oxford University Press]]|year=2010|isbn=9780199562107|edition=2nd|location=Oxford|pages=|oclc=607907330|doi=10.1093/acprof:oso/9780199562091.001.0001 |url=http://cds.cern.ch/record/1235139}}<br />
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*[[Ludwig Boltzmann|Boltzmann, L.]] (1896/1964). ''Lectures on Gas Theory'', translated by S.G. Brush, University of California Press, Berkeley.<br />
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*Borgnakke, C., Sonntag., R.E. (2009). ''Fundamentals of Thermodynamics'', seventh edition, Wiley, {{ISBN|978-0-470-04192-5}}.<br />
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*Buchdahl, H.A. (1966). ''The Concepts of Classical Thermodynamics'', Cambridge University Press, Cambridge UK.<br />
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*[[Percy Williams Bridgman|Bridgman, P.W.]] (1943). ''The Nature of Thermodynamics'', Harvard University Press, Cambridge MA.<br />
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*[[Herbert Callen|Callen, H.B.]] (1960/1985). ''Thermodynamics and an Introduction to Thermostatistics'', (1st edition 1960) 2nd edition 1985, Wiley, New York, {{ISBN|0-471-86256-8}}.<br />
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*{{cite journal|author=C. Carathéodory|author1-link=Constantin Carathéodory<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|title=Untersuchungen über die Grundlagen der Thermodynamik|year=1909|journal=Mathematische Annalen|volume=67|issue=3<br />
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|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
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|pages=355–386|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0067&DMDID=DMDLOG_0033&L=1<br />
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2010年3月15日 | 第355-386页 | http://gdz.sub.uni-goettingen.de/index.php?id=11&ppn=ppn235181684_0067&dmdid=dmdlog_0033&l=1<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). ''The Second Law of Thermodynamics'', Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können. (p.363)|doi=10.1007/bf01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). The Second Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA.<br />
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|quote=Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können.(p. 363) | doi 10.1007 / bf01450409}.译文可以在这里找到[ http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf ]。同时，在 Kestin，j。(1976).热力学第二定律，Dowden，Hutchinson & Ross，Stroudsburg PA. 。<br />
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*[[Nicolas Léonard Sadi Carnot|Carnot, S.]] (1824/1986). [http://www.worldcat.org/title/reflections-on-the-motive-power-of-fire-a-critical-edition-with-the-surviving-scientific-manuscripts-translated-and-edited-by-fox-robert/oclc/812944517&referer=brief_results ''Reflections on the motive power of fire''], Manchester University Press, Manchester UK, {{ISBN|0-7190-1741-6}}. [https://archive.org/stream/reflectionsonmot00carnrich#page/n7/mode/2up Also here.]<br />
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*[[Sydney Chapman (mathematician)|Chapman, S.]], [[Thomas George Cowling|Cowling, T.G.]] (1939/1970). ''The Mathematical Theory of Non-uniform gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases'', third edition 1970, Cambridge University Press, London.<br />
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*{{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Ueber Die Bewegende Kraft Der Wärme Und Die Gesetze, Welche Sich Daraus Für Die Wärmelehre Selbst Ableiten Lassen|journal=Annalen der Physik|year=1850|volume=79|issue=4|pages=368–397, 500–524|url=http://gallica.bnf.fr/ark:/12148/bpt6k15164w/f518.image|accessdate=26 June 2012|doi=10.1002/andp.18501550403|bibcode = 1850AnP...155..500C |hdl=2027/uc1.$b242250|hdl-access=free}} Translated into English: {{cite journal|last=Clausius|first=R.|title=On the Moving Force of Heat, and the Laws regarding the Nature of Heat itself which are deducible therefrom|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1851| volume=2|series=4th|issue=VIII|pages=1–21; 102–119 |url=https://archive.org/stream/londonedinburghd02lond#page/1/mode/1up|accessdate=26 June 2012<br />
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|doi=10.1080/14786445108646819}}<br />
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|doi=10.1080/14786445108646819}}<br />
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10.1080 / 14786445108646819}<br />
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* {{cite journal|ref=harv|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie|year=1854|journal=Annalen der Physik|volume=xciii|issue=12|pages=481–506|url=http://zfbb.thulb.uni-jena.de/servlets/MCRFileNodeServlet/jportal_derivate_00140956/18541691202_ftp.pdf|accessdate=24 March 2014|doi=10.1002/andp.18541691202<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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|bibcode = 1854AnP...169..481C }} Translated into English: {{cite journal|last=Clausius|first=R.|title=On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1856| volume=2|series=4th|page=86 |url=https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=24 March 2014<br />
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| bibcode 1854AnP... 169. 1856年7月 | 第2卷 | 第4页 | 第86页 | https://www.biodiversitylibrary.org/item/20044#page/100/mode/1up|accessdate=242014年3月<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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}} Reprinted in: {{cite book|last=Clausius|first=R.|author1-link=Rudolf Clausius|title=The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies|year=1867|publisher=John van Voorst|location=London|url=https://archive.org/details/mechanicaltheor04claugoog|quote=editions:PwR_Sbkwa8IC.|accessdate=19 June 2012<br />
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热力学理论及其在蒸汽机和物体物理性质中的应用 | 1867年 | 出版商 John van Voorst | 位置: 伦敦 | https://archive.org/details/mechanicaltheor04claugoog|quote=editions:pwr_sbkwa8ic。 2012年6月19日<br />
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*{{cite journal|ref=harv|last=Thomson|first=W.|author-link=William Thomson, 1st Baron Kelvin|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Transactions of the Royal Society of Edinburgh|year=1851|volume=XX|issue=part II|pages=261–268; 289–298|url=https://www.biodiversitylibrary.org/item/126047#page/295/mode/1up}} Also published in {{cite journal|last=Thomson|first=W.|title=On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam|journal=Philos. Mag. |date=December 1852 |volume=IV |series=4 |issue=22 |page=13|url=https://archive.org/stream/londonedinburghp04maga#page/12/mode/2up |accessdate=25 June 2012}}<br />
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*[[William Thomson, 1st Baron Kelvin|Thomson, W.]] (1852). ''On the universal tendency in nature to the dissipation of mechanical energy'' Philosophical Magazine, Ser. 4, p.&nbsp;304.<br />
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*[[László Tisza|Tisza, L.]] (1966). ''Generalized Thermodynamics'', M.I.T Press, Cambridge MA.<br />
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*[[Clifford Truesdell|Truesdell, C.]] (1980). ''The Tragicomical History of Thermodynamics 1822–1854'', Springer, New York, {{ISBN|0-387-90403-4}}.<br />
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*Uffink, J. (2001). Bluff your way in the second law of thermodynamics, ''Stud. Hist. Phil. Mod. Phys.'', '''32'''(3): 305–394.<br />
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*Uffink, J. (2003). Irreversibility and the Second Law of Thermodynamics, Chapter 7 of ''Entropy'', Greven, A., Keller, G., Warnecke (editors) (2003), Princeton University Press, Princeton NJ, {{ISBN|0-691-11338-6}}.<br />
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*[[George Uhlenbeck|Uhlenbeck, G.E.]], Ford, G.W. (1963). ''Lectures in Statistical Mechanics'', American Mathematical Society, Providence RI.<br />
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*[[Mark Zemansky|Zemansky, M.W.]] (1968). ''Heat and Thermodynamics. An Intermediate Textbook'', fifth edition, McGraw-Hill Book Company, New York.<br />
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==Further reading==<br />
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==Further reading==<br />
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进一步阅读<br />
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*Goldstein, Martin, and Inge F., 1993. ''The Refrigerator and the Universe''. Harvard Univ. Press. Chpts. 4–9 contain an introduction to the Second Law, one a bit less technical than this entry. {{ISBN|978-0-674-75324-2}}<br />
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*Leff, Harvey S., and Rex, Andrew F. (eds.) 2003. ''Maxwell's Demon 2 : Entropy, classical and quantum information, computing''. Bristol UK; Philadelphia PA: [[Institute of Physics]]. {{ISBN|978-0-585-49237-7}}<br />
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*{{Cite book | first = J.J. | last = Halliwell | title = Physical Origins of Time Asymmetry| publisher = Cambridge | year = 1994| isbn = 978-0-521-56837-1}}(technical).<br />
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*{{cite book |title=Reflections on the Motive Power of Heat and on Machines Fitted to Develop That Power |last=Carnot |first=Sadi |editor=[[Robert Henry Thurston|Thurston, Robert Henry]] |year=1890 |publisher=J. Wiley & Sons |location=New York |isbn= |pages= }} ([https://books.google.com/books?id=tgdJAAAAIAAJ full text of 1897 ed.]) ([http://www.history.rochester.edu/steam/carnot/1943/ html])<br />
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*Stephen Jay Kline (1999). ''The Low-Down on Entropy and Interpretive Thermodynamics'', La Cañada, CA: DCW Industries. {{ISBN|1-928729-01-0}}.<br />
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*{{cite book | last1 = Kostic | first1 = M | year = 2011 | title = Revisiting The Second Law of Energy Degradation and Entropy Generation: From Sadi Carnot's Ingenious Reasoning to Holistic Generalization | journal = AIP Conf. Proc. | volume = 1411 | issue = 1| pages = 327–350 | doi = 10.1063/1.3665247 | isbn = 978-0-7354-0985-9 | bibcode = 2011AIPC.1411..327K | series = AIP Conference Proceedings | citeseerx = 10.1.1.405.1945 }} also at [https://web.archive.org/web/20130420222450/http://www.kostic.niu.edu/2ndLaw/Revisiting%20The%20Second%20Law%20of%20Energy%20Degradation%20and%20Entropy%20Generation%20-%20From%20Carnot%20to%20Holistic%20Generalization-4.pdf].<br />
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==External links==<br />
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==External links==<br />
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外部链接<br />
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*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics]" – by Lawrence Sklar.<br />
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*[http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node30.html ''Second law of thermodynamics''] in the MIT Course [http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/notes.html ''Unified Thermodynamics and Propulsion''] from Prof. Z. S. Spakovszky<br />
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*[[E.T. Jaynes]], 1988, "[http://bayes.wustl.edu/etj/articles/ccarnot.pdf The evolution of Carnot's principle]," in G. J. Erickson and C. R. Smith (eds.)''Maximum-Entropy and Bayesian Methods in Science and Engineering, Vol'' 1: p.&nbsp;267.<br />
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*[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf Caratheodory, C., "Examination of the foundations of thermodynamics," trans. by D. H. Delphenich]<br />
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* [https://www.bbc.co.uk/programmes/p004y2bm The Second Law of Thermodynamics], BBC Radio 4 discussion with John Gribbin, Peter Atkins & Monica Grady (''In Our Time'', Dec. 16, 2004)<br />
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* [http://mdpi.org/entropy/papers/e6010001.pdf [[Entropy (journal)]], 2004]<br />
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* [https://www.journals.uchicago.edu/doi/abs/10.1086/663835 The Journal of the International Society for the History of Philosophy of Science, 2012]<br />
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[[Category:Concepts in physics]]<br />
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Category:Concepts in physics<br />
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分类: 物理概念<br />
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[[Category:Laws of thermodynamics|2]]<br />
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[[Category:Non-equilibrium thermodynamics]]<br />
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Category:Non-equilibrium thermodynamics<br />
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类别: 非平衡态热力学<br />
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[[Category:Philosophy of thermal and statistical physics]]<br />
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Category:Philosophy of thermal and statistical physics<br />
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类别: 热力学和统计物理学哲学<br />
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<small>This page was moved from [[wikipedia:en:Second law of thermodynamics]]. Its edit history can be viewed at [[热力学第二定律/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%9D%9E%E7%BA%BF%E6%80%A7%E7%B3%BB%E7%BB%9F&diff=10133非线性系统2020-07-11T06:26:15Z<p>Dorr：</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{Redirect|Nonlinear dynamics|the journal|Nonlinear Dynamics (journal)}}<br />
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{{About|"nonlinearity" in mathematics, physics and other sciences|video and film editing|Non-linear editing system|other uses|Nonlinearity (disambiguation)}}<br />
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In [[mathematics]] and [[science]], a '''nonlinear system''' is a [[system]] in which the change of the output is not [[proportionality (mathematics)|proportional]] to the change of the input.<ref>{{Cite news|url=https://news.mit.edu/2010/explained-linear-0226|title=Explained: Linear and nonlinear systems|work=MIT News|access-date=2018-06-30}}</ref><ref>{{Cite web|url=https://www.birmingham.ac.uk/research/activity/mathematics/applied-maths/nonlinear-systems.aspx|title=Nonlinear systems, Applied Mathematics - University of Birmingham|website=www.birmingham.ac.uk|language=en-gb|access-date=2018-06-30}}</ref> Nonlinear problems are of interest to [[engineer]]s, [[biologist]]s,<ref>{{Citation|date=2007|pages=181–276|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-540-34153-6_7|isbn=9783540341529|title = The Nonlinear Universe|series = The Frontiers Collection|chapter = Nonlinear Biology}}</ref><ref>{{cite journal|last=Korenberg|first=Michael J.|last2=Hunter|first2=Ian W.|date=March 1996|title=The identification of nonlinear biological systems: Volterra kernel approaches|journal=Annals of Biomedical Engineering|language=en|volume=24|issue=2|pages=250–268|doi=10.1007/bf02667354|issn=0090-6964}}</ref><ref>{{cite journal|last=Mosconi|first=Francesco|last2=Julou|first2=Thomas|last3=Desprat|first3=Nicolas|last4=Sinha|first4=Deepak Kumar|last5=Allemand|first5=Jean-François|last6=Vincent Croquette|last7=Bensimon|first7=David|date=2008|title=Some nonlinear challenges in biology|url=http://stacks.iop.org/0951-7715/21/i=8/a=T03|journal=Nonlinearity|language=en|volume=21|issue=8|pages=T131|doi=10.1088/0951-7715/21/8/T03|issn=0951-7715|bibcode=2008Nonli..21..131M}}</ref> [[physicist]]s,<ref>{{cite journal|last1=Gintautas|first1=V.|title=Resonant forcing of nonlinear systems of differential equations|journal=Chaos|date=2008|volume=18|issue=3|pages=033118|doi=10.1063/1.2964200|pmid=19045456|arxiv=0803.2252|bibcode=2008Chaos..18c3118G}}</ref><ref>{{cite journal|last1=Stephenson|first1=C.|last2=et.|first2=al.|title=Topological properties of a self-assembled electrical network via ab initio calculation|journal=Sci. Rep.|volume=7|pages=41621|date=2017|doi=10.1038/srep41621|pmid=28155863|pmc=5290745|bibcode=2017NatSR...741621S}}</ref> [[mathematician]]s, and many other [[scientist]]s because most systems are inherently nonlinear in nature.<ref>{{cite book|last1=de Canete|first1=Javier, Cipriano Galindo, and Inmaculada Garcia-Moral|title=System Engineering and Automation: An Interactive Educational Approach|date=2011|publisher=Springer|location=Berlin|isbn=978-3642202292|page=46|url=https://books.google.com/?id=h8rCQYXGGY8C&pg=PA46&lpg=PA46&dq=most+systems+are+inherently+nonlinear+in+nature#v=onepage&q=most%20systems%20are%20inherently%20nonlinear%20in%20nature&f=false|accessdate=20 January 2018}}</ref> Nonlinear [[dynamical system]]s, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler [[linear system]]s.<br />
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In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.<br />
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在数学及科学中，'''非线性系统 Nonlinear System'''是一种输出的变化与输入的变化不成比例的系统。大多数系统在本质上是非线性的，因而非线性问题引起了工程师、生物学家、物理学家、数学家和许多其他科学家的兴趣。描述变量随时间变化的非线性动力系统与较之简单得多的线性系统相比，可能显得混沌、不可预测或违反直觉。<br />
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Typically, the behavior of a nonlinear system is described in mathematics by a '''nonlinear system of equations''', which is a set of simultaneous [[equation]]s in which the unknowns (or the unknown functions in the case of [[differential equation]]s) appear as variables of a [[polynomial]] of degree higher than one or in the argument of a [[function (mathematics)|function]] which is not a polynomial of degree one.<br />
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Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one.<br />
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通常，非线性系统的行为在数学上是由一组非线性联立方程来描述的，其中未知数（或微分方程中的未知函数）作为一个高于一次的多项式的变量出现，或者作为一个不是一次多项式的函数的论元出现。__[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）后半句不知如何翻译<br />
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In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a [[linear combination]] of the unknown [[variable (mathematics)|variables]] or [[function (mathematics)|functions]] that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is ''linear'' if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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换句话说，在非线性方程系统中，要求解的方程不能被写成未知变量或函数的线性组合。无论方程中是否有已知的线性函数，系统都可以被定义为非线性。特别是当一个微分方程的未知函数及其导数是线性的，即使其他变量是非线性的，也称该方程是线性的。<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations ([[linearization]]). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as [[soliton]]s, [[chaos theory|chaos]],<ref>[http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm Nonlinear Dynamics I: Chaos] {{webarchive|url=https://web.archive.org/web/20080212045134/http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm |date=2008-02-12}} at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]</ref> and [[mathematical singularity|singularities]] are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble [[randomness|random]] behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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由于非线性动力学方程难以求解，通常用线性化方程来近似非线性系统（'''线性化 Linearization'''）。这种方法在一定的精度和范围对输入值效果很好，但一些有趣的现象如'''孤子 Soliton'''、'''混沌 Chaos'''和'''奇异性 Singularity'''在线性化后被隐藏。因此，非线性系统的动态行为在某些方面可能看起来违反直觉、不可预测、甚至混沌。尽管这种混沌行为可能感觉很像随机行为，但它实际上并不是随机的。例如，天气的某些方面被认为是混沌的，其系统某部分的微小扰动就会产生复杂的影响。这种非线性是目前技术无法进行精确长期预测的原因之一。<br />
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Some authors use the term '''nonlinear science''' for the study of nonlinear systems. This term is disputed by others:<br />
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Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:<br />
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有些作者用非线性科学这一术语来研究非线性系统。这一术语引起了其他人的争议:<br />
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{{quote|Using a term like nonlinear science is like referring to the bulk of zoology as the study of [[negation|non]]-elephant animals.|[[Stanislaw Ulam]]<ref>{{cite journal|last1=Campbell|first1=David K.|title=Nonlinear physics: Fresh breather|journal=Nature|date=25 November 2004|volume=432|issue=7016|pages=455–456|doi=10.1038/432455a|pmid=15565139|url=https://zenodo.org/record/1134179|language=en|issn=0028-0836|bibcode=2004Natur.432..455C}}</ref>}}<br />
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“使用‘非线性科学’这样的术语，就如同把动物学里大部分对象称作‘非大象动物’研究一样可笑。”<br />
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——'''斯塔尼斯拉夫·乌拉姆 Stanislaw Ulam'''<br />
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==Definition 定义==<br />
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In [[mathematics]], a [[linear map]] (or ''linear function'') <math>f(x)</math> is one which satisfies both of the following properties:<br />
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In mathematics, a linear map (or linear function) <math>f(x)</math> is one which satisfies both of the following properties:<br />
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在数学中，线性映射（或线性函数）<math>f (x)</math>满足以下两个性质:<br />
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*Additivity or [[superposition Principle]]: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*Homogeneity: <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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*'''可加性 Additivity'''（'''叠加性 Superposition principle'''）: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*'''齐次性 Homogeneity''': <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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Additivity implies homogeneity for any [[rational number|rational]] ''α'', and, for [[continuous function]]s, for any [[real number|real]] ''α''. For a [[complex number|complex]] ''α'', homogeneity does not follow from additivity. For example, an [[antilinear map]] is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle<br />
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Additivity implies homogeneity for any rational α, and, for continuous functions, for any real α. For a complex α, homogeneity does not follow from additivity. For example, an antilinear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle:<br />
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当α是有理数或实数，且<math>f(x)</math>是连续函数时，由可加性可以推出齐次性。但当α是复数时，可加性不能导出齐次性。例如，反线性映射是可加的，但不是齐次的。可加性和齐次性条件经常组合，称为叠加原理：<br />
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<math>f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)</math><br />
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An equation written as<br />
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<math>f(x) = C</math><br />
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is called '''linear''' if <math>f(x)</math> is a linear map (as defined above) and '''nonlinear''' otherwise. The equation is called ''homogeneous'' if <math>C = 0</math>.<br />
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is called linear if <math>f(x)</math> is a linear map (as defined above) and nonlinear otherwise. The equation is called homogeneous if <math>C = 0</math>.<br />
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的方程，若 <math> f (x) </math> 是线性映射（如上定义） ，则称其为'''线性的 Linear'''，否则称为'''非线性的 Nonlinear'''。若<math>C = 0</math>，该方程称为是齐次的。<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any [[map (mathematics)|mapping]], including integration or differentiation with associated constraints (such as [[boundary values]]). If <math>f(x)</math> contains [[derivative|differentiation]] with respect to <math>x</math>, the result will be a [[differential equation]].<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any mapping, including integration or differentiation with associated constraints (such as boundary values). If <math>f(x)</math> contains differentiation with respect to <math>x</math>, the result will be a differential equation.<br />
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定义 <math>f(x) = C</math> 是非常具有一般性的，因为 <math>x</math> 可以是任意可感知的数学对象（数字、向量、函数等），函数 <math>f(x)</math> 实际上可以是任意映射，包括有相关约束（如给定边界值）的积分或微分。若 <math>f(x)</math> 包含对 <math>x</math> 的微分运算，则该方程为微分方程。<br />
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===相关概念辨析===<br />
'''线性Linear'''、'''非线性Nonlinear'''、'''亚线性 Sublinear'''、'''超线性 Superlinear'''：描述量与量之间的一种变化关系，例如<math>y=a+b*x^n</math>，其中<math>n>1</math>。当<math>n=1</math>时，表示为线性关系；<math>n\neq 1</math>时，表示为非线性关系；当<math>0<n<1</math>时，表示为亚线性关系；当<math>n>1</math>时，表示为超线性关系。<br />
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亚线性与超线性属于非线性变化关系范畴。当两个变量之间存在亚线性关系时，其典型特性是因变量的变化速率会随着自变量的增大而减小，即其一阶导数会随着自变量的增大而减小；两个变量之间存在超线性关系时，其典型特性是因变量的变化速率会随着自变量的增大而增大，即其一阶导数会随着自变量的增大而增大。<br />
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在线性系统中，整体等于部分和，描述线性系统的方程满足叠加原理，作用的总和正好等于每一部分作用相加的代数和，这意味着每一部分作用都是独立的、互不相关的；而在普遍存在的非线性系统中，作用的总和不等于每一部分作用相加的代数和，因为系统内部要素之间存在着复杂的非线性相互作用。<br />
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==Nonlinear algebraic equations 非线性代数方程==<br />
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{{Main|Algebraic equation|System of polynomial equations}}<br />
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Nonlinear [[algebraic equation]]s, which are also called ''[[polynomial equation]]s'', are defined by equating [[polynomial]]s (of degree greater than one) to zero. For example,<br />
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Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials (of degree greater than one) to zero. For example,<br />
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非线性'''代数方程 Algebraic Equation'''，又称'''多项式方程 Polynomial Equation'''，由某多项式（次数大于1）等于零定义。例如：<br />
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For a single polynomial equation, [[root-finding algorithm]]s can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of [[algebraic geometry]], a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see [[Hilbert's Nullstellensatz]]). Nevertheless, in the case of the systems with a finite number of complex solutions, these [[systems of polynomial equations]] are now well understood and efficient methods exist for solving them.<ref>{{cite journal |last1= Lazard |first1= D. |title= Thirty years of Polynomial System Solving, and now? |doi= 10.1016/j.jsc.2008.03.004 |journal= Journal of Symbolic Computation |volume= 44 |issue= 3 |pages= 222–231 |year= 2009 |pmid= |pmc=}}</ref><br />
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For a single polynomial equation, root-finding algorithms can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of algebraic geometry, a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see Hilbert's Nullstellensatz). Nevertheless, in the case of the systems with a finite number of complex solutions, these systems of polynomial equations are now well understood and efficient methods exist for solving them.<br />
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对于一个单一的多项式方程，'''求根算法 Root-finding Algorithms'''可用于其求解（即找到满足该方程的变量的值集）。而代数方程组则相对复杂，其研究是现代数学的较难分支——'''代数几何 Algebraic Geometry'''领域的动力之一。甚至很难判断一个给定的代数系统是否有复数解（见'''希尔伯特零点定律 Hilbert's Nullstellensatz'''）。不过，对于具有有限个复数解的系统的多项式方程组，我们现在已经有了充分的理解，并且找到了有效的求解方法。<br />
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==Nonlinear recurrence relations 非线性递推关系==<br />
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A nonlinear [[recurrence relation]] defines successive terms of a [[sequence]] as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the [[logistic map]] and the relations that define the various [[Hofstadter sequence]]s. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related [[nonlinear system identification]] and analysis procedures.<ref name="SAB1">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</ref> These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related nonlinear system identification and analysis procedures. These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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非线性递归关系中，序列的连续项被定义为其前项的非线性函数。非线性递归关系的例子有 [[logistic 映射]]和定义各种'''霍夫斯塔特序列 Hofstadter Sequences''' 的关系。非线性离散模型代表了一类广泛的非线性递归关系，包括 NARMAX（外部输入非线性自回归移动平均）模型和相关的非线性系统辨识和分析程序。这些方法可用于研究时域、频域和时空域的广泛复杂非线性行为。<br />
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==Nonlinear differential equations 非线性微分方程==<br />
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A [[simultaneous equations|system]] of [[differential equation]]s is said to be nonlinear if it is not a [[linear system]]. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the [[Navier–Stokes equations]] in fluid dynamics and the [[Lotka–Volterra equation]]s in biology.<br />
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A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology.<br />
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若一个微分方程组不是线性系统，则称其为非线性的。涉及非线性微分方程的问题非常多样，对不同问题的解决或分析方法也不相同。非线性微分方程的例子有流体力学中的 '''纳维-斯托克斯方程 Navier-Stokes Equations'''和生物学中的'''洛特卡－沃尔泰拉方程 Lotka-Volterra Equations'''。<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of [[linearly independent]] solutions can be used to construct general solutions through the [[superposition principle]]. A good example of this is one-dimensional heat transport with [[Dirichlet boundary conditions]], the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle. A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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非线性问题最大的困难之一是通常不可能将已知的解组合成新的解。例如，在线性问题中，可以根据叠加原理以一族线性独立的解构造通解。一个很好的例子是带有'''狄利克雷边界条件 Dirichlet Boundary Conditions'''的一维热传导问题，其解（随时间变化）可以写成不同频率的正弦波的线性组合，这使得解非常灵活。而对非线性方程，通常可以找到几个非常特殊的解，但是此时叠加原理不适用，故无法构造新的解。<br />
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===Ordinary differential equations 常微分方程===<br />
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First order [[ordinary differential equation]]s are often exactly solvable by [[separation of variables]], especially for autonomous equations. For example, the nonlinear equation<br />
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First order ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. For example, the nonlinear equation<br />
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一阶常微分方程，尤其是自治方程，通常可以用'''分离变量法 Separation of Variables'''来精确求解。例如，非线性方程<br />
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<math>\frac{d u}{d x} = -u^2</math><br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also ''u'' = 0 as a particular solution, corresponding to the limit of the general solution when ''C'' tends to infinity). The equation is nonlinear because it may be written as<br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also u = 0 as a particular solution, corresponding to the limit of the general solution when C tends to infinity). The equation is nonlinear because it may be written as<br />
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将 <math>u=\frac{1}{x+C}</math> 作为一般解（也有特解 ''u'' = 0，对应于 ''C'' 趋于无穷时的一般解的极限)。该方程是非线性的，因为它可以改写成<br />
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<math>\frac{du}{d x} + u^2=0</math><br />
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and the left-hand side of the equation is not a linear function of ''u'' and its derivatives. Note that if the ''u''<sup>2</sup> term were replaced with ''u'', the problem would be linear (the [[exponential decay]] problem).<br />
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and the left-hand side of the equation is not a linear function of u and its derivatives. Note that if the u<sup>2</sup> term were replaced with u, the problem would be linear (the exponential decay problem).<br />
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方程的左边不是 ''u'' 及其导数的线性函数。注意，若将 ''u''<sup>2</sup> 项替换为''u''，该问题将变为线性的（'''指数衰减 Exponential Decay'''问题）。<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield [[closed-form expression|closed-form]] solutions, though implicit solutions and solutions involving [[nonelementary integral]]s are encountered.<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered.<br />
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二阶和高阶常微分方程（更一般地说，非线性方程组）很少能产生封闭解，而隐式解和非初等函数积分形式的解较为常见。<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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非线性常微分方程定性分析的常用方法包括:<br />
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*Examination of any [[conserved quantities]], especially in [[Hamiltonian system]]s<br />
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*检查是否有任意'''守恒量 Conserved Quantities'''（特别是在'''哈密顿系统 Hamiltonian System'''中）<br />
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*Examination of dissipative quantities (see [[Lyapunov function]]) analogous to conserved quantities<br />
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*检查是否有类似守恒量的耗散量（见'''李亚普诺夫函数 Lyapunov Function'''）<br />
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*Linearization via [[Taylor expansion]]<br />
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*基于'''泰勒展开 Taylor Expansion'''的线性化<br />
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*Change of variables into something easier to study<br />
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*将变量进行代换以便更好的进行研究<br />
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*[[Bifurcation theory]]<br />
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*'''分岔理论 Bifurcation Theory'''<br />
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*[[Perturbation theory|Perturbation]] methods (can be applied to algebraic equations too)<br />
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*'''摄动理论 Perturbation Theory'''（也可应用于代数方程）<br />
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===Partial differential equations 偏微分方程===<br />
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{{main|Nonlinear partial differential equation}}<br />
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{{See also|List of nonlinear partial differential equations}}<br />
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参见：非线性偏微分列表<br />
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The most common basic approach to studying nonlinear [[partial differential equation]]s is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more [[ordinary differential equation]]s, as seen in [[separation of variables]], which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
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The most common basic approach to studying nonlinear partial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
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研究非线性偏微分方程最常用的基本方法是变量代换（或转换问题），使变换后的问题更简单（甚至可能变为线性的）。有时可以将此类方程转化成一或多个常微分方程（如同分离变量法所示），此时不论得到的常微分方程是否可解，对研究问题总是有用的。<br />
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Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use [[scale analysis (mathematics)|scale analysis]] to simplify a general, natural equation in a certain specific [[boundary value problem]]. For example, the (very) nonlinear [[Navier-Stokes equations]] can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
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Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem. For example, the (very) nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
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另一个流体力学和热力学中常见的策略（虽然不是数学上的）是利用'''尺度分析 Scale Analysis'''来在某一特定边界条件下简化一般自然方程。例如，在描述圆管内一维层流的暂态时，非线性的纳维－斯托克斯方程可以简化为一个线性的偏微分方程; 尺度分析提供了层流和一维流动的条件，也产生了简化的方程。<br />
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Other methods include examining the [[method of characteristics|characteristics]] and using the methods outlined above for ordinary differential equations.<br />
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Other methods include examining the characteristics and using the methods outlined above for ordinary differential equations.<br />
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其他方法包括检查特征线法及前面所述研究常微分方程的方法。<br />
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===Pendula 摆===<br />
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{{Main|Pendulum (mathematics)}}<br />
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[[File:PendulumLayout.svg.png|thumb|图一：Illustration of a pendulum 摆图解|right|200px]]<br />
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[[File:PendulumLinearizations.png|thumb|图二：Linearizations of a pendulum 摆的线性化|right|200px]]<br />
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A classic, extensively studied nonlinear problem is the dynamics of a [[pendulum (mathematics)|pendulum]] under the influence of [[gravity]]. Using [[Lagrangian mechanics]], it may be shown<ref>[http://www.damtp.cam.ac.uk/user/tong/dynamics.html David Tong: Lectures on Classical Dynamics]</ref> that the motion of a pendulum can be described by the [[dimensionless]] nonlinear equation<br />
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A classic, extensively studied nonlinear problem is the dynamics of a pendulum under the influence of gravity. Using Lagrangian mechanics, it may be shown that the motion of a pendulum can be described by the dimensionless nonlinear equation<br />
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一个经典的被广泛研究的非线性问题是重力影响下的摆的动力学。利用'''拉格朗日力学 Lagrangian Mechanics'''，可以证明摆的运动可以用无量纲的非线性方程<br />
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<math>\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0</math><br />
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where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an [[integrating factor]], which would eventually yield<br />
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where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an integrating factor, which would eventually yield<br />
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描述，其中重力指向“下方”，<math>\theta</math> 是摆与其静止位置形成的角度，如右图所示。“解”这个方程的方法之一是用 <math>d\theta/dt</math> 作为'''积分因子 Integrating Factor'''，最终得<br />
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<math>\int{\frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)}}} = t + C_1</math><br />
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which is an implicit solution involving an [[elliptic integral]]. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the [[nonelementary integral]] (nonelementary unless <math>C_0 = 2</math>).<br />
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which is an implicit solution involving an elliptic integral. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral (nonelementary unless <math>C_0 = 2</math>).<br />
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这是一个含'''椭圆积分 Elliptic Integral'''的隐式解。这个“解”通常没什么用，因为这个解的大部分性质都隐藏在非初等函数积分中（除非<math>C_0 = 2</math>，否则是非初等的）。<br />
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Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through [[Taylor expansion]]s. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
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Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through Taylor expansions. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
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另一种解决这个问题的方法是利用泰勒展开将任意非线性项（此时为正弦函数项)在某些感兴趣的点线性化。例如，在<math>\theta = 0</math> 的点附近线性化（称为小角度近似）为<br />
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<math>\frac{d^2 \theta}{d t^2} + \theta = 0</math><br />
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since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a [[simple harmonic oscillator]] corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
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since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
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因为 <math>\theta \approx 0</math> 时，有 <math>\sin(\theta) \approx \theta</math>。这是一个'''简谐振子 Simple Harmonic Oscillator''' ，对应于摆在其路径底部附近的摆动。另一种线性化方法是在 <math>\theta = \pi</math>附近线性化，对应于运动到最高点的摆:<br />
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<math>\frac{d^2 \theta}{d t^2} + \pi - \theta = 0</math><br />
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since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves [[hyperbolic sinusoid]]s, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
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since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
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因为 <math>\theta \approx \pi</math> 时，有 <math>\sin(\theta) \approx \pi - \theta</math>。这个问题的解含双曲正弦曲线；注意到不同于小角度近似，它是不稳定的，这意味着 <math>|\theta|</math> 通常会无限增长（但解也有可能是有界的）。这就解释了摆在最高点达到平衡的困难，此时实际上是一种不稳定的状态。<br />
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One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
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One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
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一个更有趣的线性化可能是在 <math>\theta = \pi/2</math>附近，此时 <math>\sin(\theta) \approx 1</math>:<br />
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<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math><br />
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This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) [[phase portrait]]s and approximate periods.<br />
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This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) phase portraits and approximate periods.<br />
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这相当于一个自由落体问题。把这样线性化的结果合在一起看，就能得到有关摆的运动的非常有用的图像，如右图所示。利用其他方法寻找（精确的）'''相图 Phase Portrait'''和估计周期。<br />
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==Types of nonlinear dynamic behaviors 非线性动力学行为的类型==<br />
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*[[Amplitude death]] – any oscillations present in the system cease due to some kind of interaction with other system or feedback by the same system<br />
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*'''振幅死亡 Amplitude Death'''——系统内的某振荡因系统的自回馈或与其他系统的某种相互作用而停止的现象<br />
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*[[Chaos theory|Chaos]] – values of a system cannot be predicted indefinitely far into the future, and fluctuations are [[aperiodic]]<br />
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*'''混沌 Chaos'''——系统内的值无法无限期地预测到遥远的未来；波动是非周期性的<br />
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*[[Multistability]] – the presence of two or more stable states<br />
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*'''多稳态 Multistability'''——两或多个稳态的存在<br />
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*[[Soliton]]s – self-reinforcing solitary waves<br />
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*'''孤波 Soliton'''s——自增强的孤立波<br />
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*[[Limit cycle|Limit cycles]] – asymptotic periodic orbits to which destabilized fixed points are attracted. <br />
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*'''极限环 Limit Cycles'''——吸引不稳定不动点的渐近周期轨道<br />
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*[[Self-oscillation|Self-oscillations]] - feedback oscillations taking place in open dissipative physical systems.<br />
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*'''自激振荡 Self-oscillations'''——开放耗散物理系统中的反馈振荡<br />
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==Examples of nonlinear equations 非线性方程的例子==<br />
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*[[Algebraic Riccati equation]] 代数黎卡提方程<br />
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*[[Ball and beam]] system 球杆系统<br />
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*[[Bellman equation]] for optimal policy 最佳策略的贝尔曼方程<br />
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*[[Boltzmann equation]] 玻尔兹曼方程<br />
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*[[Colebrook equation]] 科尔布鲁克方程<br />
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*[[General relativity]] 广义相对论<br />
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*[[Ginzburg–Landau theory]] 金兹堡-朗道方程<br />
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*[[Ishimori equation]] 石森方程<br />
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*[[Kadomtsev–Petviashvili equation]] 卡东穆塞夫-彼得韦亚斯维利方程<br />
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*[[Korteweg–de Vries equation]] kdV方程<br />
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*[[Landau–Lifshitz–Gilbert equation]] 朗道-利夫希兹-吉尔伯特方程<br />
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*[[Liénard equation]] 林纳德方程<br />
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*[[Navier–Stokes equations]] of [[fluid dynamics]] 流体力学的纳维－斯托克斯方程<br />
<br />
<br />
<br />
*[[Nonlinear optics]] 非线性光学<br />
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<br />
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*[[Nonlinear Schrödinger equation]] 非线性薛定谔方程<br />
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*[[Power-flow study]] 功率流研究<br />
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*[[Richards equation]] for unsaturated water flow 未饱和层水流的理查氏方程<br />
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<br />
*[[Self-balancing unicycle]] 自平衡独轮车<br />
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*[[Sine-Gordon equation]] 正弦-戈尔登方程<br />
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<br />
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*[[Van der Pol oscillator]] 范德波尔振荡器<br />
<br />
<br />
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*[[Vlasov equation]] 弗拉索夫方程<br />
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{{Div col end}}<br />
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==See also 参见==<br />
<br />
<br />
*[[Aleksandr Mikhailovich Lyapunov]] 亚历山大·李亚普诺夫<br />
<br />
<br />
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*[[Dynamical system]] 动态系统<br />
<br />
<br />
<br />
*[[Feedback]] 反馈<br />
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<br />
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*[[Initial condition]] 初始条件<br />
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<br />
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*[[Interaction]] 相互作用<br />
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*[[Linear system]] 线性系统<br />
<br />
<br />
<br />
*[[Mode coupling]] 模式耦合<br />
<br />
<br />
<br />
*[[Vector soliton]] 矢量孤子<br />
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*[[Volterra series]] 沃尔泰拉级数<br />
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<br />
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==References 参考资料==<br />
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{{Reflist|35em}}<br />
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==Further reading 进一步阅读==<br />
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{{Refbegin|35em}}<br />
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*{{cite book<br />
<br />
<br />
<br />
| author= [[Diederich Hinrichsen]] and Anthony J. Pritchard<br />
<br />
| author= Diederich Hinrichsen and Anthony J. Pritchard<br />
<br />
作者 Diederich Hinrichsen 和 Anthony j. Pritchard<br />
<br />
| year= 2005<br />
<br />
| year= 2005<br />
<br />
2005年<br />
<br />
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
<br />
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
<br />
数学系统理论 i-建模，状态空间分析，稳定性和鲁棒性<br />
<br />
| publisher= Springer Verlag<br />
<br />
| publisher= Springer Verlag<br />
<br />
出版商 Springer Verlag<br />
<br />
| isbn=9783540441250<br />
<br />
| isbn=9783540441250<br />
<br />
9783540441250<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Jordan<br />
<br />
| last= Jordan<br />
<br />
最后一个乔丹<br />
<br />
| first= D. W.<br />
<br />
| first= D. W.<br />
<br />
首先是 D.w。<br />
<br />
| last2= Smith<br />
<br />
| last2= Smith<br />
<br />
最后2个史密斯<br />
<br />
| first2= P.<br />
<br />
| first2= P.<br />
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| first2 p.<br />
<br />
| year= 2007<br />
<br />
| year= 2007<br />
<br />
2007年<br />
<br />
| edition= fourth<br />
<br />
| edition= fourth<br />
<br />
第四版<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
非线性常微分方程<br />
<br />
| publisher= Oxford University Press<br />
<br />
| publisher= Oxford University Press<br />
<br />
牛津大学出版社<br />
<br />
| isbn= 978-0-19-920824-1<br />
<br />
| isbn= 978-0-19-920824-1<br />
<br />
[国际标准图书编号978-0-19-920824-1]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Khalil<br />
<br />
| last= Khalil<br />
<br />
最后的哈利勒<br />
<br />
| first= Hassan K.<br />
<br />
| first= Hassan K.<br />
<br />
首先是哈桑 · k。<br />
<br />
| year= 2001<br />
<br />
| year= 2001<br />
<br />
2001年<br />
<br />
| title= Nonlinear Systems<br />
<br />
| title= Nonlinear Systems<br />
<br />
非线性系统<br />
<br />
| publisher= Prentice Hall<br />
<br />
| publisher= Prentice Hall<br />
<br />
出版商 Prentice Hall<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
[国际标准图书编号978-0-13-067389-3]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Kreyszig<br />
<br />
| last= Kreyszig<br />
<br />
| last= Kreyszig<br />
<br />
| first= Erwin<br />
<br />
| first= Erwin<br />
<br />
首先是欧文<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
作者 / 链接 Erwin Kreyszig<br />
<br />
| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
高等工程数学<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
Https://archive.org/details/advancedengineer0008krey<br />
<br />
| url-access= registration<br />
<br />
| url-access= registration<br />
<br />
访问注册<br />
<br />
| publisher= Wiley<br />
<br />
| publisher= Wiley<br />
<br />
出版商 Wiley<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
[国际标准图书编号978-0-471-15496-9]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Sontag<br />
<br />
| last= Sontag<br />
<br />
最后一个桑塔格<br />
<br />
| first= Eduardo<br />
<br />
| first= Eduardo<br />
<br />
首先是爱德华多<br />
<br />
| authorlink= Eduardo D. Sontag<br />
<br />
| authorlink= Eduardo D. Sontag<br />
<br />
作者: Eduardo d. Sontag<br />
<br />
| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
数学控制理论: 确定性有限维系统。第二版<br />
<br />
| publisher= Springer<br />
<br />
| publisher= Springer<br />
<br />
出版商斯普林格<br />
<br />
| isbn= 978-0-387-98489-6<br />
<br />
| isbn= 978-0-387-98489-6<br />
<br />
[国际标准图书编号978-0-387-98489-6]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
{{Refend}}<br />
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<br />
<br />
<br />
<br />
<br />
==External links 外部链接==<br />
<br />
<br />
<br />
*[http://www.dodccrp.org/ Command and Control Research Program (CCRP)]<br />
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<br />
*[http://necsi.edu/guide/concepts/linearnonlinear.html New England Complex Systems Institute: Concepts in Complex Systems]<br />
<br />
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<br />
*[http://ocw.mit.edu/courses/mathematics/18-353j-nonlinear-dynamics-i-chaos-fall-2012/ Nonlinear Dynamics I: Chaos] at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]<br />
<br />
<br />
<br />
*[http://www.hedengren.net/research/models.htm Nonlinear Model Library]{{snd}} (in [[MATLAB]]) a Database of Physical Systems<br />
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<br />
*[http://cnls.lanl.gov/ The Center for Nonlinear Studies at Los Alamos National Laboratory]<br />
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<br />
{{Systems science}}<br />
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{{Differential equations topics}}<br />
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<br />
<br />
{{Authority control}}<br />
<br />
<br />
==编者推荐==<br />
[https://campus.swarma.org/ 集智学园]课程：[https://campus.swarma.org/course/697 非线性动力学与混沌]<br />
<br />
非线性动力学和混沌理论是系统发展的，从一阶微分方程及其分岔开始，然后是相平面分析，极限环和它们的分岔，最终得到Lorenz方程，混沌，迭代映射，周期倍增，重整化，分形和奇怪吸引。<br />
<br />
<br />
此系列课程由[[斯蒂文·斯特罗加茨 Steven H. Strogatz]]主持，内容包括机械振动，激光，生物节律，超导电路，昆虫爆发，化学振荡器，遗传控制系统，混沌水轮，甚至是使用混乱发送秘密信息的技术。在每种情况下，科学背景都在初级阶段进行解释，并与数学理论紧密结合。<br />
<br />
[[Category:Nonlinear systems| ]]<br />
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<br />
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[[Category:Dynamical systems]]<br />
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Category:Dynamical systems<br />
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类别: 动力系统<br />
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[[Category:Concepts in physics]]<br />
<br />
Category:Concepts in physics<br />
<br />
分类: 物理概念<br />
<br />
<noinclude><br />
<br />
<small>This page was moved from [[wikipedia:en:Nonlinear system]]. Its edit history can be viewed at [[非线性系统/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%9D%9E%E7%BA%BF%E6%80%A7%E7%B3%BB%E7%BB%9F&diff=10118非线性系统2020-07-11T06:13:32Z<p>Dorr：</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{Redirect|Nonlinear dynamics|the journal|Nonlinear Dynamics (journal)}}<br />
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{{About|"nonlinearity" in mathematics, physics and other sciences|video and film editing|Non-linear editing system|other uses|Nonlinearity (disambiguation)}}<br />
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In [[mathematics]] and [[science]], a '''nonlinear system''' is a [[system]] in which the change of the output is not [[proportionality (mathematics)|proportional]] to the change of the input.<ref>{{Cite news|url=https://news.mit.edu/2010/explained-linear-0226|title=Explained: Linear and nonlinear systems|work=MIT News|access-date=2018-06-30}}</ref><ref>{{Cite web|url=https://www.birmingham.ac.uk/research/activity/mathematics/applied-maths/nonlinear-systems.aspx|title=Nonlinear systems, Applied Mathematics - University of Birmingham|website=www.birmingham.ac.uk|language=en-gb|access-date=2018-06-30}}</ref> Nonlinear problems are of interest to [[engineer]]s, [[biologist]]s,<ref>{{Citation|date=2007|pages=181–276|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-540-34153-6_7|isbn=9783540341529|title = The Nonlinear Universe|series = The Frontiers Collection|chapter = Nonlinear Biology}}</ref><ref>{{cite journal|last=Korenberg|first=Michael J.|last2=Hunter|first2=Ian W.|date=March 1996|title=The identification of nonlinear biological systems: Volterra kernel approaches|journal=Annals of Biomedical Engineering|language=en|volume=24|issue=2|pages=250–268|doi=10.1007/bf02667354|issn=0090-6964}}</ref><ref>{{cite journal|last=Mosconi|first=Francesco|last2=Julou|first2=Thomas|last3=Desprat|first3=Nicolas|last4=Sinha|first4=Deepak Kumar|last5=Allemand|first5=Jean-François|last6=Vincent Croquette|last7=Bensimon|first7=David|date=2008|title=Some nonlinear challenges in biology|url=http://stacks.iop.org/0951-7715/21/i=8/a=T03|journal=Nonlinearity|language=en|volume=21|issue=8|pages=T131|doi=10.1088/0951-7715/21/8/T03|issn=0951-7715|bibcode=2008Nonli..21..131M}}</ref> [[physicist]]s,<ref>{{cite journal|last1=Gintautas|first1=V.|title=Resonant forcing of nonlinear systems of differential equations|journal=Chaos|date=2008|volume=18|issue=3|pages=033118|doi=10.1063/1.2964200|pmid=19045456|arxiv=0803.2252|bibcode=2008Chaos..18c3118G}}</ref><ref>{{cite journal|last1=Stephenson|first1=C.|last2=et.|first2=al.|title=Topological properties of a self-assembled electrical network via ab initio calculation|journal=Sci. Rep.|volume=7|pages=41621|date=2017|doi=10.1038/srep41621|pmid=28155863|pmc=5290745|bibcode=2017NatSR...741621S}}</ref> [[mathematician]]s, and many other [[scientist]]s because most systems are inherently nonlinear in nature.<ref>{{cite book|last1=de Canete|first1=Javier, Cipriano Galindo, and Inmaculada Garcia-Moral|title=System Engineering and Automation: An Interactive Educational Approach|date=2011|publisher=Springer|location=Berlin|isbn=978-3642202292|page=46|url=https://books.google.com/?id=h8rCQYXGGY8C&pg=PA46&lpg=PA46&dq=most+systems+are+inherently+nonlinear+in+nature#v=onepage&q=most%20systems%20are%20inherently%20nonlinear%20in%20nature&f=false|accessdate=20 January 2018}}</ref> Nonlinear [[dynamical system]]s, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler [[linear system]]s.<br />
<br />
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.<br />
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在数学及科学中，'''非线性系统 Nonlinear System'''是一种输出的变化与输入的变化不成比例的系统。大多数系统在本质上是非线性的，因而非线性问题引起了工程师、生物学家、物理学家、数学家和许多其他科学家的兴趣。描述变量随时间变化的非线性动力系统与较之简单得多的线性系统相比，可能显得混沌、不可预测或违反直觉。<br />
<br />
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）大多数系统在本质上是非线性的，因而非线性问题引起了工程师、生物学家、物理学家、数学家和许多其他科学家的兴趣。 这里调整语序很棒，但是因果关系是不是 因为大多数系统在本质上是非线性的，所以非线性问题引起了工程师、生物学家、物理学家、数学家和许多其他科学家的兴趣。<br />
--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）这两句因果关系不一样吗<br />
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Typically, the behavior of a nonlinear system is described in mathematics by a '''nonlinear system of equations''', which is a set of simultaneous [[equation]]s in which the unknowns (or the unknown functions in the case of [[differential equation]]s) appear as variables of a [[polynomial]] of degree higher than one or in the argument of a [[function (mathematics)|function]] which is not a polynomial of degree one.<br />
<br />
Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one.<br />
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通常，非线性系统的行为在数学上是由一组非线性联立方程来描述的，其中未知数（或微分方程中的未知函数）作为一个高于一次的多项式的变量出现，或者作为一个不是一次多项式的函数的论元出现。__[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）后半句不知如何翻译<br />
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In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a [[linear combination]] of the unknown [[variable (mathematics)|variables]] or [[function (mathematics)|functions]] that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is ''linear'' if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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换句话说，在非线性方程系统中，要求解的方程不能被写成未知变量或函数的线性组合。无论方程中是否有已知的线性函数，系统都可以被定义为非线性。特别是当一个微分方程的未知函数及其导数是线性的，即使其他变量是非线性的，也称该方程是线性的。<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations ([[linearization]]). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as [[soliton]]s, [[chaos theory|chaos]],<ref>[http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm Nonlinear Dynamics I: Chaos] {{webarchive|url=https://web.archive.org/web/20080212045134/http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm |date=2008-02-12}} at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]</ref> and [[mathematical singularity|singularities]] are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble [[randomness|random]] behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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由于非线性动力学方程难以求解，通常用线性化方程来近似非线性系统（'''线性化 Linearization'''）。这种方法在一定的精度和范围对输入值效果很好，但一些有趣的现象如'''孤子 Soliton'''、'''混沌 Chaos'''和'''奇异性 Singularity'''在线性化后被隐藏。因此，非线性系统的动态行为在某些方面可能看起来违反直觉、不可预测、甚至混沌。尽管这种混沌行为可能感觉很像随机行为，但它实际上并不是随机的。例如，天气的某些方面被认为是混沌的，其系统某部分的微小扰动就会产生复杂的影响。这种非线性是目前技术无法进行精确长期预测的原因之一。<br />
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Some authors use the term '''nonlinear science''' for the study of nonlinear systems. This term is disputed by others:<br />
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Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:<br />
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有些作者用非线性科学这一术语来研究非线性系统。这一术语引起了其他人的争议:<br />
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{{quote|Using a term like nonlinear science is like referring to the bulk of zoology as the study of [[negation|non]]-elephant animals.|[[Stanislaw Ulam]]<ref>{{cite journal|last1=Campbell|first1=David K.|title=Nonlinear physics: Fresh breather|journal=Nature|date=25 November 2004|volume=432|issue=7016|pages=455–456|doi=10.1038/432455a|pmid=15565139|url=https://zenodo.org/record/1134179|language=en|issn=0028-0836|bibcode=2004Natur.432..455C}}</ref>}}<br />
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“使用‘非线性科学’这样的术语，就如同把动物学里大部分对象称作‘非大象动物’研究一样可笑。”<br />
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——'''斯塔尼斯拉夫·乌拉姆 Stanislaw Ulam'''<br />
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==Definition 定义==<br />
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In [[mathematics]], a [[linear map]] (or ''linear function'') <math>f(x)</math> is one which satisfies both of the following properties:<br />
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In mathematics, a linear map (or linear function) <math>f(x)</math> is one which satisfies both of the following properties:<br />
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在数学中，线性映射（或线性函数）<math>f (x)</math>满足以下两个性质:<br />
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*Additivity or [[superposition Principle]]: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*Homogeneity: <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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*'''可加性 Additivity'''（'''叠加性 Superposition principle'''）: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*'''齐次性 Homogeneity''': <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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Additivity implies homogeneity for any [[rational number|rational]] ''α'', and, for [[continuous function]]s, for any [[real number|real]] ''α''. For a [[complex number|complex]] ''α'', homogeneity does not follow from additivity. For example, an [[antilinear map]] is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle<br />
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Additivity implies homogeneity for any rational α, and, for continuous functions, for any real α. For a complex α, homogeneity does not follow from additivity. For example, an antilinear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle:<br />
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当α是有理数或实数，且<math>f(x)</math>是连续函数时，由可加性可以推出齐次性。但当α是复数时，可加性不能导出齐次性。例如，反线性映射是可加的，但不是齐次的。可加性和齐次性条件经常组合，称为叠加原理：<br />
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<math>f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)</math><br />
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An equation written as<br />
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对一个写成<br />
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<math>f(x) = C</math><br />
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is called '''linear''' if <math>f(x)</math> is a linear map (as defined above) and '''nonlinear''' otherwise. The equation is called ''homogeneous'' if <math>C = 0</math>.<br />
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is called linear if <math>f(x)</math> is a linear map (as defined above) and nonlinear otherwise. The equation is called homogeneous if <math>C = 0</math>.<br />
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的方程，若 <math> f (x) </math> 是线性映射（如上定义） ，则称其为'''线性的 Linear'''，否则称为'''非线性的 Nonlinear'''。若<math>C = 0</math>，该方程称为是齐次的。<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any [[map (mathematics)|mapping]], including integration or differentiation with associated constraints (such as [[boundary values]]). If <math>f(x)</math> contains [[derivative|differentiation]] with respect to <math>x</math>, the result will be a [[differential equation]].<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any mapping, including integration or differentiation with associated constraints (such as boundary values). If <math>f(x)</math> contains differentiation with respect to <math>x</math>, the result will be a differential equation.<br />
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定义 <math>f(x) = C</math> 是非常具有一般性的，因为 <math>x</math> 可以是任意可感知的数学对象（数字、向量、函数等），函数 <math>f(x)</math> 实际上可以是任意映射，包括有相关约束（如给定边界值）的积分或微分。若 <math>f(x)</math> 包含对 <math>x</math> 的微分运算，则该方程为微分方程。<br />
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===相关概念辨析===<br />
'''线性Linear'''、'''非线性Nonlinear'''、'''亚线性 Sublinear'''、'''超线性 Superlinear'''：描述量与量之间的一种变化关系，例如<math>y=a+b*x^n</math>，其中<math>n>1</math>。当<math>n=1</math>时，表示为线性关系；<math>n\neq 1</math>时，表示为非线性关系；当<math>0<n<1</math>时，表示为亚线性关系；当<math>n>1</math>时，表示为超线性关系。<br />
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亚线性与超线性属于非线性变化关系范畴。当两个变量之间存在亚线性关系时，其典型特性是因变量的变化速率会随着自变量的增大而减小，即其一阶导数会随着自变量的增大而减小；两个变量之间存在超线性关系时，其典型特性是因变量的变化速率会随着自变量的增大而增大，即其一阶导数会随着自变量的增大而增大。<br />
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在线性系统中，整体等于部分和，描述线性系统的方程满足叠加原理，作用的总和正好等于每一部分作用相加的代数和，这意味着每一部分作用都是独立的、互不相关的；而在普遍存在的非线性系统中，作用的总和不等于每一部分作用相加的代数和，因为系统内部要素之间存在着复杂的非线性相互作用。<br />
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==Nonlinear algebraic equations 非线性代数方程==<br />
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{{Main|Algebraic equation|System of polynomial equations}}<br />
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Nonlinear [[algebraic equation]]s, which are also called ''[[polynomial equation]]s'', are defined by equating [[polynomial]]s (of degree greater than one) to zero. For example,<br />
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Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials (of degree greater than one) to zero. For example,<br />
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非线性'''代数方程 Algebraic Equation'''，又称'''多项式方程 Polynomial Equation'''，由某多项式（次数大于1）等于零定义。例如：<br />
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<math>x^2 + x - 1 = 0\,.</math><br />
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For a single polynomial equation, [[root-finding algorithm]]s can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of [[algebraic geometry]], a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see [[Hilbert's Nullstellensatz]]). Nevertheless, in the case of the systems with a finite number of complex solutions, these [[systems of polynomial equations]] are now well understood and efficient methods exist for solving them.<ref>{{cite journal |last1= Lazard |first1= D. |title= Thirty years of Polynomial System Solving, and now? |doi= 10.1016/j.jsc.2008.03.004 |journal= Journal of Symbolic Computation |volume= 44 |issue= 3 |pages= 222–231 |year= 2009 |pmid= |pmc=}}</ref><br />
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For a single polynomial equation, root-finding algorithms can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of algebraic geometry, a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see Hilbert's Nullstellensatz). Nevertheless, in the case of the systems with a finite number of complex solutions, these systems of polynomial equations are now well understood and efficient methods exist for solving them.<br />
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对于一个单一的多项式方程，'''求根算法 Root-finding Algorithms'''可用于其求解（即找到满足该方程的变量的值集）。而代数方程组则相对复杂，其研究是现代数学的较难分支——'''代数几何 Algebraic Geometry'''领域的动力之一。甚至很难判断一个给定的代数系统是否有复数解（见'''希尔伯特零点定律 Hilbert's Nullstellensatz'''）。不过，对于具有有限个复数解的系统的多项式方程组，我们现在已经有了充分的理解，并且找到了有效的求解方法。<br />
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==Nonlinear recurrence relations 非线性递推关系==<br />
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A nonlinear [[recurrence relation]] defines successive terms of a [[sequence]] as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the [[logistic map]] and the relations that define the various [[Hofstadter sequence]]s. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related [[nonlinear system identification]] and analysis procedures.<ref name="SAB1">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</ref> These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related nonlinear system identification and analysis procedures. These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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非线性递归关系中，序列的连续项被定义为其前项的非线性函数。非线性递归关系的例子有 [[logistic 映射]]和定义各种'''霍夫斯塔特序列 Hofstadter Sequences''' 的关系。非线性离散模型代表了一类广泛的非线性递归关系，包括 NARMAX（外部输入非线性自回归移动平均）模型和相关的非线性系统辨识和分析程序。这些方法可用于研究时域、频域和时空域的广泛复杂非线性行为。<br />
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==Nonlinear differential equations 非线性微分方程==<br />
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A [[simultaneous equations|system]] of [[differential equation]]s is said to be nonlinear if it is not a [[linear system]]. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the [[Navier–Stokes equations]] in fluid dynamics and the [[Lotka–Volterra equation]]s in biology.<br />
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A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology.<br />
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若一个微分方程组不是线性系统，则称其为非线性的。涉及非线性微分方程的问题非常多样，对不同问题的解决或分析方法也不相同。非线性微分方程的例子有流体力学中的 '''纳维-斯托克斯方程 Navier-Stokes Equations'''和生物学中的'''洛特卡－沃尔泰拉方程 Lotka-Volterra Equations'''。<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of [[linearly independent]] solutions can be used to construct general solutions through the [[superposition principle]]. A good example of this is one-dimensional heat transport with [[Dirichlet boundary conditions]], the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle. A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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非线性问题最大的困难之一是通常不可能将已知的解组合成新的解。例如，在线性问题中，可以根据叠加原理以一族线性独立的解构造通解。一个很好的例子是带有'''狄利克雷边界条件 Dirichlet Boundary Conditions'''的一维热传导问题，其解（随时间变化）可以写成不同频率的正弦波的线性组合，这使得解非常灵活。而对非线性方程，通常可以找到几个非常特殊的解，但是此时叠加原理不适用，故无法构造新的解。<br />
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===Ordinary differential equations 常微分方程===<br />
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First order [[ordinary differential equation]]s are often exactly solvable by [[separation of variables]], especially for autonomous equations. For example, the nonlinear equation<br />
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First order ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. For example, the nonlinear equation<br />
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一阶常微分方程，尤其是自治方程，通常可以用'''分离变量法 Separation of Variables'''来精确求解。例如，非线性方程<br />
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<math>\frac{d u}{d x} = -u^2</math><br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also ''u'' = 0 as a particular solution, corresponding to the limit of the general solution when ''C'' tends to infinity). The equation is nonlinear because it may be written as<br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also u = 0 as a particular solution, corresponding to the limit of the general solution when C tends to infinity). The equation is nonlinear because it may be written as<br />
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将 <math>u=\frac{1}{x+C}</math> 作为一般解（也有特解 ''u'' = 0，对应于 ''C'' 趋于无穷时的一般解的极限)。该方程是非线性的，因为它可以改写成<br />
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<math>\frac{du}{d x} + u^2=0</math><br />
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and the left-hand side of the equation is not a linear function of ''u'' and its derivatives. Note that if the ''u''<sup>2</sup> term were replaced with ''u'', the problem would be linear (the [[exponential decay]] problem).<br />
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and the left-hand side of the equation is not a linear function of u and its derivatives. Note that if the u<sup>2</sup> term were replaced with u, the problem would be linear (the exponential decay problem).<br />
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方程的左边不是 ''u'' 及其导数的线性函数。注意，若将 ''u''<sup>2</sup> 项替换为''u''，该问题将变为线性的（'''指数衰减 Exponential Decay'''问题）。<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield [[closed-form expression|closed-form]] solutions, though implicit solutions and solutions involving [[nonelementary integral]]s are encountered.<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered.<br />
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二阶和高阶常微分方程（更一般地说，非线性方程组）很少能产生封闭解，而隐式解和非初等函数积分形式的解较为常见。<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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非线性常微分方程定性分析的常用方法包括:<br />
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*Examination of any [[conserved quantities]], especially in [[Hamiltonian system]]s<br />
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*检查是否有任意'''守恒量 Conserved Quantities'''（特别是在'''哈密顿系统 Hamiltonian System'''中）<br />
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*Examination of dissipative quantities (see [[Lyapunov function]]) analogous to conserved quantities<br />
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*检查是否有类似守恒量的耗散量（见'''李亚普诺夫函数 Lyapunov Function'''）<br />
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*Linearization via [[Taylor expansion]]<br />
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*基于'''泰勒展开 Taylor Expansion'''的线性化<br />
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*Change of variables into something easier to study<br />
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*将变量进行代换以便更好的进行研究<br />
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*[[Bifurcation theory]]<br />
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*'''分岔理论 Bifurcation Theory'''<br />
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*[[Perturbation theory|Perturbation]] methods (can be applied to algebraic equations too)<br />
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*'''摄动理论 Perturbation Theory'''（也可应用于代数方程）<br />
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===Partial differential equations 偏微分方程===<br />
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{{main|Nonlinear partial differential equation}}<br />
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{{See also|List of nonlinear partial differential equations}}<br />
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参见：非线性偏微分列表<br />
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The most common basic approach to studying nonlinear [[partial differential equation]]s is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more [[ordinary differential equation]]s, as seen in [[separation of variables]], which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
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The most common basic approach to studying nonlinear partial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
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研究非线性偏微分方程最常用的基本方法是变量代换（或转换问题），使变换后的问题更简单（甚至可能变为线性的）。有时可以将此类方程转化成一或多个常微分方程（如同分离变量法所示），此时不论得到的常微分方程是否可解，对研究问题总是有用的。<br />
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Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use [[scale analysis (mathematics)|scale analysis]] to simplify a general, natural equation in a certain specific [[boundary value problem]]. For example, the (very) nonlinear [[Navier-Stokes equations]] can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
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Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem. For example, the (very) nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
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另一个流体力学和热力学中常见的策略（虽然不是数学上的）是利用'''尺度分析 Scale Analysis'''来在某一特定边界条件下简化一般自然方程。例如，在描述圆管内一维层流的暂态时，非线性的纳维－斯托克斯方程可以简化为一个线性的偏微分方程; 尺度分析提供了层流和一维流动的条件，也产生了简化的方程。<br />
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Other methods include examining the [[method of characteristics|characteristics]] and using the methods outlined above for ordinary differential equations.<br />
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Other methods include examining the characteristics and using the methods outlined above for ordinary differential equations.<br />
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其他方法包括检查特征线法及前面所述研究常微分方程的方法。<br />
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===Pendula 摆===<br />
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{{Main|Pendulum (mathematics)}}<br />
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[[File:PendulumLayout.svg.png|thumb|图一：Illustration of a pendulum 摆图解|right|200px]]<br />
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[[File:PendulumLinearizations.png|thumb|图二：Linearizations of a pendulum 摆的线性化|right|200px]]<br />
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A classic, extensively studied nonlinear problem is the dynamics of a [[pendulum (mathematics)|pendulum]] under the influence of [[gravity]]. Using [[Lagrangian mechanics]], it may be shown<ref>[http://www.damtp.cam.ac.uk/user/tong/dynamics.html David Tong: Lectures on Classical Dynamics]</ref> that the motion of a pendulum can be described by the [[dimensionless]] nonlinear equation<br />
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A classic, extensively studied nonlinear problem is the dynamics of a pendulum under the influence of gravity. Using Lagrangian mechanics, it may be shown that the motion of a pendulum can be described by the dimensionless nonlinear equation<br />
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一个经典的被广泛研究的非线性问题是重力影响下的摆的动力学。利用'''拉格朗日力学 Lagrangian Mechanics'''，可以证明摆的运动可以用无量纲的非线性方程<br />
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<math>\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0</math><br />
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where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an [[integrating factor]], which would eventually yield<br />
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where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an integrating factor, which would eventually yield<br />
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描述，其中重力指向“下方”，<math>\theta</math> 是摆与其静止位置形成的角度，如右图所示。“解”这个方程的方法之一是用 <math>d\theta/dt</math> 作为'''积分因子 Integrating Factor'''，最终得<br />
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<math>\int{\frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)}}} = t + C_1</math><br />
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which is an implicit solution involving an [[elliptic integral]]. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the [[nonelementary integral]] (nonelementary unless <math>C_0 = 2</math>).<br />
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which is an implicit solution involving an elliptic integral. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral (nonelementary unless <math>C_0 = 2</math>).<br />
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这是一个含'''椭圆积分 Elliptic Integral'''的隐式解。这个“解”通常没什么用，因为这个解的大部分性质都隐藏在非初等函数积分中（除非<math>C_0 = 2</math>，否则是非初等的）。<br />
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Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through [[Taylor expansion]]s. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
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Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through Taylor expansions. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
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另一种解决这个问题的方法是利用泰勒展开将任意非线性项（此时为正弦函数项)在某些感兴趣的点线性化。例如，在<math>\theta = 0</math> 的点附近线性化（称为小角度近似）为<br />
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<math>\frac{d^2 \theta}{d t^2} + \theta = 0</math><br />
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since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a [[simple harmonic oscillator]] corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
<br />
since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
<br />
因为 <math>\theta \approx 0</math> 时，有 <math>\sin(\theta) \approx \theta</math>。这是一个'''简谐振子 Simple Harmonic Oscillator''' ，对应于摆在其路径底部附近的摆动。另一种线性化方法是在 <math>\theta = \pi</math>附近线性化，对应于运动到最高点的摆:<br />
<br />
<math>\frac{d^2 \theta}{d t^2} + \pi - \theta = 0</math><br />
<br />
<br />
<br />
<br />
<br />
since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves [[hyperbolic sinusoid]]s, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
<br />
since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
<br />
因为 <math>\theta \approx \pi</math> 时，有 <math>\sin(\theta) \approx \pi - \theta</math>。这个问题的解含双曲正弦曲线；注意到不同于小角度近似，它是不稳定的，这意味着 <math>|\theta|</math> 通常会无限增长（但解也有可能是有界的）。这就解释了摆在最高点达到平衡的困难，此时实际上是一种不稳定的状态。<br />
<br />
<br />
<br />
<br />
<br />
One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
<br />
One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
<br />
一个更有趣的线性化可能是在 <math>\theta = \pi/2</math>附近，此时 <math>\sin(\theta) \approx 1</math>:<br />
<br />
<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math><br />
<br />
<br />
<br />
<br />
<br />
This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) [[phase portrait]]s and approximate periods.<br />
<br />
This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) phase portraits and approximate periods.<br />
<br />
这相当于一个自由落体问题。把这样线性化的结果合在一起看，就能得到有关摆的运动的非常有用的图像，如右图所示。利用其他方法寻找（精确的）'''相图 Phase portrait'''和估计周期。<br />
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==Types of nonlinear dynamic behaviors 非线性动力学行为的类型==<br />
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<br />
<br />
*[[Amplitude death]] – any oscillations present in the system cease due to some kind of interaction with other system or feedback by the same system<br />
<br />
*'''振幅死亡 Amplitude Death'''——系统内的某振荡因系统的自回馈或与其他系统的某种相互作用而停止的现象<br />
<br />
*[[Chaos theory|Chaos]] – values of a system cannot be predicted indefinitely far into the future, and fluctuations are [[aperiodic]]<br />
<br />
*'''混沌 Chaos'''——系统内的值无法无限期地预测到遥远的未来；波动是非周期性的<br />
<br />
*[[Multistability]] – the presence of two or more stable states<br />
<br />
*'''多稳态 Multistability'''——两或多个稳态的存在<br />
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*[[Soliton]]s – self-reinforcing solitary waves<br />
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*'''孤波 Soliton'''s——自增强的孤立波<br />
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*[[Limit cycle|Limit cycles]] – asymptotic periodic orbits to which destabilized fixed points are attracted. <br />
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*'''极限环 Limit Cycles'''——吸引不稳定不动点的渐近周期轨道<br />
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*[[Self-oscillation|Self-oscillations]] - feedback oscillations taking place in open dissipative physical systems.<br />
<br />
*'''自激振荡 Self-oscillations'''——开放耗散物理系统中的反馈振荡<br />
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==Examples of nonlinear equations 非线性方程的例子==<br />
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{{Div col|colwidth=25em}}<br />
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<br />
<br />
*[[Algebraic Riccati equation]] 代数黎卡提方程<br />
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<br />
<br />
*[[Ball and beam]] system 球杆系统<br />
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<br />
<br />
*[[Bellman equation]] for optimal policy 最佳策略的贝尔曼方程<br />
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*[[Boltzmann equation]] 玻尔兹曼方程<br />
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<br />
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*[[Colebrook equation]] 科尔布鲁克方程<br />
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*[[General relativity]] 广义相对论<br />
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<br />
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*[[Ginzburg–Landau theory]] 金兹堡-朗道方程<br />
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*[[Ishimori equation]] 石森方程<br />
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<br />
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*[[Kadomtsev–Petviashvili equation]] 卡东穆塞夫-彼得韦亚斯维利方程<br />
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<br />
<br />
*[[Korteweg–de Vries equation]] kdV方程<br />
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*[[Landau–Lifshitz–Gilbert equation]] 朗道-利夫希兹-吉尔伯特方程<br />
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<br />
<br />
*[[Liénard equation]] 林纳德方程<br />
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<br />
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*[[Navier–Stokes equations]] of [[fluid dynamics]] 流体力学的纳维－斯托克斯方程<br />
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*[[Nonlinear optics]] 非线性光学<br />
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*[[Nonlinear Schrödinger equation]] 非线性薛定谔方程<br />
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*[[Power-flow study]] 功率流研究<br />
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*[[Richards equation]] for unsaturated water flow 未饱和层水流的理查氏方程<br />
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<br />
<br />
*[[Self-balancing unicycle]] 自平衡独轮车<br />
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*[[Sine-Gordon equation]] 正弦-戈尔登方程<br />
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<br />
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*[[Van der Pol oscillator]] 范德波尔振荡器<br />
<br />
<br />
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*[[Vlasov equation]] 弗拉索夫方程<br />
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{{Div col end}}<br />
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==See also 参见==<br />
<br />
<br />
*[[Aleksandr Mikhailovich Lyapunov]] 亚历山大·李亚普诺夫<br />
<br />
<br />
<br />
*[[Dynamical system]] 动态系统<br />
<br />
<br />
<br />
*[[Feedback]] 反馈<br />
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<br />
<br />
*[[Initial condition]] 初始条件<br />
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<br />
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*[[Interaction]] 相互作用<br />
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*[[Linear system]] 线性系统<br />
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<br />
<br />
*[[Mode coupling]] 模式耦合<br />
<br />
<br />
<br />
*[[Vector soliton]] 矢量孤子<br />
<br />
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<br />
*[[Volterra series]] 沃尔泰拉级数<br />
<br />
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<br />
<br />
==References 参考资料==<br />
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<br />
{{Reflist|35em}}<br />
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==Further reading 进一步阅读==<br />
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{{Refbegin|35em}}<br />
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*{{cite book<br />
<br />
<br />
<br />
| author= [[Diederich Hinrichsen]] and Anthony J. Pritchard<br />
<br />
| author= Diederich Hinrichsen and Anthony J. Pritchard<br />
<br />
作者 Diederich Hinrichsen 和 Anthony j. Pritchard<br />
<br />
| year= 2005<br />
<br />
| year= 2005<br />
<br />
2005年<br />
<br />
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
<br />
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
<br />
数学系统理论 i-建模，状态空间分析，稳定性和鲁棒性<br />
<br />
| publisher= Springer Verlag<br />
<br />
| publisher= Springer Verlag<br />
<br />
出版商 Springer Verlag<br />
<br />
| isbn=9783540441250<br />
<br />
| isbn=9783540441250<br />
<br />
9783540441250<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Jordan<br />
<br />
| last= Jordan<br />
<br />
最后一个乔丹<br />
<br />
| first= D. W.<br />
<br />
| first= D. W.<br />
<br />
首先是 D.w。<br />
<br />
| last2= Smith<br />
<br />
| last2= Smith<br />
<br />
最后2个史密斯<br />
<br />
| first2= P.<br />
<br />
| first2= P.<br />
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| first2 p.<br />
<br />
| year= 2007<br />
<br />
| year= 2007<br />
<br />
2007年<br />
<br />
| edition= fourth<br />
<br />
| edition= fourth<br />
<br />
第四版<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
非线性常微分方程<br />
<br />
| publisher= Oxford University Press<br />
<br />
| publisher= Oxford University Press<br />
<br />
牛津大学出版社<br />
<br />
| isbn= 978-0-19-920824-1<br />
<br />
| isbn= 978-0-19-920824-1<br />
<br />
[国际标准图书编号978-0-19-920824-1]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Khalil<br />
<br />
| last= Khalil<br />
<br />
最后的哈利勒<br />
<br />
| first= Hassan K.<br />
<br />
| first= Hassan K.<br />
<br />
首先是哈桑 · k。<br />
<br />
| year= 2001<br />
<br />
| year= 2001<br />
<br />
2001年<br />
<br />
| title= Nonlinear Systems<br />
<br />
| title= Nonlinear Systems<br />
<br />
非线性系统<br />
<br />
| publisher= Prentice Hall<br />
<br />
| publisher= Prentice Hall<br />
<br />
出版商 Prentice Hall<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
[国际标准图书编号978-0-13-067389-3]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Kreyszig<br />
<br />
| last= Kreyszig<br />
<br />
| last= Kreyszig<br />
<br />
| first= Erwin<br />
<br />
| first= Erwin<br />
<br />
首先是欧文<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
作者 / 链接 Erwin Kreyszig<br />
<br />
| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
高等工程数学<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
Https://archive.org/details/advancedengineer0008krey<br />
<br />
| url-access= registration<br />
<br />
| url-access= registration<br />
<br />
访问注册<br />
<br />
| publisher= Wiley<br />
<br />
| publisher= Wiley<br />
<br />
出版商 Wiley<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
[国际标准图书编号978-0-471-15496-9]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Sontag<br />
<br />
| last= Sontag<br />
<br />
最后一个桑塔格<br />
<br />
| first= Eduardo<br />
<br />
| first= Eduardo<br />
<br />
首先是爱德华多<br />
<br />
| authorlink= Eduardo D. Sontag<br />
<br />
| authorlink= Eduardo D. Sontag<br />
<br />
作者: Eduardo d. Sontag<br />
<br />
| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
数学控制理论: 确定性有限维系统。第二版<br />
<br />
| publisher= Springer<br />
<br />
| publisher= Springer<br />
<br />
出版商斯普林格<br />
<br />
| isbn= 978-0-387-98489-6<br />
<br />
| isbn= 978-0-387-98489-6<br />
<br />
[国际标准图书编号978-0-387-98489-6]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
{{Refend}}<br />
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<br />
<br />
<br />
<br />
<br />
==External links 外部链接==<br />
<br />
<br />
<br />
*[http://www.dodccrp.org/ Command and Control Research Program (CCRP)]<br />
<br />
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<br />
*[http://necsi.edu/guide/concepts/linearnonlinear.html New England Complex Systems Institute: Concepts in Complex Systems]<br />
<br />
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<br />
*[http://ocw.mit.edu/courses/mathematics/18-353j-nonlinear-dynamics-i-chaos-fall-2012/ Nonlinear Dynamics I: Chaos] at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]<br />
<br />
<br />
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*[http://www.hedengren.net/research/models.htm Nonlinear Model Library]{{snd}} (in [[MATLAB]]) a Database of Physical Systems<br />
<br />
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<br />
*[http://cnls.lanl.gov/ The Center for Nonlinear Studies at Los Alamos National Laboratory]<br />
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{{Systems science}}<br />
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{{Differential equations topics}}<br />
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<br />
{{Authority control}}<br />
<br />
<br />
==编者推荐==<br />
[https://campus.swarma.org/ 集智学园]课程：[https://campus.swarma.org/course/697 非线性动力学与混沌]<br />
<br />
非线性动力学和混沌理论是系统发展的，从一阶微分方程及其分岔开始，然后是相平面分析，极限环和它们的分岔，最终得到Lorenz方程，混沌，迭代映射，周期倍增，重整化，分形和奇怪吸引。<br />
<br />
<br />
此系列课程由[[斯蒂文·斯特罗加茨 Steven H. Strogatz]]主持，内容包括机械振动，激光，生物节律，超导电路，昆虫爆发，化学振荡器，遗传控制系统，混沌水轮，甚至是使用混乱发送秘密信息的技术。在每种情况下，科学背景都在初级阶段进行解释，并与数学理论紧密结合。<br />
<br />
[[Category:Nonlinear systems| ]]<br />
<br />
<br />
<br />
[[Category:Dynamical systems]]<br />
<br />
Category:Dynamical systems<br />
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类别: 动力系统<br />
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[[Category:Concepts in physics]]<br />
<br />
Category:Concepts in physics<br />
<br />
分类: 物理概念<br />
<br />
<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Nonlinear system]]. Its edit history can be viewed at [[非线性系统/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%9D%9E%E7%BA%BF%E6%80%A7%E7%B3%BB%E7%BB%9F&diff=10116非线性系统2020-07-11T06:07:17Z<p>Dorr：</p>
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{{About|"nonlinearity" in mathematics, physics and other sciences|video and film editing|Non-linear editing system|other uses|Nonlinearity (disambiguation)}}<br />
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In [[mathematics]] and [[science]], a '''nonlinear system''' is a [[system]] in which the change of the output is not [[proportionality (mathematics)|proportional]] to the change of the input.<ref>{{Cite news|url=https://news.mit.edu/2010/explained-linear-0226|title=Explained: Linear and nonlinear systems|work=MIT News|access-date=2018-06-30}}</ref><ref>{{Cite web|url=https://www.birmingham.ac.uk/research/activity/mathematics/applied-maths/nonlinear-systems.aspx|title=Nonlinear systems, Applied Mathematics - University of Birmingham|website=www.birmingham.ac.uk|language=en-gb|access-date=2018-06-30}}</ref> Nonlinear problems are of interest to [[engineer]]s, [[biologist]]s,<ref>{{Citation|date=2007|pages=181–276|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-540-34153-6_7|isbn=9783540341529|title = The Nonlinear Universe|series = The Frontiers Collection|chapter = Nonlinear Biology}}</ref><ref>{{cite journal|last=Korenberg|first=Michael J.|last2=Hunter|first2=Ian W.|date=March 1996|title=The identification of nonlinear biological systems: Volterra kernel approaches|journal=Annals of Biomedical Engineering|language=en|volume=24|issue=2|pages=250–268|doi=10.1007/bf02667354|issn=0090-6964}}</ref><ref>{{cite journal|last=Mosconi|first=Francesco|last2=Julou|first2=Thomas|last3=Desprat|first3=Nicolas|last4=Sinha|first4=Deepak Kumar|last5=Allemand|first5=Jean-François|last6=Vincent Croquette|last7=Bensimon|first7=David|date=2008|title=Some nonlinear challenges in biology|url=http://stacks.iop.org/0951-7715/21/i=8/a=T03|journal=Nonlinearity|language=en|volume=21|issue=8|pages=T131|doi=10.1088/0951-7715/21/8/T03|issn=0951-7715|bibcode=2008Nonli..21..131M}}</ref> [[physicist]]s,<ref>{{cite journal|last1=Gintautas|first1=V.|title=Resonant forcing of nonlinear systems of differential equations|journal=Chaos|date=2008|volume=18|issue=3|pages=033118|doi=10.1063/1.2964200|pmid=19045456|arxiv=0803.2252|bibcode=2008Chaos..18c3118G}}</ref><ref>{{cite journal|last1=Stephenson|first1=C.|last2=et.|first2=al.|title=Topological properties of a self-assembled electrical network via ab initio calculation|journal=Sci. Rep.|volume=7|pages=41621|date=2017|doi=10.1038/srep41621|pmid=28155863|pmc=5290745|bibcode=2017NatSR...741621S}}</ref> [[mathematician]]s, and many other [[scientist]]s because most systems are inherently nonlinear in nature.<ref>{{cite book|last1=de Canete|first1=Javier, Cipriano Galindo, and Inmaculada Garcia-Moral|title=System Engineering and Automation: An Interactive Educational Approach|date=2011|publisher=Springer|location=Berlin|isbn=978-3642202292|page=46|url=https://books.google.com/?id=h8rCQYXGGY8C&pg=PA46&lpg=PA46&dq=most+systems+are+inherently+nonlinear+in+nature#v=onepage&q=most%20systems%20are%20inherently%20nonlinear%20in%20nature&f=false|accessdate=20 January 2018}}</ref> Nonlinear [[dynamical system]]s, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler [[linear system]]s.<br />
<br />
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.<br />
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在数学及科学中，'''非线性系统 Nonlinear System'''是一种输出的变化与输入的变化不成比例的系统。大多数系统在本质上是非线性的，因而非线性问题引起了工程师、生物学家、物理学家、数学家和许多其他科学家的兴趣。描述变量随时间变化的非线性动力系统与较之简单得多的线性系统相比，可能显得混沌、不可预测或违反直觉。<br />
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--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）大多数系统在本质上是非线性的，因而非线性问题引起了工程师、生物学家、物理学家、数学家和许多其他科学家的兴趣。 这里调整语序很棒，但是因果关系是不是<br />
因为大多数系统在本质上是非线性的，所以非线性问题引起了工程师、生物学家、物理学家、数学家和许多其他科学家的兴趣。<br />
__[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）这两句因果关系不一样吗<br />
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Typically, the behavior of a nonlinear system is described in mathematics by a '''nonlinear system of equations''', which is a set of simultaneous [[equation]]s in which the unknowns (or the unknown functions in the case of [[differential equation]]s) appear as variables of a [[polynomial]] of degree higher than one or in the argument of a [[function (mathematics)|function]] which is not a polynomial of degree one.<br />
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Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one.<br />
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通常，非线性系统的行为在数学上是由一组非线性联立方程来描述的，其中未知数（或微分方程中的未知函数）作为一个高于一次的多项式的变量出现，或者作为一个不是一次多项式的函数的论元出现。__[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）后半句不知如何翻译<br />
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In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a [[linear combination]] of the unknown [[variable (mathematics)|variables]] or [[function (mathematics)|functions]] that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is ''linear'' if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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换句话说，在非线性方程系统中，要求解的方程不能被写成未知变量或函数的线性组合。无论方程中是否有已知的线性函数，系统都可以被定义为非线性。特别是当一个微分方程的未知函数及其导数是线性的，即使其他变量是非线性的，也称该方程是线性的。<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations ([[linearization]]). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as [[soliton]]s, [[chaos theory|chaos]],<ref>[http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm Nonlinear Dynamics I: Chaos] {{webarchive|url=https://web.archive.org/web/20080212045134/http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm |date=2008-02-12}} at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]</ref> and [[mathematical singularity|singularities]] are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble [[randomness|random]] behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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由于非线性动力学方程难以求解，通常用线性化方程来近似非线性系统（'''线性化 Linearization'''）。这种方法在一定的精度和范围对输入值效果很好，但一些有趣的现象如'''孤子 Soliton'''、'''混沌 Chaos'''和'''奇异性 Singularity'''在线性化后被隐藏。因此，非线性系统的动态行为在某些方面可能看起来违反直觉、不可预测、甚至混沌。尽管这种混沌行为可能感觉很像随机行为，但它实际上并不是随机的。例如，天气的某些方面被认为是混沌的，其系统某部分的微小扰动就会产生复杂的影响。这种非线性是目前技术无法进行精确长期预测的原因之一。<br />
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Some authors use the term '''nonlinear science''' for the study of nonlinear systems. This term is disputed by others:<br />
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Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:<br />
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有些作者用非线性科学这一术语来研究非线性系统。这一术语引起了其他人的争议:<br />
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{{quote|Using a term like nonlinear science is like referring to the bulk of zoology as the study of [[negation|non]]-elephant animals.|[[Stanislaw Ulam]]<ref>{{cite journal|last1=Campbell|first1=David K.|title=Nonlinear physics: Fresh breather|journal=Nature|date=25 November 2004|volume=432|issue=7016|pages=455–456|doi=10.1038/432455a|pmid=15565139|url=https://zenodo.org/record/1134179|language=en|issn=0028-0836|bibcode=2004Natur.432..455C}}</ref>}}<br />
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“使用‘非线性科学’这样的术语，就如同把动物学里大部分对象称作‘非大象动物’研究一样可笑。”<br />
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——'''斯塔尼斯拉夫·乌拉姆 Stanislaw Ulam'''<br />
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==Definition 定义==<br />
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In [[mathematics]], a [[linear map]] (or ''linear function'') <math>f(x)</math> is one which satisfies both of the following properties:<br />
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In mathematics, a linear map (or linear function) <math>f(x)</math> is one which satisfies both of the following properties:<br />
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在数学中，线性映射（或线性函数）<math>f (x)</math>满足以下两个性质:<br />
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*Additivity or [[superposition principle]]: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*Homogeneity: <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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*'''可加性 Additivity'''（'''叠加性 Superposition principle'''）: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*'''齐次性 Homogeneity''': <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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Additivity implies homogeneity for any [[rational number|rational]] ''α'', and, for [[continuous function]]s, for any [[real number|real]] ''α''. For a [[complex number|complex]] ''α'', homogeneity does not follow from additivity. For example, an [[antilinear map]] is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle<br />
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Additivity implies homogeneity for any rational α, and, for continuous functions, for any real α. For a complex α, homogeneity does not follow from additivity. For example, an antilinear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle:<br />
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当α是有理数或实数，且<math>f(x)</math>是连续函数时，由可加性可以推出齐次性。但当α是复数时，可加性不能导出齐次性。例如，反线性映射是可加的，但不是齐次的。可加性和齐次性条件经常组合，称为叠加原理：<br />
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<math>f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)</math><br />
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An equation written as<br />
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对一个写成<br />
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<math>f(x) = C</math><br />
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is called '''linear''' if <math>f(x)</math> is a linear map (as defined above) and '''nonlinear''' otherwise. The equation is called ''homogeneous'' if <math>C = 0</math>.<br />
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is called linear if <math>f(x)</math> is a linear map (as defined above) and nonlinear otherwise. The equation is called homogeneous if <math>C = 0</math>.<br />
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的方程，若 <math> f (x) </math> 是线性映射（如上定义） ，则称其为'''线性的 Linear'''，否则称为'''非线性的 Nonlinear'''。若<math>C = 0</math>，该方程称为是齐次的。<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any [[map (mathematics)|mapping]], including integration or differentiation with associated constraints (such as [[boundary values]]). If <math>f(x)</math> contains [[derivative|differentiation]] with respect to <math>x</math>, the result will be a [[differential equation]].<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any mapping, including integration or differentiation with associated constraints (such as boundary values). If <math>f(x)</math> contains differentiation with respect to <math>x</math>, the result will be a differential equation.<br />
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定义 <math>f(x) = C</math> 是非常具有一般性的，因为 <math>x</math> 可以是任意可感知的数学对象（数字、向量、函数等），函数 <math>f(x)</math> 实际上可以是任意映射，包括有相关约束（如给定边界值）的积分或微分。若 <math>f(x)</math> 包含对 <math>x</math> 的微分运算，则该方程为微分方程。<br />
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===相关概念辨析===<br />
'''线性Linear'''、'''非线性Nonlinear'''、'''亚线性 Sublinear'''、'''超线性 Superlinear'''：描述量与量之间的一种变化关系，例如<math>y=a+b*x^n</math>，其中<math>n>1</math>。当<math>n=1</math>时，表示为线性关系；<math>n\neq 1</math>时，表示为非线性关系；当<math>0<n<1</math>时，表示为亚线性关系；当<math>n>1</math>时，表示为超线性关系。<br />
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亚线性与超线性属于非线性变化关系范畴。当两个变量之间存在亚线性关系时，其典型特性是因变量的变化速率会随着自变量的增大而减小，即其一阶导数会随着自变量的增大而减小；两个变量之间存在超线性关系时，其典型特性是因变量的变化速率会随着自变量的增大而增大，即其一阶导数会随着自变量的增大而增大。<br />
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在线性系统中，整体等于部分和，描述线性系统的方程满足叠加原理，作用的总和正好等于每一部分作用相加的代数和，这意味着每一部分作用都是独立的、互不相关的；而在普遍存在的非线性系统中，作用的总和不等于每一部分作用相加的代数和，因为系统内部要素之间存在着复杂的非线性相互作用。<br />
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==Nonlinear algebraic equations 非线性代数方程==<br />
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{{Main|Algebraic equation|System of polynomial equations}}<br />
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Nonlinear [[algebraic equation]]s, which are also called ''[[polynomial equation]]s'', are defined by equating [[polynomial]]s (of degree greater than one) to zero. For example,<br />
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Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials (of degree greater than one) to zero. For example,<br />
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非线性'''代数方程 Algebraic Equation'''，又称'''多项式方程 Polynomial Equation'''，由某多项式（次数大于1）等于零定义。例如：<br />
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<math>x^2 + x - 1 = 0\,.</math><br />
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For a single polynomial equation, [[root-finding algorithm]]s can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of [[algebraic geometry]], a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see [[Hilbert's Nullstellensatz]]). Nevertheless, in the case of the systems with a finite number of complex solutions, these [[systems of polynomial equations]] are now well understood and efficient methods exist for solving them.<ref>{{cite journal |last1= Lazard |first1= D. |title= Thirty years of Polynomial System Solving, and now? |doi= 10.1016/j.jsc.2008.03.004 |journal= Journal of Symbolic Computation |volume= 44 |issue= 3 |pages= 222–231 |year= 2009 |pmid= |pmc=}}</ref><br />
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For a single polynomial equation, root-finding algorithms can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of algebraic geometry, a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see Hilbert's Nullstellensatz). Nevertheless, in the case of the systems with a finite number of complex solutions, these systems of polynomial equations are now well understood and efficient methods exist for solving them.<br />
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对于一个单一的多项式方程，'''求根算法 Root-finding Algorithms'''可用于其求解（即找到满足该方程的变量的值集）。而代数方程组则相对复杂，其研究是现代数学的较难分支——'''代数几何 Algebraic Geometry'''领域的动力之一。甚至很难判断一个给定的代数系统是否有复数解（见'''希尔伯特零点定律 Hilbert's Nullstellensatz'''）。不过，对于具有有限个复数解的系统的多项式方程组，我们现在已经有了充分的理解，并且找到了有效的求解方法。<br />
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==Nonlinear recurrence relations 非线性递推关系==<br />
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A nonlinear [[recurrence relation]] defines successive terms of a [[sequence]] as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the [[logistic map]] and the relations that define the various [[Hofstadter sequence]]s. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related [[nonlinear system identification]] and analysis procedures.<ref name="SAB1">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</ref> These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related nonlinear system identification and analysis procedures. These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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非线性递归关系中，序列的连续项被定义为其前项的非线性函数。非线性递归关系的例子有 [[logistic 映射]]和定义各种'''霍夫斯塔特序列 Hofstadter Sequences''' 的关系。非线性离散模型代表了一类广泛的非线性递归关系，包括 NARMAX（外部输入非线性自回归移动平均）模型和相关的非线性系统辨识和分析程序。这些方法可用于研究时域、频域和时空域的广泛复杂非线性行为。<br />
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==Nonlinear differential equations 非线性微分方程==<br />
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A [[simultaneous equations|system]] of [[differential equation]]s is said to be nonlinear if it is not a [[linear system]]. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the [[Navier–Stokes equations]] in fluid dynamics and the [[Lotka–Volterra equation]]s in biology.<br />
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A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology.<br />
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若一个微分方程组不是线性系统，则称其为非线性的。涉及非线性微分方程的问题非常多样，对不同问题的解决或分析方法也不相同。非线性微分方程的例子有流体力学中的 '''纳维-斯托克斯方程 Navier-Stokes Equations'''和生物学中的'''洛特卡－沃尔泰拉方程 Lotka-Volterra Equations'''。<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of [[linearly independent]] solutions can be used to construct general solutions through the [[superposition principle]]. A good example of this is one-dimensional heat transport with [[Dirichlet boundary conditions]], the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle. A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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非线性问题最大的困难之一是通常不可能将已知的解组合成新的解。例如，在线性问题中，可以根据叠加原理以一族线性独立的解构造通解。一个很好的例子是带有'''狄利克雷边界条件 Dirichlet Boundary Conditions'''的一维热传导问题，其解（随时间变化）可以写成不同频率的正弦波的线性组合，这使得解非常灵活。而对非线性方程，通常可以找到几个非常特殊的解，但是此时叠加原理不适用，故无法构造新的解。<br />
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===Ordinary differential equations 常微分方程===<br />
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First order [[ordinary differential equation]]s are often exactly solvable by [[separation of variables]], especially for autonomous equations. For example, the nonlinear equation<br />
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First order ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. For example, the nonlinear equation<br />
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一阶常微分方程，尤其是自治方程，通常可以用'''分离变量法 Separation of Variables'''来精确求解。例如，非线性方程<br />
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<math>\frac{d u}{d x} = -u^2</math><br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also ''u'' = 0 as a particular solution, corresponding to the limit of the general solution when ''C'' tends to infinity). The equation is nonlinear because it may be written as<br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also u = 0 as a particular solution, corresponding to the limit of the general solution when C tends to infinity). The equation is nonlinear because it may be written as<br />
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将 <math>u=\frac{1}{x+C}</math> 作为一般解（也有特解 ''u'' = 0，对应于 ''C'' 趋于无穷时的一般解的极限)。该方程是非线性的，因为它可以改写成<br />
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<math>\frac{du}{d x} + u^2=0</math><br />
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and the left-hand side of the equation is not a linear function of ''u'' and its derivatives. Note that if the ''u''<sup>2</sup> term were replaced with ''u'', the problem would be linear (the [[exponential decay]] problem).<br />
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and the left-hand side of the equation is not a linear function of u and its derivatives. Note that if the u<sup>2</sup> term were replaced with u, the problem would be linear (the exponential decay problem).<br />
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方程的左边不是 ''u'' 及其导数的线性函数。注意，若将 ''u''<sup>2</sup> 项替换为''u''，该问题将变为线性的（'''指数衰减 Exponential Decay'''问题）。<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield [[closed-form expression|closed-form]] solutions, though implicit solutions and solutions involving [[nonelementary integral]]s are encountered.<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered.<br />
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二阶和高阶常微分方程（更一般地说，非线性方程组）很少能产生封闭解，而隐式解和非初等函数积分形式的解较为常见。<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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非线性常微分方程定性分析的常用方法包括:<br />
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*Examination of any [[conserved quantities]], especially in [[Hamiltonian system]]s<br />
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*检查是否有任意'''守恒量 Conserved Quantities'''（特别是在'''哈密顿系统 Hamiltonian System'''中）<br />
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*Examination of dissipative quantities (see [[Lyapunov function]]) analogous to conserved quantities<br />
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*检查是否有类似守恒量的耗散量（见'''李亚普诺夫函数 Lyapunov Function'''）<br />
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*Linearization via [[Taylor expansion]]<br />
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*基于'''泰勒展开 Taylor Expansion'''的线性化<br />
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*Change of variables into something easier to study<br />
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*将变量进行代换以便更好的进行研究<br />
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*[[Bifurcation theory]]<br />
<br />
*'''分岔理论 Bifurcation Theory'''<br />
<br />
*[[Perturbation theory|Perturbation]] methods (can be applied to algebraic equations too)<br />
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*'''摄动理论 Perturbation Theory'''（也可应用于代数方程）<br />
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===Partial differential equations 偏微分方程===<br />
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{{main|Nonlinear partial differential equation}}<br />
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{{See also|List of nonlinear partial differential equations}}<br />
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参见：非线性偏微分列表<br />
<br />
The most common basic approach to studying nonlinear [[partial differential equation]]s is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more [[ordinary differential equation]]s, as seen in [[separation of variables]], which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
<br />
The most common basic approach to studying nonlinear partial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
<br />
研究非线性偏微分方程最常用的基本方法是变换变量（或转换问题），使变换后的问题更简单（甚至可能变为线性的）。有时可以将此类方程转化成一或多个常微分方程（如同分离变量法所示），此时不论得到的常微分方程是否可解，对研究问题总是有用的。<br />
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）在数学上 变量代换这个语言方式更加的常见 可以考虑一下是否将变换变量 变为变量代换 符合中文表达<br />
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<br />
Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use [[scale analysis (mathematics)|scale analysis]] to simplify a general, natural equation in a certain specific [[boundary value problem]]. For example, the (very) nonlinear [[Navier-Stokes equations]] can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
<br />
Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem. For example, the (very) nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
<br />
另一个流体力学和热力学中常见的策略（虽然不是数学上的）是利用'''尺度分析 Scale Analysis'''来在某一特定边界条件下简化一般自然方程。例如，在描述圆管内一维层流的暂态时，非线性的纳维－斯托克斯方程可以简化为一个线性的偏微分方程; 尺度分析提供了层流和一维流动的条件，也产生了简化的方程。<br />
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Other methods include examining the [[method of characteristics|characteristics]] and using the methods outlined above for ordinary differential equations.<br />
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Other methods include examining the characteristics and using the methods outlined above for ordinary differential equations.<br />
<br />
其他方法包括检查特征线法及前面所述研究常微分方程的方法。<br />
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===Pendula 摆===<br />
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{{Main|Pendulum (mathematics)}}<br />
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[[File:PendulumLayout.svg.png|thumb|图一：Illustration of a pendulum 摆图解|right|200px]]<br />
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[[File:PendulumLinearizations.png|thumb|图二：Linearizations of a pendulum 摆的线性化|right|200px]]<br />
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A classic, extensively studied nonlinear problem is the dynamics of a [[pendulum (mathematics)|pendulum]] under the influence of [[gravity]]. Using [[Lagrangian mechanics]], it may be shown<ref>[http://www.damtp.cam.ac.uk/user/tong/dynamics.html David Tong: Lectures on Classical Dynamics]</ref> that the motion of a pendulum can be described by the [[dimensionless]] nonlinear equation<br />
<br />
A classic, extensively studied nonlinear problem is the dynamics of a pendulum under the influence of gravity. Using Lagrangian mechanics, it may be shown that the motion of a pendulum can be described by the dimensionless nonlinear equation<br />
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一个经典的被广泛研究的非线性问题是重力影响下的摆的动力学。利用'''拉格朗日力学 Lagrangian Mechanics'''，可以证明摆的运动可以用无量纲的非线性方程<br />
<br />
<math>\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0</math><br />
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where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an [[integrating factor]], which would eventually yield<br />
<br />
where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an integrating factor, which would eventually yield<br />
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描述，其中重力指向“下方”，<math>\theta</math> 是摆与其静止位置形成的角度，如右图所示。“解”这个方程的方法之一是用 <math>d\theta/dt</math> 作为'''积分因子 Integrating Factor'''，最终得<br />
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<math>\int{\frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)}}} = t + C_1</math><br />
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which is an implicit solution involving an [[elliptic integral]]. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the [[nonelementary integral]] (nonelementary unless <math>C_0 = 2</math>).<br />
<br />
which is an implicit solution involving an elliptic integral. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral (nonelementary unless <math>C_0 = 2</math>).<br />
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这是一个含'''椭圆积分 Elliptic Integral'''的隐式解。这个“解”通常没什么用，因为这个解的大部分性质都隐藏在非初等函数积分中（除非<math>C_0 = 2</math>，否则是非初等的）。<br />
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Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through [[Taylor expansion]]s. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
<br />
Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through Taylor expansions. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
<br />
另一种解决这个问题的方法是利用泰勒展开将任意非线性项（此时为正弦函数项)在某些感兴趣的点线性化。例如，在<math>\theta = 0</math> 的点附近线性化（称为小角度近似）为<br />
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<math>\frac{d^2 \theta}{d t^2} + \theta = 0</math><br />
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since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a [[simple harmonic oscillator]] corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
<br />
since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
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因为 <math>\theta \approx 0</math> 时，有 <math>\sin(\theta) \approx \theta</math>。这是一个'''简谐振子 Simple Harmonic Oscillator''' ，对应于摆在其路径底部附近的摆动。另一种线性化方法是在 <math>\theta = \pi</math>附近线性化，对应于运动到最高点的摆:<br />
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<math>\frac{d^2 \theta}{d t^2} + \pi - \theta = 0</math><br />
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since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves [[hyperbolic sinusoid]]s, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
<br />
since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
<br />
因为 <math>\theta \approx \pi</math> 时，有 <math>\sin(\theta) \approx \pi - \theta</math>。这个问题的解含双曲正弦曲线；注意到不同于小角度近似，它是不稳定的，这意味着 <math>|\theta|</math> 通常会无限增长（但解也有可能是有界的）。这就解释了摆在最高点达到平衡的困难，此时实际上是一种不稳定的状态。<br />
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One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
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One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
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一个更有趣的线性化可能是在 <math>\theta = \pi/2</math>附近，此时 <math>\sin(\theta) \approx 1</math>:<br />
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<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math><br />
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This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) [[phase portrait]]s and approximate periods.<br />
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This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) phase portraits and approximate periods.<br />
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这相当于一个自由落体问题。把这样线性化的结果合在一起看，就能得到有关摆的运动的非常有用的图像，如右图所示。利用其他方法寻找（精确的）'''相图 Phase portrait'''和估计周期。<br />
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==Types of nonlinear dynamic behaviors 非线性动力学行为的类型==<br />
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*[[Amplitude death]] – any oscillations present in the system cease due to some kind of interaction with other system or feedback by the same system<br />
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*'''振幅死亡 Amplitude Death'''——系统内的某振荡因系统的自回馈或与其他系统的某种相互作用而停止的现象<br />
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*[[Chaos theory|Chaos]] – values of a system cannot be predicted indefinitely far into the future, and fluctuations are [[aperiodic]]<br />
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*'''混沌 Chaos'''——系统内的值无法无限期地预测到遥远的未来；波动是非周期性的<br />
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*[[Multistability]] – the presence of two or more stable states<br />
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*'''多稳态 Multistability'''——两或多个稳态的存在<br />
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*[[Soliton]]s – self-reinforcing solitary waves<br />
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*'''孤波 Soliton'''s——自增强的孤立波<br />
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*[[Limit cycle|Limit cycles]] – asymptotic periodic orbits to which destabilized fixed points are attracted. <br />
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*'''极限环 Limit Cycles'''——吸引不稳定不动点的渐近周期轨道<br />
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*[[Self-oscillation|Self-oscillations]] - feedback oscillations taking place in open dissipative physical systems.<br />
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*'''自激振荡 Self-oscillations'''——开放耗散物理系统中的反馈振荡<br />
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==Examples of nonlinear equations 非线性方程的例子==<br />
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{{Div col|colwidth=25em}}<br />
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*[[Algebraic Riccati equation]] 代数黎卡提方程<br />
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*[[Ball and beam]] system 球杆系统<br />
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*[[Bellman equation]] for optimal policy 最佳策略的贝尔曼方程<br />
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*[[Boltzmann equation]] 玻尔兹曼方程<br />
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*[[Colebrook equation]] 科尔布鲁克方程<br />
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*[[General relativity]] 广义相对论<br />
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*[[Ginzburg–Landau theory]] 金兹堡-朗道方程<br />
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*[[Ishimori equation]] 石森方程<br />
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*[[Kadomtsev–Petviashvili equation]] 卡东穆塞夫-彼得韦亚斯维利方程<br />
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*[[Korteweg–de Vries equation]] kdV方程<br />
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*[[Landau–Lifshitz–Gilbert equation]] 朗道-利夫希兹-吉尔伯特方程<br />
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*[[Liénard equation]] 林纳德方程<br />
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<br />
*[[Navier–Stokes equations]] of [[fluid dynamics]] 流体力学的纳维－斯托克斯方程<br />
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<br />
*[[Nonlinear optics]] 非线性光学<br />
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*[[Nonlinear Schrödinger equation]] 非线性薛定谔方程<br />
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*[[Power-flow study]] 功率流研究<br />
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*[[Richards equation]] for unsaturated water flow 未饱和层水流的理查氏方程<br />
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*[[Self-balancing unicycle]]自平衡独轮车<br />
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*[[Sine-Gordon equation]] 正弦-戈尔登方程<br />
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*[[Van der Pol oscillator]] 范德波尔振荡器<br />
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*[[Vlasov equation]] 弗拉索夫方程<br />
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<br />
{{Div col end}}<br />
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<br />
==See also 参见==<br />
<br />
<br />
*[[Aleksandr Mikhailovich Lyapunov]] 亚历山大·李亚普诺夫<br />
<br />
<br />
<br />
*[[Dynamical system]] 动态系统<br />
<br />
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<br />
*[[Feedback]] 反馈<br />
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<br />
<br />
*[[Initial condition]] 初始条件<br />
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<br />
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*[[Interaction]] 相互作用<br />
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*[[Linear system]] 线性系统<br />
<br />
<br />
<br />
*[[Mode coupling]] 模式耦合<br />
<br />
<br />
<br />
*[[Vector soliton]] 矢量孤子<br />
<br />
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<br />
*[[Volterra series]] 沃尔泰拉级数<br />
<br />
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<br />
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<br />
<br />
<br />
==References 参考资料==<br />
<br />
<br />
<br />
{{Reflist|35em}}<br />
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==Further reading 进一步阅读==<br />
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<br />
<br />
{{Refbegin|35em}}<br />
<br />
<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| author= [[Diederich Hinrichsen]] and Anthony J. Pritchard<br />
<br />
| author= Diederich Hinrichsen and Anthony J. Pritchard<br />
<br />
作者 Diederich Hinrichsen 和 Anthony j. Pritchard<br />
<br />
| year= 2005<br />
<br />
| year= 2005<br />
<br />
2005年<br />
<br />
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
<br />
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
<br />
数学系统理论 i-建模，状态空间分析，稳定性和鲁棒性<br />
<br />
| publisher= Springer Verlag<br />
<br />
| publisher= Springer Verlag<br />
<br />
出版商 Springer Verlag<br />
<br />
| isbn=9783540441250<br />
<br />
| isbn=9783540441250<br />
<br />
9783540441250<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Jordan<br />
<br />
| last= Jordan<br />
<br />
最后一个乔丹<br />
<br />
| first= D. W.<br />
<br />
| first= D. W.<br />
<br />
首先是 D.w。<br />
<br />
| last2= Smith<br />
<br />
| last2= Smith<br />
<br />
最后2个史密斯<br />
<br />
| first2= P.<br />
<br />
| first2= P.<br />
<br />
| first2 p.<br />
<br />
| year= 2007<br />
<br />
| year= 2007<br />
<br />
2007年<br />
<br />
| edition= fourth<br />
<br />
| edition= fourth<br />
<br />
第四版<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
非线性常微分方程<br />
<br />
| publisher= Oxford University Press<br />
<br />
| publisher= Oxford University Press<br />
<br />
牛津大学出版社<br />
<br />
| isbn= 978-0-19-920824-1<br />
<br />
| isbn= 978-0-19-920824-1<br />
<br />
[国际标准图书编号978-0-19-920824-1]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Khalil<br />
<br />
| last= Khalil<br />
<br />
最后的哈利勒<br />
<br />
| first= Hassan K.<br />
<br />
| first= Hassan K.<br />
<br />
首先是哈桑 · k。<br />
<br />
| year= 2001<br />
<br />
| year= 2001<br />
<br />
2001年<br />
<br />
| title= Nonlinear Systems<br />
<br />
| title= Nonlinear Systems<br />
<br />
非线性系统<br />
<br />
| publisher= Prentice Hall<br />
<br />
| publisher= Prentice Hall<br />
<br />
出版商 Prentice Hall<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
[国际标准图书编号978-0-13-067389-3]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Kreyszig<br />
<br />
| last= Kreyszig<br />
<br />
| last= Kreyszig<br />
<br />
| first= Erwin<br />
<br />
| first= Erwin<br />
<br />
首先是欧文<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
作者 / 链接 Erwin Kreyszig<br />
<br />
| year= 1998<br />
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| year= 1998<br />
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1998年<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
高等工程数学<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
Https://archive.org/details/advancedengineer0008krey<br />
<br />
| url-access= registration<br />
<br />
| url-access= registration<br />
<br />
访问注册<br />
<br />
| publisher= Wiley<br />
<br />
| publisher= Wiley<br />
<br />
出版商 Wiley<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
[国际标准图书编号978-0-471-15496-9]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Sontag<br />
<br />
| last= Sontag<br />
<br />
最后一个桑塔格<br />
<br />
| first= Eduardo<br />
<br />
| first= Eduardo<br />
<br />
首先是爱德华多<br />
<br />
| authorlink= Eduardo D. Sontag<br />
<br />
| authorlink= Eduardo D. Sontag<br />
<br />
作者: Eduardo d. Sontag<br />
<br />
| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
数学控制理论: 确定性有限维系统。第二版<br />
<br />
| publisher= Springer<br />
<br />
| publisher= Springer<br />
<br />
出版商斯普林格<br />
<br />
| isbn= 978-0-387-98489-6<br />
<br />
| isbn= 978-0-387-98489-6<br />
<br />
[国际标准图书编号978-0-387-98489-6]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
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{{Refend}}<br />
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==External links 外部链接==<br />
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*[http://www.dodccrp.org/ Command and Control Research Program (CCRP)]<br />
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*[http://necsi.edu/guide/concepts/linearnonlinear.html New England Complex Systems Institute: Concepts in Complex Systems]<br />
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*[http://ocw.mit.edu/courses/mathematics/18-353j-nonlinear-dynamics-i-chaos-fall-2012/ Nonlinear Dynamics I: Chaos] at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]<br />
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*[http://www.hedengren.net/research/models.htm Nonlinear Model Library]{{snd}} (in [[MATLAB]]) a Database of Physical Systems<br />
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*[http://cnls.lanl.gov/ The Center for Nonlinear Studies at Los Alamos National Laboratory]<br />
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{{Systems science}}<br />
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==编者推荐==<br />
[https://campus.swarma.org/ 集智学园]课程：[https://campus.swarma.org/course/697 非线性动力学与混沌]<br />
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非线性动力学和混沌理论是系统发展的，从一阶微分方程及其分岔开始，然后是相平面分析，极限环和它们的分岔，最终得到Lorenz方程，混沌，迭代映射，周期倍增，重整化，分形和奇怪吸引。<br />
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此系列课程由[[斯蒂文·斯特罗加茨 Steven H. Strogatz]]主持，内容包括机械振动，激光，生物节律，超导电路，昆虫爆发，化学振荡器，遗传控制系统，混沌水轮，甚至是使用混乱发送秘密信息的技术。在每种情况下，科学背景都在初级阶段进行解释，并与数学理论紧密结合。<br />
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[[Category:Nonlinear systems| ]]<br />
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[[Category:Dynamical systems]]<br />
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Category:Dynamical systems<br />
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类别: 动力系统<br />
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[[Category:Concepts in physics]]<br />
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Category:Concepts in physics<br />
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分类: 物理概念<br />
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<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Nonlinear system]]. Its edit history can be viewed at [[非线性系统/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%9D%9E%E7%BA%BF%E6%80%A7%E7%B3%BB%E7%BB%9F&diff=10100非线性系统2020-07-11T02:17:45Z<p>Dorr：</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{Redirect|Nonlinear dynamics|the journal|Nonlinear Dynamics (journal)}}<br />
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{{About|"nonlinearity" in mathematics, physics and other sciences|video and film editing|Non-linear editing system|other uses|Nonlinearity (disambiguation)}}<br />
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In [[mathematics]] and [[science]], a '''nonlinear system''' is a [[system]] in which the change of the output is not [[proportionality (mathematics)|proportional]] to the change of the input.<ref>{{Cite news|url=https://news.mit.edu/2010/explained-linear-0226|title=Explained: Linear and nonlinear systems|work=MIT News|access-date=2018-06-30}}</ref><ref>{{Cite web|url=https://www.birmingham.ac.uk/research/activity/mathematics/applied-maths/nonlinear-systems.aspx|title=Nonlinear systems, Applied Mathematics - University of Birmingham|website=www.birmingham.ac.uk|language=en-gb|access-date=2018-06-30}}</ref> Nonlinear problems are of interest to [[engineer]]s, [[biologist]]s,<ref>{{Citation|date=2007|pages=181–276|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-540-34153-6_7|isbn=9783540341529|title = The Nonlinear Universe|series = The Frontiers Collection|chapter = Nonlinear Biology}}</ref><ref>{{cite journal|last=Korenberg|first=Michael J.|last2=Hunter|first2=Ian W.|date=March 1996|title=The identification of nonlinear biological systems: Volterra kernel approaches|journal=Annals of Biomedical Engineering|language=en|volume=24|issue=2|pages=250–268|doi=10.1007/bf02667354|issn=0090-6964}}</ref><ref>{{cite journal|last=Mosconi|first=Francesco|last2=Julou|first2=Thomas|last3=Desprat|first3=Nicolas|last4=Sinha|first4=Deepak Kumar|last5=Allemand|first5=Jean-François|last6=Vincent Croquette|last7=Bensimon|first7=David|date=2008|title=Some nonlinear challenges in biology|url=http://stacks.iop.org/0951-7715/21/i=8/a=T03|journal=Nonlinearity|language=en|volume=21|issue=8|pages=T131|doi=10.1088/0951-7715/21/8/T03|issn=0951-7715|bibcode=2008Nonli..21..131M}}</ref> [[physicist]]s,<ref>{{cite journal|last1=Gintautas|first1=V.|title=Resonant forcing of nonlinear systems of differential equations|journal=Chaos|date=2008|volume=18|issue=3|pages=033118|doi=10.1063/1.2964200|pmid=19045456|arxiv=0803.2252|bibcode=2008Chaos..18c3118G}}</ref><ref>{{cite journal|last1=Stephenson|first1=C.|last2=et.|first2=al.|title=Topological properties of a self-assembled electrical network via ab initio calculation|journal=Sci. Rep.|volume=7|pages=41621|date=2017|doi=10.1038/srep41621|pmid=28155863|pmc=5290745|bibcode=2017NatSR...741621S}}</ref> [[mathematician]]s, and many other [[scientist]]s because most systems are inherently nonlinear in nature.<ref>{{cite book|last1=de Canete|first1=Javier, Cipriano Galindo, and Inmaculada Garcia-Moral|title=System Engineering and Automation: An Interactive Educational Approach|date=2011|publisher=Springer|location=Berlin|isbn=978-3642202292|page=46|url=https://books.google.com/?id=h8rCQYXGGY8C&pg=PA46&lpg=PA46&dq=most+systems+are+inherently+nonlinear+in+nature#v=onepage&q=most%20systems%20are%20inherently%20nonlinear%20in%20nature&f=false|accessdate=20 January 2018}}</ref> Nonlinear [[dynamical system]]s, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler [[linear system]]s.<br />
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In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.<br />
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在数学及科学中，非线性系统是一种输出的变化与输入的变化不成比例的系统。大多数系统在本质上是非线性的，因而非线性问题引起了工程师、生物学家、物理学家、数学家和许多其他科学家的兴趣。描述变量随时间变化的非线性动力系统与较之简单得多的线性系统相比，可能显得混沌、不可预测或违反直觉。<br />
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Typically, the behavior of a nonlinear system is described in mathematics by a '''nonlinear system of equations''', which is a set of simultaneous [[equation]]s in which the unknowns (or the unknown functions in the case of [[differential equation]]s) appear as variables of a [[polynomial]] of degree higher than one or in the argument of a [[function (mathematics)|function]] which is not a polynomial of degree one.<br />
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Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one.<br />
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通常，非线性系统的行为在数学上是由一组非线性联立方程来描述的，其中未知数（或微分方程中的未知函数）作为一个高于一次的多项式的变量出现，或者作为一个不是一次多项式的函数的论元出现。__[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）+后半句不知如何翻译<br />
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In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a [[linear combination]] of the unknown [[variable (mathematics)|variables]] or [[function (mathematics)|functions]] that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is ''linear'' if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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换句话说，在非线性方程系统中，要求解的方程不能被写成未知变量或函数的线性组合。无论方程中是否有已知的线性函数，系统都可以被定义为非线性。特别是，如果一个微分方程的未知函数及其导数是线性的，即使其他变量是非线性的，也称该方程是线性的。<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations ([[linearization]]). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as [[soliton]]s, [[chaos theory|chaos]],<ref>[http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm Nonlinear Dynamics I: Chaos] {{webarchive|url=https://web.archive.org/web/20080212045134/http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm |date=2008-02-12}} at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]</ref> and [[mathematical singularity|singularities]] are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble [[randomness|random]] behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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由于非线性动力学方程难以求解，通常用线性化方程来近似非线性系统（'''线性化 Linearization'''）。这种方法在一定的精度和范围对输入值效果很好，但一些有趣的现象如'''孤子 Soliton'''、'''混沌 Chaos'''和'''奇异性 Singularity'''在线性化后被隐藏。因此，非线性系统的动态行为在某些方面可能看起来违反直觉、不可预测，甚至混沌。尽管这种混沌行为可能感觉很像随机行为，但它实际上并不是随机的。例如，天气的某些方面被认为是混沌的，其系统某部分的微小扰动就会产生复杂的影响。这种非线性是目前技术无法进行精确长期预测的原因之一。<br />
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Some authors use the term '''nonlinear science''' for the study of nonlinear systems. This term is disputed by others:<br />
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Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:<br />
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有些作者用非线性科学这一术语来研究非线性系统。这一术语引起了其他人的争议:<br />
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{{quote|Using a term like nonlinear science is like referring to the bulk of zoology as the study of [[negation|non]]-elephant animals.|[[Stanislaw Ulam]]<ref>{{cite journal|last1=Campbell|first1=David K.|title=Nonlinear physics: Fresh breather|journal=Nature|date=25 November 2004|volume=432|issue=7016|pages=455–456|doi=10.1038/432455a|pmid=15565139|url=https://zenodo.org/record/1134179|language=en|issn=0028-0836|bibcode=2004Natur.432..455C}}</ref>}}<br />
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“使用‘非线性科学’这样的术语，就如同把动物学里大部分对象称作‘非大象动物’研究一样可笑。”<br />
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——'''斯塔尼斯拉夫·乌拉姆 Stanislaw Ulam'''<br />
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==Definition 定义==<br />
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In [[mathematics]], a [[linear map]] (or ''linear function'') <math>f(x)</math> is one which satisfies both of the following properties:<br />
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In mathematics, a linear map (or linear function) <math>f(x)</math> is one which satisfies both of the following properties:<br />
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在数学中，线性映射（或线性函数）<math>f (x)</math>满足以下两个性质:<br />
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*Additivity or [[superposition principle]]: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*Homogeneity: <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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*'''可加性 Additivity'''（'''叠加性 Superposition principle'''）: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*'''齐次性 Homogeneity''': <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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Additivity implies homogeneity for any [[rational number|rational]] ''α'', and, for [[continuous function]]s, for any [[real number|real]] ''α''. For a [[complex number|complex]] ''α'', homogeneity does not follow from additivity. For example, an [[antilinear map]] is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle<br />
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Additivity implies homogeneity for any rational α, and, for continuous functions, for any real α. For a complex α, homogeneity does not follow from additivity. For example, an antilinear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle:<br />
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α是有理数，或α是实数且<math>f(x)</math>是连续函数时，由可加性可以推出齐次性。但当α是复数时，可加性不能导出齐次性。例如，反线性映射是可加的，但不是齐次的。可加性和齐次性条件经常组合，称为叠加原理：<br />
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<math>f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)</math><br />
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An equation written as<br />
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<math>f(x) = C</math><br />
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is called '''linear''' if <math>f(x)</math> is a linear map (as defined above) and '''nonlinear''' otherwise. The equation is called ''homogeneous'' if <math>C = 0</math>.<br />
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is called linear if <math>f(x)</math> is a linear map (as defined above) and nonlinear otherwise. The equation is called homogeneous if <math>C = 0</math>.<br />
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的方程，若 <math> f (x) </math> 是线性映射（如上定义） ，则称其为'''线性的 Linear'''，否则称为'''非线性的 Nonlinear'''。若<math>C = 0</math>，该方程称为是齐次的。<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any [[map (mathematics)|mapping]], including integration or differentiation with associated constraints (such as [[boundary values]]). If <math>f(x)</math> contains [[derivative|differentiation]] with respect to <math>x</math>, the result will be a [[differential equation]].<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any mapping, including integration or differentiation with associated constraints (such as boundary values). If <math>f(x)</math> contains differentiation with respect to <math>x</math>, the result will be a differential equation.<br />
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定义 <math>f(x) = C</math> 是非常具有一般性的，因为 <math>x</math> 可以是任意可感知的数学对象（数字、向量、函数等），函数 <math>f(x)</math> 实际上可以是任意映射，包括有相关约束（如给定边界值）的积分或微分。若 <math>f(x)</math> 包含对 <math>x</math> 的微分运算，则该方程为微分方程。<br />
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===相关概念辨析===<br />
'''线性Linear'''、'''非线性Nonlinear'''、'''亚线性 Sublinear'''、'''超线性 Superlinear'''：描述量与量之间的一种变化关系，例如<math>y=a+b*x^n</math>，其中<math>n>1</math>。当<math>n=1</math>时，表示为线性关系；<math>n\neq 1</math>时，表示为非线性关系；当<math>0<n<1</math>时，表示为亚线性关系；当<math>n>1</math>时，表示为超线性关系。<br />
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亚线性与超线性属于非线性变化关系范畴。当两个变量之间存在亚线性关系时，其典型特性是因变量的变化速率会随着自变量的增大而减小，即其一阶导数会随着自变量的增大而减小；两个变量之间存在超线性关系时，其典型特性是因变量的变化速率会随着自变量的增大而增大，即其一阶导数会随着自变量的增大而增大。<br />
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在线性系统中，整体等于部分和，描述线性系统的方程满足叠加原理，作用的总和正好等于每一部分作用相加的代数和，这意味着每一部分作用都是独立的、互不相关的；而在普遍存在的非线性系统中，作用的总和不等于每一部分作用相加的代数和，因为系统内部要素之间存在着复杂的非线性相互作用。<br />
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==Nonlinear algebraic equations 非线性代数方程==<br />
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{{Main|Algebraic equation|System of polynomial equations}}<br />
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Nonlinear [[algebraic equation]]s, which are also called ''[[polynomial equation]]s'', are defined by equating [[polynomial]]s (of degree greater than one) to zero. For example,<br />
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Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials (of degree greater than one) to zero. For example,<br />
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非线性'''代数方程 Algebraic equation'''，又称'''多项式方程 Polynomial equation'''，由某多项式（次数大于1）等于零定义。例如：<br />
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For a single polynomial equation, [[root-finding algorithm]]s can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of [[algebraic geometry]], a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see [[Hilbert's Nullstellensatz]]). Nevertheless, in the case of the systems with a finite number of complex solutions, these [[systems of polynomial equations]] are now well understood and efficient methods exist for solving them.<ref>{{cite journal |last1= Lazard |first1= D. |title= Thirty years of Polynomial System Solving, and now? |doi= 10.1016/j.jsc.2008.03.004 |journal= Journal of Symbolic Computation |volume= 44 |issue= 3 |pages= 222–231 |year= 2009 |pmid= |pmc=}}</ref><br />
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For a single polynomial equation, root-finding algorithms can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of algebraic geometry, a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see Hilbert's Nullstellensatz). Nevertheless, in the case of the systems with a finite number of complex solutions, these systems of polynomial equations are now well understood and efficient methods exist for solving them.<br />
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对于一个单一的多项式方程，'''求根算法 Root-finding algorithms'''可用于其求解（即找到满足该方程的变量的值集）。而代数方程组则相对复杂，其研究是现代数学的较难分支——'''代数几何 Algebraic geometry'''领域的动力之一。甚至很难判断一个给定的代数系统是否有复数解（见'''希尔伯特零点定律 Hilbert's Nullstellensatz'''）。不过，对于具有有限个复数解的系统的多项式方程组，我们现在已经有了充分的理解，并且找到了有效的求解方法。<br />
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==Nonlinear recurrence relations 非线性递推关系==<br />
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A nonlinear [[recurrence relation]] defines successive terms of a [[sequence]] as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the [[logistic map]] and the relations that define the various [[Hofstadter sequence]]s. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related [[nonlinear system identification]] and analysis procedures.<ref name="SAB1">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</ref> These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related nonlinear system identification and analysis procedures. These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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非线性递归关系中，序列的连续项被定义为其前项的非线性函数。非线性递归关系的例子有 logistic 映射和定义各种'''霍夫斯塔特序列 Hofstadter sequences''' 的关系。非线性离散模型代表了一类广泛的非线性递归关系，包括 NARMAX（外部输入非线性自回归移动平均）模型和相关的非线性系统辨识和分析程序。这些方法可用于研究时域、频域和时空域的广泛复杂非线性行为。<br />
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==Nonlinear differential equations 非线性微分方程==<br />
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A [[simultaneous equations|system]] of [[differential equation]]s is said to be nonlinear if it is not a [[linear system]]. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the [[Navier–Stokes equations]] in fluid dynamics and the [[Lotka–Volterra equation]]s in biology.<br />
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A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology.<br />
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若一个微分方程组不是线性系统，则称其为非线性的。涉及非线性微分方程的问题非常多样，对不同问题的解决或分析方法也不相同。非线性微分方程的例子有流体力学中的 '''纳维-斯托克斯方程 Navier-Stokes equations'''和生物学中的'''洛特卡－沃尔泰拉方程 Lotka-Volterra equations'''。<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of [[linearly independent]] solutions can be used to construct general solutions through the [[superposition principle]]. A good example of this is one-dimensional heat transport with [[Dirichlet boundary conditions]], the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle. A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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非线性问题最大的困难之一是通常不可能将已知的解组合成新的解。例如，在线性问题中，可以根据叠加原理以一族线性独立的解构造通解。一个很好的例子是带有'''狄利克雷边界条件 Dirichlet boundary conditions'''的一维热传导问题，其解（随时间变化）可以写成不同频率的正弦波的线性组合，这使得解非常灵活。而对非线性方程，通常可以找到几个非常特殊的解，但是此时叠加原理不适用，故无法构造新的解。<br />
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===Ordinary differential equations 常微分方程===<br />
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First order [[ordinary differential equation]]s are often exactly solvable by [[separation of variables]], especially for autonomous equations. For example, the nonlinear equation<br />
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First order ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. For example, the nonlinear equation<br />
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一阶常微分方程，尤其是自治方程，通常可以用'''分离变量法 Separation of variables'''来精确求解。例如，非线性方程<br />
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<math>\frac{d u}{d x} = -u^2</math><br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also ''u'' = 0 as a particular solution, corresponding to the limit of the general solution when ''C'' tends to infinity). The equation is nonlinear because it may be written as<br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also u = 0 as a particular solution, corresponding to the limit of the general solution when C tends to infinity). The equation is nonlinear because it may be written as<br />
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将 <math>u=\frac{1}{x+C}</math> 作为一般解（也有特解 ''u'' = 0，对应于 ''C'' 趋于无穷时的一般解的极限)。该方程是非线性的，因为它可以改写成<br />
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<math>\frac{du}{d x} + u^2=0</math><br />
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and the left-hand side of the equation is not a linear function of ''u'' and its derivatives. Note that if the ''u''<sup>2</sup> term were replaced with ''u'', the problem would be linear (the [[exponential decay]] problem).<br />
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and the left-hand side of the equation is not a linear function of u and its derivatives. Note that if the u<sup>2</sup> term were replaced with u, the problem would be linear (the exponential decay problem).<br />
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方程的左边不是 ''u'' 及其导数的线性函数。注意，若将 ''u''<sup>2</sup> 项替换为''u''，该问题将变为线性的（'''指数衰减 Exponential decay'''问题）。<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield [[closed-form expression|closed-form]] solutions, though implicit solutions and solutions involving [[nonelementary integral]]s are encountered.<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered.<br />
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二阶和高阶常微分方程（更一般地说，非线性方程组）很少能产生封闭解，而隐式解和非初等函数积分形式的解较为常见。<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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非线性常微分方程定性分析的常用方法包括:<br />
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*Examination of any [[conserved quantities]], especially in [[Hamiltonian system]]s<br />
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*检查是否有任意'''守恒量 Conserved quantities'''（特别是在'''哈密顿系统 Hamiltonian system'''中）<br />
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*Examination of dissipative quantities (see [[Lyapunov function]]) analogous to conserved quantities<br />
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*检查是否有类似守恒量的耗散量（见'''李亚普诺夫函数 Lyapunov function'''）<br />
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*Linearization via [[Taylor expansion]]<br />
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*基于'''泰勒展开 Taylor expansion'''的线性化<br />
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*Change of variables into something easier to study<br />
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*将变量转化得更易于研究<br />
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*[[Bifurcation theory]]<br />
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*'''分岔理论 Bifurcation theory'''<br />
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*[[Perturbation theory|Perturbation]] methods (can be applied to algebraic equations too)<br />
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*'''摄动理论 Perturbation theory'''（也可应用于代数方程）<br />
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===Partial differential equations 偏微分方程===<br />
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{{main|Nonlinear partial differential equation}}<br />
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{{See also|List of nonlinear partial differential equations}}<br />
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参见：非线性偏微分列表<br />
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The most common basic approach to studying nonlinear [[partial differential equation]]s is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more [[ordinary differential equation]]s, as seen in [[separation of variables]], which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
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The most common basic approach to studying nonlinear partial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
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研究非线性偏微分方程最常用的基本方法是变换变量（或转换问题），使变换后的问题更简单（甚至可能变为线性的）。有时可以将此类方程转化成一或多个常微分方程（如同分离变量法所示），此时不论得到的常微分方程是否可解，对研究问题总是有用的。<br />
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Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use [[scale analysis (mathematics)|scale analysis]] to simplify a general, natural equation in a certain specific [[boundary value problem]]. For example, the (very) nonlinear [[Navier-Stokes equations]] can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
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Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem. For example, the (very) nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
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另一个流体力学和热力学中常见的策略（虽然不是数学上的）是利用'''尺度分析 Scale analysis'''来在某一特定边界条件下简化一般自然方程。例如，在描述圆管内一维层流的暂态时，非线性的纳维－斯托克斯方程可以简化为一个线性的偏微分方程; 尺度分析提供了层流和一维流动的条件，也产生了简化的方程。<br />
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Other methods include examining the [[method of characteristics|characteristics]] and using the methods outlined above for ordinary differential equations.<br />
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Other methods include examining the characteristics and using the methods outlined above for ordinary differential equations.<br />
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其他方法包括检查特征线法及前面所述研究常微分方程的方法。<br />
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===Pendula 摆===<br />
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{{Main|Pendulum (mathematics)}}<br />
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[[File:PendulumLayout.svg.png|thumb|图一：Illustration of a pendulum 摆图解|right|200px]]<br />
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[[File:PendulumLinearizations.png|thumb|图二：Linearizations of a pendulum 摆的线性化|right|200px]]<br />
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A classic, extensively studied nonlinear problem is the dynamics of a [[pendulum (mathematics)|pendulum]] under the influence of [[gravity]]. Using [[Lagrangian mechanics]], it may be shown<ref>[http://www.damtp.cam.ac.uk/user/tong/dynamics.html David Tong: Lectures on Classical Dynamics]</ref> that the motion of a pendulum can be described by the [[dimensionless]] nonlinear equation<br />
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A classic, extensively studied nonlinear problem is the dynamics of a pendulum under the influence of gravity. Using Lagrangian mechanics, it may be shown that the motion of a pendulum can be described by the dimensionless nonlinear equation<br />
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一个经典的被广泛研究的非线性问题是重力影响下的摆的动力学。利用'''拉格朗日力学 Lagrangian mechanics'''，可以证明摆的运动可以用无量纲的非线性方程<br />
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<math>\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0</math><br />
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where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an [[integrating factor]], which would eventually yield<br />
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where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an integrating factor, which would eventually yield<br />
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描述，其中重力指向“下方”，<math>\theta</math> 是摆与其静止位置形成的角度，如右图所示。“解”这个方程的方法之一是用 <math>d\theta/dt</math> 作为'''积分因子 Integrating factor'''，最终得<br />
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<math>\int{\frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)}}} = t + C_1</math><br />
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which is an implicit solution involving an [[elliptic integral]]. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the [[nonelementary integral]] (nonelementary unless <math>C_0 = 2</math>).<br />
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which is an implicit solution involving an elliptic integral. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral (nonelementary unless <math>C_0 = 2</math>).<br />
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这是一个含'''椭圆积分 Elliptic integral'''的隐式解。这个“解”通常没什么用，因为这个解的大部分性质都隐藏在非初等函数积分中（除非<math>C_0 = 2</math>，否则是非初等的）。<br />
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Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through [[Taylor expansion]]s. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
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Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through Taylor expansions. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
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另一种解决这个问题的方法是利用泰勒展开将任意非线性项（此时为正弦函数项)在某些感兴趣的点线性化。例如，在<math>\theta = 0</math> 的点附近线性化（称为小角度近似）为<br />
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<math>\frac{d^2 \theta}{d t^2} + \theta = 0</math><br />
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since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a [[simple harmonic oscillator]] corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
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since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
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因为 <math>\theta \approx 0</math> 时，有 <math>\sin(\theta) \approx \theta</math>。这是一个'''简谐振子 Simple harmonic oscillator''' ，对应于摆在其路径底部附近的摆动。另一种线性化方法是在 <math>\theta = \pi</math>附近线性化，对应于运动到最高点的摆:<br />
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<math>\frac{d^2 \theta}{d t^2} + \pi - \theta = 0</math><br />
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since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves [[hyperbolic sinusoid]]s, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
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since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
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因为 <math>\theta \approx \pi</math> 时，有 <math>\sin(\theta) \approx \pi - \theta</math>。这个问题的解含双曲正弦曲线；注意到不同于小角度近似，它是不稳定的，这意味着 <math>|\theta|</math> 通常会无限增长（但解也有可能是有界的）。这就解释了摆在最高点达到平衡的困难，此时实际上是一种不稳定的状态。<br />
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One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
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One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
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一个更有趣的线性化可能是在 <math>\theta = \pi/2</math>附近，此时 <math>\sin(\theta) \approx 1</math>:<br />
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<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math><br />
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This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) [[phase portrait]]s and approximate periods.<br />
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This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) phase portraits and approximate periods.<br />
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这相当于一个自由落体问题。把这样线性化的结果合在一起看，就能得到有关摆的运动的非常有用的图像，如右图所示。利用其他方法寻找（精确的）'''相图 Phase portrait'''和估计周期。<br />
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==Types of nonlinear dynamic behaviors 非线性动力学行为的类型==<br />
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*[[Amplitude death]] – any oscillations present in the system cease due to some kind of interaction with other system or feedback by the same system<br />
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*'''振幅死亡 Amplitude death'''——系统内的某振荡因系统的自回馈或与其他系统的某种相互作用而停止的现象<br />
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*[[Chaos theory|Chaos]] – values of a system cannot be predicted indefinitely far into the future, and fluctuations are [[aperiodic]]<br />
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*'''混沌 Chaos'''——系统内的值无法无限期地预测到遥远的未来；波动是非周期性的<br />
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*[[Multistability]] – the presence of two or more stable states<br />
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*'''多稳态 Multistability'''——两或多个稳态的存在<br />
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*[[Soliton]]s – self-reinforcing solitary waves<br />
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*'''孤波 Soliton'''s——自增强的孤立波<br />
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*[[Limit cycle|Limit cycles]] – asymptotic periodic orbits to which destabilized fixed points are attracted. <br />
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*'''极限环 Limit cycles'''——吸引不稳定不动点的渐近周期轨道<br />
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*[[Self-oscillation|Self-oscillations]] - feedback oscillations taking place in open dissipative physical systems.<br />
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*'''自激振荡 Self-oscillations'''——开放耗散物理系统中的反馈振荡<br />
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==Examples of nonlinear equations 非线性方程的例子==<br />
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*[[Algebraic Riccati equation]] 代数黎卡提方程<br />
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*[[Ball and beam]] system 球杆系统<br />
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*[[Bellman equation]] for optimal policy 最佳策略的贝尔曼方程<br />
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*[[Boltzmann equation]] 玻尔兹曼方程<br />
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*[[Colebrook equation]] 科尔布鲁克方程<br />
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*[[General relativity]] 广义相对论<br />
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*[[Ginzburg–Landau theory]] 金兹堡-朗道方程<br />
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*[[Ishimori equation]] 石森方程<br />
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*[[Kadomtsev–Petviashvili equation]] 卡东穆塞夫-彼得韦亚斯维利方程<br />
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*[[Korteweg–de Vries equation]] kdV方程<br />
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*[[Landau–Lifshitz–Gilbert equation]] 朗道-利夫希兹-吉尔伯特方程<br />
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*[[Liénard equation]] 林纳德方程<br />
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*[[Navier–Stokes equations]] of [[fluid dynamics]] 流体力学的纳维－斯托克斯方程<br />
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*[[Nonlinear optics]] 非线性光学<br />
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*[[Nonlinear Schrödinger equation]] 非线性薛定谔方程<br />
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*[[Power-flow study]] 功率流研究<br />
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*[[Richards equation]] for unsaturated water flow 未饱和层水流的理查氏方程<br />
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*[[Self-balancing unicycle]]<br />
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*[[Sine-Gordon equation]] 正弦-戈尔登方程<br />
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*[[Van der Pol oscillator]] 范德波尔振荡器<br />
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*[[Vlasov equation]] 弗拉索夫方程<br />
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==See also 参见==<br />
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*[[Aleksandr Mikhailovich Lyapunov]] 亚历山大·李亚普诺夫<br />
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*[[Dynamical system]] 动态系统<br />
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*[[Feedback]] 反馈<br />
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*[[Initial condition]] 初始条件<br />
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*[[Interaction]] 相互作用<br />
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*[[Linear system]] 线性系统<br />
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*[[Mode coupling]] 模式耦合<br />
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*[[Vector soliton]] 矢量孤子<br />
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*[[Volterra series]] 沃尔泰拉级数<br />
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==References 参考资料==<br />
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{{Reflist|35em}}<br />
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==Further reading 进一步阅读==<br />
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{{Refbegin|35em}}<br />
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*{{cite book<br />
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| author= [[Diederich Hinrichsen]] and Anthony J. Pritchard<br />
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| author= Diederich Hinrichsen and Anthony J. Pritchard<br />
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作者 Diederich Hinrichsen 和 Anthony j. Pritchard<br />
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| year= 2005<br />
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| year= 2005<br />
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2005年<br />
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| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
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| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
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数学系统理论 i-建模，状态空间分析，稳定性和鲁棒性<br />
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| publisher= Springer Verlag<br />
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| publisher= Springer Verlag<br />
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出版商 Springer Verlag<br />
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| isbn=9783540441250<br />
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| isbn=9783540441250<br />
<br />
9783540441250<br />
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}}<br />
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}}<br />
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}}<br />
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*{{cite book<br />
<br />
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<br />
| last= Jordan<br />
<br />
| last= Jordan<br />
<br />
最后一个乔丹<br />
<br />
| first= D. W.<br />
<br />
| first= D. W.<br />
<br />
首先是 D.w。<br />
<br />
| last2= Smith<br />
<br />
| last2= Smith<br />
<br />
最后2个史密斯<br />
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| first2= P.<br />
<br />
| first2= P.<br />
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| first2 p.<br />
<br />
| year= 2007<br />
<br />
| year= 2007<br />
<br />
2007年<br />
<br />
| edition= fourth<br />
<br />
| edition= fourth<br />
<br />
第四版<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
非线性常微分方程<br />
<br />
| publisher= Oxford University Press<br />
<br />
| publisher= Oxford University Press<br />
<br />
牛津大学出版社<br />
<br />
| isbn= 978-0-19-920824-1<br />
<br />
| isbn= 978-0-19-920824-1<br />
<br />
[国际标准图书编号978-0-19-920824-1]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Khalil<br />
<br />
| last= Khalil<br />
<br />
最后的哈利勒<br />
<br />
| first= Hassan K.<br />
<br />
| first= Hassan K.<br />
<br />
首先是哈桑 · k。<br />
<br />
| year= 2001<br />
<br />
| year= 2001<br />
<br />
2001年<br />
<br />
| title= Nonlinear Systems<br />
<br />
| title= Nonlinear Systems<br />
<br />
非线性系统<br />
<br />
| publisher= Prentice Hall<br />
<br />
| publisher= Prentice Hall<br />
<br />
出版商 Prentice Hall<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
[国际标准图书编号978-0-13-067389-3]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
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*{{cite book<br />
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| last= Kreyszig<br />
<br />
| last= Kreyszig<br />
<br />
| last= Kreyszig<br />
<br />
| first= Erwin<br />
<br />
| first= Erwin<br />
<br />
首先是欧文<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
作者 / 链接 Erwin Kreyszig<br />
<br />
| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
高等工程数学<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
Https://archive.org/details/advancedengineer0008krey<br />
<br />
| url-access= registration<br />
<br />
| url-access= registration<br />
<br />
访问注册<br />
<br />
| publisher= Wiley<br />
<br />
| publisher= Wiley<br />
<br />
出版商 Wiley<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
[国际标准图书编号978-0-471-15496-9]<br />
<br />
}}<br />
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}}<br />
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}}<br />
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*{{cite book<br />
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| last= Sontag<br />
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| last= Sontag<br />
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最后一个桑塔格<br />
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| first= Eduardo<br />
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| first= Eduardo<br />
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首先是爱德华多<br />
<br />
| authorlink= Eduardo D. Sontag<br />
<br />
| authorlink= Eduardo D. Sontag<br />
<br />
作者: Eduardo d. Sontag<br />
<br />
| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
数学控制理论: 确定性有限维系统。第二版<br />
<br />
| publisher= Springer<br />
<br />
| publisher= Springer<br />
<br />
出版商斯普林格<br />
<br />
| isbn= 978-0-387-98489-6<br />
<br />
| isbn= 978-0-387-98489-6<br />
<br />
[国际标准图书编号978-0-387-98489-6]<br />
<br />
}}<br />
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}}<br />
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}}<br />
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{{Refend}}<br />
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<br />
<br />
==External links 外部链接==<br />
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*[http://www.dodccrp.org/ Command and Control Research Program (CCRP)]<br />
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*[http://necsi.edu/guide/concepts/linearnonlinear.html New England Complex Systems Institute: Concepts in Complex Systems]<br />
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*[http://ocw.mit.edu/courses/mathematics/18-353j-nonlinear-dynamics-i-chaos-fall-2012/ Nonlinear Dynamics I: Chaos] at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]<br />
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*[http://www.hedengren.net/research/models.htm Nonlinear Model Library]{{snd}} (in [[MATLAB]]) a Database of Physical Systems<br />
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*[http://cnls.lanl.gov/ The Center for Nonlinear Studies at Los Alamos National Laboratory]<br />
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{{Systems science}}<br />
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{{Differential equations topics}}<br />
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{{Authority control}}<br />
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==编者推荐==<br />
[https://campus.swarma.org/ 集智学园]课程：[https://campus.swarma.org/course/697 非线性动力学与混沌]<br />
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非线性动力学和混沌理论是系统发展的，从一阶微分方程及其分岔开始，然后是相平面分析，极限环和它们的分岔，最终得到Lorenz方程，混沌，迭代映射，周期倍增，重整化，分形和奇怪吸引。<br />
<br />
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此系列课程由[[斯蒂文·斯特罗加茨 Steven H. Strogatz]]主持，内容包括机械振动，激光，生物节律，超导电路，昆虫爆发，化学振荡器，遗传控制系统，混沌水轮，甚至是使用混乱发送秘密信息的技术。在每种情况下，科学背景都在初级阶段进行解释，并与数学理论紧密结合。<br />
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[[Category:Nonlinear systems| ]]<br />
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[[Category:Dynamical systems]]<br />
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Category:Dynamical systems<br />
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类别: 动力系统<br />
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[[Category:Concepts in physics]]<br />
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Category:Concepts in physics<br />
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分类: 物理概念<br />
<br />
<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Nonlinear system]]. Its edit history can be viewed at [[非线性系统/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%9D%9E%E7%BA%BF%E6%80%A7%E7%B3%BB%E7%BB%9F&diff=10065非线性系统2020-07-10T04:07:40Z<p>Dorr：/* 相关概念辨析 */</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{Redirect|Nonlinear dynamics|the journal|Nonlinear Dynamics (journal)}}<br />
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{{About|"nonlinearity" in mathematics, physics and other sciences|video and film editing|Non-linear editing system|other uses|Nonlinearity (disambiguation)}}<br />
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In [[mathematics]] and [[science]], a '''nonlinear system''' is a [[system]] in which the change of the output is not [[proportionality (mathematics)|proportional]] to the change of the input.<ref>{{Cite news|url=https://news.mit.edu/2010/explained-linear-0226|title=Explained: Linear and nonlinear systems|work=MIT News|access-date=2018-06-30}}</ref><ref>{{Cite web|url=https://www.birmingham.ac.uk/research/activity/mathematics/applied-maths/nonlinear-systems.aspx|title=Nonlinear systems, Applied Mathematics - University of Birmingham|website=www.birmingham.ac.uk|language=en-gb|access-date=2018-06-30}}</ref> Nonlinear problems are of interest to [[engineer]]s, [[biologist]]s,<ref>{{Citation|date=2007|pages=181–276|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-540-34153-6_7|isbn=9783540341529|title = The Nonlinear Universe|series = The Frontiers Collection|chapter = Nonlinear Biology}}</ref><ref>{{cite journal|last=Korenberg|first=Michael J.|last2=Hunter|first2=Ian W.|date=March 1996|title=The identification of nonlinear biological systems: Volterra kernel approaches|journal=Annals of Biomedical Engineering|language=en|volume=24|issue=2|pages=250–268|doi=10.1007/bf02667354|issn=0090-6964}}</ref><ref>{{cite journal|last=Mosconi|first=Francesco|last2=Julou|first2=Thomas|last3=Desprat|first3=Nicolas|last4=Sinha|first4=Deepak Kumar|last5=Allemand|first5=Jean-François|last6=Vincent Croquette|last7=Bensimon|first7=David|date=2008|title=Some nonlinear challenges in biology|url=http://stacks.iop.org/0951-7715/21/i=8/a=T03|journal=Nonlinearity|language=en|volume=21|issue=8|pages=T131|doi=10.1088/0951-7715/21/8/T03|issn=0951-7715|bibcode=2008Nonli..21..131M}}</ref> [[physicist]]s,<ref>{{cite journal|last1=Gintautas|first1=V.|title=Resonant forcing of nonlinear systems of differential equations|journal=Chaos|date=2008|volume=18|issue=3|pages=033118|doi=10.1063/1.2964200|pmid=19045456|arxiv=0803.2252|bibcode=2008Chaos..18c3118G}}</ref><ref>{{cite journal|last1=Stephenson|first1=C.|last2=et.|first2=al.|title=Topological properties of a self-assembled electrical network via ab initio calculation|journal=Sci. Rep.|volume=7|pages=41621|date=2017|doi=10.1038/srep41621|pmid=28155863|pmc=5290745|bibcode=2017NatSR...741621S}}</ref> [[mathematician]]s, and many other [[scientist]]s because most systems are inherently nonlinear in nature.<ref>{{cite book|last1=de Canete|first1=Javier, Cipriano Galindo, and Inmaculada Garcia-Moral|title=System Engineering and Automation: An Interactive Educational Approach|date=2011|publisher=Springer|location=Berlin|isbn=978-3642202292|page=46|url=https://books.google.com/?id=h8rCQYXGGY8C&pg=PA46&lpg=PA46&dq=most+systems+are+inherently+nonlinear+in+nature#v=onepage&q=most%20systems%20are%20inherently%20nonlinear%20in%20nature&f=false|accessdate=20 January 2018}}</ref> Nonlinear [[dynamical system]]s, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler [[linear system]]s.<br />
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In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.<br />
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在数学及科学中，非线性系统是一种输出的变化与输入的变化不成比例的系统。大多数系统在本质上是非线性的，因而非线性问题引起了工程师、生物学家、物理学家、数学家和许多其他科学家的兴趣。描述变量随时间变化的非线性动力系统与较之简单得多的线性系统相比，可能显得混沌、不可预测或违反直觉。<br />
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Typically, the behavior of a nonlinear system is described in mathematics by a '''nonlinear system of equations''', which is a set of simultaneous [[equation]]s in which the unknowns (or the unknown functions in the case of [[differential equation]]s) appear as variables of a [[polynomial]] of degree higher than one or in the argument of a [[function (mathematics)|function]] which is not a polynomial of degree one.<br />
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Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one.<br />
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通常，非线性系统的行为在数学上是由一组非线性联立方程来描述的，其中未知数（或微分方程中的未知函数）作为一个高于一次的多项式的变量出现，或者作为一个不是一次多项式的函数的论元出现。__[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）+后半句不知如何翻译<br />
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In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a [[linear combination]] of the unknown [[variable (mathematics)|variables]] or [[function (mathematics)|functions]] that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is ''linear'' if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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换句话说，在非线性方程系统中，要求解的方程不能被写成未知变量或函数的线性组合。无论方程中是否有已知的线性函数，系统都可以被定义为非线性。特别是，如果一个微分方程的未知函数及其导数是线性的，即使其他变量是非线性的，也称该方程是线性的。<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations ([[linearization]]). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as [[soliton]]s, [[chaos theory|chaos]],<ref>[http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm Nonlinear Dynamics I: Chaos] {{webarchive|url=https://web.archive.org/web/20080212045134/http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm |date=2008-02-12}} at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]</ref> and [[mathematical singularity|singularities]] are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble [[randomness|random]] behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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由于非线性动力学方程难以求解，通常用线性化方程来近似非线性系统（'''线性化 Linearization'''）。这种方法在一定的精度和范围对输入值效果很好，但一些有趣的现象如'''孤子 Solitons'''、'''混沌 Chaos'''和'''奇异性 Singularities'''在线性化后被隐藏。因此，非线性系统的动态行为在某些方面可能看起来违反直觉、不可预测，甚至混沌。尽管这种混沌行为可能感觉很像随机行为，但它实际上并不是随机的。例如，天气的某些方面被认为是混沌的，其系统某部分的微小扰动就会产生复杂的影响。这种非线性是目前技术无法进行精确长期预测的原因之一。<br />
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Some authors use the term '''nonlinear science''' for the study of nonlinear systems. This term is disputed by others:<br />
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Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:<br />
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有些作者用非线性科学这一术语来研究非线性系统。这一术语引起了其他人的争议:<br />
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{{quote|Using a term like nonlinear science is like referring to the bulk of zoology as the study of [[negation|non]]-elephant animals.|[[Stanislaw Ulam]]<ref>{{cite journal|last1=Campbell|first1=David K.|title=Nonlinear physics: Fresh breather|journal=Nature|date=25 November 2004|volume=432|issue=7016|pages=455–456|doi=10.1038/432455a|pmid=15565139|url=https://zenodo.org/record/1134179|language=en|issn=0028-0836|bibcode=2004Natur.432..455C}}</ref>}}<br />
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“使用‘非线性科学’这样的术语，就如同把动物学里大部分对象称作‘非大象动物’研究一样可笑。”<br />
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——'''斯塔尼斯拉夫·乌拉姆 Stanislaw Ulam'''<br />
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==Definition 定义==<br />
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In [[mathematics]], a [[linear map]] (or ''linear function'') <math>f(x)</math> is one which satisfies both of the following properties:<br />
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In mathematics, a linear map (or linear function) <math>f(x)</math> is one which satisfies both of the following properties:<br />
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在数学中，线性映射（或线性函数）<math>f (x)</math>满足以下两个性质:<br />
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*Additivity or [[superposition principle]]: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*Homogeneity: <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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*'''可加性 Additivity'''（'''叠加性 Superposition principle'''）: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*'''齐次性 Homogeneity''': <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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Additivity implies homogeneity for any [[rational number|rational]] ''α'', and, for [[continuous function]]s, for any [[real number|real]] ''α''. For a [[complex number|complex]] ''α'', homogeneity does not follow from additivity. For example, an [[antilinear map]] is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle<br />
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Additivity implies homogeneity for any rational α, and, for continuous functions, for any real α. For a complex α, homogeneity does not follow from additivity. For example, an antilinear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle:<br />
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α是有理数，或α是实数且<math>f(x)</math>是连续函数时，由可加性可以推出齐次性。但当α是复数时，可加性不能导出齐次性。例如，反线性映射是可加的，但不是齐次的。可加性和齐次性条件经常组合，称为叠加原理：<br />
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<math>f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)</math><br />
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An equation written as<br />
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<math>f(x) = C</math><br />
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is called '''linear''' if <math>f(x)</math> is a linear map (as defined above) and '''nonlinear''' otherwise. The equation is called ''homogeneous'' if <math>C = 0</math>.<br />
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is called linear if <math>f(x)</math> is a linear map (as defined above) and nonlinear otherwise. The equation is called homogeneous if <math>C = 0</math>.<br />
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的方程，若 <math> f (x) </math> 是线性映射（如上定义） ，则称其为'''线性的 Linear'''，否则称为'''非线性的 Nonlinear'''。若<math>C = 0</math>，该方程称为是齐次的。<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any [[map (mathematics)|mapping]], including integration or differentiation with associated constraints (such as [[boundary values]]). If <math>f(x)</math> contains [[derivative|differentiation]] with respect to <math>x</math>, the result will be a [[differential equation]].<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any mapping, including integration or differentiation with associated constraints (such as boundary values). If <math>f(x)</math> contains differentiation with respect to <math>x</math>, the result will be a differential equation.<br />
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定义 <math>f(x) = C</math> 是非常具有一般性的，因为 <math>x</math> 可以是任意可感知的数学对象（数字、向量、函数等），函数 <math>f(x)</math> 实际上可以是任意映射，包括有相关约束（如给定边界值）的积分或微分。若 <math>f(x)</math> 包含对 <math>x</math> 的微分运算，则该方程为微分方程。<br />
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===相关概念辨析===<br />
'''线性Linear'''、'''非线性Nonlinear'''、'''亚线性 Sublinear'''、'''超线性 Superlinear'''：描述量与量之间的一种变化关系，例如<math>y=a+b*x^n</math>，其中<math>n>1</math>。当<math>n=1</math>时，表示为线性关系；<math>n\neq 1</math>时，表示为非线性关系；当<math>0<n<1</math>时，表示为亚线性关系；当<math>n>1</math>时，表示为超线性关系。<br />
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亚线性与超线性属于非线性变化关系范畴。当两个变量之间存在亚线性关系时，其典型特性是因变量的变化速率会随着自变量的增大而减小，即其一阶导数会随着自变量的增大而减小；两个变量之间存在超线性关系时，其典型特性是因变量的变化速率会随着自变量的增大而增大，即其一阶导数会随着自变量的增大而增大。<br />
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在线性系统中，整体等于部分和，描述线性系统的方程满足叠加原理，作用的总和正好等于每一部分作用相加的代数和，这意味着每一部分作用都是独立的、互不相关的；而在普遍存在的非线性系统中，作用的总和不等于每一部分作用相加的代数和，因为系统内部要素之间存在着复杂的非线性相互作用。<br />
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==Nonlinear algebraic equations 非线性代数方程==<br />
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{{Main|Algebraic equation|System of polynomial equations}}<br />
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Nonlinear [[algebraic equation]]s, which are also called ''[[polynomial equation]]s'', are defined by equating [[polynomial]]s (of degree greater than one) to zero. For example,<br />
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Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials (of degree greater than one) to zero. For example,<br />
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非线性'''代数方程 Algebraic equation'''，又称'''多项式方程 Polynomial equation'''，由某多项式（次数大于1）等于零定义。例如：<br />
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<math>x^2 + x - 1 = 0\,.</math><br />
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For a single polynomial equation, [[root-finding algorithm]]s can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of [[algebraic geometry]], a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see [[Hilbert's Nullstellensatz]]). Nevertheless, in the case of the systems with a finite number of complex solutions, these [[systems of polynomial equations]] are now well understood and efficient methods exist for solving them.<ref>{{cite journal |last1= Lazard |first1= D. |title= Thirty years of Polynomial System Solving, and now? |doi= 10.1016/j.jsc.2008.03.004 |journal= Journal of Symbolic Computation |volume= 44 |issue= 3 |pages= 222–231 |year= 2009 |pmid= |pmc=}}</ref><br />
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For a single polynomial equation, root-finding algorithms can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of algebraic geometry, a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see Hilbert's Nullstellensatz). Nevertheless, in the case of the systems with a finite number of complex solutions, these systems of polynomial equations are now well understood and efficient methods exist for solving them.<br />
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对于一个单一的多项式方程，'''求根算法 Root-finding algorithms'''可用于其求解（即找到满足该方程的变量的值集）。而代数方程组则相对复杂，其研究是现代数学的较难分支——'''代数几何 Algebraic geometry'''领域的动力之一。甚至很难判断一个给定的代数系统是否有复数解（见'''希尔伯特零点定律 Hilbert's Nullstellensatz'''）。不过，对于具有有限个复数解的系统的多项式方程组，我们现在已经有了充分的理解，并且找到了有效的求解方法。<br />
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==Nonlinear recurrence relations 非线性递推关系==<br />
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A nonlinear [[recurrence relation]] defines successive terms of a [[sequence]] as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the [[logistic map]] and the relations that define the various [[Hofstadter sequence]]s. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related [[nonlinear system identification]] and analysis procedures.<ref name="SAB1">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</ref> These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related nonlinear system identification and analysis procedures. These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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非线性递归关系中，序列的连续项被定义为其前项的非线性函数。非线性递归关系的例子有 logistic 映射和定义各种'''霍夫斯塔特序列 Hofstadter sequences''' 的关系。非线性离散模型代表了一类广泛的非线性递归关系，包括 NARMAX（外部输入非线性自回归移动平均）模型和相关的非线性系统辨识和分析程序。这些方法可用于研究时域、频域和时空域的广泛复杂非线性行为。<br />
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==Nonlinear differential equations 非线性微分方程==<br />
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A [[simultaneous equations|system]] of [[differential equation]]s is said to be nonlinear if it is not a [[linear system]]. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the [[Navier–Stokes equations]] in fluid dynamics and the [[Lotka–Volterra equation]]s in biology.<br />
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A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology.<br />
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若一个微分方程组不是线性系统，则称其为非线性的。涉及非线性微分方程的问题非常多样，对不同问题的解决或分析方法也不相同。非线性微分方程的例子有流体力学中的 '''纳维-斯托克斯方程 Navier-Stokes equations'''和生物学中的'''洛特卡－沃尔泰拉方程 Lotka-Volterra equations'''。<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of [[linearly independent]] solutions can be used to construct general solutions through the [[superposition principle]]. A good example of this is one-dimensional heat transport with [[Dirichlet boundary conditions]], the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle. A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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非线性问题最大的困难之一是通常不可能将已知的解组合成新的解。例如，在线性问题中，可以根据叠加原理以一族线性独立的解构造通解。一个很好的例子是带有'''狄利克雷边界条件 Dirichlet boundary conditions'''的一维热传导问题，其解（随时间变化）可以写成不同频率的正弦波的线性组合，这使得解非常灵活。而对非线性方程，通常可以找到几个非常特殊的解，但是此时叠加原理不适用，故无法构造新的解。<br />
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===Ordinary differential equations 常微分方程===<br />
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First order [[ordinary differential equation]]s are often exactly solvable by [[separation of variables]], especially for autonomous equations. For example, the nonlinear equation<br />
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First order ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. For example, the nonlinear equation<br />
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一阶常微分方程，尤其是自治方程，通常可以用'''分离变量法 Separation of variables'''来精确求解。例如，非线性方程<br />
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<math>\frac{d u}{d x} = -u^2</math><br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also ''u'' = 0 as a particular solution, corresponding to the limit of the general solution when ''C'' tends to infinity). The equation is nonlinear because it may be written as<br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also u = 0 as a particular solution, corresponding to the limit of the general solution when C tends to infinity). The equation is nonlinear because it may be written as<br />
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将 <math>u=\frac{1}{x+C}</math> 作为一般解（也有特解 ''u'' = 0，对应于 ''C'' 趋于无穷时的一般解的极限)。该方程是非线性的，因为它可以改写成<br />
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<math>\frac{du}{d x} + u^2=0</math><br />
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and the left-hand side of the equation is not a linear function of ''u'' and its derivatives. Note that if the ''u''<sup>2</sup> term were replaced with ''u'', the problem would be linear (the [[exponential decay]] problem).<br />
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and the left-hand side of the equation is not a linear function of u and its derivatives. Note that if the u<sup>2</sup> term were replaced with u, the problem would be linear (the exponential decay problem).<br />
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方程的左边不是 ''u'' 及其导数的线性函数。注意，若将 ''u''<sup>2</sup> 项替换为''u''，该问题将变为线性的（'''指数衰减 Exponential decay'''问题）。<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield [[closed-form expression|closed-form]] solutions, though implicit solutions and solutions involving [[nonelementary integral]]s are encountered.<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered.<br />
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二阶和高阶常微分方程（更一般地说，非线性方程组）很少能产生封闭解，而隐式解和非初等函数积分形式的解较为常见。<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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非线性常微分方程定性分析的常用方法包括:<br />
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*Examination of any [[conserved quantities]], especially in [[Hamiltonian system]]s<br />
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*检查是否有任意'''守恒量 Conserved quantities'''（特别是在'''哈密顿系统 Hamiltonian system'''中）<br />
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*Examination of dissipative quantities (see [[Lyapunov function]]) analogous to conserved quantities<br />
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*检查是否有类似守恒量的耗散量（见'''李亚普诺夫函数 Lyapunov function'''）<br />
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*Linearization via [[Taylor expansion]]<br />
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*基于'''泰勒展开 Taylor expansion'''的线性化<br />
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*Change of variables into something easier to study<br />
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*将变量转化得更易于研究<br />
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*[[Bifurcation theory]]<br />
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*'''分岔理论 Bifurcation theory'''<br />
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*[[Perturbation theory|Perturbation]] methods (can be applied to algebraic equations too)<br />
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*'''摄动理论 Perturbation theory'''（也可应用于代数方程）<br />
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===Partial differential equations 偏微分方程===<br />
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{{main|Nonlinear partial differential equation}}<br />
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{{See also|List of nonlinear partial differential equations}}<br />
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参见：非线性偏微分列表<br />
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The most common basic approach to studying nonlinear [[partial differential equation]]s is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more [[ordinary differential equation]]s, as seen in [[separation of variables]], which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
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The most common basic approach to studying nonlinear partial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
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研究非线性偏微分方程最常用的基本方法是变换变量（或转换问题），使变换后的问题更简单（甚至可能变为线性的）。有时可以将此类方程转化成一或多个常微分方程（如同分离变量法所示），此时不论得到的常微分方程是否可解，对研究问题总是有用的。<br />
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Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use [[scale analysis (mathematics)|scale analysis]] to simplify a general, natural equation in a certain specific [[boundary value problem]]. For example, the (very) nonlinear [[Navier-Stokes equations]] can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
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Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem. For example, the (very) nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
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另一个流体力学和热力学中常见的策略（虽然不是数学上的）是利用'''尺度分析 Scale analysis'''来在某一特定边界条件下简化一般自然方程。例如，在描述圆管内一维层流的暂态时，非线性的纳维－斯托克斯方程可以简化为一个线性的偏微分方程; 尺度分析提供了层流和一维流动的条件，也产生了简化的方程。<br />
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Other methods include examining the [[method of characteristics|characteristics]] and using the methods outlined above for ordinary differential equations.<br />
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Other methods include examining the characteristics and using the methods outlined above for ordinary differential equations.<br />
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其他方法包括检查特征线法及前面所述研究常微分方程的方法。<br />
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===Pendula 摆===<br />
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{{Main|Pendulum (mathematics)}}<br />
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[[File:PendulumLayout.svg.png|thumb|图一：Illustration of a pendulum 摆图解|right|200px]]<br />
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[[File:PendulumLinearizations.png|thumb|图二：Linearizations of a pendulum 摆的线性化|right|200px]]<br />
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A classic, extensively studied nonlinear problem is the dynamics of a [[pendulum (mathematics)|pendulum]] under the influence of [[gravity]]. Using [[Lagrangian mechanics]], it may be shown<ref>[http://www.damtp.cam.ac.uk/user/tong/dynamics.html David Tong: Lectures on Classical Dynamics]</ref> that the motion of a pendulum can be described by the [[dimensionless]] nonlinear equation<br />
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A classic, extensively studied nonlinear problem is the dynamics of a pendulum under the influence of gravity. Using Lagrangian mechanics, it may be shown that the motion of a pendulum can be described by the dimensionless nonlinear equation<br />
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一个经典的被广泛研究的非线性问题是重力影响下的摆的动力学。利用'''拉格朗日力学 Lagrangian mechanics'''，可以证明摆的运动可以用无量纲的非线性方程<br />
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<math>\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0</math><br />
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where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an [[integrating factor]], which would eventually yield<br />
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where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an integrating factor, which would eventually yield<br />
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描述，其中重力指向“下方”，<math>\theta</math> 是摆与其静止位置形成的角度，如右图所示。“解”这个方程的方法之一是用 <math>d\theta/dt</math> 作为'''积分因子 Integrating factor'''，最终得<br />
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<math>\int{\frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)}}} = t + C_1</math><br />
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which is an implicit solution involving an [[elliptic integral]]. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the [[nonelementary integral]] (nonelementary unless <math>C_0 = 2</math>).<br />
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which is an implicit solution involving an elliptic integral. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral (nonelementary unless <math>C_0 = 2</math>).<br />
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这是一个含'''椭圆积分 Elliptic integral'''的隐式解。这个“解”通常没什么用，因为这个解的大部分性质都隐藏在非初等函数积分中（除非<math>C_0 = 2</math>，否则是非初等的）。<br />
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Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through [[Taylor expansion]]s. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
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Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through Taylor expansions. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
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另一种解决这个问题的方法是利用泰勒展开将任意非线性项（此时为正弦函数项)在某些感兴趣的点线性化。例如，在<math>\theta = 0</math> 的点附近线性化（称为小角度近似）为<br />
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<math>\frac{d^2 \theta}{d t^2} + \theta = 0</math><br />
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since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a [[simple harmonic oscillator]] corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
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since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
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因为 <math>\theta \approx 0</math> 时，有 <math>\sin(\theta) \approx \theta</math>。这是一个'''简谐振子 Simple harmonic oscillator''' ，对应于摆在其路径底部附近的摆动。另一种线性化方法是在 <math>\theta = \pi</math>附近线性化，对应于运动到最高点的摆:<br />
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<math>\frac{d^2 \theta}{d t^2} + \pi - \theta = 0</math><br />
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since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves [[hyperbolic sinusoid]]s, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
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since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
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因为 <math>\theta \approx \pi</math> 时，有 <math>\sin(\theta) \approx \pi - \theta</math>。这个问题的解含双曲正弦曲线；注意到不同于小角度近似，它是不稳定的，这意味着 <math>|\theta|</math> 通常会无限增长（但解也有可能是有界的）。这就解释了摆在最高点达到平衡的困难，此时实际上是一种不稳定的状态。<br />
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One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
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One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
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一个更有趣的线性化可能是在 <math>\theta = \pi/2</math>附近，此时 <math>\sin(\theta) \approx 1</math>:<br />
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<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math><br />
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This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) [[phase portrait]]s and approximate periods.<br />
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This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) phase portraits and approximate periods.<br />
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这相当于一个自由落体问题。把这样线性化的结果合在一起看，就能得到有关摆的运动的非常有用的图像，如右图所示。利用其他方法寻找（精确的）'''相图 Phase portrait'''和估计周期。<br />
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==Types of nonlinear dynamic behaviors 非线性动力学行为的类型==<br />
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*[[Amplitude death]] – any oscillations present in the system cease due to some kind of interaction with other system or feedback by the same system<br />
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*'''振幅死亡 Amplitude death'''——系统内的某振荡因系统的自回馈或与其他系统的某种相互作用而停止的现象<br />
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*[[Chaos theory|Chaos]] – values of a system cannot be predicted indefinitely far into the future, and fluctuations are [[aperiodic]]<br />
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*'''混沌 Chaos'''——系统内的值无法无限期地预测到遥远的未来；波动是非周期性的<br />
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*[[Multistability]] – the presence of two or more stable states<br />
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*'''多稳态 Multistability'''——两或多个稳态的存在<br />
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*[[Soliton]]s – self-reinforcing solitary waves<br />
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*'''孤波 Soliton'''s——自增强的孤立波<br />
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*[[Limit cycle|Limit cycles]] – asymptotic periodic orbits to which destabilized fixed points are attracted. <br />
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*'''极限环 Limit cycles'''——吸引不稳定不动点的渐近周期轨道<br />
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*[[Self-oscillation|Self-oscillations]] - feedback oscillations taking place in open dissipative physical systems.<br />
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*'''自激振荡 Self-oscillations'''——开放耗散物理系统中的反馈振荡<br />
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==Examples of nonlinear equations 非线性方程的例子==<br />
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*[[Algebraic Riccati equation]] 代数黎卡提方程<br />
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*[[Ball and beam]] system 球杆系统<br />
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*[[Bellman equation]] for optimal policy 最佳策略的贝尔曼方程<br />
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*[[Boltzmann equation]] 玻尔兹曼方程<br />
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*[[Colebrook equation]] 科尔布鲁克方程<br />
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*[[General relativity]] 广义相对论<br />
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*[[Ginzburg–Landau theory]] 金兹堡-朗道方程<br />
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*[[Ishimori equation]] 石森方程<br />
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*[[Kadomtsev–Petviashvili equation]] 卡东穆塞夫-彼得韦亚斯维利方程<br />
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*[[Korteweg–de Vries equation]] kdV方程<br />
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*[[Landau–Lifshitz–Gilbert equation]] 朗道-利夫希兹-吉尔伯特方程<br />
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*[[Liénard equation]] 林纳德方程<br />
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*[[Navier–Stokes equations]] of [[fluid dynamics]] 流体力学的纳维－斯托克斯方程<br />
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*[[Nonlinear optics]] 非线性光学<br />
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*[[Nonlinear Schrödinger equation]] 非线性薛定谔方程<br />
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*[[Power-flow study]] 功率流研究<br />
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*[[Richards equation]] for unsaturated water flow 未饱和层水流的理查氏方程<br />
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*[[Self-balancing unicycle]]<br />
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*[[Sine-Gordon equation]] 正弦-戈尔登方程<br />
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*[[Van der Pol oscillator]] 范德波尔振荡器<br />
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*[[Vlasov equation]] 弗拉索夫方程<br />
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{{Div col end}}<br />
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==See also 参见==<br />
<br />
<br />
*[[Aleksandr Mikhailovich Lyapunov]] 亚历山大·李亚普诺夫<br />
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*[[Dynamical system]] 动态系统<br />
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*[[Feedback]] 反馈<br />
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*[[Initial condition]] 初始条件<br />
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*[[Interaction]] 相互作用<br />
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*[[Linear system]] 线性系统<br />
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<br />
<br />
*[[Mode coupling]] 模式耦合<br />
<br />
<br />
<br />
*[[Vector soliton]] 矢量孤子<br />
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*[[Volterra series]] 沃尔泰拉级数<br />
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==References 参考资料==<br />
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{{Reflist|35em}}<br />
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==Further reading 进一步阅读==<br />
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{{Refbegin|35em}}<br />
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*{{cite book<br />
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| author= [[Diederich Hinrichsen]] and Anthony J. Pritchard<br />
<br />
| author= Diederich Hinrichsen and Anthony J. Pritchard<br />
<br />
作者 Diederich Hinrichsen 和 Anthony j. Pritchard<br />
<br />
| year= 2005<br />
<br />
| year= 2005<br />
<br />
2005年<br />
<br />
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
<br />
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
<br />
数学系统理论 i-建模，状态空间分析，稳定性和鲁棒性<br />
<br />
| publisher= Springer Verlag<br />
<br />
| publisher= Springer Verlag<br />
<br />
出版商 Springer Verlag<br />
<br />
| isbn=9783540441250<br />
<br />
| isbn=9783540441250<br />
<br />
9783540441250<br />
<br />
}}<br />
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}}<br />
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}}<br />
<br />
*{{cite book<br />
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<br />
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| last= Jordan<br />
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| last= Jordan<br />
<br />
最后一个乔丹<br />
<br />
| first= D. W.<br />
<br />
| first= D. W.<br />
<br />
首先是 D.w。<br />
<br />
| last2= Smith<br />
<br />
| last2= Smith<br />
<br />
最后2个史密斯<br />
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| first2= P.<br />
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| first2= P.<br />
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| first2 p.<br />
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| year= 2007<br />
<br />
| year= 2007<br />
<br />
2007年<br />
<br />
| edition= fourth<br />
<br />
| edition= fourth<br />
<br />
第四版<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
非线性常微分方程<br />
<br />
| publisher= Oxford University Press<br />
<br />
| publisher= Oxford University Press<br />
<br />
牛津大学出版社<br />
<br />
| isbn= 978-0-19-920824-1<br />
<br />
| isbn= 978-0-19-920824-1<br />
<br />
[国际标准图书编号978-0-19-920824-1]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Khalil<br />
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| last= Khalil<br />
<br />
最后的哈利勒<br />
<br />
| first= Hassan K.<br />
<br />
| first= Hassan K.<br />
<br />
首先是哈桑 · k。<br />
<br />
| year= 2001<br />
<br />
| year= 2001<br />
<br />
2001年<br />
<br />
| title= Nonlinear Systems<br />
<br />
| title= Nonlinear Systems<br />
<br />
非线性系统<br />
<br />
| publisher= Prentice Hall<br />
<br />
| publisher= Prentice Hall<br />
<br />
出版商 Prentice Hall<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
[国际标准图书编号978-0-13-067389-3]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Kreyszig<br />
<br />
| last= Kreyszig<br />
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| last= Kreyszig<br />
<br />
| first= Erwin<br />
<br />
| first= Erwin<br />
<br />
首先是欧文<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
作者 / 链接 Erwin Kreyszig<br />
<br />
| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
高等工程数学<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
Https://archive.org/details/advancedengineer0008krey<br />
<br />
| url-access= registration<br />
<br />
| url-access= registration<br />
<br />
访问注册<br />
<br />
| publisher= Wiley<br />
<br />
| publisher= Wiley<br />
<br />
出版商 Wiley<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
[国际标准图书编号978-0-471-15496-9]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Sontag<br />
<br />
| last= Sontag<br />
<br />
最后一个桑塔格<br />
<br />
| first= Eduardo<br />
<br />
| first= Eduardo<br />
<br />
首先是爱德华多<br />
<br />
| authorlink= Eduardo D. Sontag<br />
<br />
| authorlink= Eduardo D. Sontag<br />
<br />
作者: Eduardo d. Sontag<br />
<br />
| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
数学控制理论: 确定性有限维系统。第二版<br />
<br />
| publisher= Springer<br />
<br />
| publisher= Springer<br />
<br />
出版商斯普林格<br />
<br />
| isbn= 978-0-387-98489-6<br />
<br />
| isbn= 978-0-387-98489-6<br />
<br />
[国际标准图书编号978-0-387-98489-6]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
{{Refend}}<br />
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<br />
<br />
==External links 外部链接==<br />
<br />
<br />
<br />
*[http://www.dodccrp.org/ Command and Control Research Program (CCRP)]<br />
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<br />
*[http://necsi.edu/guide/concepts/linearnonlinear.html New England Complex Systems Institute: Concepts in Complex Systems]<br />
<br />
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*[http://ocw.mit.edu/courses/mathematics/18-353j-nonlinear-dynamics-i-chaos-fall-2012/ Nonlinear Dynamics I: Chaos] at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]<br />
<br />
<br />
<br />
*[http://www.hedengren.net/research/models.htm Nonlinear Model Library]{{snd}} (in [[MATLAB]]) a Database of Physical Systems<br />
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<br />
*[http://cnls.lanl.gov/ The Center for Nonlinear Studies at Los Alamos National Laboratory]<br />
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{{Systems science}}<br />
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{{Differential equations topics}}<br />
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{{Authority control}}<br />
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==编者推荐==<br />
集智学园：[https://campus.swarma.org/course/697 非线性动力学与混沌]<br />
<br />
非线性动力学和混沌理论是系统发展的，从一阶微分方程及其分岔开始，然后是相平面分析，极限环和它们的分岔，最终得到Lorenz方程，混沌，迭代映射，周期倍增，重整化，分形和奇怪吸引。<br />
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此系列课程由[[斯蒂文·斯特罗加茨 Steven H. Strogatz]]主持，内容包括机械振动，激光，生物节律，超导电路，昆虫爆发，化学振荡器，遗传控制系统，混沌水轮，甚至是使用混乱发送秘密信息的技术。在每种情况下，科学背景都在初级阶段进行解释，并与数学理论紧密结合。<br />
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[[Category:Nonlinear systems| ]]<br />
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<br />
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[[Category:Dynamical systems]]<br />
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Category:Dynamical systems<br />
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类别: 动力系统<br />
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[[Category:Concepts in physics]]<br />
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Category:Concepts in physics<br />
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分类: 物理概念<br />
<br />
<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Nonlinear system]]. Its edit history can be viewed at [[非线性系统/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%9D%9E%E7%BA%BF%E6%80%A7%E7%B3%BB%E7%BB%9F&diff=10064非线性系统2020-07-10T04:05:17Z<p>Dorr：</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{Redirect|Nonlinear dynamics|the journal|Nonlinear Dynamics (journal)}}<br />
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{{About|"nonlinearity" in mathematics, physics and other sciences|video and film editing|Non-linear editing system|other uses|Nonlinearity (disambiguation)}}<br />
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In [[mathematics]] and [[science]], a '''nonlinear system''' is a [[system]] in which the change of the output is not [[proportionality (mathematics)|proportional]] to the change of the input.<ref>{{Cite news|url=https://news.mit.edu/2010/explained-linear-0226|title=Explained: Linear and nonlinear systems|work=MIT News|access-date=2018-06-30}}</ref><ref>{{Cite web|url=https://www.birmingham.ac.uk/research/activity/mathematics/applied-maths/nonlinear-systems.aspx|title=Nonlinear systems, Applied Mathematics - University of Birmingham|website=www.birmingham.ac.uk|language=en-gb|access-date=2018-06-30}}</ref> Nonlinear problems are of interest to [[engineer]]s, [[biologist]]s,<ref>{{Citation|date=2007|pages=181–276|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-540-34153-6_7|isbn=9783540341529|title = The Nonlinear Universe|series = The Frontiers Collection|chapter = Nonlinear Biology}}</ref><ref>{{cite journal|last=Korenberg|first=Michael J.|last2=Hunter|first2=Ian W.|date=March 1996|title=The identification of nonlinear biological systems: Volterra kernel approaches|journal=Annals of Biomedical Engineering|language=en|volume=24|issue=2|pages=250–268|doi=10.1007/bf02667354|issn=0090-6964}}</ref><ref>{{cite journal|last=Mosconi|first=Francesco|last2=Julou|first2=Thomas|last3=Desprat|first3=Nicolas|last4=Sinha|first4=Deepak Kumar|last5=Allemand|first5=Jean-François|last6=Vincent Croquette|last7=Bensimon|first7=David|date=2008|title=Some nonlinear challenges in biology|url=http://stacks.iop.org/0951-7715/21/i=8/a=T03|journal=Nonlinearity|language=en|volume=21|issue=8|pages=T131|doi=10.1088/0951-7715/21/8/T03|issn=0951-7715|bibcode=2008Nonli..21..131M}}</ref> [[physicist]]s,<ref>{{cite journal|last1=Gintautas|first1=V.|title=Resonant forcing of nonlinear systems of differential equations|journal=Chaos|date=2008|volume=18|issue=3|pages=033118|doi=10.1063/1.2964200|pmid=19045456|arxiv=0803.2252|bibcode=2008Chaos..18c3118G}}</ref><ref>{{cite journal|last1=Stephenson|first1=C.|last2=et.|first2=al.|title=Topological properties of a self-assembled electrical network via ab initio calculation|journal=Sci. Rep.|volume=7|pages=41621|date=2017|doi=10.1038/srep41621|pmid=28155863|pmc=5290745|bibcode=2017NatSR...741621S}}</ref> [[mathematician]]s, and many other [[scientist]]s because most systems are inherently nonlinear in nature.<ref>{{cite book|last1=de Canete|first1=Javier, Cipriano Galindo, and Inmaculada Garcia-Moral|title=System Engineering and Automation: An Interactive Educational Approach|date=2011|publisher=Springer|location=Berlin|isbn=978-3642202292|page=46|url=https://books.google.com/?id=h8rCQYXGGY8C&pg=PA46&lpg=PA46&dq=most+systems+are+inherently+nonlinear+in+nature#v=onepage&q=most%20systems%20are%20inherently%20nonlinear%20in%20nature&f=false|accessdate=20 January 2018}}</ref> Nonlinear [[dynamical system]]s, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler [[linear system]]s.<br />
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In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.<br />
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在数学及科学中，非线性系统是一种输出的变化与输入的变化不成比例的系统。大多数系统在本质上是非线性的，因而非线性问题引起了工程师、生物学家、物理学家、数学家和许多其他科学家的兴趣。描述变量随时间变化的非线性动力系统与较之简单得多的线性系统相比，可能显得混沌、不可预测或违反直觉。<br />
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Typically, the behavior of a nonlinear system is described in mathematics by a '''nonlinear system of equations''', which is a set of simultaneous [[equation]]s in which the unknowns (or the unknown functions in the case of [[differential equation]]s) appear as variables of a [[polynomial]] of degree higher than one or in the argument of a [[function (mathematics)|function]] which is not a polynomial of degree one.<br />
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Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one.<br />
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通常，非线性系统的行为在数学上是由一组非线性联立方程来描述的，其中未知数（或微分方程中的未知函数）作为一个高于一次的多项式的变量出现，或者作为一个不是一次多项式的函数的论元出现。__[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）+后半句不知如何翻译<br />
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In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a [[linear combination]] of the unknown [[variable (mathematics)|variables]] or [[function (mathematics)|functions]] that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is ''linear'' if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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换句话说，在非线性方程系统中，要求解的方程不能被写成未知变量或函数的线性组合。无论方程中是否有已知的线性函数，系统都可以被定义为非线性。特别是，如果一个微分方程的未知函数及其导数是线性的，即使其他变量是非线性的，也称该方程是线性的。<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations ([[linearization]]). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as [[soliton]]s, [[chaos theory|chaos]],<ref>[http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm Nonlinear Dynamics I: Chaos] {{webarchive|url=https://web.archive.org/web/20080212045134/http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm |date=2008-02-12}} at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]</ref> and [[mathematical singularity|singularities]] are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble [[randomness|random]] behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
<br />
As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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由于非线性动力学方程难以求解，通常用线性化方程来近似非线性系统（'''线性化 Linearization'''）。这种方法在一定的精度和范围对输入值效果很好，但一些有趣的现象如'''孤子 Solitons'''、'''混沌 Chaos'''和'''奇异性 Singularities'''在线性化后被隐藏。因此，非线性系统的动态行为在某些方面可能看起来违反直觉、不可预测，甚至混沌。尽管这种混沌行为可能感觉很像随机行为，但它实际上并不是随机的。例如，天气的某些方面被认为是混沌的，其系统某部分的微小扰动就会产生复杂的影响。这种非线性是目前技术无法进行精确长期预测的原因之一。<br />
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Some authors use the term '''nonlinear science''' for the study of nonlinear systems. This term is disputed by others:<br />
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Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:<br />
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有些作者用非线性科学这一术语来研究非线性系统。这一术语引起了其他人的争议:<br />
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{{quote|Using a term like nonlinear science is like referring to the bulk of zoology as the study of [[negation|non]]-elephant animals.|[[Stanislaw Ulam]]<ref>{{cite journal|last1=Campbell|first1=David K.|title=Nonlinear physics: Fresh breather|journal=Nature|date=25 November 2004|volume=432|issue=7016|pages=455–456|doi=10.1038/432455a|pmid=15565139|url=https://zenodo.org/record/1134179|language=en|issn=0028-0836|bibcode=2004Natur.432..455C}}</ref>}}<br />
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“使用‘非线性科学’这样的术语，就如同把动物学里大部分对象称作‘非大象动物’研究一样可笑。”<br />
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——'''斯塔尼斯拉夫·乌拉姆 Stanislaw Ulam'''<br />
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==Definition 定义==<br />
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In [[mathematics]], a [[linear map]] (or ''linear function'') <math>f(x)</math> is one which satisfies both of the following properties:<br />
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In mathematics, a linear map (or linear function) <math>f(x)</math> is one which satisfies both of the following properties:<br />
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在数学中，线性映射（或线性函数）<math>f (x)</math>满足以下两个性质:<br />
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*Additivity or [[superposition principle]]: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*Homogeneity: <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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*'''可加性 Additivity'''（'''叠加性 Superposition principle'''）: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*'''齐次性 Homogeneity''': <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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Additivity implies homogeneity for any [[rational number|rational]] ''α'', and, for [[continuous function]]s, for any [[real number|real]] ''α''. For a [[complex number|complex]] ''α'', homogeneity does not follow from additivity. For example, an [[antilinear map]] is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle<br />
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Additivity implies homogeneity for any rational α, and, for continuous functions, for any real α. For a complex α, homogeneity does not follow from additivity. For example, an antilinear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle:<br />
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α是有理数，或α是实数且<math>f(x)</math>是连续函数时，由可加性可以推出齐次性。但当α是复数时，可加性不能导出齐次性。例如，反线性映射是可加的，但不是齐次的。可加性和齐次性条件经常组合，称为叠加原理：<br />
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<math>f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)</math><br />
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An equation written as<br />
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对一个写成<br />
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<math>f(x) = C</math><br />
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is called '''linear''' if <math>f(x)</math> is a linear map (as defined above) and '''nonlinear''' otherwise. The equation is called ''homogeneous'' if <math>C = 0</math>.<br />
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is called linear if <math>f(x)</math> is a linear map (as defined above) and nonlinear otherwise. The equation is called homogeneous if <math>C = 0</math>.<br />
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的方程，若 <math> f (x) </math> 是线性映射（如上定义） ，则称其为'''线性的 Linear'''，否则称为'''非线性的 Nonlinear'''。若<math>C = 0</math>，该方程称为是齐次的。<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any [[map (mathematics)|mapping]], including integration or differentiation with associated constraints (such as [[boundary values]]). If <math>f(x)</math> contains [[derivative|differentiation]] with respect to <math>x</math>, the result will be a [[differential equation]].<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any mapping, including integration or differentiation with associated constraints (such as boundary values). If <math>f(x)</math> contains differentiation with respect to <math>x</math>, the result will be a differential equation.<br />
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定义 <math>f(x) = C</math> 是非常具有一般性的，因为 <math>x</math> 可以是任意可感知的数学对象（数字、向量、函数等），函数 <math>f(x)</math> 实际上可以是任意映射，包括有相关约束（如给定边界值）的积分或微分。若 <math>f(x)</math> 包含对 <math>x</math> 的微分运算，则该方程为微分方程。<br />
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===相关概念辨析===<br />
线性、非线性、亚线性 Sublinear、超线性 Superlinear：描述量与量之间的一种变化关系，例如<math>y=a+b*x^n</math>，其中<math>n>1</math>。当<math>n=1</math>时，表示为线性关系；<math>n\neq 1</math>时，表示为非线性关系；当<math>0<n<1</math>时，表示为亚线性关系；当<math>n>1</math>时，表示为超线性关系。<br />
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亚线性与超线性属于非线性变化关系范畴。当两个变量之间存在亚线性关系时，其典型特性是因变量的变化速率会随着自变量的增大而减小，即其一阶导数会随着自变量的增大而减小；两个变量之间存在超线性关系时，其典型特性是因变量的变化速率会随着自变量的增大而增大，即其一阶导数会随着自变量的增大而增大。<br />
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在线性系统中，整体等于部分和，描述线性系统的方程满足叠加原理，作用的总和正好等于每一部分作用相加的代数和，这意味着每一部分作用都是独立的、互不相关的；而在普遍存在的非线性系统中，作用的总和不等于每一部分作用相加的代数和，因为系统内部要素之间存在着复杂的非线性相互作用。<br />
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==Nonlinear algebraic equations 非线性代数方程==<br />
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{{Main|Algebraic equation|System of polynomial equations}}<br />
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Nonlinear [[algebraic equation]]s, which are also called ''[[polynomial equation]]s'', are defined by equating [[polynomial]]s (of degree greater than one) to zero. For example,<br />
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Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials (of degree greater than one) to zero. For example,<br />
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非线性'''代数方程 Algebraic equation'''，又称'''多项式方程 Polynomial equation'''，由某多项式（次数大于1）等于零定义。例如：<br />
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<math>x^2 + x - 1 = 0\,.</math><br />
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For a single polynomial equation, [[root-finding algorithm]]s can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of [[algebraic geometry]], a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see [[Hilbert's Nullstellensatz]]). Nevertheless, in the case of the systems with a finite number of complex solutions, these [[systems of polynomial equations]] are now well understood and efficient methods exist for solving them.<ref>{{cite journal |last1= Lazard |first1= D. |title= Thirty years of Polynomial System Solving, and now? |doi= 10.1016/j.jsc.2008.03.004 |journal= Journal of Symbolic Computation |volume= 44 |issue= 3 |pages= 222–231 |year= 2009 |pmid= |pmc=}}</ref><br />
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For a single polynomial equation, root-finding algorithms can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of algebraic geometry, a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see Hilbert's Nullstellensatz). Nevertheless, in the case of the systems with a finite number of complex solutions, these systems of polynomial equations are now well understood and efficient methods exist for solving them.<br />
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对于一个单一的多项式方程，'''求根算法 Root-finding algorithms'''可用于其求解（即找到满足该方程的变量的值集）。而代数方程组则相对复杂，其研究是现代数学的较难分支——'''代数几何 Algebraic geometry'''领域的动力之一。甚至很难判断一个给定的代数系统是否有复数解（见'''希尔伯特零点定律 Hilbert's Nullstellensatz'''）。不过，对于具有有限个复数解的系统的多项式方程组，我们现在已经有了充分的理解，并且找到了有效的求解方法。<br />
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==Nonlinear recurrence relations 非线性递推关系==<br />
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A nonlinear [[recurrence relation]] defines successive terms of a [[sequence]] as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the [[logistic map]] and the relations that define the various [[Hofstadter sequence]]s. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related [[nonlinear system identification]] and analysis procedures.<ref name="SAB1">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</ref> These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related nonlinear system identification and analysis procedures. These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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非线性递归关系中，序列的连续项被定义为其前项的非线性函数。非线性递归关系的例子有 logistic 映射和定义各种'''霍夫斯塔特序列 Hofstadter sequences''' 的关系。非线性离散模型代表了一类广泛的非线性递归关系，包括 NARMAX（外部输入非线性自回归移动平均）模型和相关的非线性系统辨识和分析程序。这些方法可用于研究时域、频域和时空域的广泛复杂非线性行为。<br />
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==Nonlinear differential equations 非线性微分方程==<br />
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A [[simultaneous equations|system]] of [[differential equation]]s is said to be nonlinear if it is not a [[linear system]]. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the [[Navier–Stokes equations]] in fluid dynamics and the [[Lotka–Volterra equation]]s in biology.<br />
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A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology.<br />
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若一个微分方程组不是线性系统，则称其为非线性的。涉及非线性微分方程的问题非常多样，对不同问题的解决或分析方法也不相同。非线性微分方程的例子有流体力学中的 '''纳维-斯托克斯方程 Navier-Stokes equations'''和生物学中的'''洛特卡－沃尔泰拉方程 Lotka-Volterra equations'''。<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of [[linearly independent]] solutions can be used to construct general solutions through the [[superposition principle]]. A good example of this is one-dimensional heat transport with [[Dirichlet boundary conditions]], the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle. A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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非线性问题最大的困难之一是通常不可能将已知的解组合成新的解。例如，在线性问题中，可以根据叠加原理以一族线性独立的解构造通解。一个很好的例子是带有'''狄利克雷边界条件 Dirichlet boundary conditions'''的一维热传导问题，其解（随时间变化）可以写成不同频率的正弦波的线性组合，这使得解非常灵活。而对非线性方程，通常可以找到几个非常特殊的解，但是此时叠加原理不适用，故无法构造新的解。<br />
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===Ordinary differential equations 常微分方程===<br />
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First order [[ordinary differential equation]]s are often exactly solvable by [[separation of variables]], especially for autonomous equations. For example, the nonlinear equation<br />
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First order ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. For example, the nonlinear equation<br />
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一阶常微分方程，尤其是自治方程，通常可以用'''分离变量法 Separation of variables'''来精确求解。例如，非线性方程<br />
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<math>\frac{d u}{d x} = -u^2</math><br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also ''u'' = 0 as a particular solution, corresponding to the limit of the general solution when ''C'' tends to infinity). The equation is nonlinear because it may be written as<br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also u = 0 as a particular solution, corresponding to the limit of the general solution when C tends to infinity). The equation is nonlinear because it may be written as<br />
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将 <math>u=\frac{1}{x+C}</math> 作为一般解（也有特解 ''u'' = 0，对应于 ''C'' 趋于无穷时的一般解的极限)。该方程是非线性的，因为它可以改写成<br />
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<math>\frac{du}{d x} + u^2=0</math><br />
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and the left-hand side of the equation is not a linear function of ''u'' and its derivatives. Note that if the ''u''<sup>2</sup> term were replaced with ''u'', the problem would be linear (the [[exponential decay]] problem).<br />
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and the left-hand side of the equation is not a linear function of u and its derivatives. Note that if the u<sup>2</sup> term were replaced with u, the problem would be linear (the exponential decay problem).<br />
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方程的左边不是 ''u'' 及其导数的线性函数。注意，若将 ''u''<sup>2</sup> 项替换为''u''，该问题将变为线性的（'''指数衰减 Exponential decay'''问题）。<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield [[closed-form expression|closed-form]] solutions, though implicit solutions and solutions involving [[nonelementary integral]]s are encountered.<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered.<br />
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二阶和高阶常微分方程（更一般地说，非线性方程组）很少能产生封闭解，而隐式解和非初等函数积分形式的解较为常见。<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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非线性常微分方程定性分析的常用方法包括:<br />
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*Examination of any [[conserved quantities]], especially in [[Hamiltonian system]]s<br />
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*检查是否有任意'''守恒量 Conserved quantities'''（特别是在'''哈密顿系统 Hamiltonian system'''中）<br />
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*Examination of dissipative quantities (see [[Lyapunov function]]) analogous to conserved quantities<br />
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*检查是否有类似守恒量的耗散量（见'''李亚普诺夫函数 Lyapunov function'''）<br />
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*Linearization via [[Taylor expansion]]<br />
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*基于'''泰勒展开 Taylor expansion'''的线性化<br />
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*Change of variables into something easier to study<br />
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*将变量转化得更易于研究<br />
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*[[Bifurcation theory]]<br />
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*'''分岔理论 Bifurcation theory'''<br />
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*[[Perturbation theory|Perturbation]] methods (can be applied to algebraic equations too)<br />
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*'''摄动理论 Perturbation theory'''（也可应用于代数方程）<br />
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===Partial differential equations 偏微分方程===<br />
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{{main|Nonlinear partial differential equation}}<br />
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{{See also|List of nonlinear partial differential equations}}<br />
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参见：非线性偏微分列表<br />
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The most common basic approach to studying nonlinear [[partial differential equation]]s is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more [[ordinary differential equation]]s, as seen in [[separation of variables]], which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
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The most common basic approach to studying nonlinear partial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
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研究非线性偏微分方程最常用的基本方法是变换变量（或转换问题），使变换后的问题更简单（甚至可能变为线性的）。有时可以将此类方程转化成一或多个常微分方程（如同分离变量法所示），此时不论得到的常微分方程是否可解，对研究问题总是有用的。<br />
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Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use [[scale analysis (mathematics)|scale analysis]] to simplify a general, natural equation in a certain specific [[boundary value problem]]. For example, the (very) nonlinear [[Navier-Stokes equations]] can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
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Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem. For example, the (very) nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
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另一个流体力学和热力学中常见的策略（虽然不是数学上的）是利用'''尺度分析 Scale analysis'''来在某一特定边界条件下简化一般自然方程。例如，在描述圆管内一维层流的暂态时，非线性的纳维－斯托克斯方程可以简化为一个线性的偏微分方程; 尺度分析提供了层流和一维流动的条件，也产生了简化的方程。<br />
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Other methods include examining the [[method of characteristics|characteristics]] and using the methods outlined above for ordinary differential equations.<br />
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Other methods include examining the characteristics and using the methods outlined above for ordinary differential equations.<br />
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其他方法包括检查特征线法及前面所述研究常微分方程的方法。<br />
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===Pendula 摆===<br />
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{{Main|Pendulum (mathematics)}}<br />
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[[File:PendulumLayout.svg.png|thumb|图一：Illustration of a pendulum 摆图解|right|200px]]<br />
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[[File:PendulumLinearizations.png|thumb|图二：Linearizations of a pendulum 摆的线性化|right|200px]]<br />
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A classic, extensively studied nonlinear problem is the dynamics of a [[pendulum (mathematics)|pendulum]] under the influence of [[gravity]]. Using [[Lagrangian mechanics]], it may be shown<ref>[http://www.damtp.cam.ac.uk/user/tong/dynamics.html David Tong: Lectures on Classical Dynamics]</ref> that the motion of a pendulum can be described by the [[dimensionless]] nonlinear equation<br />
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A classic, extensively studied nonlinear problem is the dynamics of a pendulum under the influence of gravity. Using Lagrangian mechanics, it may be shown that the motion of a pendulum can be described by the dimensionless nonlinear equation<br />
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一个经典的被广泛研究的非线性问题是重力影响下的摆的动力学。利用'''拉格朗日力学 Lagrangian mechanics'''，可以证明摆的运动可以用无量纲的非线性方程<br />
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<math>\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0</math><br />
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where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an [[integrating factor]], which would eventually yield<br />
<br />
where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an integrating factor, which would eventually yield<br />
<br />
描述，其中重力指向“下方”，<math>\theta</math> 是摆与其静止位置形成的角度，如右图所示。“解”这个方程的方法之一是用 <math>d\theta/dt</math> 作为'''积分因子 Integrating factor'''，最终得<br />
<br />
<math>\int{\frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)}}} = t + C_1</math><br />
<br />
<br />
<br />
<br />
<br />
which is an implicit solution involving an [[elliptic integral]]. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the [[nonelementary integral]] (nonelementary unless <math>C_0 = 2</math>).<br />
<br />
which is an implicit solution involving an elliptic integral. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral (nonelementary unless <math>C_0 = 2</math>).<br />
<br />
这是一个含'''椭圆积分 Elliptic integral'''的隐式解。这个“解”通常没什么用，因为这个解的大部分性质都隐藏在非初等函数积分中（除非<math>C_0 = 2</math>，否则是非初等的）。<br />
<br />
<br />
<br />
<br />
<br />
Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through [[Taylor expansion]]s. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
<br />
Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through Taylor expansions. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
<br />
另一种解决这个问题的方法是利用泰勒展开将任意非线性项（此时为正弦函数项)在某些感兴趣的点线性化。例如，在<math>\theta = 0</math> 的点附近线性化（称为小角度近似）为<br />
<br />
<math>\frac{d^2 \theta}{d t^2} + \theta = 0</math><br />
<br />
<br />
<br />
<br />
<br />
since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a [[simple harmonic oscillator]] corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
<br />
since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
<br />
因为 <math>\theta \approx 0</math> 时，有 <math>\sin(\theta) \approx \theta</math>。这是一个'''简谐振子 Simple harmonic oscillator''' ，对应于摆在其路径底部附近的摆动。另一种线性化方法是在 <math>\theta = \pi</math>附近线性化，对应于运动到最高点的摆:<br />
<br />
<math>\frac{d^2 \theta}{d t^2} + \pi - \theta = 0</math><br />
<br />
<br />
<br />
<br />
<br />
since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves [[hyperbolic sinusoid]]s, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
<br />
since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
<br />
因为 <math>\theta \approx \pi</math> 时，有 <math>\sin(\theta) \approx \pi - \theta</math>。这个问题的解含双曲正弦曲线；注意到不同于小角度近似，它是不稳定的，这意味着 <math>|\theta|</math> 通常会无限增长（但解也有可能是有界的）。这就解释了摆在最高点达到平衡的困难，此时实际上是一种不稳定的状态。<br />
<br />
<br />
<br />
<br />
<br />
One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
<br />
One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
<br />
一个更有趣的线性化可能是在 <math>\theta = \pi/2</math>附近，此时 <math>\sin(\theta) \approx 1</math>:<br />
<br />
<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math><br />
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<br />
<br />
<br />
This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) [[phase portrait]]s and approximate periods.<br />
<br />
This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) phase portraits and approximate periods.<br />
<br />
这相当于一个自由落体问题。把这样线性化的结果合在一起看，就能得到有关摆的运动的非常有用的图像，如右图所示。利用其他方法寻找（精确的）'''相图 Phase portrait'''和估计周期。<br />
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==Types of nonlinear dynamic behaviors 非线性动力学行为的类型==<br />
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*[[Amplitude death]] – any oscillations present in the system cease due to some kind of interaction with other system or feedback by the same system<br />
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*'''振幅死亡 Amplitude death'''——系统内的某振荡因系统的自回馈或与其他系统的某种相互作用而停止的现象<br />
<br />
*[[Chaos theory|Chaos]] – values of a system cannot be predicted indefinitely far into the future, and fluctuations are [[aperiodic]]<br />
<br />
*'''混沌 Chaos'''——系统内的值无法无限期地预测到遥远的未来；波动是非周期性的<br />
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*[[Multistability]] – the presence of two or more stable states<br />
<br />
*'''多稳态 Multistability'''——两或多个稳态的存在<br />
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*[[Soliton]]s – self-reinforcing solitary waves<br />
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*'''孤波 Soliton'''s——自增强的孤立波<br />
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*[[Limit cycle|Limit cycles]] – asymptotic periodic orbits to which destabilized fixed points are attracted. <br />
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*'''极限环 Limit cycles'''——吸引不稳定不动点的渐近周期轨道<br />
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*[[Self-oscillation|Self-oscillations]] - feedback oscillations taking place in open dissipative physical systems.<br />
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*'''自激振荡 Self-oscillations'''——开放耗散物理系统中的反馈振荡<br />
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==Examples of nonlinear equations 非线性方程的例子==<br />
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{{Div col|colwidth=25em}}<br />
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*[[Algebraic Riccati equation]] 代数黎卡提方程<br />
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*[[Ball and beam]] system 球杆系统<br />
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*[[Bellman equation]] for optimal policy 最佳策略的贝尔曼方程<br />
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*[[Boltzmann equation]] 玻尔兹曼方程<br />
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*[[Colebrook equation]] 科尔布鲁克方程<br />
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*[[General relativity]] 广义相对论<br />
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*[[Ginzburg–Landau theory]] 金兹堡-朗道方程<br />
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<br />
*[[Ishimori equation]] 石森方程<br />
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<br />
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*[[Kadomtsev–Petviashvili equation]] 卡东穆塞夫-彼得韦亚斯维利方程<br />
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<br />
<br />
*[[Korteweg–de Vries equation]] kdV方程<br />
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<br />
*[[Landau–Lifshitz–Gilbert equation]] 朗道-利夫希兹-吉尔伯特方程<br />
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<br />
<br />
*[[Liénard equation]] 林纳德方程<br />
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<br />
<br />
*[[Navier–Stokes equations]] of [[fluid dynamics]] 流体力学的纳维－斯托克斯方程<br />
<br />
<br />
<br />
*[[Nonlinear optics]] 非线性光学<br />
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*[[Nonlinear Schrödinger equation]] 非线性薛定谔方程<br />
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*[[Power-flow study]] 功率流研究<br />
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<br />
<br />
*[[Richards equation]] for unsaturated water flow 未饱和层水流的理查氏方程<br />
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*[[Self-balancing unicycle]]<br />
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*[[Sine-Gordon equation]] 正弦-戈尔登方程<br />
<br />
<br />
<br />
*[[Van der Pol oscillator]] 范德波尔振荡器<br />
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*[[Vlasov equation]] 弗拉索夫方程<br />
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<br />
{{Div col end}}<br />
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==See also 参见==<br />
<br />
<br />
*[[Aleksandr Mikhailovich Lyapunov]] 亚历山大·李亚普诺夫<br />
<br />
<br />
<br />
*[[Dynamical system]] 动态系统<br />
<br />
<br />
<br />
*[[Feedback]] 反馈<br />
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*[[Initial condition]] 初始条件<br />
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<br />
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*[[Interaction]] 相互作用<br />
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*[[Linear system]] 线性系统<br />
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<br />
<br />
*[[Mode coupling]] 模式耦合<br />
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<br />
*[[Vector soliton]] 矢量孤子<br />
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*[[Volterra series]] 沃尔泰拉级数<br />
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<br />
==References 参考资料==<br />
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<br />
{{Reflist|35em}}<br />
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==Further reading 进一步阅读==<br />
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<br />
{{Refbegin|35em}}<br />
<br />
<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| author= [[Diederich Hinrichsen]] and Anthony J. Pritchard<br />
<br />
| author= Diederich Hinrichsen and Anthony J. Pritchard<br />
<br />
作者 Diederich Hinrichsen 和 Anthony j. Pritchard<br />
<br />
| year= 2005<br />
<br />
| year= 2005<br />
<br />
2005年<br />
<br />
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
<br />
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
<br />
数学系统理论 i-建模，状态空间分析，稳定性和鲁棒性<br />
<br />
| publisher= Springer Verlag<br />
<br />
| publisher= Springer Verlag<br />
<br />
出版商 Springer Verlag<br />
<br />
| isbn=9783540441250<br />
<br />
| isbn=9783540441250<br />
<br />
9783540441250<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Jordan<br />
<br />
| last= Jordan<br />
<br />
最后一个乔丹<br />
<br />
| first= D. W.<br />
<br />
| first= D. W.<br />
<br />
首先是 D.w。<br />
<br />
| last2= Smith<br />
<br />
| last2= Smith<br />
<br />
最后2个史密斯<br />
<br />
| first2= P.<br />
<br />
| first2= P.<br />
<br />
| first2 p.<br />
<br />
| year= 2007<br />
<br />
| year= 2007<br />
<br />
2007年<br />
<br />
| edition= fourth<br />
<br />
| edition= fourth<br />
<br />
第四版<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
非线性常微分方程<br />
<br />
| publisher= Oxford University Press<br />
<br />
| publisher= Oxford University Press<br />
<br />
牛津大学出版社<br />
<br />
| isbn= 978-0-19-920824-1<br />
<br />
| isbn= 978-0-19-920824-1<br />
<br />
[国际标准图书编号978-0-19-920824-1]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Khalil<br />
<br />
| last= Khalil<br />
<br />
最后的哈利勒<br />
<br />
| first= Hassan K.<br />
<br />
| first= Hassan K.<br />
<br />
首先是哈桑 · k。<br />
<br />
| year= 2001<br />
<br />
| year= 2001<br />
<br />
2001年<br />
<br />
| title= Nonlinear Systems<br />
<br />
| title= Nonlinear Systems<br />
<br />
非线性系统<br />
<br />
| publisher= Prentice Hall<br />
<br />
| publisher= Prentice Hall<br />
<br />
出版商 Prentice Hall<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
[国际标准图书编号978-0-13-067389-3]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Kreyszig<br />
<br />
| last= Kreyszig<br />
<br />
| last= Kreyszig<br />
<br />
| first= Erwin<br />
<br />
| first= Erwin<br />
<br />
首先是欧文<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
作者 / 链接 Erwin Kreyszig<br />
<br />
| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
高等工程数学<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
Https://archive.org/details/advancedengineer0008krey<br />
<br />
| url-access= registration<br />
<br />
| url-access= registration<br />
<br />
访问注册<br />
<br />
| publisher= Wiley<br />
<br />
| publisher= Wiley<br />
<br />
出版商 Wiley<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
[国际标准图书编号978-0-471-15496-9]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Sontag<br />
<br />
| last= Sontag<br />
<br />
最后一个桑塔格<br />
<br />
| first= Eduardo<br />
<br />
| first= Eduardo<br />
<br />
首先是爱德华多<br />
<br />
| authorlink= Eduardo D. Sontag<br />
<br />
| authorlink= Eduardo D. Sontag<br />
<br />
作者: Eduardo d. Sontag<br />
<br />
| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
数学控制理论: 确定性有限维系统。第二版<br />
<br />
| publisher= Springer<br />
<br />
| publisher= Springer<br />
<br />
出版商斯普林格<br />
<br />
| isbn= 978-0-387-98489-6<br />
<br />
| isbn= 978-0-387-98489-6<br />
<br />
[国际标准图书编号978-0-387-98489-6]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
{{Refend}}<br />
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<br />
<br />
<br />
==External links 外部链接==<br />
<br />
<br />
<br />
*[http://www.dodccrp.org/ Command and Control Research Program (CCRP)]<br />
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<br />
*[http://necsi.edu/guide/concepts/linearnonlinear.html New England Complex Systems Institute: Concepts in Complex Systems]<br />
<br />
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<br />
*[http://ocw.mit.edu/courses/mathematics/18-353j-nonlinear-dynamics-i-chaos-fall-2012/ Nonlinear Dynamics I: Chaos] at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]<br />
<br />
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<br />
*[http://www.hedengren.net/research/models.htm Nonlinear Model Library]{{snd}} (in [[MATLAB]]) a Database of Physical Systems<br />
<br />
<br />
<br />
*[http://cnls.lanl.gov/ The Center for Nonlinear Studies at Los Alamos National Laboratory]<br />
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{{Systems science}}<br />
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{{Differential equations topics}}<br />
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<br />
{{Authority control}}<br />
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==编者推荐==<br />
集智学园：[https://campus.swarma.org/course/697 非线性动力学与混沌]<br />
<br />
非线性动力学和混沌理论是系统发展的，从一阶微分方程及其分岔开始，然后是相平面分析，极限环和它们的分岔，最终得到Lorenz方程，混沌，迭代映射，周期倍增，重整化，分形和奇怪吸引。<br />
<br />
<br />
此系列课程由[[斯蒂文·斯特罗加茨 Steven H. Strogatz]]主持，内容包括机械振动，激光，生物节律，超导电路，昆虫爆发，化学振荡器，遗传控制系统，混沌水轮，甚至是使用混乱发送秘密信息的技术。在每种情况下，科学背景都在初级阶段进行解释，并与数学理论紧密结合。<br />
<br />
[[Category:Nonlinear systems| ]]<br />
<br />
<br />
<br />
[[Category:Dynamical systems]]<br />
<br />
Category:Dynamical systems<br />
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类别: 动力系统<br />
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[[Category:Concepts in physics]]<br />
<br />
Category:Concepts in physics<br />
<br />
分类: 物理概念<br />
<br />
<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Nonlinear system]]. Its edit history can be viewed at [[非线性系统/edithistory]]</small></noinclude><br />
<br />
[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%9D%9E%E7%BA%BF%E6%80%A7%E7%B3%BB%E7%BB%9F&diff=10061非线性系统2020-07-10T03:34:17Z<p>Dorr：</p>
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{{About|"nonlinearity" in mathematics, physics and other sciences|video and film editing|Non-linear editing system|other uses|Nonlinearity (disambiguation)}}<br />
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In [[mathematics]] and [[science]], a '''nonlinear system''' is a [[system]] in which the change of the output is not [[proportionality (mathematics)|proportional]] to the change of the input.<ref>{{Cite news|url=https://news.mit.edu/2010/explained-linear-0226|title=Explained: Linear and nonlinear systems|work=MIT News|access-date=2018-06-30}}</ref><ref>{{Cite web|url=https://www.birmingham.ac.uk/research/activity/mathematics/applied-maths/nonlinear-systems.aspx|title=Nonlinear systems, Applied Mathematics - University of Birmingham|website=www.birmingham.ac.uk|language=en-gb|access-date=2018-06-30}}</ref> Nonlinear problems are of interest to [[engineer]]s, [[biologist]]s,<ref>{{Citation|date=2007|pages=181–276|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-540-34153-6_7|isbn=9783540341529|title = The Nonlinear Universe|series = The Frontiers Collection|chapter = Nonlinear Biology}}</ref><ref>{{cite journal|last=Korenberg|first=Michael J.|last2=Hunter|first2=Ian W.|date=March 1996|title=The identification of nonlinear biological systems: Volterra kernel approaches|journal=Annals of Biomedical Engineering|language=en|volume=24|issue=2|pages=250–268|doi=10.1007/bf02667354|issn=0090-6964}}</ref><ref>{{cite journal|last=Mosconi|first=Francesco|last2=Julou|first2=Thomas|last3=Desprat|first3=Nicolas|last4=Sinha|first4=Deepak Kumar|last5=Allemand|first5=Jean-François|last6=Vincent Croquette|last7=Bensimon|first7=David|date=2008|title=Some nonlinear challenges in biology|url=http://stacks.iop.org/0951-7715/21/i=8/a=T03|journal=Nonlinearity|language=en|volume=21|issue=8|pages=T131|doi=10.1088/0951-7715/21/8/T03|issn=0951-7715|bibcode=2008Nonli..21..131M}}</ref> [[physicist]]s,<ref>{{cite journal|last1=Gintautas|first1=V.|title=Resonant forcing of nonlinear systems of differential equations|journal=Chaos|date=2008|volume=18|issue=3|pages=033118|doi=10.1063/1.2964200|pmid=19045456|arxiv=0803.2252|bibcode=2008Chaos..18c3118G}}</ref><ref>{{cite journal|last1=Stephenson|first1=C.|last2=et.|first2=al.|title=Topological properties of a self-assembled electrical network via ab initio calculation|journal=Sci. Rep.|volume=7|pages=41621|date=2017|doi=10.1038/srep41621|pmid=28155863|pmc=5290745|bibcode=2017NatSR...741621S}}</ref> [[mathematician]]s, and many other [[scientist]]s because most systems are inherently nonlinear in nature.<ref>{{cite book|last1=de Canete|first1=Javier, Cipriano Galindo, and Inmaculada Garcia-Moral|title=System Engineering and Automation: An Interactive Educational Approach|date=2011|publisher=Springer|location=Berlin|isbn=978-3642202292|page=46|url=https://books.google.com/?id=h8rCQYXGGY8C&pg=PA46&lpg=PA46&dq=most+systems+are+inherently+nonlinear+in+nature#v=onepage&q=most%20systems%20are%20inherently%20nonlinear%20in%20nature&f=false|accessdate=20 January 2018}}</ref> Nonlinear [[dynamical system]]s, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler [[linear system]]s.<br />
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In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.<br />
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在数学及科学中，非线性系统是一种输出的变化与输入的变化不成比例的系统。大多数系统在本质上是非线性的，因而非线性问题引起了工程师、生物学家、物理学家、数学家和许多其他科学家的兴趣。描述变量随时间变化的非线性动力系统与较之简单得多的线性系统相比，可能显得混沌、不可预测或违反直觉。<br />
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Typically, the behavior of a nonlinear system is described in mathematics by a '''nonlinear system of equations''', which is a set of simultaneous [[equation]]s in which the unknowns (or the unknown functions in the case of [[differential equation]]s) appear as variables of a [[polynomial]] of degree higher than one or in the argument of a [[function (mathematics)|function]] which is not a polynomial of degree one.<br />
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Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one.<br />
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通常，非线性系统的行为在数学上是由一组非线性联立方程来描述的，其中未知数（或微分方程中的未知函数）作为一个高于一次的多项式的变量出现，或者作为一个不是一次多项式的函数的论元出现。__[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）+后半句不知如何翻译<br />
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In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a [[linear combination]] of the unknown [[variable (mathematics)|variables]] or [[function (mathematics)|functions]] that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is ''linear'' if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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换句话说，在非线性方程系统中，要求解的方程不能被写成未知变量或函数的线性组合。无论方程中是否有已知的线性函数，系统都可以被定义为非线性。特别是，如果一个微分方程的未知函数及其导数是线性的，即使其他变量是非线性的，也称该方程是线性的。<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations ([[linearization]]). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as [[soliton]]s, [[chaos theory|chaos]],<ref>[http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm Nonlinear Dynamics I: Chaos] {{webarchive|url=https://web.archive.org/web/20080212045134/http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm |date=2008-02-12}} at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]</ref> and [[mathematical singularity|singularities]] are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble [[randomness|random]] behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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由于非线性动力学方程难以求解，通常用线性化方程来近似非线性系统（'''线性化 Linearization'''）。这种方法在一定的精度和范围对输入值效果很好，但一些有趣的现象如'''孤子 Solitons'''、'''混沌 Chaos'''和'''奇异性 Singularities'''在线性化后被隐藏。因此，非线性系统的动态行为在某些方面可能看起来违反直觉、不可预测，甚至混沌。尽管这种混沌行为可能感觉很像随机行为，但它实际上并不是随机的。例如，天气的某些方面被认为是混沌的，其系统某部分的微小扰动就会产生复杂的影响。这种非线性是目前技术无法进行精确长期预测的原因之一。<br />
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Some authors use the term '''nonlinear science''' for the study of nonlinear systems. This term is disputed by others:<br />
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Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:<br />
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有些作者用非线性科学这一术语来研究非线性系统。这一术语引起了其他人的争议:<br />
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{{quote|Using a term like nonlinear science is like referring to the bulk of zoology as the study of [[negation|non]]-elephant animals.|[[Stanislaw Ulam]]<ref>{{cite journal|last1=Campbell|first1=David K.|title=Nonlinear physics: Fresh breather|journal=Nature|date=25 November 2004|volume=432|issue=7016|pages=455–456|doi=10.1038/432455a|pmid=15565139|url=https://zenodo.org/record/1134179|language=en|issn=0028-0836|bibcode=2004Natur.432..455C}}</ref>}}<br />
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“使用‘非线性科学’这样的术语，就如同把动物学里大部分对象称作‘非大象动物’研究一样可笑。”<br />
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——'''斯塔尼斯拉夫·乌拉姆 Stanislaw Ulam'''<br />
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==Definition 定义==<br />
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In [[mathematics]], a [[linear map]] (or ''linear function'') <math>f(x)</math> is one which satisfies both of the following properties:<br />
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In mathematics, a linear map (or linear function) <math>f(x)</math> is one which satisfies both of the following properties:<br />
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在数学中，线性映射（或线性函数）<math>f (x)</math>满足以下两个性质:<br />
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*Additivity or [[superposition principle]]: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*Homogeneity: <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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*'''可加性 Additivity'''（'''叠加性 Superposition principle'''）: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*'''齐次性 Homogeneity''': <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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Additivity implies homogeneity for any [[rational number|rational]] ''α'', and, for [[continuous function]]s, for any [[real number|real]] ''α''. For a [[complex number|complex]] ''α'', homogeneity does not follow from additivity. For example, an [[antilinear map]] is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle<br />
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Additivity implies homogeneity for any rational α, and, for continuous functions, for any real α. For a complex α, homogeneity does not follow from additivity. For example, an antilinear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle:<br />
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α是有理数，或α是实数且<math>f(x)</math>是连续函数时，由可加性可以推出齐次性。但当α是复数时，可加性不能导出齐次性。例如，反线性映射是可加的，但不是齐次的。可加性和齐次性条件经常组合，称为叠加原理：<br />
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<math>f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)</math><br />
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An equation written as<br />
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对一个写成<br />
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<math>f(x) = C</math><br />
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is called '''linear''' if <math>f(x)</math> is a linear map (as defined above) and '''nonlinear''' otherwise. The equation is called ''homogeneous'' if <math>C = 0</math>.<br />
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is called linear if <math>f(x)</math> is a linear map (as defined above) and nonlinear otherwise. The equation is called homogeneous if <math>C = 0</math>.<br />
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的方程，若 <math> f (x) </math> 是线性映射（如上定义） ，则称其为'''线性的 Linear'''，否则称为'''非线性的 Nonlinear'''。若<math>C = 0</math>，该方程称为是齐次的。<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any [[map (mathematics)|mapping]], including integration or differentiation with associated constraints (such as [[boundary values]]). If <math>f(x)</math> contains [[derivative|differentiation]] with respect to <math>x</math>, the result will be a [[differential equation]].<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any mapping, including integration or differentiation with associated constraints (such as boundary values). If <math>f(x)</math> contains differentiation with respect to <math>x</math>, the result will be a differential equation.<br />
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定义 <math>f(x) = C</math> 是非常具有一般性的，因为 <math>x</math> 可以是任意可感知的数学对象（数字、向量、函数等），函数 <math>f(x)</math> 实际上可以是任意映射，包括有相关约束（如给定边界值）的积分或微分。若 <math>f(x)</math> 包含对 <math>x</math> 的微分运算，则该方程为微分方程。<br />
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==Nonlinear algebraic equations 非线性代数方程==<br />
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{{Main|Algebraic equation|System of polynomial equations}}<br />
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Nonlinear [[algebraic equation]]s, which are also called ''[[polynomial equation]]s'', are defined by equating [[polynomial]]s (of degree greater than one) to zero. For example,<br />
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Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials (of degree greater than one) to zero. For example,<br />
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非线性'''代数方程 Algebraic equation'''，又称'''多项式方程 Polynomial equation'''，由某多项式（次数大于1）等于零定义。例如：<br />
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<math>x^2 + x - 1 = 0\,.</math><br />
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For a single polynomial equation, [[root-finding algorithm]]s can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of [[algebraic geometry]], a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see [[Hilbert's Nullstellensatz]]). Nevertheless, in the case of the systems with a finite number of complex solutions, these [[systems of polynomial equations]] are now well understood and efficient methods exist for solving them.<ref>{{cite journal |last1= Lazard |first1= D. |title= Thirty years of Polynomial System Solving, and now? |doi= 10.1016/j.jsc.2008.03.004 |journal= Journal of Symbolic Computation |volume= 44 |issue= 3 |pages= 222–231 |year= 2009 |pmid= |pmc=}}</ref><br />
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For a single polynomial equation, root-finding algorithms can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of algebraic geometry, a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see Hilbert's Nullstellensatz). Nevertheless, in the case of the systems with a finite number of complex solutions, these systems of polynomial equations are now well understood and efficient methods exist for solving them.<br />
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对于一个单一的多项式方程，'''求根算法 Root-finding algorithms'''可用于其求解（即找到满足该方程的变量的值集）。而代数方程组则相对复杂，其研究是现代数学的较难分支——'''代数几何 Algebraic geometry'''领域的动力之一。甚至很难判断一个给定的代数系统是否有复数解（见'''希尔伯特零点定律 Hilbert's Nullstellensatz'''）。不过，对于具有有限个复数解的系统的多项式方程组，我们现在已经有了充分的理解，并且找到了有效的求解方法。<br />
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==Nonlinear recurrence relations 非线性递推关系==<br />
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A nonlinear [[recurrence relation]] defines successive terms of a [[sequence]] as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the [[logistic map]] and the relations that define the various [[Hofstadter sequence]]s. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related [[nonlinear system identification]] and analysis procedures.<ref name="SAB1">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</ref> These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related nonlinear system identification and analysis procedures. These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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非线性递归关系中，序列的连续项被定义为其前项的非线性函数。非线性递归关系的例子有 logistic 映射和定义各种'''霍夫斯塔特序列 Hofstadter sequences''' 的关系。非线性离散模型代表了一类广泛的非线性递归关系，包括 NARMAX（外部输入非线性自回归移动平均）模型和相关的非线性系统辨识和分析程序。这些方法可用于研究时域、频域和时空域的广泛复杂非线性行为。<br />
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==Nonlinear differential equations 非线性微分方程==<br />
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A [[simultaneous equations|system]] of [[differential equation]]s is said to be nonlinear if it is not a [[linear system]]. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the [[Navier–Stokes equations]] in fluid dynamics and the [[Lotka–Volterra equation]]s in biology.<br />
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A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology.<br />
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若一个微分方程组不是线性系统，则称其为非线性的。涉及非线性微分方程的问题非常多样，对不同问题的解决或分析方法也不相同。非线性微分方程的例子有流体力学中的 '''纳维-斯托克斯方程 Navier-Stokes equations'''和生物学中的'''洛特卡－沃尔泰拉方程 Lotka-Volterra equations'''。<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of [[linearly independent]] solutions can be used to construct general solutions through the [[superposition principle]]. A good example of this is one-dimensional heat transport with [[Dirichlet boundary conditions]], the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle. A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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非线性问题最大的困难之一是通常不可能将已知的解组合成新的解。例如，在线性问题中，可以根据叠加原理以一族线性独立的解构造通解。一个很好的例子是带有'''狄利克雷边界条件 Dirichlet boundary conditions'''的一维热传导问题，其解（随时间变化）可以写成不同频率的正弦波的线性组合，这使得解非常灵活。而对非线性方程，通常可以找到几个非常特殊的解，但是此时叠加原理不适用，故无法构造新的解。<br />
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===Ordinary differential equations 常微分方程===<br />
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First order [[ordinary differential equation]]s are often exactly solvable by [[separation of variables]], especially for autonomous equations. For example, the nonlinear equation<br />
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First order ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. For example, the nonlinear equation<br />
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一阶常微分方程，尤其是自治方程，通常可以用'''分离变量法 Separation of variables'''来精确求解。例如，非线性方程<br />
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<math>\frac{d u}{d x} = -u^2</math><br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also ''u'' = 0 as a particular solution, corresponding to the limit of the general solution when ''C'' tends to infinity). The equation is nonlinear because it may be written as<br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also u = 0 as a particular solution, corresponding to the limit of the general solution when C tends to infinity). The equation is nonlinear because it may be written as<br />
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将 <math>u=\frac{1}{x+C}</math> 作为一般解（也有特解 ''u'' = 0，对应于 ''C'' 趋于无穷时的一般解的极限)。该方程是非线性的，因为它可以改写成<br />
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<math>\frac{du}{d x} + u^2=0</math><br />
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and the left-hand side of the equation is not a linear function of ''u'' and its derivatives. Note that if the ''u''<sup>2</sup> term were replaced with ''u'', the problem would be linear (the [[exponential decay]] problem).<br />
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and the left-hand side of the equation is not a linear function of u and its derivatives. Note that if the u<sup>2</sup> term were replaced with u, the problem would be linear (the exponential decay problem).<br />
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方程的左边不是 ''u'' 及其导数的线性函数。注意，若将 ''u''<sup>2</sup> 项替换为''u''，该问题将变为线性的（'''指数衰减 Exponential decay'''问题）。<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield [[closed-form expression|closed-form]] solutions, though implicit solutions and solutions involving [[nonelementary integral]]s are encountered.<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered.<br />
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二阶和高阶常微分方程（更一般地说，非线性方程组）很少能产生封闭解，而隐式解和非初等函数积分形式的解较为常见。<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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非线性常微分方程定性分析的常用方法包括:<br />
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*Examination of any [[conserved quantities]], especially in [[Hamiltonian system]]s<br />
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*检查是否有任意'''守恒量 Conserved quantities'''（特别是在'''哈密顿系统 Hamiltonian system'''中）<br />
<br />
*Examination of dissipative quantities (see [[Lyapunov function]]) analogous to conserved quantities<br />
<br />
*检查是否有类似守恒量的耗散量（见'''李亚普诺夫函数 Lyapunov function'''）<br />
<br />
*Linearization via [[Taylor expansion]]<br />
<br />
*基于'''泰勒展开 Taylor expansion'''的线性化<br />
<br />
*Change of variables into something easier to study<br />
<br />
*将变量转化得更易于研究<br />
<br />
*[[Bifurcation theory]]<br />
<br />
*'''分岔理论 Bifurcation theory'''<br />
<br />
*[[Perturbation theory|Perturbation]] methods (can be applied to algebraic equations too)<br />
<br />
*'''摄动理论 Perturbation theory'''（也可应用于代数方程）<br />
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<br />
<br />
===Partial differential equations 偏微分方程===<br />
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<br />
{{main|Nonlinear partial differential equation}}<br />
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<br />
{{See also|List of nonlinear partial differential equations}}<br />
<br />
参见：非线性偏微分列表<br />
<br />
The most common basic approach to studying nonlinear [[partial differential equation]]s is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more [[ordinary differential equation]]s, as seen in [[separation of variables]], which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
<br />
The most common basic approach to studying nonlinear partial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
<br />
研究非线性偏微分方程最常用的基本方法是变换变量（或转换问题），使变换后的问题更简单（甚至可能变为线性的）。有时可以将此类方程转化成一或多个常微分方程（如同分离变量法所示），此时不论得到的常微分方程是否可解，对研究问题总是有用的。<br />
<br />
<br />
<br />
<br />
<br />
Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use [[scale analysis (mathematics)|scale analysis]] to simplify a general, natural equation in a certain specific [[boundary value problem]]. For example, the (very) nonlinear [[Navier-Stokes equations]] can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
<br />
Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem. For example, the (very) nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
<br />
另一个流体力学和热力学中常见的策略（虽然不是数学上的）是利用'''尺度分析 Scale analysis'''来在某一特定边界条件下简化一般自然方程。例如，在描述圆管内一维层流的暂态时，非线性的纳维－斯托克斯方程可以简化为一个线性的偏微分方程; 尺度分析提供了层流和一维流动的条件，也产生了简化的方程。<br />
<br />
<br />
<br />
<br />
<br />
Other methods include examining the [[method of characteristics|characteristics]] and using the methods outlined above for ordinary differential equations.<br />
<br />
Other methods include examining the characteristics and using the methods outlined above for ordinary differential equations.<br />
<br />
其他方法包括检查特征线法及前面所述研究常微分方程的方法。<br />
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<br />
===Pendula 摆===<br />
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{{Main|Pendulum (mathematics)}}<br />
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[[File:PendulumLayout.svg.png|thumb|图一：Illustration of a pendulum 摆图解|right|200px]]<br />
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[[File:PendulumLinearizations.png|thumb|图二：Linearizations of a pendulum 摆的线性化|right|200px]]<br />
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A classic, extensively studied nonlinear problem is the dynamics of a [[pendulum (mathematics)|pendulum]] under the influence of [[gravity]]. Using [[Lagrangian mechanics]], it may be shown<ref>[http://www.damtp.cam.ac.uk/user/tong/dynamics.html David Tong: Lectures on Classical Dynamics]</ref> that the motion of a pendulum can be described by the [[dimensionless]] nonlinear equation<br />
<br />
A classic, extensively studied nonlinear problem is the dynamics of a pendulum under the influence of gravity. Using Lagrangian mechanics, it may be shown that the motion of a pendulum can be described by the dimensionless nonlinear equation<br />
<br />
一个经典的被广泛研究的非线性问题是重力影响下的摆的动力学。利用'''拉格朗日力学 Lagrangian mechanics'''，可以证明摆的运动可以用无量纲的非线性方程<br />
<br />
<math>\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0</math><br />
<br />
<br />
<br />
<br />
<br />
where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an [[integrating factor]], which would eventually yield<br />
<br />
where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an integrating factor, which would eventually yield<br />
<br />
描述，其中重力指向“下方”，<math>\theta</math> 是摆与其静止位置形成的角度，如右图所示。“解”这个方程的方法之一是用 <math>d\theta/dt</math> 作为'''积分因子 Integrating factor'''，最终得<br />
<br />
<math>\int{\frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)}}} = t + C_1</math><br />
<br />
<br />
<br />
<br />
<br />
which is an implicit solution involving an [[elliptic integral]]. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the [[nonelementary integral]] (nonelementary unless <math>C_0 = 2</math>).<br />
<br />
which is an implicit solution involving an elliptic integral. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral (nonelementary unless <math>C_0 = 2</math>).<br />
<br />
这是一个含'''椭圆积分 Elliptic integral'''的隐式解。这个“解”通常没什么用，因为这个解的大部分性质都隐藏在非初等函数积分中（除非<math>C_0 = 2</math>，否则是非初等的）。<br />
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Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through [[Taylor expansion]]s. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
<br />
Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through Taylor expansions. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
<br />
另一种解决这个问题的方法是利用泰勒展开将任意非线性项（此时为正弦函数项)在某些感兴趣的点线性化。例如，在<math>\theta = 0</math> 的点附近线性化（称为小角度近似）为<br />
<br />
<math>\frac{d^2 \theta}{d t^2} + \theta = 0</math><br />
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<br />
<br />
<br />
<br />
since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a [[simple harmonic oscillator]] corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
<br />
since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
<br />
因为 <math>\theta \approx 0</math> 时，有 <math>\sin(\theta) \approx \theta</math>。这是一个'''简谐振子 Simple harmonic oscillator''' ，对应于摆在其路径底部附近的摆动。另一种线性化方法是在 <math>\theta = \pi</math>附近线性化，对应于运动到最高点的摆:<br />
<br />
<math>\frac{d^2 \theta}{d t^2} + \pi - \theta = 0</math><br />
<br />
<br />
<br />
<br />
<br />
since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves [[hyperbolic sinusoid]]s, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
<br />
since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
<br />
因为 <math>\theta \approx \pi</math> 时，有 <math>\sin(\theta) \approx \pi - \theta</math>。这个问题的解含双曲正弦曲线；注意到不同于小角度近似，它是不稳定的，这意味着 <math>|\theta|</math> 通常会无限增长（但解也有可能是有界的）。这就解释了摆在最高点达到平衡的困难，此时实际上是一种不稳定的状态。<br />
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<br />
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One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
<br />
One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
<br />
一个更有趣的线性化可能是在 <math>\theta = \pi/2</math>附近，此时 <math>\sin(\theta) \approx 1</math>:<br />
<br />
<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math><br />
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<br />
<br />
<br />
This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) [[phase portrait]]s and approximate periods.<br />
<br />
This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) phase portraits and approximate periods.<br />
<br />
这相当于一个自由落体问题。把这样线性化的结果合在一起看，就能得到有关摆的运动的非常有用的图像，如右图所示。利用其他方法寻找（精确的）'''相图 Phase portrait'''和估计周期。<br />
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<br />
<br />
<br />
==Types of nonlinear dynamic behaviors 非线性动力学行为的类型==<br />
<br />
<br />
<br />
*[[Amplitude death]] – any oscillations present in the system cease due to some kind of interaction with other system or feedback by the same system<br />
<br />
*'''振幅死亡 Amplitude death'''——系统内的某振荡因系统的自回馈或与其他系统的某种相互作用而停止的现象<br />
<br />
*[[Chaos theory|Chaos]] – values of a system cannot be predicted indefinitely far into the future, and fluctuations are [[aperiodic]]<br />
<br />
*'''混沌 Chaos'''——系统内的值无法无限期地预测到遥远的未来；波动是非周期性的<br />
<br />
*[[Multistability]] – the presence of two or more stable states<br />
<br />
*'''多稳态 Multistability'''——两或多个稳态的存在<br />
<br />
*[[Soliton]]s – self-reinforcing solitary waves<br />
<br />
*'''孤波 Soliton'''s——自增强的孤立波<br />
<br />
*[[Limit cycle|Limit cycles]] – asymptotic periodic orbits to which destabilized fixed points are attracted. <br />
<br />
*'''极限环 Limit cycles'''——吸引不稳定不动点的渐近周期轨道<br />
<br />
*[[Self-oscillation|Self-oscillations]] - feedback oscillations taking place in open dissipative physical systems.<br />
<br />
*'''自激振荡 Self-oscillations'''——开放耗散物理系统中的反馈振荡<br />
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==Examples of nonlinear equations 非线性方程的例子==<br />
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{{Div col|colwidth=25em}}<br />
<br />
<br />
<br />
*[[Algebraic Riccati equation]] 代数黎卡提方程<br />
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<br />
<br />
*[[Ball and beam]] system 球杆系统<br />
<br />
<br />
<br />
*[[Bellman equation]] for optimal policy 最佳策略的贝尔曼方程<br />
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<br />
<br />
*[[Boltzmann equation]] 玻尔兹曼方程<br />
<br />
<br />
<br />
*[[Colebrook equation]] 科尔布鲁克方程<br />
<br />
<br />
<br />
*[[General relativity]] 广义相对论<br />
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<br />
<br />
*[[Ginzburg–Landau theory]] 金兹堡-朗道方程<br />
<br />
<br />
<br />
*[[Ishimori equation]] 石森方程<br />
<br />
<br />
<br />
*[[Kadomtsev–Petviashvili equation]] 卡东穆塞夫-彼得韦亚斯维利方程<br />
<br />
<br />
<br />
*[[Korteweg–de Vries equation]] kdV方程<br />
<br />
<br />
<br />
*[[Landau–Lifshitz–Gilbert equation]] 朗道-利夫希兹-吉尔伯特方程<br />
<br />
<br />
<br />
*[[Liénard equation]] 林纳德方程<br />
<br />
<br />
<br />
*[[Navier–Stokes equations]] of [[fluid dynamics]] 流体力学的纳维－斯托克斯方程<br />
<br />
<br />
<br />
*[[Nonlinear optics]] 非线性光学<br />
<br />
<br />
<br />
*[[Nonlinear Schrödinger equation]] 非线性薛定谔方程<br />
<br />
<br />
<br />
<br />
*[[Power-flow study]] 功率流研究<br />
<br />
<br />
<br />
*[[Richards equation]] for unsaturated water flow 未饱和层水流的理查氏方程<br />
<br />
<br />
<br />
*[[Self-balancing unicycle]]<br />
<br />
<br />
<br />
*[[Sine-Gordon equation]] 正弦-戈尔登方程<br />
<br />
<br />
<br />
*[[Van der Pol oscillator]] 范德波尔振荡器<br />
<br />
<br />
<br />
*[[Vlasov equation]] 弗拉索夫方程<br />
<br />
<br />
<br />
{{Div col end}}<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
==See also 参见==<br />
<br />
<br />
*[[Aleksandr Mikhailovich Lyapunov]] 亚历山大·李亚普诺夫<br />
<br />
<br />
<br />
*[[Dynamical system]] 动态系统<br />
<br />
<br />
<br />
*[[Feedback]] 反馈<br />
<br />
<br />
<br />
*[[Initial condition]] 初始条件<br />
<br />
<br />
<br />
*[[Interaction]] 相互作用<br />
<br />
<br />
<br />
*[[Linear system]] 线性系统<br />
<br />
<br />
<br />
*[[Mode coupling]] 模式耦合<br />
<br />
<br />
<br />
*[[Vector soliton]] 矢量孤子<br />
<br />
<br />
<br />
*[[Volterra series]] 沃尔泰拉级数<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
==References 参考资料==<br />
<br />
<br />
<br />
{{Reflist|35em}}<br />
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<br />
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<br />
<br />
<br />
==Further reading 进一步阅读==<br />
<br />
<br />
<br />
{{Refbegin|35em}}<br />
<br />
<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| author= [[Diederich Hinrichsen]] and Anthony J. Pritchard<br />
<br />
| author= Diederich Hinrichsen and Anthony J. Pritchard<br />
<br />
作者 Diederich Hinrichsen 和 Anthony j. Pritchard<br />
<br />
| year= 2005<br />
<br />
| year= 2005<br />
<br />
2005年<br />
<br />
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
<br />
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
<br />
数学系统理论 i-建模，状态空间分析，稳定性和鲁棒性<br />
<br />
| publisher= Springer Verlag<br />
<br />
| publisher= Springer Verlag<br />
<br />
出版商 Springer Verlag<br />
<br />
| isbn=9783540441250<br />
<br />
| isbn=9783540441250<br />
<br />
9783540441250<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Jordan<br />
<br />
| last= Jordan<br />
<br />
最后一个乔丹<br />
<br />
| first= D. W.<br />
<br />
| first= D. W.<br />
<br />
首先是 D.w。<br />
<br />
| last2= Smith<br />
<br />
| last2= Smith<br />
<br />
最后2个史密斯<br />
<br />
| first2= P.<br />
<br />
| first2= P.<br />
<br />
| first2 p.<br />
<br />
| year= 2007<br />
<br />
| year= 2007<br />
<br />
2007年<br />
<br />
| edition= fourth<br />
<br />
| edition= fourth<br />
<br />
第四版<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
非线性常微分方程<br />
<br />
| publisher= Oxford University Press<br />
<br />
| publisher= Oxford University Press<br />
<br />
牛津大学出版社<br />
<br />
| isbn= 978-0-19-920824-1<br />
<br />
| isbn= 978-0-19-920824-1<br />
<br />
[国际标准图书编号978-0-19-920824-1]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Khalil<br />
<br />
| last= Khalil<br />
<br />
最后的哈利勒<br />
<br />
| first= Hassan K.<br />
<br />
| first= Hassan K.<br />
<br />
首先是哈桑 · k。<br />
<br />
| year= 2001<br />
<br />
| year= 2001<br />
<br />
2001年<br />
<br />
| title= Nonlinear Systems<br />
<br />
| title= Nonlinear Systems<br />
<br />
非线性系统<br />
<br />
| publisher= Prentice Hall<br />
<br />
| publisher= Prentice Hall<br />
<br />
出版商 Prentice Hall<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
[国际标准图书编号978-0-13-067389-3]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Kreyszig<br />
<br />
| last= Kreyszig<br />
<br />
| last= Kreyszig<br />
<br />
| first= Erwin<br />
<br />
| first= Erwin<br />
<br />
首先是欧文<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
作者 / 链接 Erwin Kreyszig<br />
<br />
| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
高等工程数学<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
Https://archive.org/details/advancedengineer0008krey<br />
<br />
| url-access= registration<br />
<br />
| url-access= registration<br />
<br />
访问注册<br />
<br />
| publisher= Wiley<br />
<br />
| publisher= Wiley<br />
<br />
出版商 Wiley<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
[国际标准图书编号978-0-471-15496-9]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Sontag<br />
<br />
| last= Sontag<br />
<br />
最后一个桑塔格<br />
<br />
| first= Eduardo<br />
<br />
| first= Eduardo<br />
<br />
首先是爱德华多<br />
<br />
| authorlink= Eduardo D. Sontag<br />
<br />
| authorlink= Eduardo D. Sontag<br />
<br />
作者: Eduardo d. Sontag<br />
<br />
| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
数学控制理论: 确定性有限维系统。第二版<br />
<br />
| publisher= Springer<br />
<br />
| publisher= Springer<br />
<br />
出版商斯普林格<br />
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| isbn= 978-0-387-98489-6<br />
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| isbn= 978-0-387-98489-6<br />
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[国际标准图书编号978-0-387-98489-6]<br />
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==External links 外部链接==<br />
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*[http://www.dodccrp.org/ Command and Control Research Program (CCRP)]<br />
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*[http://necsi.edu/guide/concepts/linearnonlinear.html New England Complex Systems Institute: Concepts in Complex Systems]<br />
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*[http://ocw.mit.edu/courses/mathematics/18-353j-nonlinear-dynamics-i-chaos-fall-2012/ Nonlinear Dynamics I: Chaos] at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]<br />
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*[http://www.hedengren.net/research/models.htm Nonlinear Model Library]{{snd}} (in [[MATLAB]]) a Database of Physical Systems<br />
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*[http://cnls.lanl.gov/ The Center for Nonlinear Studies at Los Alamos National Laboratory]<br />
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{{Systems science}}<br />
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==编者推荐==<br />
集智学园：[https://campus.swarma.org/course/697 非线性动力学与混沌]<br />
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非线性动力学和混沌理论是系统发展的，从一阶微分方程及其分岔开始，然后是相平面分析，极限环和它们的分岔，最终得到Lorenz方程，混沌，迭代映射，周期倍增，重整化，分形和奇怪吸引。<br />
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此系列课程由[[斯蒂文·斯特罗加茨 Steven H. Strogatz]]主持，内容包括机械振动，激光，生物节律，超导电路，昆虫爆发，化学振荡器，遗传控制系统，混沌水轮，甚至是使用混乱发送秘密信息的技术。在每种情况下，科学背景都在初级阶段进行解释，并与数学理论紧密结合。<br />
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[[Category:Nonlinear systems| ]]<br />
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[[Category:Dynamical systems]]<br />
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Category:Dynamical systems<br />
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类别: 动力系统<br />
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[[Category:Concepts in physics]]<br />
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Category:Concepts in physics<br />
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分类: 物理概念<br />
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<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Nonlinear system]]. Its edit history can be viewed at [[非线性系统/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%9D%9E%E7%BA%BF%E6%80%A7%E7%B3%BB%E7%BB%9F&diff=10057非线性系统2020-07-10T03:13:12Z<p>Dorr：</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{Redirect|Nonlinear dynamics|the journal|Nonlinear Dynamics (journal)}}<br />
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{{About|"nonlinearity" in mathematics, physics and other sciences|video and film editing|Non-linear editing system|other uses|Nonlinearity (disambiguation)}}<br />
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In [[mathematics]] and [[science]], a '''nonlinear system''' is a [[system]] in which the change of the output is not [[proportionality (mathematics)|proportional]] to the change of the input.<ref>{{Cite news|url=https://news.mit.edu/2010/explained-linear-0226|title=Explained: Linear and nonlinear systems|work=MIT News|access-date=2018-06-30}}</ref><ref>{{Cite web|url=https://www.birmingham.ac.uk/research/activity/mathematics/applied-maths/nonlinear-systems.aspx|title=Nonlinear systems, Applied Mathematics - University of Birmingham|website=www.birmingham.ac.uk|language=en-gb|access-date=2018-06-30}}</ref> Nonlinear problems are of interest to [[engineer]]s, [[biologist]]s,<ref>{{Citation|date=2007|pages=181–276|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-540-34153-6_7|isbn=9783540341529|title = The Nonlinear Universe|series = The Frontiers Collection|chapter = Nonlinear Biology}}</ref><ref>{{cite journal|last=Korenberg|first=Michael J.|last2=Hunter|first2=Ian W.|date=March 1996|title=The identification of nonlinear biological systems: Volterra kernel approaches|journal=Annals of Biomedical Engineering|language=en|volume=24|issue=2|pages=250–268|doi=10.1007/bf02667354|issn=0090-6964}}</ref><ref>{{cite journal|last=Mosconi|first=Francesco|last2=Julou|first2=Thomas|last3=Desprat|first3=Nicolas|last4=Sinha|first4=Deepak Kumar|last5=Allemand|first5=Jean-François|last6=Vincent Croquette|last7=Bensimon|first7=David|date=2008|title=Some nonlinear challenges in biology|url=http://stacks.iop.org/0951-7715/21/i=8/a=T03|journal=Nonlinearity|language=en|volume=21|issue=8|pages=T131|doi=10.1088/0951-7715/21/8/T03|issn=0951-7715|bibcode=2008Nonli..21..131M}}</ref> [[physicist]]s,<ref>{{cite journal|last1=Gintautas|first1=V.|title=Resonant forcing of nonlinear systems of differential equations|journal=Chaos|date=2008|volume=18|issue=3|pages=033118|doi=10.1063/1.2964200|pmid=19045456|arxiv=0803.2252|bibcode=2008Chaos..18c3118G}}</ref><ref>{{cite journal|last1=Stephenson|first1=C.|last2=et.|first2=al.|title=Topological properties of a self-assembled electrical network via ab initio calculation|journal=Sci. Rep.|volume=7|pages=41621|date=2017|doi=10.1038/srep41621|pmid=28155863|pmc=5290745|bibcode=2017NatSR...741621S}}</ref> [[mathematician]]s, and many other [[scientist]]s because most systems are inherently nonlinear in nature.<ref>{{cite book|last1=de Canete|first1=Javier, Cipriano Galindo, and Inmaculada Garcia-Moral|title=System Engineering and Automation: An Interactive Educational Approach|date=2011|publisher=Springer|location=Berlin|isbn=978-3642202292|page=46|url=https://books.google.com/?id=h8rCQYXGGY8C&pg=PA46&lpg=PA46&dq=most+systems+are+inherently+nonlinear+in+nature#v=onepage&q=most%20systems%20are%20inherently%20nonlinear%20in%20nature&f=false|accessdate=20 January 2018}}</ref> Nonlinear [[dynamical system]]s, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler [[linear system]]s.<br />
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In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.<br />
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在数学及科学中，非线性系统是一种输出的变化与输入的变化不成比例的系统。大多数系统在本质上是非线性的，因而非线性问题引起了工程师、生物学家、物理学家、数学家和许多其他科学家的兴趣。描述变量随时间变化的非线性动力系统与较之简单得多的线性系统相比，可能显得混沌、不可预测或违反直觉。<br />
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Typically, the behavior of a nonlinear system is described in mathematics by a '''nonlinear system of equations''', which is a set of simultaneous [[equation]]s in which the unknowns (or the unknown functions in the case of [[differential equation]]s) appear as variables of a [[polynomial]] of degree higher than one or in the argument of a [[function (mathematics)|function]] which is not a polynomial of degree one.<br />
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Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one.<br />
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通常，非线性系统的行为在数学上是由一组非线性联立方程来描述的，其中未知数(或微分方程中的未知函数)作为一个高于一次的多项式的变量出现，或者作为一个不是一次多项式的函数的论元出现。<br />
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In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a [[linear combination]] of the unknown [[variable (mathematics)|variables]] or [[function (mathematics)|functions]] that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is ''linear'' if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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换句话说，在非线性方程系统中，要求解的方程不能被写成未知变量或函数的线性组合。无论方程中是否有已知的线性函数，系统都可以被定义为非线性。特别是，如果一个微分方程的未知函数及其导数是线性的，即使其他变量是非线性的，也称其是线性的。<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations ([[linearization]]). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as [[soliton]]s, [[chaos theory|chaos]],<ref>[http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm Nonlinear Dynamics I: Chaos] {{webarchive|url=https://web.archive.org/web/20080212045134/http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm |date=2008-02-12}} at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]</ref> and [[mathematical singularity|singularities]] are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble [[randomness|random]] behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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由于非线性动力学方程难以求解，通常用线性化方程来近似非线性系统（线性化）。这种方法在一定的精度和范围对输入值效果很好，但一些有趣的现象如'''孤子 Solitons'''、'''混沌 Chaos'''和'''奇异性 Singularities'''在线性化后被隐藏。因此，非线性系统的动态行为在某些方面可能看起来违反直觉、不可预测，甚至混沌。尽管这种混沌行为可能感觉很像随机行为，但它实际上并不是随机的。例如，天气的某些方面被认为是混沌的，其系统某部分的微小扰动就会产生复杂的影响。这种非线性是目前技术无法进行精确长期预测的原因之一。<br />
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Some authors use the term '''nonlinear science''' for the study of nonlinear systems. This term is disputed by others:<br />
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Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:<br />
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有些作者用非线性科学这一术语来研究非线性系统。这一术语引起了其他人的争议:<br />
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{{quote|Using a term like nonlinear science is like referring to the bulk of zoology as the study of [[negation|non]]-elephant animals.|[[Stanislaw Ulam]]<ref>{{cite journal|last1=Campbell|first1=David K.|title=Nonlinear physics: Fresh breather|journal=Nature|date=25 November 2004|volume=432|issue=7016|pages=455–456|doi=10.1038/432455a|pmid=15565139|url=https://zenodo.org/record/1134179|language=en|issn=0028-0836|bibcode=2004Natur.432..455C}}</ref>}}<br />
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“使用‘非线性科学’这样的术语，就如同把动物学里大部分对象称作‘非大象动物’研究一样可笑。”<br />
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——'''斯塔尼斯拉夫·乌拉姆 Stanislaw Ulam'''<br />
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==Definition 定义==<br />
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In [[mathematics]], a [[linear map]] (or ''linear function'') <math>f(x)</math> is one which satisfies both of the following properties:<br />
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In mathematics, a linear map (or linear function) <math>f(x)</math> is one which satisfies both of the following properties:<br />
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在数学中，线性映射（或线性函数）<math>f (x)</math>满足以下两个性质:<br />
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*Additivity or [[superposition principle]]: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*Homogeneity: <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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*'''可加性 Additivity'''（'''叠加性 Superposition principle'''）: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*'''齐次性 Homogeneity''': <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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Additivity implies homogeneity for any [[rational number|rational]] ''α'', and, for [[continuous function]]s, for any [[real number|real]] ''α''. For a [[complex number|complex]] ''α'', homogeneity does not follow from additivity. For example, an [[antilinear map]] is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle<br />
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Additivity implies homogeneity for any rational α, and, for continuous functions, for any real α. For a complex α, homogeneity does not follow from additivity. For example, an antilinear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle:<br />
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α是有理数，或α是实数且<math>f(x)</math>是连续函数时，由可加性可以推出齐次性。但当α是复数时，可加性不能导出齐次性。例如，反线性映射是可加的，但不是齐次的。可加性和齐次性条件经常组合，称为叠加原理：<br />
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<math>f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)</math><br />
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An equation written as<br />
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<math>f(x) = C</math><br />
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is called '''linear''' if <math>f(x)</math> is a linear map (as defined above) and '''nonlinear''' otherwise. The equation is called ''homogeneous'' if <math>C = 0</math>.<br />
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is called linear if <math>f(x)</math> is a linear map (as defined above) and nonlinear otherwise. The equation is called homogeneous if <math>C = 0</math>.<br />
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的方程，若 <math> f (x) </math> 是线性映射(如上定义) ，则称其为'''线性的 Linear'''，否则称为'''非线性的 Nonlinear'''。若<math>C = 0</math>，该方程称为是齐次的。<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any [[map (mathematics)|mapping]], including integration or differentiation with associated constraints (such as [[boundary values]]). If <math>f(x)</math> contains [[derivative|differentiation]] with respect to <math>x</math>, the result will be a [[differential equation]].<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any mapping, including integration or differentiation with associated constraints (such as boundary values). If <math>f(x)</math> contains differentiation with respect to <math>x</math>, the result will be a differential equation.<br />
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定义 <math>f(x) = C</math> 是非常具有一般性的，因为 <math>x</math> 可以是任意可感知的数学对象（数字、向量、函数等），函数 <math>f(x)</math> 实际上可以是任意映射，包括有相关约束(如给定边界值)的积分或微分。若 <math>f(x)</math> 包含对 <math>x</math> 的微分运算，则该方程为微分方程。<br />
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==Nonlinear algebraic equations 非线性代数方程==<br />
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{{Main|Algebraic equation|System of polynomial equations}}<br />
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Nonlinear [[algebraic equation]]s, which are also called ''[[polynomial equation]]s'', are defined by equating [[polynomial]]s (of degree greater than one) to zero. For example,<br />
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Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials (of degree greater than one) to zero. For example,<br />
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非线性代数方程，又称多项式方程，由某多项式（次数大于1）等于零定义。例如：<br />
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For a single polynomial equation, [[root-finding algorithm]]s can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of [[algebraic geometry]], a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see [[Hilbert's Nullstellensatz]]). Nevertheless, in the case of the systems with a finite number of complex solutions, these [[systems of polynomial equations]] are now well understood and efficient methods exist for solving them.<ref>{{cite journal |last1= Lazard |first1= D. |title= Thirty years of Polynomial System Solving, and now? |doi= 10.1016/j.jsc.2008.03.004 |journal= Journal of Symbolic Computation |volume= 44 |issue= 3 |pages= 222–231 |year= 2009 |pmid= |pmc=}}</ref><br />
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For a single polynomial equation, root-finding algorithms can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of algebraic geometry, a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see Hilbert's Nullstellensatz). Nevertheless, in the case of the systems with a finite number of complex solutions, these systems of polynomial equations are now well understood and efficient methods exist for solving them.<br />
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对于一个单一的多项式方程，'''求根算法 Root-finding algorithms'''可用于其求解（即找到满足该方程的变量的值集）。而代数方程组则相对复杂，其研究是现代数学的较难分支——'''代数几何 Algebraic geometry'''领域的动力之一。甚至很难判断一个给定的代数系统是否有复数解（见'''希尔伯特零点定律 Hilbert's Nullstellensatz'''）。不过，对于具有有限个复数解的系统的多项式方程组，我们现在已经有了充分的理解，并且找到了有效的求解方法。<br />
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==Nonlinear recurrence relations 非线性递推关系==<br />
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A nonlinear [[recurrence relation]] defines successive terms of a [[sequence]] as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the [[logistic map]] and the relations that define the various [[Hofstadter sequence]]s. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related [[nonlinear system identification]] and analysis procedures.<ref name="SAB1">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</ref> These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related nonlinear system identification and analysis procedures. These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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非线性递归关系中，序列的连续项被定义为其前项的非线性函数。非线性递归关系的例子有 logistic 映射和定义各种'''霍夫斯塔特序列 Hofstadter sequences''' 的关系。非线性离散模型代表了一类广泛的非线性递归关系，包括 NARMAX（外部输入非线性自回归移动平均）模型和相关的非线性系统辨识和分析程序。这些方法可用于研究时域、频域和时空域的广泛复杂非线性行为。<br />
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==Nonlinear differential equations 非线性微分方程==<br />
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A [[simultaneous equations|system]] of [[differential equation]]s is said to be nonlinear if it is not a [[linear system]]. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the [[Navier–Stokes equations]] in fluid dynamics and the [[Lotka–Volterra equation]]s in biology.<br />
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A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology.<br />
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若一个微分方程组不是线性系统，则称其为非线性的。涉及非线性微分方程的问题非常多样，对不同问题的解决或分析方法也不相同。非线性微分方程的例子有流体力学中的 '''纳维-斯托克斯方程 Navier-Stokes equations'''和生物学中的'''洛特卡－沃尔泰拉方程 Lotka-Volterra equations'''。<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of [[linearly independent]] solutions can be used to construct general solutions through the [[superposition principle]]. A good example of this is one-dimensional heat transport with [[Dirichlet boundary conditions]], the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle. A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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非线性问题最大的困难之一是通常不可能将已知的解组合成新的解。例如，在线性问题中，可以根据叠加原理以一族线性独立的解构造通解。一个很好的例子是带有'''狄利克雷边界条件 Dirichlet boundary conditions'''的一维热传导问题，其解（随时间变化）可以写成不同频率的正弦波的线性组合，这使得解非常灵活。而对非线性方程，通常可以找到几个非常特殊的解，但是此时叠加原理不适用，故无法构造新的解。<br />
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===Ordinary differential equations 常微分方程===<br />
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First order [[ordinary differential equation]]s are often exactly solvable by [[separation of variables]], especially for autonomous equations. For example, the nonlinear equation<br />
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First order ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. For example, the nonlinear equation<br />
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一阶常微分方程，尤其是自治方程，通常可以用'''分离变量法 Separation of variables'''来精确求解。例如，非线性方程<br />
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<math>\frac{d u}{d x} = -u^2</math><br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also ''u'' = 0 as a particular solution, corresponding to the limit of the general solution when ''C'' tends to infinity). The equation is nonlinear because it may be written as<br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also u = 0 as a particular solution, corresponding to the limit of the general solution when C tends to infinity). The equation is nonlinear because it may be written as<br />
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将 <math>u=\frac{1}{x+C}</math> 作为一般解（也有特解 ''u'' = 0，对应于 ''C'' 趋于无穷时的一般解的极限)。该方程是非线性的，因为它可以改写成<br />
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<math>\frac{du}{d x} + u^2=0</math><br />
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and the left-hand side of the equation is not a linear function of ''u'' and its derivatives. Note that if the ''u''<sup>2</sup> term were replaced with ''u'', the problem would be linear (the [[exponential decay]] problem).<br />
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and the left-hand side of the equation is not a linear function of u and its derivatives. Note that if the u<sup>2</sup> term were replaced with u, the problem would be linear (the exponential decay problem).<br />
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方程的左边不是 ''u'' 及其导数的线性函数。注意，若将 ''u''<sup>2</sup> 项替换为''u''，该问题将变为线性的（'''指数衰减 Exponential decay'''问题）。<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield [[closed-form expression|closed-form]] solutions, though implicit solutions and solutions involving [[nonelementary integral]]s are encountered.<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered.<br />
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二阶和高阶常微分方程（更一般地说，非线性方程组）很少能产生封闭解，而隐式解和非初等函数积分形式的解较为常见。<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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非线性常微分方程定性分析的常用方法包括:<br />
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*Examination of any [[conserved quantities]], especially in [[Hamiltonian system]]s<br />
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*检查是否有任意'''守恒量 Conserved quantities'''（特别是在'''哈密顿系统 Hamiltonian system'''中）<br />
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*Examination of dissipative quantities (see [[Lyapunov function]]) analogous to conserved quantities<br />
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*检查是否有类似守恒量的耗散量（见'''李亚普诺夫函数 Lyapunov function'''）<br />
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*Linearization via [[Taylor expansion]]<br />
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*基于'''泰勒展开 Taylor expansion'''的线性化<br />
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*Change of variables into something easier to study<br />
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*将变量转化得更易于研究<br />
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*[[Bifurcation theory]]<br />
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*'''分岔理论 Bifurcation theory'''<br />
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*[[Perturbation theory|Perturbation]] methods (can be applied to algebraic equations too)<br />
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*'''摄动理论 Perturbation theory'''（也可应用于代数方程）<br />
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===Partial differential equations 偏微分方程===<br />
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{{main|Nonlinear partial differential equation}}<br />
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{{See also|List of nonlinear partial differential equations}}<br />
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参见：非线性偏微分列表<br />
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The most common basic approach to studying nonlinear [[partial differential equation]]s is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more [[ordinary differential equation]]s, as seen in [[separation of variables]], which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
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The most common basic approach to studying nonlinear partial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
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研究非线性偏微分方程最常用的基本方法是变换变量（或转换问题），使变换后的问题更简单（甚至可能变为线性的）。有时可以将此类方程转化成一或多个常微分方程（如同分离变量法所示），此时不论得到的常微分方程是否可解，对研究问题总是有用的。<br />
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Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use [[scale analysis (mathematics)|scale analysis]] to simplify a general, natural equation in a certain specific [[boundary value problem]]. For example, the (very) nonlinear [[Navier-Stokes equations]] can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
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Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem. For example, the (very) nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
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另一个流体力学和热力学中常见的策略（虽然不是数学上的）是利用'''尺度分析 Scale analysis'''来在某一特定边界条件下简化一般自然方程。例如，在描述圆管内一维层流的暂态时，非线性的纳维－斯托克斯方程可以简化为一个线性的偏微分方程; 尺度分析提供了层流和一维流动的条件，也产生了简化的方程。<br />
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Other methods include examining the [[method of characteristics|characteristics]] and using the methods outlined above for ordinary differential equations.<br />
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Other methods include examining the characteristics and using the methods outlined above for ordinary differential equations.<br />
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其他方法包括检查特征线法及前面所述研究常微分方程的方法。<br />
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===Pendula 摆===<br />
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{{Main|Pendulum (mathematics)}}<br />
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[[File:PendulumLayout.svg.png|thumb|Illustration of a pendulum 摆图解|right|200px]]<br />
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[[File:PendulumLinearizations.png|thumb|Linearizations of a pendulum 摆的线性化|right|200px]]<br />
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A classic, extensively studied nonlinear problem is the dynamics of a [[pendulum (mathematics)|pendulum]] under the influence of [[gravity]]. Using [[Lagrangian mechanics]], it may be shown<ref>[http://www.damtp.cam.ac.uk/user/tong/dynamics.html David Tong: Lectures on Classical Dynamics]</ref> that the motion of a pendulum can be described by the [[dimensionless]] nonlinear equation<br />
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A classic, extensively studied nonlinear problem is the dynamics of a pendulum under the influence of gravity. Using Lagrangian mechanics, it may be shown that the motion of a pendulum can be described by the dimensionless nonlinear equation<br />
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一个经典的被广泛研究的非线性问题是重力影响下的摆的动力学。利用'''拉格朗日力学 Lagrangian mechanics'''，可以证明摆的运动可以用无量纲的非线性方程<br />
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<math>\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0</math><br />
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where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an [[integrating factor]], which would eventually yield<br />
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where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an integrating factor, which would eventually yield<br />
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描述，其中重力指向“下方”，<math>\theta</math> 是摆与其静止位置形成的角度，如右图所示。“解”这个方程的方法之一是用 <math>d\theta/dt</math> 作为'''积分因子 Integrating factor'''，最终得<br />
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<math>\int{\frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)}}} = t + C_1</math><br />
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which is an implicit solution involving an [[elliptic integral]]. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the [[nonelementary integral]] (nonelementary unless <math>C_0 = 2</math>).<br />
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which is an implicit solution involving an elliptic integral. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral (nonelementary unless <math>C_0 = 2</math>).<br />
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这是一个含'''椭圆积分 Elliptic integral'''的隐式解。这个“解”通常没什么用，因为这个解的大部分性质都隐藏在非初等函数积分中（除非<math>C_0 = 2</math>，否则是非初等的）。<br />
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Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through [[Taylor expansion]]s. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
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Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through Taylor expansions. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
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另一种解决这个问题的方法是利用泰勒展开将任意非线性项（此时为正弦函数项)在某些感兴趣的点线性化。例如，在<math>\theta = 0</math> 的点附近线性化（称为小角度近似）为<br />
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<math>\frac{d^2 \theta}{d t^2} + \theta = 0</math><br />
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since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a [[simple harmonic oscillator]] corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
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since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
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因为 <math>\theta \approx 0</math> 时，有 <math>\sin(\theta) \approx \theta</math>。这是一个'''简谐振子 Simple harmonic oscillator''' ，对应于摆在其路径底部附近的摆动。另一种线性化方法是在 <math>\theta = \pi</math>附近线性化，对应于运动到最高点的摆:<br />
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<math>\frac{d^2 \theta}{d t^2} + \pi - \theta = 0</math><br />
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since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves [[hyperbolic sinusoid]]s, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
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since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
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因为 <math>\theta \approx \pi</math> 时，有 <math>\sin(\theta) \approx \pi - \theta</math>。这个问题的解含双曲正弦曲线；注意到不同于小角度近似，它是不稳定的，这意味着 <math>|\theta|</math> 通常会无限增长（但解也有可能是有界的）。这就解释了摆在最高点达到平衡的困难，此时实际上是一种不稳定的状态。<br />
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One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
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One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
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一个更有趣的线性化可能是在 <math>\theta = \pi/2</math>附近，此时 <math>\sin(\theta) \approx 1</math>:<br />
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<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math><br />
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This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) [[phase portrait]]s and approximate periods.<br />
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This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) phase portraits and approximate periods.<br />
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这相当于一个自由落体问题。把这样线性化的结果合在一起看，就能得到有关摆的运动的非常有用的图像，如右图所示。利用其他方法寻找（精确的）'''相图 Phase portrait'''和估计周期。<br />
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==Types of nonlinear dynamic behaviors 非线性动力学行为的类型==<br />
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*[[Amplitude death]] – any oscillations present in the system cease due to some kind of interaction with other system or feedback by the same system<br />
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*'''振幅死亡 Amplitude death'''——系统内的某振荡因系统的自回馈或与其他系统的某种相互作用而停止的现象<br />
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*[[Chaos theory|Chaos]] – values of a system cannot be predicted indefinitely far into the future, and fluctuations are [[aperiodic]]<br />
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*'''混沌 Chaos'''——系统内的值无法无限期地预测到遥远的未来；波动是非周期性的<br />
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*[[Multistability]] – the presence of two or more stable states<br />
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*'''多稳态 Multistability'''——两或多个稳态的存在<br />
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*[[Soliton]]s – self-reinforcing solitary waves<br />
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*'''孤波 Soliton'''s——自增强的孤立波<br />
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*[[Limit cycle|Limit cycles]] – asymptotic periodic orbits to which destabilized fixed points are attracted. <br />
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*'''极限环 Limit cycles'''——吸引不稳定不动点的渐近周期轨道<br />
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*[[Self-oscillation|Self-oscillations]] - feedback oscillations taking place in open dissipative physical systems.<br />
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*'''自激振荡 Self-oscillations'''——开放耗散物理系统中的反馈振荡<br />
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==Examples of nonlinear equations 非线性方程的例子==<br />
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*[[Algebraic Riccati equation]] 代数黎卡提方程<br />
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*[[Ball and beam]] system 球杆系统<br />
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*[[Bellman equation]] for optimal policy 最佳策略的贝尔曼方程<br />
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*[[Boltzmann equation]] 玻尔兹曼方程<br />
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*[[Colebrook equation]] 科尔布鲁克方程<br />
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*[[General relativity]] 广义相对论<br />
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*[[Ginzburg–Landau theory]] 金兹堡-朗道方程<br />
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*[[Ishimori equation]] 石森方程<br />
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*[[Kadomtsev–Petviashvili equation]] 卡东穆塞夫-彼得韦亚斯维利方程<br />
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*[[Korteweg–de Vries equation]] kdV方程<br />
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*[[Landau–Lifshitz–Gilbert equation]] 朗道-利夫希兹-吉尔伯特方程<br />
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*[[Liénard equation]] 林纳德方程<br />
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*[[Navier–Stokes equations]] of [[fluid dynamics]] 流体力学的纳维－斯托克斯方程<br />
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*[[Nonlinear optics]] 非线性光学<br />
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*[[Nonlinear Schrödinger equation]] 非线性薛定谔方程<br />
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*[[Power-flow study]] 功率流研究<br />
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*[[Richards equation]] for unsaturated water flow 未饱和层水流的理查氏方程<br />
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*[[Self-balancing unicycle]]<br />
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*[[Sine-Gordon equation]] 正弦-戈尔登方程<br />
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*[[Van der Pol oscillator]] 范德波尔振荡器<br />
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*[[Vlasov equation]] 弗拉索夫方程<br />
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==See also 参见==<br />
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*[[Aleksandr Mikhailovich Lyapunov]] 亚历山大·李亚普诺夫<br />
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*[[Dynamical system]] 动态系统<br />
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*[[Feedback]] 反馈<br />
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*[[Initial condition]] 初始条件<br />
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*[[Interaction]] 相互作用<br />
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*[[Linear system]] 线性系统<br />
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*[[Mode coupling]] 模式耦合<br />
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*[[Vector soliton]] 矢量孤子<br />
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*[[Volterra series]] 沃尔泰拉级数<br />
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==References 参考资料==<br />
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{{Reflist|35em}}<br />
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==Further reading 进一步阅读==<br />
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{{Refbegin|35em}}<br />
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*{{cite book<br />
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| author= [[Diederich Hinrichsen]] and Anthony J. Pritchard<br />
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| author= Diederich Hinrichsen and Anthony J. Pritchard<br />
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作者 Diederich Hinrichsen 和 Anthony j. Pritchard<br />
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| year= 2005<br />
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| year= 2005<br />
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2005年<br />
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| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
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| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
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数学系统理论 i-建模，状态空间分析，稳定性和鲁棒性<br />
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| publisher= Springer Verlag<br />
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| publisher= Springer Verlag<br />
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出版商 Springer Verlag<br />
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| isbn=9783540441250<br />
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| isbn=9783540441250<br />
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9783540441250<br />
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}}<br />
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}}<br />
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}}<br />
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*{{cite book<br />
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| last= Jordan<br />
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| last= Jordan<br />
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最后一个乔丹<br />
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| first= D. W.<br />
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| first= D. W.<br />
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首先是 D.w。<br />
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| last2= Smith<br />
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| last2= Smith<br />
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最后2个史密斯<br />
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| first2= P.<br />
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| first2= P.<br />
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| first2 p.<br />
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| year= 2007<br />
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| year= 2007<br />
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2007年<br />
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| edition= fourth<br />
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| edition= fourth<br />
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第四版<br />
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| title= Nonlinear Ordinary Differential Equations<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
非线性常微分方程<br />
<br />
| publisher= Oxford University Press<br />
<br />
| publisher= Oxford University Press<br />
<br />
牛津大学出版社<br />
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| isbn= 978-0-19-920824-1<br />
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| isbn= 978-0-19-920824-1<br />
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[国际标准图书编号978-0-19-920824-1]<br />
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}}<br />
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}}<br />
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}}<br />
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*{{cite book<br />
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| last= Khalil<br />
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| last= Khalil<br />
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最后的哈利勒<br />
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| first= Hassan K.<br />
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| first= Hassan K.<br />
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首先是哈桑 · k。<br />
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| year= 2001<br />
<br />
| year= 2001<br />
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2001年<br />
<br />
| title= Nonlinear Systems<br />
<br />
| title= Nonlinear Systems<br />
<br />
非线性系统<br />
<br />
| publisher= Prentice Hall<br />
<br />
| publisher= Prentice Hall<br />
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出版商 Prentice Hall<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
[国际标准图书编号978-0-13-067389-3]<br />
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}}<br />
<br />
}}<br />
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}}<br />
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*{{cite book<br />
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| last= Kreyszig<br />
<br />
| last= Kreyszig<br />
<br />
| last= Kreyszig<br />
<br />
| first= Erwin<br />
<br />
| first= Erwin<br />
<br />
首先是欧文<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
作者 / 链接 Erwin Kreyszig<br />
<br />
| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
高等工程数学<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
Https://archive.org/details/advancedengineer0008krey<br />
<br />
| url-access= registration<br />
<br />
| url-access= registration<br />
<br />
访问注册<br />
<br />
| publisher= Wiley<br />
<br />
| publisher= Wiley<br />
<br />
出版商 Wiley<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
[国际标准图书编号978-0-471-15496-9]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
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*{{cite book<br />
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| last= Sontag<br />
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| last= Sontag<br />
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最后一个桑塔格<br />
<br />
| first= Eduardo<br />
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| first= Eduardo<br />
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首先是爱德华多<br />
<br />
| authorlink= Eduardo D. Sontag<br />
<br />
| authorlink= Eduardo D. Sontag<br />
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作者: Eduardo d. Sontag<br />
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| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
数学控制理论: 确定性有限维系统。第二版<br />
<br />
| publisher= Springer<br />
<br />
| publisher= Springer<br />
<br />
出版商斯普林格<br />
<br />
| isbn= 978-0-387-98489-6<br />
<br />
| isbn= 978-0-387-98489-6<br />
<br />
[国际标准图书编号978-0-387-98489-6]<br />
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}}<br />
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}}<br />
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}}<br />
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{{Refend}}<br />
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==External links 外部链接==<br />
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*[http://www.dodccrp.org/ Command and Control Research Program (CCRP)]<br />
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*[http://necsi.edu/guide/concepts/linearnonlinear.html New England Complex Systems Institute: Concepts in Complex Systems]<br />
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*[http://ocw.mit.edu/courses/mathematics/18-353j-nonlinear-dynamics-i-chaos-fall-2012/ Nonlinear Dynamics I: Chaos] at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]<br />
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*[http://www.hedengren.net/research/models.htm Nonlinear Model Library]{{snd}} (in [[MATLAB]]) a Database of Physical Systems<br />
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*[http://cnls.lanl.gov/ The Center for Nonlinear Studies at Los Alamos National Laboratory]<br />
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[[Category:Nonlinear systems| ]]<br />
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[[Category:Dynamical systems]]<br />
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Category:Dynamical systems<br />
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类别: 动力系统<br />
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[[Category:Concepts in physics]]<br />
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Category:Concepts in physics<br />
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分类: 物理概念<br />
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<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Nonlinear system]]. Its edit history can be viewed at [[非线性系统/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%9D%9E%E7%BA%BF%E6%80%A7%E7%B3%BB%E7%BB%9F&diff=10044非线性系统2020-07-09T15:37:03Z<p>Dorr：</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{Redirect|Nonlinear dynamics|the journal|Nonlinear Dynamics (journal)}}<br />
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{{About|"nonlinearity" in mathematics, physics and other sciences|video and film editing|Non-linear editing system|other uses|Nonlinearity (disambiguation)}}<br />
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In [[mathematics]] and [[science]], a '''nonlinear system''' is a [[system]] in which the change of the output is not [[proportionality (mathematics)|proportional]] to the change of the input.<ref>{{Cite news|url=https://news.mit.edu/2010/explained-linear-0226|title=Explained: Linear and nonlinear systems|work=MIT News|access-date=2018-06-30}}</ref><ref>{{Cite web|url=https://www.birmingham.ac.uk/research/activity/mathematics/applied-maths/nonlinear-systems.aspx|title=Nonlinear systems, Applied Mathematics - University of Birmingham|website=www.birmingham.ac.uk|language=en-gb|access-date=2018-06-30}}</ref> Nonlinear problems are of interest to [[engineer]]s, [[biologist]]s,<ref>{{Citation|date=2007|pages=181–276|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-540-34153-6_7|isbn=9783540341529|title = The Nonlinear Universe|series = The Frontiers Collection|chapter = Nonlinear Biology}}</ref><ref>{{cite journal|last=Korenberg|first=Michael J.|last2=Hunter|first2=Ian W.|date=March 1996|title=The identification of nonlinear biological systems: Volterra kernel approaches|journal=Annals of Biomedical Engineering|language=en|volume=24|issue=2|pages=250–268|doi=10.1007/bf02667354|issn=0090-6964}}</ref><ref>{{cite journal|last=Mosconi|first=Francesco|last2=Julou|first2=Thomas|last3=Desprat|first3=Nicolas|last4=Sinha|first4=Deepak Kumar|last5=Allemand|first5=Jean-François|last6=Vincent Croquette|last7=Bensimon|first7=David|date=2008|title=Some nonlinear challenges in biology|url=http://stacks.iop.org/0951-7715/21/i=8/a=T03|journal=Nonlinearity|language=en|volume=21|issue=8|pages=T131|doi=10.1088/0951-7715/21/8/T03|issn=0951-7715|bibcode=2008Nonli..21..131M}}</ref> [[physicist]]s,<ref>{{cite journal|last1=Gintautas|first1=V.|title=Resonant forcing of nonlinear systems of differential equations|journal=Chaos|date=2008|volume=18|issue=3|pages=033118|doi=10.1063/1.2964200|pmid=19045456|arxiv=0803.2252|bibcode=2008Chaos..18c3118G}}</ref><ref>{{cite journal|last1=Stephenson|first1=C.|last2=et.|first2=al.|title=Topological properties of a self-assembled electrical network via ab initio calculation|journal=Sci. Rep.|volume=7|pages=41621|date=2017|doi=10.1038/srep41621|pmid=28155863|pmc=5290745|bibcode=2017NatSR...741621S}}</ref> [[mathematician]]s, and many other [[scientist]]s because most systems are inherently nonlinear in nature.<ref>{{cite book|last1=de Canete|first1=Javier, Cipriano Galindo, and Inmaculada Garcia-Moral|title=System Engineering and Automation: An Interactive Educational Approach|date=2011|publisher=Springer|location=Berlin|isbn=978-3642202292|page=46|url=https://books.google.com/?id=h8rCQYXGGY8C&pg=PA46&lpg=PA46&dq=most+systems+are+inherently+nonlinear+in+nature#v=onepage&q=most%20systems%20are%20inherently%20nonlinear%20in%20nature&f=false|accessdate=20 January 2018}}</ref> Nonlinear [[dynamical system]]s, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler [[linear system]]s.<br />
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In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.<br />
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在数学及科学中，非线性系统是一种输出的变化与输入的变化不成比例的系统。大多数系统在本质上是非线性的，因而非线性问题引起了工程师、生物学家、物理学家、数学家和许多其他科学家的兴趣。描述变量随时间变化的非线性动力系统与较之简单得多的线性系统相比，可能显得混沌、不可预测或违反直觉。<br />
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Typically, the behavior of a nonlinear system is described in mathematics by a '''nonlinear system of equations''', which is a set of simultaneous [[equation]]s in which the unknowns (or the unknown functions in the case of [[differential equation]]s) appear as variables of a [[polynomial]] of degree higher than one or in the argument of a [[function (mathematics)|function]] which is not a polynomial of degree one.<br />
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Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one.<br />
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通常，非线性系统的行为在数学上是由一组非线性联立方程来描述的，其中未知数(或微分方程中的未知函数)作为一个高于一次的多项式的变量出现，或者作为一个不是一次多项式的函数的论元出现。<br />
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In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a [[linear combination]] of the unknown [[variable (mathematics)|variables]] or [[function (mathematics)|functions]] that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is ''linear'' if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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换句话说，在非线性方程系统中，要求解的方程不能被写成未知变量或函数的线性组合。无论方程中是否有已知的线性函数，系统都可以被定义为非线性。特别是，如果一个微分方程的未知函数及其导数是线性的，即使其他变量是非线性的，也称其是线性的。<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations ([[linearization]]). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as [[soliton]]s, [[chaos theory|chaos]],<ref>[http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm Nonlinear Dynamics I: Chaos] {{webarchive|url=https://web.archive.org/web/20080212045134/http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm |date=2008-02-12}} at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]</ref> and [[mathematical singularity|singularities]] are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble [[randomness|random]] behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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由于非线性动力学方程难以求解，通常用线性化方程来近似非线性系统（线性化）。这种方法在一定的精度和范围对输入值效果很好，但一些有趣的现象如孤子、混沌和奇异性在线性化后被隐藏。因此，非线性系统的动态行为在某些方面可能看起来违反直觉、不可预测，甚至混沌。尽管这种混沌行为可能感觉很像随机行为，但它实际上并不是随机的。例如，天气的某些方面被认为是混沌的，其系统某部分的微小扰动就会产生复杂的影响。这种非线性是目前技术无法进行精确长期预测的原因之一。<br />
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Some authors use the term '''nonlinear science''' for the study of nonlinear systems. This term is disputed by others:<br />
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Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:<br />
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有些作者用非线性科学这一术语来研究非线性系统。这一术语引起了其他人的争议:<br />
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{{quote|Using a term like nonlinear science is like referring to the bulk of zoology as the study of [[negation|non]]-elephant animals.|[[Stanislaw Ulam]]<ref>{{cite journal|last1=Campbell|first1=David K.|title=Nonlinear physics: Fresh breather|journal=Nature|date=25 November 2004|volume=432|issue=7016|pages=455–456|doi=10.1038/432455a|pmid=15565139|url=https://zenodo.org/record/1134179|language=en|issn=0028-0836|bibcode=2004Natur.432..455C}}</ref>}}<br />
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“使用‘非线性科学’这样的术语，就如同把动物学里大部分对象称作‘非大象动物’研究一样可笑。”<br />
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——斯塔尼斯拉夫·乌拉姆<br />
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==Definition 定义==<br />
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In [[mathematics]], a [[linear map]] (or ''linear function'') <math>f(x)</math> is one which satisfies both of the following properties:<br />
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In mathematics, a linear map (or linear function) <math>f(x)</math> is one which satisfies both of the following properties:<br />
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在数学中，线性映射(或线性函数)的数学<math>f (x)</math>满足以下两个性质:<br />
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*Additivity or [[superposition principle]]: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*Homogeneity: <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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*可加性（叠加性）: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*齐次性: <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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Additivity implies homogeneity for any [[rational number|rational]] ''α'', and, for [[continuous function]]s, for any [[real number|real]] ''α''. For a [[complex number|complex]] ''α'', homogeneity does not follow from additivity. For example, an [[antilinear map]] is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle<br />
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Additivity implies homogeneity for any rational α, and, for continuous functions, for any real α. For a complex α, homogeneity does not follow from additivity. For example, an antilinear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle:<br />
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α是有理数，或α是实数且<math>f(x)</math>是连续函数时，由可加性可以推出齐次性。但当α是复数时，可加性不能导出齐次性。例如，反线性映射是可加的，但不是齐次的。可加性和齐次性条件经常组合，称为叠加原理：<br />
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<math>f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)</math><br />
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An equation written as<br />
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<math>f(x) = C</math><br />
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is called '''linear''' if <math>f(x)</math> is a linear map (as defined above) and '''nonlinear''' otherwise. The equation is called ''homogeneous'' if <math>C = 0</math>.<br />
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is called linear if <math>f(x)</math> is a linear map (as defined above) and nonlinear otherwise. The equation is called homogeneous if <math>C = 0</math>.<br />
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的方程，若 <math> f (x) </math> 是线性映射(如上定义) ，则称其为线性的，否则称为非线性的。若<math>C = 0</math>，该方程称为是齐次的。<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any [[map (mathematics)|mapping]], including integration or differentiation with associated constraints (such as [[boundary values]]). If <math>f(x)</math> contains [[derivative|differentiation]] with respect to <math>x</math>, the result will be a [[differential equation]].<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any mapping, including integration or differentiation with associated constraints (such as boundary values). If <math>f(x)</math> contains differentiation with respect to <math>x</math>, the result will be a differential equation.<br />
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定义 <math>f(x) = C</math> 是非常具有一般性的，因为 <math>x</math> 可以是任意可感知的数学对象（数字、向量、函数等），函数 <math>f(x)</math> 实际上可以是任意映射，包括有相关约束(如给定边界值)的积分或微分。若 <math>f(x)</math> 包含对 <math>x</math> 的微分运算，则该方程为微分方程。<br />
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==Nonlinear algebraic equations 非线性代数方程==<br />
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{{Main|Algebraic equation|System of polynomial equations}}<br />
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Nonlinear [[algebraic equation]]s, which are also called ''[[polynomial equation]]s'', are defined by equating [[polynomial]]s (of degree greater than one) to zero. For example,<br />
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Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials (of degree greater than one) to zero. For example,<br />
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非线性代数方程，又称多项式方程，由某多项式（次数大于1）等于零定义。例如：<br />
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<math>x^2 + x - 1 = 0\,.</math><br />
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For a single polynomial equation, [[root-finding algorithm]]s can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of [[algebraic geometry]], a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see [[Hilbert's Nullstellensatz]]). Nevertheless, in the case of the systems with a finite number of complex solutions, these [[systems of polynomial equations]] are now well understood and efficient methods exist for solving them.<ref>{{cite journal |last1= Lazard |first1= D. |title= Thirty years of Polynomial System Solving, and now? |doi= 10.1016/j.jsc.2008.03.004 |journal= Journal of Symbolic Computation |volume= 44 |issue= 3 |pages= 222–231 |year= 2009 |pmid= |pmc=}}</ref><br />
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For a single polynomial equation, root-finding algorithms can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of algebraic geometry, a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see Hilbert's Nullstellensatz). Nevertheless, in the case of the systems with a finite number of complex solutions, these systems of polynomial equations are now well understood and efficient methods exist for solving them.<br />
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对于一个单一的多项式方程，求根算法可用于其求解（即找到满足该方程的变量的值集）。而代数方程组则相对复杂，其研究是现代数学的较难分支——代数几何领域的动力之一。甚至很难判断一个给定的代数系统是否有复数解（见希尔伯特的零点定律）。不过，对于具有有限个复数解的系统的多项式方程组，我们现在已经有了充分的理解，并且找到了有效的求解方法。<br />
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==Nonlinear recurrence relations 非线性递推关系==<br />
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A nonlinear [[recurrence relation]] defines successive terms of a [[sequence]] as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the [[logistic map]] and the relations that define the various [[Hofstadter sequence]]s. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related [[nonlinear system identification]] and analysis procedures.<ref name="SAB1">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</ref> These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related nonlinear system identification and analysis procedures. These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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非线性递归关系中，序列的连续项被定义为其前项的非线性函数。非线性递归关系的例子有 logistic 映射和定义各种霍夫斯塔特序列 Hofstadter sequences 的关系。非线性离散模型代表了一类广泛的非线性递归关系，包括 NARMAX（外部输入非线性自回归移动平均）模型和相关的非线性系统辨识和分析程序。这些方法可用于研究时域、频域和时空域的广泛复杂非线性行为。<br />
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==Nonlinear differential equations 非线性微分方程==<br />
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A [[simultaneous equations|system]] of [[differential equation]]s is said to be nonlinear if it is not a [[linear system]]. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the [[Navier–Stokes equations]] in fluid dynamics and the [[Lotka–Volterra equation]]s in biology.<br />
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A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology.<br />
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若一个微分方程组不是线性系统，则称其为非线性的。涉及非线性微分方程的问题非常多样，对不同问题的解决或分析方法也不相同。非线性微分方程的例子有流体力学中的 纳维-斯托克斯 Navier-Stokes 方程和生物学中的洛特卡－沃尔泰拉 Lotka-Volterra 方程。<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of [[linearly independent]] solutions can be used to construct general solutions through the [[superposition principle]]. A good example of this is one-dimensional heat transport with [[Dirichlet boundary conditions]], the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle. A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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非线性问题最大的困难之一是通常不可能将已知的解组合成新的解。例如，在线性问题中，可以根据叠加原理以一族线性独立的解构造通解。一个很好的例子是带有狄利克雷 Dirichlet 边界条件的一维热传导问题，其解（随时间变化）可以写成不同频率的正弦波的线性组合，这使得解非常灵活。而对非线性方程，通常可以找到几个非常特殊的解，但是此时叠加原理不适用，故无法构造新的解。<br />
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===Ordinary differential equations 常微分方程===<br />
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First order [[ordinary differential equation]]s are often exactly solvable by [[separation of variables]], especially for autonomous equations. For example, the nonlinear equation<br />
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First order ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. For example, the nonlinear equation<br />
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一阶常微分方程，尤其是自治方程，通常可以用分离变量法来精确求解。例如，非线性方程<br />
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<math>\frac{d u}{d x} = -u^2</math><br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also ''u'' = 0 as a particular solution, corresponding to the limit of the general solution when ''C'' tends to infinity). The equation is nonlinear because it may be written as<br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also u = 0 as a particular solution, corresponding to the limit of the general solution when C tends to infinity). The equation is nonlinear because it may be written as<br />
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将 <math>u=\frac{1}{x+C}</math> 作为一般解（也有特解 ''u'' = 0，对应于 ''C'' 趋于无穷时的一般解的极限)。该方程是非线性的，因为它可以改写成<br />
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<math>\frac{du}{d x} + u^2=0</math><br />
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and the left-hand side of the equation is not a linear function of ''u'' and its derivatives. Note that if the ''u''<sup>2</sup> term were replaced with ''u'', the problem would be linear (the [[exponential decay]] problem).<br />
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and the left-hand side of the equation is not a linear function of u and its derivatives. Note that if the u<sup>2</sup> term were replaced with u, the problem would be linear (the exponential decay problem).<br />
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方程的左边不是 ''u'' 及其导数的线性函数。注意，若将 ''u''<sup>2</sup> 项替换为''u''，该问题将变为线性的(指数衰减问题)。<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield [[closed-form expression|closed-form]] solutions, though implicit solutions and solutions involving [[nonelementary integral]]s are encountered.<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered.<br />
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二阶和高阶常微分方程（更一般地说，非线性方程组）很少能产生封闭解，而隐式解和非初等函数积分形式的解较为常见。<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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非线性常微分方程定性分析的常用方法包括:<br />
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*Examination of any [[conserved quantities]], especially in [[Hamiltonian system]]s<br />
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*检查是否有任意守恒量（特别是在哈密顿系统中）<br />
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*Examination of dissipative quantities (see [[Lyapunov function]]) analogous to conserved quantities<br />
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*检查是否有类似守恒量的耗散量（见李亚普诺夫函数）<br />
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*Linearization via [[Taylor expansion]]<br />
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*基于泰勒展开的线性化<br />
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*Change of variables into something easier to study<br />
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*将变量转化得更易于研究<br />
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*[[Bifurcation theory]]<br />
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*分岔理论<br />
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*[[Perturbation theory|Perturbation]] methods (can be applied to algebraic equations too)<br />
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*摄动理论（也可应用于代数方程）<br />
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===Partial differential equations 偏微分方程===<br />
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{{main|Nonlinear partial differential equation}}<br />
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{{See also|List of nonlinear partial differential equations}}<br />
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参见：非线性偏微分列表<br />
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The most common basic approach to studying nonlinear [[partial differential equation]]s is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more [[ordinary differential equation]]s, as seen in [[separation of variables]], which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
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The most common basic approach to studying nonlinear partial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
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研究非线性偏微分方程最常用的基本方法是变换变量（或转换问题），使变换后的问题更简单（甚至可能变为线性的）。有时可以将此类方程转化成一或多个常微分方程（如同分离变量法所示），此时不论得到的常微分方程是否可解，对研究问题总是有用的。<br />
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Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use [[scale analysis (mathematics)|scale analysis]] to simplify a general, natural equation in a certain specific [[boundary value problem]]. For example, the (very) nonlinear [[Navier-Stokes equations]] can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
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Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem. For example, the (very) nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
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另一个流体力学和热力学中常见的策略（虽然不是数学上的）是利用尺度分析来在某一特定边界条件下简化一般自然方程。例如，在描述圆管内一维层流的暂态时，非线性的纳维－斯托克斯方程可以简化为一个线性的偏微分方程; 尺度分析提供了层流和一维流动的条件，也产生了简化的方程。<br />
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Other methods include examining the [[method of characteristics|characteristics]] and using the methods outlined above for ordinary differential equations.<br />
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Other methods include examining the characteristics and using the methods outlined above for ordinary differential equations.<br />
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其他方法包括检查特征线法及前面所述研究常微分方程的方法。<br />
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===Pendula 摆===<br />
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{{Main|Pendulum (mathematics)}}<br />
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[[File:PendulumLayout.svg.png|thumb|Illustration of a pendulum 摆图解|right|200px]]<br />
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[[File:PendulumLinearizations.png|thumb|Linearizations of a pendulum 摆的线性化|right|200px]]<br />
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A classic, extensively studied nonlinear problem is the dynamics of a [[pendulum (mathematics)|pendulum]] under the influence of [[gravity]]. Using [[Lagrangian mechanics]], it may be shown<ref>[http://www.damtp.cam.ac.uk/user/tong/dynamics.html David Tong: Lectures on Classical Dynamics]</ref> that the motion of a pendulum can be described by the [[dimensionless]] nonlinear equation<br />
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A classic, extensively studied nonlinear problem is the dynamics of a pendulum under the influence of gravity. Using Lagrangian mechanics, it may be shown that the motion of a pendulum can be described by the dimensionless nonlinear equation<br />
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一个经典的被广泛研究的非线性问题是重力影响下的摆的动力学。利用拉格朗日力学，可以证明摆的运动可以用无量纲的非线性方程<br />
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<math>\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0</math><br />
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where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an [[integrating factor]], which would eventually yield<br />
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where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an integrating factor, which would eventually yield<br />
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描述，其中重力指向“下方”，<math>\theta</math> 是摆与其静止位置形成的角度，如右图所示。“解”这个方程的方法之一是用 <math>d\theta/dt</math> 作为积分因子，最终得<br />
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<math>\int{\frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)}}} = t + C_1</math><br />
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which is an implicit solution involving an [[elliptic integral]]. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the [[nonelementary integral]] (nonelementary unless <math>C_0 = 2</math>).<br />
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which is an implicit solution involving an elliptic integral. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral (nonelementary unless <math>C_0 = 2</math>).<br />
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这是一个含椭圆积分的隐式解。这个“解”通常没什么用，因为这个解的大部分性质都隐藏在非初等函数积分中（除非<math>C_0 = 2</math>，否则是非初等的）。<br />
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Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through [[Taylor expansion]]s. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
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Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through Taylor expansions. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
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另一种解决这个问题的方法是利用泰勒展开将任意非线性项（此时为正弦函数项)在某些感兴趣的点线性化。例如，在<math>\theta = 0</math> 的点附近线性化（称为小角度近似）为<br />
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<math>\frac{d^2 \theta}{d t^2} + \theta = 0</math><br />
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since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a [[simple harmonic oscillator]] corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
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since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
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因为 <math>\theta \approx 0</math> 时，有 <math>\sin(\theta) \approx \theta</math>。这是一个简谐振子，对应于摆在其路径底部附近的摆动。另一种线性化方法是在 <math>\theta = \pi</math>附近线性化，对应于运动到最高点的摆:<br />
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<math>\frac{d^2 \theta}{d t^2} + \pi - \theta = 0</math><br />
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since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves [[hyperbolic sinusoid]]s, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
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since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
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因为 <math>\theta \approx \pi</math> 时，有 <math>\sin(\theta) \approx \pi - \theta</math>。这个问题的解含双曲正弦曲线；注意到不同于小角度近似，它是不稳定的，这意味着 <math>|\theta|</math> 通常会无限增长，尽管有界解是可能的。这相当于平衡一个直立的钟摆的难度，它实际上是一种不稳定的状态。<br />
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One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
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One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
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一个更有趣的线性化可能围绕着 math theta pi / 2 / math，围绕着 math sin ( theta) approx1 / math:<br />
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:<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math><br />
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<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math><br />
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Math frac ^ 2 theta } d t ^ 2} + 10. / math<br />
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This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) [[phase portrait]]s and approximate periods.<br />
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This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) phase portraits and approximate periods.<br />
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这相当于一个自由落体问题。一个非常有用的定性图片的钟摆的动态可以得到一起拼凑这样的线性化，如图中所示。其他技术可用于寻找(精确的)相位图和近似周期。<br />
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==Types of nonlinear dynamic behaviors==<br />
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==Types of nonlinear dynamic behaviors==<br />
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非线性动力学行为的类型<br />
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*[[Amplitude death]] – any oscillations present in the system cease due to some kind of interaction with other system or feedback by the same system<br />
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*[[Chaos theory|Chaos]] – values of a system cannot be predicted indefinitely far into the future, and fluctuations are [[aperiodic]]<br />
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*[[Multistability]] – the presence of two or more stable states<br />
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*[[Soliton]]s – self-reinforcing solitary waves<br />
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*[[Limit cycle|Limit cycles]] – asymptotic periodic orbits to which destabilized fixed points are attracted.<br />
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*[[Self-oscillation|Self-oscillations]] - feedback oscillations taking place in open dissipative physical systems.<br />
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==Examples of nonlinear equations==<br />
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==Examples of nonlinear equations==<br />
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非线性方程的例子<br />
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*[[Algebraic Riccati equation]]<br />
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*[[Ball and beam]] system<br />
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*[[Bellman equation]] for optimal policy<br />
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*[[Boltzmann equation]]<br />
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*[[Colebrook equation]]<br />
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*[[General relativity]]<br />
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*[[Ginzburg–Landau theory]]<br />
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*[[Ishimori equation]]<br />
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*[[Kadomtsev–Petviashvili equation]]<br />
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*[[Korteweg–de Vries equation]]<br />
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*[[Landau–Lifshitz–Gilbert equation]]<br />
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*[[Liénard equation]]<br />
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*[[Navier–Stokes equations]] of [[fluid dynamics]]<br />
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*[[Nonlinear optics]]<br />
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*[[Nonlinear Schrödinger equation]]<br />
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*[[Power-flow study]]<br />
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*[[Richards equation]] for unsaturated water flow<br />
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*[[Self-balancing unicycle]]<br />
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*[[Sine-Gordon equation]]<br />
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*[[Van der Pol oscillator]]<br />
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*[[Vlasov equation]]<br />
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==See also==<br />
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==See also==<br />
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参见<br />
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*[[Aleksandr Mikhailovich Lyapunov]]<br />
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*[[Dynamical system]]<br />
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*[[Feedback]]<br />
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*[[Initial condition]]<br />
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*[[Interaction]]<br />
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*[[Linear system]]<br />
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*[[Mode coupling]]<br />
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*[[Vector soliton]]<br />
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*[[Volterra series]]<br />
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==References==<br />
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==References==<br />
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参考资料<br />
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{{Reflist|35em}}<br />
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==Further reading==<br />
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==Further reading==<br />
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进一步阅读<br />
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{{Refbegin|35em}}<br />
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*{{cite book<br />
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| author= [[Diederich Hinrichsen]] and Anthony J. Pritchard<br />
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| author= Diederich Hinrichsen and Anthony J. Pritchard<br />
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作者 Diederich Hinrichsen 和 Anthony j. Pritchard<br />
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| year= 2005<br />
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| year= 2005<br />
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2005年<br />
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| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
<br />
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
<br />
数学系统理论 i-建模，状态空间分析，稳定性和鲁棒性<br />
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| publisher= Springer Verlag<br />
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| publisher= Springer Verlag<br />
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出版商 Springer Verlag<br />
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| isbn=9783540441250<br />
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| isbn=9783540441250<br />
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9783540441250<br />
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}}<br />
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*{{cite book<br />
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| last= Jordan<br />
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| last= Jordan<br />
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最后一个乔丹<br />
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| first= D. W.<br />
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| first= D. W.<br />
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首先是 D.w。<br />
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| last2= Smith<br />
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| last2= Smith<br />
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最后2个史密斯<br />
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| first2= P.<br />
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| first2= P.<br />
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| first2 p.<br />
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| year= 2007<br />
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| year= 2007<br />
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2007年<br />
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| edition= fourth<br />
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| edition= fourth<br />
<br />
第四版<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
非线性常微分方程<br />
<br />
| publisher= Oxford University Press<br />
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| publisher= Oxford University Press<br />
<br />
牛津大学出版社<br />
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| isbn= 978-0-19-920824-1<br />
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| isbn= 978-0-19-920824-1<br />
<br />
[国际标准图书编号978-0-19-920824-1]<br />
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}}<br />
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}}<br />
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*{{cite book<br />
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| last= Khalil<br />
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| last= Khalil<br />
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最后的哈利勒<br />
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| first= Hassan K.<br />
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| first= Hassan K.<br />
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首先是哈桑 · k。<br />
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| year= 2001<br />
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| year= 2001<br />
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2001年<br />
<br />
| title= Nonlinear Systems<br />
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| title= Nonlinear Systems<br />
<br />
非线性系统<br />
<br />
| publisher= Prentice Hall<br />
<br />
| publisher= Prentice Hall<br />
<br />
出版商 Prentice Hall<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
[国际标准图书编号978-0-13-067389-3]<br />
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}}<br />
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}}<br />
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}}<br />
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*{{cite book<br />
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| last= Kreyszig<br />
<br />
| last= Kreyszig<br />
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| last= Kreyszig<br />
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| first= Erwin<br />
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| first= Erwin<br />
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首先是欧文<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
作者 / 链接 Erwin Kreyszig<br />
<br />
| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
高等工程数学<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
Https://archive.org/details/advancedengineer0008krey<br />
<br />
| url-access= registration<br />
<br />
| url-access= registration<br />
<br />
访问注册<br />
<br />
| publisher= Wiley<br />
<br />
| publisher= Wiley<br />
<br />
出版商 Wiley<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
[国际标准图书编号978-0-471-15496-9]<br />
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}}<br />
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}}<br />
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}}<br />
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*{{cite book<br />
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| last= Sontag<br />
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| last= Sontag<br />
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最后一个桑塔格<br />
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| first= Eduardo<br />
<br />
| first= Eduardo<br />
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首先是爱德华多<br />
<br />
| authorlink= Eduardo D. Sontag<br />
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| authorlink= Eduardo D. Sontag<br />
<br />
作者: Eduardo d. Sontag<br />
<br />
| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
数学控制理论: 确定性有限维系统。第二版<br />
<br />
| publisher= Springer<br />
<br />
| publisher= Springer<br />
<br />
出版商斯普林格<br />
<br />
| isbn= 978-0-387-98489-6<br />
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| isbn= 978-0-387-98489-6<br />
<br />
[国际标准图书编号978-0-387-98489-6]<br />
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{{Refend}}<br />
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==External links==<br />
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==External links==<br />
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外部链接<br />
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*[http://www.dodccrp.org/ Command and Control Research Program (CCRP)]<br />
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*[http://necsi.edu/guide/concepts/linearnonlinear.html New England Complex Systems Institute: Concepts in Complex Systems]<br />
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*[http://ocw.mit.edu/courses/mathematics/18-353j-nonlinear-dynamics-i-chaos-fall-2012/ Nonlinear Dynamics I: Chaos] at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]<br />
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*[http://www.hedengren.net/research/models.htm Nonlinear Model Library]{{snd}} (in [[MATLAB]]) a Database of Physical Systems<br />
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*[http://cnls.lanl.gov/ The Center for Nonlinear Studies at Los Alamos National Laboratory]<br />
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[[Category:Nonlinear systems| ]]<br />
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[[Category:Dynamical systems]]<br />
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Category:Dynamical systems<br />
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类别: 动力系统<br />
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[[Category:Concepts in physics]]<br />
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Category:Concepts in physics<br />
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分类: 物理概念<br />
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<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Nonlinear system]]. Its edit history can be viewed at [[非线性系统/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E6%96%87%E4%BB%B6:PendulumLayout.svg.png&diff=10041文件:PendulumLayout.svg.png2020-07-09T14:47:24Z<p>Dorr：</p>
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<div></div>Dorrhttps://wiki.swarma.org/index.php?title=%E6%96%87%E4%BB%B6:PendulumLinearizations.png&diff=10040文件:PendulumLinearizations.png2020-07-09T14:43:51Z<p>Dorr：</p>
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<div></div>Dorrhttps://wiki.swarma.org/index.php?title=%E9%9D%9E%E7%BA%BF%E6%80%A7%E7%B3%BB%E7%BB%9F&diff=10037非线性系统2020-07-09T10:53:54Z<p>Dorr：</p>
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<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{Redirect|Nonlinear dynamics|the journal|Nonlinear Dynamics (journal)}}<br />
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{{About|"nonlinearity" in mathematics, physics and other sciences|video and film editing|Non-linear editing system|other uses|Nonlinearity (disambiguation)}}<br />
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In [[mathematics]] and [[science]], a '''nonlinear system''' is a [[system]] in which the change of the output is not [[proportionality (mathematics)|proportional]] to the change of the input.<ref>{{Cite news|url=https://news.mit.edu/2010/explained-linear-0226|title=Explained: Linear and nonlinear systems|work=MIT News|access-date=2018-06-30}}</ref><ref>{{Cite web|url=https://www.birmingham.ac.uk/research/activity/mathematics/applied-maths/nonlinear-systems.aspx|title=Nonlinear systems, Applied Mathematics - University of Birmingham|website=www.birmingham.ac.uk|language=en-gb|access-date=2018-06-30}}</ref> Nonlinear problems are of interest to [[engineer]]s, [[biologist]]s,<ref>{{Citation|date=2007|pages=181–276|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-540-34153-6_7|isbn=9783540341529|title = The Nonlinear Universe|series = The Frontiers Collection|chapter = Nonlinear Biology}}</ref><ref>{{cite journal|last=Korenberg|first=Michael J.|last2=Hunter|first2=Ian W.|date=March 1996|title=The identification of nonlinear biological systems: Volterra kernel approaches|journal=Annals of Biomedical Engineering|language=en|volume=24|issue=2|pages=250–268|doi=10.1007/bf02667354|issn=0090-6964}}</ref><ref>{{cite journal|last=Mosconi|first=Francesco|last2=Julou|first2=Thomas|last3=Desprat|first3=Nicolas|last4=Sinha|first4=Deepak Kumar|last5=Allemand|first5=Jean-François|last6=Vincent Croquette|last7=Bensimon|first7=David|date=2008|title=Some nonlinear challenges in biology|url=http://stacks.iop.org/0951-7715/21/i=8/a=T03|journal=Nonlinearity|language=en|volume=21|issue=8|pages=T131|doi=10.1088/0951-7715/21/8/T03|issn=0951-7715|bibcode=2008Nonli..21..131M}}</ref> [[physicist]]s,<ref>{{cite journal|last1=Gintautas|first1=V.|title=Resonant forcing of nonlinear systems of differential equations|journal=Chaos|date=2008|volume=18|issue=3|pages=033118|doi=10.1063/1.2964200|pmid=19045456|arxiv=0803.2252|bibcode=2008Chaos..18c3118G}}</ref><ref>{{cite journal|last1=Stephenson|first1=C.|last2=et.|first2=al.|title=Topological properties of a self-assembled electrical network via ab initio calculation|journal=Sci. Rep.|volume=7|pages=41621|date=2017|doi=10.1038/srep41621|pmid=28155863|pmc=5290745|bibcode=2017NatSR...741621S}}</ref> [[mathematician]]s, and many other [[scientist]]s because most systems are inherently nonlinear in nature.<ref>{{cite book|last1=de Canete|first1=Javier, Cipriano Galindo, and Inmaculada Garcia-Moral|title=System Engineering and Automation: An Interactive Educational Approach|date=2011|publisher=Springer|location=Berlin|isbn=978-3642202292|page=46|url=https://books.google.com/?id=h8rCQYXGGY8C&pg=PA46&lpg=PA46&dq=most+systems+are+inherently+nonlinear+in+nature#v=onepage&q=most%20systems%20are%20inherently%20nonlinear%20in%20nature&f=false|accessdate=20 January 2018}}</ref> Nonlinear [[dynamical system]]s, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler [[linear system]]s.<br />
<br />
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.<br />
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在数学及科学中，非线性系统是一种输出的变化与输入的变化不成比例的系统。大多数系统在本质上是非线性的，因而非线性问题引起了工程师、生物学家、物理学家、数学家和许多其他科学家的兴趣。描述变量随时间变化的非线性动力系统与较之简单得多的线性系统相比，可能显得混沌、不可预测或违反直觉。<br />
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Typically, the behavior of a nonlinear system is described in mathematics by a '''nonlinear system of equations''', which is a set of simultaneous [[equation]]s in which the unknowns (or the unknown functions in the case of [[differential equation]]s) appear as variables of a [[polynomial]] of degree higher than one or in the argument of a [[function (mathematics)|function]] which is not a polynomial of degree one.<br />
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Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one.<br />
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通常，非线性系统的行为在数学上是由一组非线性联立方程来描述的，其中未知数(或微分方程中的未知函数)作为一个高于一次的多项式的变量出现，或者作为一个不是一次多项式的函数的论元出现。<br />
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In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a [[linear combination]] of the unknown [[variable (mathematics)|variables]] or [[function (mathematics)|functions]] that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is ''linear'' if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.<br />
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换句话说，在非线性方程系统中，要求解的方程不能被写成未知变量或函数的线性组合。无论方程中是否有已知的线性函数，系统都可以被定义为非线性。特别是，如果一个微分方程的未知函数及其导数是线性的，即使其他变量是非线性的，也称其是线性的。<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations ([[linearization]]). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as [[soliton]]s, [[chaos theory|chaos]],<ref>[http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm Nonlinear Dynamics I: Chaos] {{webarchive|url=https://web.archive.org/web/20080212045134/http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm |date=2008-02-12}} at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]</ref> and [[mathematical singularity|singularities]] are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble [[randomness|random]] behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.<br />
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由于非线性动力学方程难以求解，通常用线性化方程来近似非线性系统（线性化）。这种方法在一定的精度和范围对输入值效果很好，但一些有趣的现象如孤子、混沌和奇异性在线性化后被隐藏。因此，非线性系统的动态行为在某些方面可能看起来违反直觉、不可预测，甚至混沌。尽管这种混沌行为可能感觉很像随机行为，但它实际上并不是随机的。例如，天气的某些方面被认为是混沌的，其系统某部分的微小扰动就会产生复杂的影响。这种非线性是目前技术无法进行精确长期预测的原因之一。<br />
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Some authors use the term '''nonlinear science''' for the study of nonlinear systems. This term is disputed by others:<br />
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Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:<br />
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有些作者用非线性科学这一术语来研究非线性系统。这一术语引起了其他人的争议:<br />
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{{quote|Using a term like nonlinear science is like referring to the bulk of zoology as the study of [[negation|non]]-elephant animals.|[[Stanislaw Ulam]]<ref>{{cite journal|last1=Campbell|first1=David K.|title=Nonlinear physics: Fresh breather|journal=Nature|date=25 November 2004|volume=432|issue=7016|pages=455–456|doi=10.1038/432455a|pmid=15565139|url=https://zenodo.org/record/1134179|language=en|issn=0028-0836|bibcode=2004Natur.432..455C}}</ref>}}<br />
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“使用‘非线性科学’这样的术语，就如同把动物学里大部分对象称作‘非大象动物’研究一样可笑。”<br />
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——斯塔尼斯拉夫．乌拉姆<br />
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==Definition==<br />
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==Definition==<br />
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定义<br />
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In [[mathematics]], a [[linear map]] (or ''linear function'') <math>f(x)</math> is one which satisfies both of the following properties:<br />
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In mathematics, a linear map (or linear function) <math>f(x)</math> is one which satisfies both of the following properties:<br />
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在数学中，线性映射(或线性函数)的数学<math>f (x)</math>满足以下两个性质:<br />
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*Additivity or [[superposition principle]]: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*Homogeneity: <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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*可加性（叠加性）: <math>\textstyle f(x + y) = f(x) + f(y);</math><br />
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*齐次性: <math>\textstyle f(\alpha x) = \alpha f(x).</math><br />
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Additivity implies homogeneity for any [[rational number|rational]] ''α'', and, for [[continuous function]]s, for any [[real number|real]] ''α''. For a [[complex number|complex]] ''α'', homogeneity does not follow from additivity. For example, an [[antilinear map]] is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle<br />
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Additivity implies homogeneity for any rational α, and, for continuous functions, for any real α. For a complex α, homogeneity does not follow from additivity. For example, an antilinear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle:<br />
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α是有理数，或α是实数且<math>f(x)</math>是连续函数时，由可加性可以推出齐次性。但当α是复数时，可加性不能导出齐次性。例如，反线性映射是可加的，但不是齐次的。可加性和齐次性条件经常组合，称为叠加原理：<br />
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<math>f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)</math><br />
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An equation written as<br />
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写成<br />
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<math>f(x) = C</math><br />
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is called '''linear''' if <math>f(x)</math> is a linear map (as defined above) and '''nonlinear''' otherwise. The equation is called ''homogeneous'' if <math>C = 0</math>.<br />
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is called linear if <math>f(x)</math> is a linear map (as defined above) and nonlinear otherwise. The equation is called homogeneous if <math>C = 0</math>.<br />
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的一个等式称为是线性的，如果 <math> f (x) </math> 是线性映射(如上所定义的) ，则称为线性映射，否则称为非线性映射。如果用数学的话，这个方程叫齐次方程。<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any [[map (mathematics)|mapping]], including integration or differentiation with associated constraints (such as [[boundary values]]). If <math>f(x)</math> contains [[derivative|differentiation]] with respect to <math>x</math>, the result will be a [[differential equation]].<br />
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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any mapping, including integration or differentiation with associated constraints (such as boundary values). If <math>f(x)</math> contains differentiation with respect to <math>x</math>, the result will be a differential equation.<br />
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定义 math f (x) c / math 是非常通用的，因为 math x / math 可以是任何可感知的数学对象(数字、向量、函数等等)。) ，函数 math f (x) / math 实际上可以是任何映射，包括与相关约束(如边界值)的集成或微分。如果 math f (x) / math 包含了与 math x / math 相关的微分，那么结果将是一个微分方程。<br />
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==Nonlinear algebraic equations==<br />
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==Nonlinear algebraic equations==<br />
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非线性代数方程<br />
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{{Main|Algebraic equation|System of polynomial equations}}<br />
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Nonlinear [[algebraic equation]]s, which are also called ''[[polynomial equation]]s'', are defined by equating [[polynomial]]s (of degree greater than one) to zero. For example,<br />
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Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials (of degree greater than one) to zero. For example,<br />
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非线性代数方程，也称为多项式方程，是用多项式(次数大于1)等于零来定义的。比如说,<br />
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:<math>x^2 + x - 1 = 0\,.</math><br />
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<math>x^2 + x - 1 = 0\,.</math><br />
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数学 x ^ 2 + x-10 / 数学<br />
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For a single polynomial equation, [[root-finding algorithm]]s can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of [[algebraic geometry]], a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see [[Hilbert's Nullstellensatz]]). Nevertheless, in the case of the systems with a finite number of complex solutions, these [[systems of polynomial equations]] are now well understood and efficient methods exist for solving them.<ref>{{cite journal |last1= Lazard |first1= D. |title= Thirty years of Polynomial System Solving, and now? |doi= 10.1016/j.jsc.2008.03.004 |journal= Journal of Symbolic Computation |volume= 44 |issue= 3 |pages= 222–231 |year= 2009 |pmid= |pmc=}}</ref><br />
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For a single polynomial equation, root-finding algorithms can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of algebraic geometry, a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see Hilbert's Nullstellensatz). Nevertheless, in the case of the systems with a finite number of complex solutions, these systems of polynomial equations are now well understood and efficient methods exist for solving them.<br />
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对于一个单一的多项式方程，求根算法可用于求解该方程(即满足该方程的变量的值集)。然而，代数方程组更加复杂; 他们的研究是代数几何领域的一个动力，这是现代数学的一个难以理解的分支。甚至很难判断一个给定的代数系统是否有复解(见希尔伯特的零点定律)。然而，对于具有有限个复解的系统，这些多项式方程组现在已经被很好地理解，并且存在有效的方法来解决它们。<br />
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==Nonlinear recurrence relations==<br />
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==Nonlinear recurrence relations==<br />
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非线性递推关系<br />
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A nonlinear [[recurrence relation]] defines successive terms of a [[sequence]] as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the [[logistic map]] and the relations that define the various [[Hofstadter sequence]]s. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related [[nonlinear system identification]] and analysis procedures.<ref name="SAB1">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</ref> These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related nonlinear system identification and analysis procedures. These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.<br />
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非线性递回关系式将序列的连续项定义为前面项的非线性函数。非线性递归关系的例子有 logistic 映射和定义各种 Hofstadter 序列的关系。非线性离散模型是一类广泛的非线性递归关系，包括 NARMAX (非线性自回归移动平均加外生输入)模型和相关的非线性系统辨识和分析程序。这些方法可用于研究一类广泛的复杂非线性行为的时间，频率和时空域。<br />
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==Nonlinear differential equations==<br />
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==Nonlinear differential equations==<br />
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非线性微分方程<br />
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A [[simultaneous equations|system]] of [[differential equation]]s is said to be nonlinear if it is not a [[linear system]]. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the [[Navier–Stokes equations]] in fluid dynamics and the [[Lotka–Volterra equation]]s in biology.<br />
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A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology.<br />
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如果一个微分方程组不是一个线性系统，我们就说它是非线性的。涉及非线性微分方程的问题是非常多样的，解决或分析的方法是依赖于问题的。非线性微分方程的例子有流体力学中的 Navier-Stokes 方程和生物学中的 Lotka-Volterra 方程。<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of [[linearly independent]] solutions can be used to construct general solutions through the [[superposition principle]]. A good example of this is one-dimensional heat transport with [[Dirichlet boundary conditions]], the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle. A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.<br />
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非线性问题最大的困难之一是通常不可能将已知的解组合成新的解。例如，在线性问题中，一族线性无关的解可以通过叠加原理来构造通解。一个很好的例子是一维带有狄利克雷边界条件的热传输，其解可以写成频率不同的正弦波随时间变化的线性组合，这使得解非常灵活。通常可以找到几个非常具体的非线性方程的解，但是由于缺乏叠加原理，无法构造新的解。<br />
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===Ordinary differential equations===<br />
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===Ordinary differential equations===<br />
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常微分方程<br />
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First order [[ordinary differential equation]]s are often exactly solvable by [[separation of variables]], especially for autonomous equations. For example, the nonlinear equation<br />
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First order ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. For example, the nonlinear equation<br />
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一阶常微分方程，尤其是自治方程，通常可以用分离变量法精确地求解。例如，非线性方程<br />
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:<math>\frac{d u}{d x} = -u^2</math><br />
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<math>\frac{d u}{d x} = -u^2</math><br />
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数学问题-数学问题<br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also ''u'' = 0 as a particular solution, corresponding to the limit of the general solution when ''C'' tends to infinity). The equation is nonlinear because it may be written as<br />
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has <math>u=\frac{1}{x+C}</math> as a general solution (and also u = 0 as a particular solution, corresponding to the limit of the general solution when C tends to infinity). The equation is nonlinear because it may be written as<br />
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将 math u frac { x + c } / math 作为一般解(也将 u 0作为特殊解，对应于 c 趋于无穷时的一般解的极限)。这个方程是非线性的，因为它可以写成<br />
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:<math>\frac{du}{d x} + u^2=0</math><br />
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<math>\frac{du}{d x} + u^2=0</math><br />
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[数学][数学][数学]<br />
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and the left-hand side of the equation is not a linear function of ''u'' and its derivatives. Note that if the ''u''<sup>2</sup> term were replaced with ''u'', the problem would be linear (the [[exponential decay]] problem).<br />
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and the left-hand side of the equation is not a linear function of u and its derivatives. Note that if the u<sup>2</sup> term were replaced with u, the problem would be linear (the exponential decay problem).<br />
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方程的左边不是 u 及其导数的线性函数。注意，如果 u sup 2 / sup 项替换为 u，问题将是线性的(指数衰减问题)。<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield [[closed-form expression|closed-form]] solutions, though implicit solutions and solutions involving [[nonelementary integral]]s are encountered.<br />
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Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered.<br />
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二阶和高阶常微分方程(更一般地说，非线性方程组)很少能产生闭合形式的解，尽管遇到了隐式解和涉及非初等积分的解。<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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Common methods for the qualitative analysis of nonlinear ordinary differential equations include:<br />
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非线性常微分方程定性分析的常用方法包括:<br />
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*Examination of any [[conserved quantities]], especially in [[Hamiltonian system]]s<br />
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*Examination of dissipative quantities (see [[Lyapunov function]]) analogous to conserved quantities<br />
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*Linearization via [[Taylor expansion]]<br />
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*Change of variables into something easier to study<br />
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*[[Bifurcation theory]]<br />
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*[[Perturbation theory|Perturbation]] methods (can be applied to algebraic equations too)<br />
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===Partial differential equations===<br />
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===Partial differential equations===<br />
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偏微分方程<br />
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{{main|Nonlinear partial differential equation}}<br />
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{{See also|List of nonlinear partial differential equations}}<br />
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The most common basic approach to studying nonlinear [[partial differential equation]]s is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more [[ordinary differential equation]]s, as seen in [[separation of variables]], which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
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The most common basic approach to studying nonlinear partial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable.<br />
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研究非线性偏微分方程最常用的基本方法是改变变量(或转换问题) ，使得问题更简单(甚至可能是线性的)。有时候，这个方程可以转化成一个或多个常微分方程，就像在分离变量法中看到的那样，不管得到的常微分方程是否可解，这个常微分方程都是有用的。<br />
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Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use [[scale analysis (mathematics)|scale analysis]] to simplify a general, natural equation in a certain specific [[boundary value problem]]. For example, the (very) nonlinear [[Navier-Stokes equations]] can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
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Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem. For example, the (very) nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.<br />
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另一个常见的策略(虽然不是数学上的) ，通常见于流体力学和热力学，是使用尺度分析来简化某一特定边值问题的一般自然方程。例如，非线性的纳维－斯托克斯方程可以简化为一个线性的偏微分方程，在圆管中的瞬态、层流、一维流动; 尺度分析提供了层流和一维流动的条件，也产生了简化的方程。<br />
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Other methods include examining the [[method of characteristics|characteristics]] and using the methods outlined above for ordinary differential equations.<br />
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Other methods include examining the characteristics and using the methods outlined above for ordinary differential equations.<br />
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其他方法包括研究常微分方程的特性和使用上述方法。<br />
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===Pendula===<br />
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===Pendula===<br />
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钟摆属<br />
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{{Main|Pendulum (mathematics)}}<br />
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[[File:PendulumLayout.svg|thumb|Illustration of a pendulum|right|200px]]<br />
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[[File:PendulumLinearizations.png|thumb|Linearizations of a pendulum|right|200px]]<br />
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A classic, extensively studied nonlinear problem is the dynamics of a [[pendulum (mathematics)|pendulum]] under the influence of [[gravity]]. Using [[Lagrangian mechanics]], it may be shown<ref>[http://www.damtp.cam.ac.uk/user/tong/dynamics.html David Tong: Lectures on Classical Dynamics]</ref> that the motion of a pendulum can be described by the [[dimensionless]] nonlinear equation<br />
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A classic, extensively studied nonlinear problem is the dynamics of a pendulum under the influence of gravity. Using Lagrangian mechanics, it may be shown that the motion of a pendulum can be described by the dimensionless nonlinear equation<br />
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一个经典的，广泛研究的非线性问题是重力影响下的摆的动力学。利用拉格朗日力学，可以证明摆的运动可以用无量纲的非线性方程来描述<br />
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:<math>\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0</math><br />
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<math>\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0</math><br />
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Math frac ^ 2 theta } d ^ 2} + sin ( theta)0 / math<br />
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where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an [[integrating factor]], which would eventually yield<br />
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where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an integrating factor, which would eventually yield<br />
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其中重力指向“下方”和 math theta / math 是摆与其静止位置形成的角度，如右图所示。“解决”这个方程式的一个方法是用 math d theta / dt / math 作为积分因子，它最终会产生<br />
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:<math>\int{\frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)}}} = t + C_1</math><br />
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<math>\int{\frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)}}} = t + C_1</math><br />
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Math int { frac { d theta }{ sqrt { c0 + 2 cos ( theta)}} t + c1 / math<br />
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which is an implicit solution involving an [[elliptic integral]]. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the [[nonelementary integral]] (nonelementary unless <math>C_0 = 2</math>).<br />
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which is an implicit solution involving an elliptic integral. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral (nonelementary unless <math>C_0 = 2</math>).<br />
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这是一个隐式的解决方案，涉及到一个椭圆积分。这个“解”通常没有多少用处，因为这个解的大部分性质都隐藏在非初等积分中(除非是数学 c02 / math)。<br />
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Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through [[Taylor expansion]]s. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
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Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through Taylor expansions. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is<br />
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另一种解决这个问题的方法是通过泰勒展开将任何非线性(在这种情况下是正弦函数项)在各个感兴趣的点线性化。例如，math theta 0 / math 的线性化，称为小角度近似<br />
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:<math>\frac{d^2 \theta}{d t^2} + \theta = 0</math><br />
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<math>\frac{d^2 \theta}{d t^2} + \theta = 0</math><br />
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Math frac ^ 2 theta } d ^ 2} + theta 0 / math<br />
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since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a [[simple harmonic oscillator]] corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
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since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:<br />
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从 math sin theta approx theta / math theta approx 0 / math 开始。这是一个简单的谐振子，对应于摆在其路径底部附近的摆动。另一种线性化方法是 math theta pi / math，对应于钟摆是笔直向上的:<br />
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:<math>\frac{d^2 \theta}{d t^2} + \pi - \theta = 0</math><br />
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<math>\frac{d^2 \theta}{d t^2} + \pi - \theta = 0</math><br />
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Math frac ^ 2 theta } d ^ 2} + pi- theta 0 / math<br />
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since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves [[hyperbolic sinusoid]]s, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
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since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.<br />
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自从 math sin ( theta) approx pi theta / math 对 math theta π / math。这个问题的解决方案涉及到双曲正弦曲线，并且注意到这个近似不同于小角度近似，它是不稳定的，这意味着 math | theta | math 通常会无限增长，尽管有界解是可能的。这相当于平衡一个直立的钟摆的难度，它实际上是一种不稳定的状态。<br />
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One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
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One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:<br />
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一个更有趣的线性化可能围绕着 math theta pi / 2 / math，围绕着 math sin ( theta) approx1 / math:<br />
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:<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math><br />
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<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math><br />
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Math frac ^ 2 theta } d t ^ 2} + 10. / math<br />
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This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) [[phase portrait]]s and approximate periods.<br />
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This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) phase portraits and approximate periods.<br />
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这相当于一个自由落体问题。一个非常有用的定性图片的钟摆的动态可以得到一起拼凑这样的线性化，如图中所示。其他技术可用于寻找(精确的)相位图和近似周期。<br />
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==Types of nonlinear dynamic behaviors==<br />
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==Types of nonlinear dynamic behaviors==<br />
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*[[Amplitude death]] – any oscillations present in the system cease due to some kind of interaction with other system or feedback by the same system<br />
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*[[Chaos theory|Chaos]] – values of a system cannot be predicted indefinitely far into the future, and fluctuations are [[aperiodic]]<br />
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*[[Multistability]] – the presence of two or more stable states<br />
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*[[Soliton]]s – self-reinforcing solitary waves<br />
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*[[Limit cycle|Limit cycles]] – asymptotic periodic orbits to which destabilized fixed points are attracted.<br />
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*[[Self-oscillation|Self-oscillations]] - feedback oscillations taking place in open dissipative physical systems.<br />
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==Examples of nonlinear equations==<br />
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==Examples of nonlinear equations==<br />
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非线性方程的例子<br />
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*[[Algebraic Riccati equation]]<br />
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*[[Bellman equation]] for optimal policy<br />
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*[[Boltzmann equation]]<br />
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*[[Colebrook equation]]<br />
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*[[General relativity]]<br />
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*[[Navier–Stokes equations]] of [[fluid dynamics]]<br />
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*[[Nonlinear optics]]<br />
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*[[Power-flow study]]<br />
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*[[Richards equation]] for unsaturated water flow<br />
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*[[Self-balancing unicycle]]<br />
<br />
<br />
<br />
*[[Sine-Gordon equation]]<br />
<br />
<br />
<br />
*[[Van der Pol oscillator]]<br />
<br />
<br />
<br />
*[[Vlasov equation]]<br />
<br />
<br />
<br />
{{Div col end}}<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
==See also==<br />
<br />
==See also==<br />
<br />
参见<br />
<br />
*[[Aleksandr Mikhailovich Lyapunov]]<br />
<br />
<br />
<br />
*[[Dynamical system]]<br />
<br />
<br />
<br />
*[[Feedback]]<br />
<br />
<br />
<br />
*[[Initial condition]]<br />
<br />
<br />
<br />
*[[Interaction]]<br />
<br />
<br />
<br />
*[[Linear system]]<br />
<br />
<br />
<br />
*[[Mode coupling]]<br />
<br />
<br />
<br />
*[[Vector soliton]]<br />
<br />
<br />
<br />
*[[Volterra series]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
==References==<br />
<br />
==References==<br />
<br />
参考资料<br />
<br />
{{Reflist|35em}}<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
==Further reading==<br />
<br />
==Further reading==<br />
<br />
进一步阅读<br />
<br />
{{Refbegin|35em}}<br />
<br />
<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| author= [[Diederich Hinrichsen]] and Anthony J. Pritchard<br />
<br />
| author= Diederich Hinrichsen and Anthony J. Pritchard<br />
<br />
作者 Diederich Hinrichsen 和 Anthony j. Pritchard<br />
<br />
| year= 2005<br />
<br />
| year= 2005<br />
<br />
2005年<br />
<br />
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
<br />
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness<br />
<br />
数学系统理论 i-建模，状态空间分析，稳定性和鲁棒性<br />
<br />
| publisher= Springer Verlag<br />
<br />
| publisher= Springer Verlag<br />
<br />
出版商 Springer Verlag<br />
<br />
| isbn=9783540441250<br />
<br />
| isbn=9783540441250<br />
<br />
9783540441250<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Jordan<br />
<br />
| last= Jordan<br />
<br />
最后一个乔丹<br />
<br />
| first= D. W.<br />
<br />
| first= D. W.<br />
<br />
首先是 D.w。<br />
<br />
| last2= Smith<br />
<br />
| last2= Smith<br />
<br />
最后2个史密斯<br />
<br />
| first2= P.<br />
<br />
| first2= P.<br />
<br />
| first2 p.<br />
<br />
| year= 2007<br />
<br />
| year= 2007<br />
<br />
2007年<br />
<br />
| edition= fourth<br />
<br />
| edition= fourth<br />
<br />
第四版<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
| title= Nonlinear Ordinary Differential Equations<br />
<br />
非线性常微分方程<br />
<br />
| publisher= Oxford University Press<br />
<br />
| publisher= Oxford University Press<br />
<br />
牛津大学出版社<br />
<br />
| isbn= 978-0-19-920824-1<br />
<br />
| isbn= 978-0-19-920824-1<br />
<br />
[国际标准图书编号978-0-19-920824-1]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Khalil<br />
<br />
| last= Khalil<br />
<br />
最后的哈利勒<br />
<br />
| first= Hassan K.<br />
<br />
| first= Hassan K.<br />
<br />
首先是哈桑 · k。<br />
<br />
| year= 2001<br />
<br />
| year= 2001<br />
<br />
2001年<br />
<br />
| title= Nonlinear Systems<br />
<br />
| title= Nonlinear Systems<br />
<br />
非线性系统<br />
<br />
| publisher= Prentice Hall<br />
<br />
| publisher= Prentice Hall<br />
<br />
出版商 Prentice Hall<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
| isbn= 978-0-13-067389-3<br />
<br />
[国际标准图书编号978-0-13-067389-3]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Kreyszig<br />
<br />
| last= Kreyszig<br />
<br />
| last= Kreyszig<br />
<br />
| first= Erwin<br />
<br />
| first= Erwin<br />
<br />
首先是欧文<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
| authorlink= Erwin Kreyszig<br />
<br />
作者 / 链接 Erwin Kreyszig<br />
<br />
| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
| title= Advanced Engineering Mathematics<br />
<br />
高等工程数学<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
| url= https://archive.org/details/advancedengineer0008krey<br />
<br />
Https://archive.org/details/advancedengineer0008krey<br />
<br />
| url-access= registration<br />
<br />
| url-access= registration<br />
<br />
访问注册<br />
<br />
| publisher= Wiley<br />
<br />
| publisher= Wiley<br />
<br />
出版商 Wiley<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
| isbn= 978-0-471-15496-9<br />
<br />
[国际标准图书编号978-0-471-15496-9]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
*{{cite book<br />
<br />
<br />
<br />
| last= Sontag<br />
<br />
| last= Sontag<br />
<br />
最后一个桑塔格<br />
<br />
| first= Eduardo<br />
<br />
| first= Eduardo<br />
<br />
首先是爱德华多<br />
<br />
| authorlink= Eduardo D. Sontag<br />
<br />
| authorlink= Eduardo D. Sontag<br />
<br />
作者: Eduardo d. Sontag<br />
<br />
| year= 1998<br />
<br />
| year= 1998<br />
<br />
1998年<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition<br />
<br />
数学控制理论: 确定性有限维系统。第二版<br />
<br />
| publisher= Springer<br />
<br />
| publisher= Springer<br />
<br />
出版商斯普林格<br />
<br />
| isbn= 978-0-387-98489-6<br />
<br />
| isbn= 978-0-387-98489-6<br />
<br />
[国际标准图书编号978-0-387-98489-6]<br />
<br />
}}<br />
<br />
}}<br />
<br />
}}<br />
<br />
{{Refend}}<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
==External links==<br />
<br />
==External links==<br />
<br />
外部链接<br />
<br />
*[http://www.dodccrp.org/ Command and Control Research Program (CCRP)]<br />
<br />
<br />
<br />
*[http://necsi.edu/guide/concepts/linearnonlinear.html New England Complex Systems Institute: Concepts in Complex Systems]<br />
<br />
<br />
<br />
*[http://ocw.mit.edu/courses/mathematics/18-353j-nonlinear-dynamics-i-chaos-fall-2012/ Nonlinear Dynamics I: Chaos] at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]<br />
<br />
<br />
<br />
*[http://www.hedengren.net/research/models.htm Nonlinear Model Library]{{snd}} (in [[MATLAB]]) a Database of Physical Systems<br />
<br />
<br />
<br />
*[http://cnls.lanl.gov/ The Center for Nonlinear Studies at Los Alamos National Laboratory]<br />
<br />
<br />
<br />
<br />
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{{Systems science}}<br />
<br />
<br />
<br />
{{Differential equations topics}}<br />
<br />
<br />
<br />
{{Authority control}}<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
[[Category:Nonlinear systems| ]]<br />
<br />
<br />
<br />
[[Category:Dynamical systems]]<br />
<br />
Category:Dynamical systems<br />
<br />
类别: 动力系统<br />
<br />
[[Category:Concepts in physics]]<br />
<br />
Category:Concepts in physics<br />
<br />
分类: 物理概念<br />
<br />
<noinclude><br />
<br />
<small>This page was moved from [[wikipedia:en:Nonlinear system]]. Its edit history can be viewed at [[非线性系统/edithistory]]</small></noinclude><br />
<br />
[[Category:待整理页面]]</div>Dorrhttps://wiki.swarma.org/index.php?title=%E6%95%B0%E6%8D%AE%E7%A7%91%E5%AD%A6&diff=7629数据科学2020-05-10T14:06:12Z<p>Dorr：</p>
<hr />
<div>此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。<br />
<br />
* 词条预计填充内容<br />
1.foundations 背景(了解的一些基础知识)；<br><br />
<br />
2.术语内涵衍变（该术语如何产生及目前为止用法的一些不同）；<br><br />
<br />
3.数据科学的研究内容<br><br />
<br />
3.1数据科学基础理论<br><br />
<br />
3.2 数据预处理<br><br />
<br />
3.3数据计算<br><br />
<br />
3.4数据管理<br><br />
<br />
4.在数据科学方面的职业和工作；<br><br />
<br />
5.数据科学的影响；<br><br />
<br />
6.数据科学中所涉及的一些技术和应用软件；<br><br />
<br />
7.数据科学、人工智能、机器学习之间的差别<br><br />
<br />
找到两篇博文供参考https://blog.csdn.net/fengdu78/article/details/105154546 https://blog.csdn.net/dev_csdn/article/details/79127658 <br><br />
<br />
8.与统计学的关系 <br><br />
<br />
<br />
其中，第2部分是需要搜集补充的内容，第7部分有一些参考资料（后续还会再找一些），第8部分可进行补充。<br />
<br />
有英文翻译的部分'''引言、1，2，4，5，6，8'''，没有英文内容'''3、7'''<br />
<br />
*任务分配<br />
'''任务一：引言，1背景、2术语内涵、3研究内容'''【负责人[[用户:乐多多|乐多多]] 】<br />
其中'''背景'''部分文字需要进行翻译；'''引言、术语内涵'''已有参考资料和初期的人工翻译文本，'''研究内容'''需要找到资料进行填充；<br><br />
<br />
'''任务二：4相关职业【[[Dorr]]已修订】、5数据科学的影响'''【负责人： 】<br />
其中并没有初期的人工翻译文本，可进一步搜集资料，使其更加完善；<br><br />
<br />
'''任务三：6相关应用软件、7与机器学习人工智能的差别、8与统计学的关系'''【负责人：[[趣木木]] 】<br />
其中7、8需要搜集资料进行填充，8已有参考资料和初期的人工翻译文本；<br><br />
<br />
<br />
*附言<br />
#任务完成上交为5月11号下午六点前<br />
#有些部分的内容过少，需要大家自行斟酌进行一下填充<br />
#任务领取方式 在对应的任务一二三后附上自己的名字或者昵称<br />
#大家有相关的参考资料也可以共享出来，并发给[[趣木木]]以便后期编者推荐时挑选进行运用<br />
#觉得还需要再添加什么模块，或者遇到什么问题可及时微信私聊[[趣木木]]<br />
<br />
<br />
<br />
旧版有这个词条，感觉可以在方法论层面再充实一下<br />
<br />
<br />
<br />
<br />
<br />
'''Data science''' is an [[inter-disciplinary]] field that uses scientific methods, processes, algorithms and systems to extract [[knowledge]] and insights from many structural and [[unstructured data]].<ref>{{Cite journal | last1 = Dhar | first1 = V. | title = Data science and prediction | doi = 10.1145/2500499 | journal = Communications of the ACM | volume = 56 | issue = 12 | pages = 64–73 | year = 2013 | pmid = | pmc = | url = http://cacm.acm.org/magazines/2013/12/169933-data-science-and-prediction/fulltext | access-date = 2 September 2015 | archive-url = https://web.archive.org/web/20141109113411/http://cacm.acm.org/magazines/2013/12/169933-data-science-and-prediction/fulltext | archive-date = 9 November 2014 | url-status = live }}</ref><ref>{{cite web | url=http://simplystatistics.org/2013/12/12/the-key-word-in-data-science-is-not-data-it-is-science/ | title=The key word in "Data Science" is not Data, it is Science | publisher=Simply Statistics | date=2013-12-12 | author=[[Jeffrey T. Leek|Jeff Leek]] | access-date=1 January 2014 | archive-url=https://web.archive.org/web/20140102194117/http://simplystatistics.org/2013/12/12/the-key-word-in-data-science-is-not-data-it-is-science/ | archive-date=2 January 2014 | url-status=live }}</ref> Data science is related to [[data mining]] and [[big data]].<br />
<br />
Data science is an inter-disciplinary field that uses scientific methods, processes, algorithms and systems to extract knowledge and insights from many structural and unstructured data. Data science is related to data mining and big data.<br />
<br />
<br />
<br />
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）下为旧版相对应的引言内容的参考 可进行一下整及或填充<br />
<br />
<br />
Data science is a "concept to unify [[statistics]], [[data analysis]], [[machine learning]] and their related methods" in order to "understand and analyze actual phenomena" with data.<ref>{{Cite book|chapter-url=https://www.springer.com/book/9784431702085|title=Data Science, Classification, and Related Methods|last=Hayashi|first=Chikio|date=1998-01-01|publisher=Springer Japan|isbn=9784431702085|editor-last=Hayashi|editor-first=Chikio|series=Studies in Classification, Data Analysis, and Knowledge Organization|location=|pages=40–51|language=en|chapter=What is Data Science? Fundamental Concepts and a Heuristic Example|doi=10.1007/978-4-431-65950-1_3|editor-last2=Yajima|editor-first2=Keiji|editor-last3=Bock|editor-first3=Hans-Hermann|editor-last4=Ohsumi|editor-first4=Noboru|editor-last5=Tanaka|editor-first5=Yutaka|editor-last6=Baba|editor-first6=Yasumasa}}</ref> It uses techniques and theories drawn from many fields within the context of [[mathematics]], [[statistics]], [[computer science]], and [[information science]]. [[Turing award]] winner [[Jim Gray (computer scientist)|Jim Gray]] imagined data science as a "fourth paradigm" of science ([[Empirical research|empirical]], [[Basic research|theoretical]], [[computational science|computational]] and now data-driven) and asserted that "everything about science is changing because of the impact of information technology" and the [[information explosion|data deluge]].<ref name="TansleyTolle2009">{{cite book|author1=Stewart Tansley|author2=Kristin Michele Tolle|title=The Fourth Paradigm: Data-intensive Scientific Discovery|url=https://books.google.com/?id=oGs_AQAAIAAJ|year=2009|publisher=Microsoft Research|isbn=978-0-9825442-0-4|access-date=16 December 2016|archive-url=https://web.archive.org/web/20170320193019/https://books.google.com/books?id=oGs_AQAAIAAJ|archive-date=20 March 2017|url-status=live}}</ref><ref name="BellHey2009">{{cite journal|last1=Bell|first1=G.|last2=Hey|first2=T.|last3=Szalay|first3=A.|title=COMPUTER SCIENCE: Beyond the Data Deluge|journal=Science|volume=323|issue=5919|year=2009|pages=1297–1298|issn=0036-8075|doi=10.1126/science.1170411|pmid=19265007}}</ref><br />
<br />
Data science is a "concept to unify statistics, data analysis, machine learning and their related methods" in order to "understand and analyze actual phenomena" with data. It uses techniques and theories drawn from many fields within the context of mathematics, statistics, computer science, and information science. Turing award winner Jim Gray imagined data science as a "fourth paradigm" of science (empirical, theoretical, computational and now data-driven) and asserted that "everything about science is changing because of the impact of information technology" and the data deluge.<br />
<br />
<br />
数据科学类似于[https://en.wikipedia.org/wiki/Data_mining 数据挖掘]，是一个通过科学的方法、过程、算法和系统，从有结构或无结构的各种形式的[https://en.wikipedia.org/wiki/Data 数据]中提炼[https://en.wikipedia.org/wiki/Knowledge 知识]和见解的跨学科领域。<br />
<ref name=":0"><br />
{{Cite journal <br />
| last1 = Dhar <br />
| first1 = V. <br />
| title = Data science and prediction <br />
| doi : 10.1145/2500499 <br />
| journal = Communications of the ACM <br />
| volume = 56 <br />
| issue = 12 <br />
| pages = 64 <br />
| year = 2013 <br />
| pmid = <br />
| pmc = <br />
| url = http://cacm.acm.org/magazines/2013/12/169933-data-science-and-prediction/fulltext<br />
}}</ref><br />
<ref><br />
Jeff Leek<br />
2013-12-12. <br />
[http://simplystatistics.org/2013/12/12/the-key-word-in-data-science-is-not-data-it-is-science/ The key word in "Data Science" is not Data, it is Science.]<br />
Simply Statistics. <br />
</ref> <br />
数据科学的概念结合了统计学、数据分析、机器学习等相关方法，以便于借助数据理解和分析实际现象。<br />
<ref name="Hayashi" /><br />
它使用了来自[https://en.wikipedia.org/wiki/Mathematics 数学]、[https://en.wikipedia.org/wiki/Statistics 统计学]、[https://en.wikipedia.org/wiki/Information_science 信息科学]、[https://en.wikipedia.org/wiki/Computer_science 计算机科学]等许多学科领域的技术与理论。<br />
<br />
<br />
[https://en.wikipedia.org/wiki/Turing_award 图灵奖]得主[https://en.wikipedia.org/wiki/Jim_Gray_(computer_scientist) 吉姆·格雷] Jim Gray 将数据科学设想为一种科学的“第四范式”（[https://en.wikipedia.org/wiki/Empirical_research 经验主义]、[https://en.wikipedia.org/wiki/Basic_research 理论研究]、计算机辅助，现在是数据驱动），并且断言所有关于科学的事物由于信息技术和[https://en.wikipedia.org/wiki/Information_explosion 数据洪流]的影响在不断地发生改变。<br />
<ref name="TansleyTolle2009"><br />
{{cite book<br />
|author1=Stewart Tansley<br />
|author2=Kristin Michele Tolle<br />
|title=The Fourth Paradigm: Data-intensive Scientific Discovery<br />
|url=https://books.google.com/books?id=oGs_AQAAIAAJ<br />
|year=2009<br />
|publisher=Microsoft Research<br />
|isbn:978-0-9825442-0-4<br />
}}</ref><br />
<ref name="BellHey2009"><br />
{{cite journal<br />
|last1=Bell<br />
|first1=G.<br />
|last2=Hey<br />
|first2=T.<br />
|last3=Szalay<br />
|first3=A.<br />
|title=COMPUTER SCIENCE: Beyond the Data Deluge<br />
|journal=Science<br />
|volume=323<br />
|issue=5919<br />
|year=2009<br />
|pages=1297–1298<br />
|issn:0036-8075<br />
|doi:10.1126/science.1170411<br />
}}</ref><br />
在2012年[https://en.wikipedia.org/wiki/Harvard_Business_Review 《哈佛商业评论》]称其为“21世纪最富有魅力的工作”后<br />
<ref name="Harvard" /><br />
，“数据科学”成了一个[https://en.wikipedia.org/wiki/Buzzword 流行术语]。它现在经常与早期概念互换使用，例如[https://en.wikipedia.org/wiki/Business_analytics 商业分析]<br />
<ref name="GilPress" /><br />
、[https://en.wikipedia.org/wiki/Business_intelligence 商业智能]、[https://en.wikipedia.org/wiki/Predictive_modelling 预测模型]和[https://en.wikipedia.org/wiki/Statistics 统计学]。“数据科学富有魅力”的观点甚至被汉斯·罗斯林 Hans Rosling博士在2011年BBC纪录片中转述为“统计学是当今世界最具吸引力的学科”。内特·西尔弗 Nate Silver<br />
<ref name="NateSilver" /><br />
则将数据科学描述为一种对于统计学家更具吸引力的词语。<br />
<br />
<br />
在许多场合，为了博人眼球，一些早期的解决方案现在被简单地打上了“数据科学”的旗号，而这可能冲淡这个术语的效用。<br />
<ref><br />
Warden, Pete(2011-05-09).<br />
[http://radar.oreilly.com/2011/05/data-science-terminology.html "Why the term "data science" is flawed but useful"]<br />
''O'Reilly Radar''. Retrieved 2018-05-20.<br />
</ref> <br />
虽然现在许多大学的项目都提供数据科学学位，然而它们对数据科学的定义或者合适的课程内容都没有达成一致。<br />
<ref name="GilPress" /> <br />
数据科学学位分量大跌，究其原因是许多数据科学和[https://en.wikipedia.org/wiki/Big_data 大数据]项目没能给出有用的结果，而这通常是糟糕的管理和资源利用造成的。<br />
<ref><br />
[https://hbr.org/2018/01/are-you-setting-your-data-scientists-up-to-fail "Are You Setting Your Data Scientists Up to Fail?"]. <br />
''Harvard Business Review''.2018-01-25. Retrieved 2018-05-26.<br />
</ref><br />
<ref><br />
[https://www.consultancy.uk/news/16839/70-of-big-data-projects-in-uk-fail-to-realise-full-potential "70% of Big Data projects in UK fail to realise full potential"]<br />
''www.consultancy.uk.'' Retrieved 2018-05-26.<br />
</ref><br />
<ref><br />
[http://analytics-magazine.org/the-data-economy-why-do-so-many-analytics-projects-fail/ "The Data Economy: Why do so many analytics projects fail? - Analytics Magazine"].<br />
''Analytics Magazine''. 2014-07-07. Retrieved 2018-05-26.<br />
</ref><br />
<ref><br />
[https://www.kdnuggets.com/2018/05/data-science-4-reasons-failing-deliver.html "Data Science: 4 Reasons Why Most Are Failing to Deliver"]. ''www.kdnuggets.com''. Retrieved 2018-05-26.<br />
</ref><br />
<br />
<br />
== Foundations背景 ==<br />
<br />
<br />
<br />
Data science is an interdisciplinary field focused on extracting knowledge from data sets, which are typically large (see [[big data]]).<ref>{{Cite web|url=http://www.datascienceassn.org/about-data-science|title=About Data Science {{!}} Data Science Association|website=www.datascienceassn.org|access-date=2020-04-03}}</ref> The field encompasses analysis, preparing data for analysis, and presenting findings to inform high-level decisions in an organization. As such, it incorporates skills from computer science, mathematics, statistics, [[information visualization]], graphic design, and business.<ref>{{Cite web|url=https://www.oreilly.com/library/view/doing-data-science/9781449363871/ch01.html|title=1. Introduction: What Is Data Science? - Doing Data Science [Book]|website=www.oreilly.com|language=en|access-date=2020-04-03}}</ref><ref>{{Cite web|url=https://medriscoll.com/post/4740157098/the-three-sexy-skills-of-data-geeks|title=the three sexy skills of data geeks|website=m.e.driscoll: data utopian|language=en|access-date=2020-04-03}}</ref> Statistician [[Nathan Yau]], drawing on [[Ben Fry]], also links data science to [[Human–computer interaction|human-computer interaction]]: users should be able to intuitively control and explore data.<ref>{{Cite web|url=https://flowingdata.com/2009/06/04/rise-of-the-data-scientist/|title=Rise of the Data Scientist|last=Yau|first=Nathan|date=2009-06-04|website=FlowingData|language=en|access-date=2020-04-03}}</ref><ref>{{Cite web|url=https://benfry.com/phd/dissertation/2.html|title=Basic Example|last=|first=|date=|website=benfry.com|url-status=live|archive-url=|archive-date=|access-date=2020-04-03}}</ref> In 2015, the [[American Statistical Association]] identified [[Database|database management]], statistics and [[machine learning]], and [[Distributed computing|distributed and parallel systems]] as the three emerging foundational professional communities.<ref>{{Cite web|url=https://magazine.amstat.org/blog/2015/10/01/asa-statement-on-the-role-of-statistics-in-data-science/|title=ASA Statement on the Role of Statistics in Data Science|date=2015-10-01|website=AMSTATNEWS|publisher=[[American Statistical Association]]|access-date=2019-05-29|archive-url=https://web.archive.org/web/20190620184935/https://magazine.amstat.org/blog/2015/10/01/asa-statement-on-the-role-of-statistics-in-data-science/|archive-date=20 June 2019|url-status=live}}</ref><br />
<br />
Data science is an interdisciplinary field focused on extracting knowledge from data sets, which are typically large (see big data). The field encompasses analysis, preparing data for analysis, and presenting findings to inform high-level decisions in an organization. As such, it incorporates skills from computer science, mathematics, statistics, information visualization, graphic design, and business. Statistician Nathan Yau, drawing on Ben Fry, also links data science to human-computer interaction: users should be able to intuitively control and explore data. In 2015, the American Statistical Association identified database management, statistics and machine learning, and distributed and parallel systems as the three emerging foundational professional communities.<br />
<br />
数据科学是一个跨学科的领域，致力于从数据集中提取知识，这些数据集通常都很大(请参阅[[大数据]])。<ref><br />
{{Cite web|url=http://www.datascienceassn.org/about-data-science|title=About Data Science {{!}} Data Science Association|website=www.datascienceassn.org|access-date=2020-04-03}}<br />
</ref> 该领域包括分析，准备分析数据，及呈现结果，以此为组织的高层决策提供依据。因此，它融合了来自计算机科学、数学、统计学、信息可视化、图形设计和商业技能。<ref>{{Cite web|url=https://www.oreilly.com/library/view/doing-data-science/9781449363871/ch01.html|title=1. Introduction: What Is Data Science? - Doing Data Science [Book]|website=www.oreilly.com|language=en|access-date=2020-04-03}}</ref><ref>{{Cite web|url=https://medriscoll.com/post/4740157098/the-three-sexy-skills-of-data-geeks|title=the three sexy skills of data geeks|website=m.e.driscoll: data utopian|language=en|access-date=2020-04-03}}<br />
</ref>统计学家丘南森 Nathan Yau 借鉴本•弗莱 Ben Fry 的观点，将数据科学和[[人机交互]]联系起来: 用户应该能够直观地控制和探索数据。<ref><br />
{{Cite web|url=https://flowingdata.com/2009/06/04/rise-of-the-data-scientist/|title=Rise of the Data Scientist|last=Yau|first=Nathan|date=2009-06-04|website=FlowingData|language=en|access-date=2020-04-03}}</ref><ref>{{Cite web|url=https://benfry.com/phd/dissertation/2.html|title=Basic Example|last=|first=|date=|website=benfry.com|url-status=live|archive-url=|archive-date=|access-date=2020-04-03}}<br />
</ref> 2015年，美国统计协会 the American Statistical Association 确定数据库管理、统计和机器学习，以及分布式和并行系统为三个新兴的基础专业社区。<ref>{{Cite web|url=https://magazine.amstat.org/blog/2015/10/01/asa-statement-on-the-role-of-statistics-in-data-science/|title=ASA Statement on the Role of Statistics in Data Science|date=2015-10-01|website=AMSTATNEWS|publisher=[[American Statistical Association]]|access-date=2019-05-29|archive-url=https://web.archive.org/web/20190620184935/https://magazine.amstat.org/blog/2015/10/01/asa-statement-on-the-role-of-statistics-in-data-science/|archive-date=20 June 2019|url-status=live}}</ref><br />
<br />
== Etymology 术语词义衍变==<br />
<br />
词源学<br />
<br />
<br />
<br />
<br />
<br />
<br />
=== Early usage ===<br />
<br />
早期使用<br />
<br />
<br />
<br />
<br />
<br />
In 1962, [[John Tukey]] described a field he called “data analysis,” which resembles modern data science.<ref name=":7">{{Cite web|url=http://courses.csail.mit.edu/18.337/2015/docs/50YearsDataScience.pdf|title=50 years of Data Science|last=Donoho|first=David|date=September 18, 2015|website=|url-status=live|archive-url=|archive-date=|access-date=April 2, 2020}}</ref> Later, attendees at a 1992 statistics symposium at the [[Montpellier 2 University|University of Montpellier II]] acknowledged the emergence of a new discipline focused on data of various origins and forms, combining established concepts and principles of statistics and data analysis with computing.<ref>{{Cite book|title=Data science and its applications = La @science des données et ses applications|date=1995|publisher=Academic Press/Harcourt Brace|others=Escoufier, Yves., Hayashi, Chikio (1918-....)., Fichet, Bernard.|year=1995|isbn=0-12-241770-4|location=Tokyo|pages=|oclc=489990740}}</ref><ref>{{Cite journal|last=Murtagh|first=Fionn|last2=Devlin|first2=Keith|date=2018|title=The Development of Data Science: Implications for Education, Employment, Research, and the Data Revolution for Sustainable Development|url=https://www.mdpi.com/2504-2289/2/2/14|journal=Big Data and Cognitive Computing|language=en|volume=2|issue=2|pages=14|doi=10.3390/bdcc2020014|via=|doi-access=free}}</ref> <br />
<br />
In 1962, John Tukey described a field he called “data analysis,” which resembles modern data science. Later, attendees at a 1992 statistics symposium at the University of Montpellier II acknowledged the emergence of a new discipline focused on data of various origins and forms, combining established concepts and principles of statistics and data analysis with computing. <br />
<br />
<br />
<br />
The term “data science” has been traced back to 1974, when [[Peter Naur]] proposed it as an alternative name for computer science.<ref name=":0">{{Cite journal|last=CaoLongbing|date=2017-06-29|title=Data Science|journal=ACM Computing Surveys (CSUR)|volume=50|issue=3|pages=1–42|language=EN|doi=10.1145/3076253|doi-access=free}}</ref> In 1996, the International Federation of Classification Societies became the first conference to specifically feature data science as a topic.<ref name=":0" /> However, the definition was still in flux. In 1997, [[C.F. Jeff Wu]] suggested that statistics should be renamed data science. He reasoned that a new name would help statistics shed inaccurate stereotypes, such as being synonymous with accounting, or limited to describing data.<ref>{{Cite web|url=http://www2.isye.gatech.edu/~jeffwu/presentations/datascience.pdf|title=Statistics=Data Science?|last=Wu|first=C.F. Jeff|date=|website=|url-status=live|archive-url=|archive-date=|access-date=April 2, 2020}}</ref> In 1998, Chikio Hayashi argued for data science as a new, interdisciplinary concept, with three aspects: data design, collection, and analysis.<ref>{{Cite journal|last=Murtagh|first=Fionn|last2=Devlin|first2=Keith|date=2018|title=The Development of Data Science: Implications for Education, Employment, Research, and the Data Revolution for Sustainable Development|url=https://www.mdpi.com/2504-2289/2/2/14|journal=Big Data and Cognitive Computing|language=en|volume=2|issue=2|pages=14|doi=10.3390/bdcc2020014|via=|doi-access=free}}</ref> <br />
<br />
The term “data science” has been traced back to 1974, when Peter Naur proposed it as an alternative name for computer science. In 1996, the International Federation of Classification Societies became the first conference to specifically feature data science as a topic. In 1998, Chikio Hayashi argued for data science as a new, interdisciplinary concept, with three aspects: data design, collection, and analysis. <br />
<br />
<br />
<br />
<br />
<br />
During the 1990s, popular terms for the process of finding patterns in datasets (which were increasingly large) included “knowledge discovery” and “data mining.”<ref name=":1">{{Cite web|url=https://www.forbes.com/sites/gilpress/2013/05/28/a-very-short-history-of-data-science/|title=A Very Short History Of Data Science|last=Press|first=Gil|website=Forbes|language=en|access-date=2020-04-03}}</ref><ref name=":0" /> <br />
<br />
During the 1990s, popular terms for the process of finding patterns in datasets (which were increasingly large) included “knowledge discovery” and “data mining.” <br />
<br />
“数据科学”这一术语在过去的三十年里已经出现在各种语境中，但直到最近才成为一个确定的术语。在早期，1960年它被[https://en.wikipedia.org/wiki/Peter_Naur 彼得·诺尔]（Peter Naur）用作[https://en.wikipedia.org/wiki/Computer_science 计算机科学]的代名词。诺尔后来引入了[https://en.wikipedia.org/wiki/Datalogy “数据学”](datalogy)这一术语。<br />
<ref><br />
{{cite journal<br />
|last1=Naur<br />
|first1=Peter<br />
|title=The science of datalogy<br />
|journal=Communications of the ACM<br />
|date=1 July 1966<br />
|volume=9<br />
|issue=7<br />
|pages=485<br />
|doi:10.1145/365719.366510<br />
}}</ref><br />
<br />
<br />
1962年，John Tukey 描述了一个他称之为“数据分析”的领域，类似于现代的数据科学。<ref name=":7"><br />
{{Cite web|url=http://courses.csail.mit.edu/18.337/2015/docs/50YearsDataScience.pdf|title=50 years of Data Science|last=Donoho|first=David|date=September 18, 2015|website=|url-status=live|archive-url=|archive-date=|access-date=April 2, 2020}}<br />
</ref> 在之后的阶段，参加1992年第二届蒙彼利埃大学统计研讨会的与会者承认了一个新的学科的出现，这个学科专注于各种缘起和形式的数据，将统计和数据分析的既定概念和原则与计算结合起来。<ref>{{Cite book|title=Data science and its applications = La @science des données et ses applications|date=1995|publisher=Academic Press/Harcourt Brace|others=Escoufier, Yves., Hayashi, Chikio (1918-....)., Fichet, Bernard.|year=1995|isbn=0-12-241770-4|location=Tokyo|pages=|oclc=489990740}}</ref><ref>{{Cite journal|last=Murtagh|first=Fionn|last2=Devlin|first2=Keith|date=2018|title=The Development of Data Science: Implications for Education, Employment, Research, and the Data Revolution for Sustainable Development|url=https://www.mdpi.com/2504-2289/2/2/14|journal=Big Data and Cognitive Computing|language=en|volume=2|issue=2|pages=14|doi=10.3390/bdcc2020014|via=|doi-access=free}}</ref> <br />
<br />
<br />
“数据科学”这一术语可以追溯到1974年，在1974年，彼得·诺尔 Peter Naur出版了《计算机方法简明调查》，在这本书对同时代被广泛应用的数据处理方法的调查中，他很自然地使用了“数据科学”这一术语，<ref name=":0"><br />
{{Cite journal|last=CaoLongbing|date=2017-06-29|title=Data Science|journal=ACM Computing Surveys (CSUR)|volume=50|issue=3|pages=1–42|language=EN|doi=10.1145/3076253|doi-access=free}}<br />
</ref> 1996年，国际船级社联合会 the International Federation of Classification Societies成为第一个以数据科学为专题的会议。<ref><br />
{{Cite web|url=http://www2.isye.gatech.edu/~jeffwu/presentations/datascience.pdf|title=Statistics=Data Science?|last=Wu|first=C.F. Jeff|date=|website=|url-status=live|archive-url=|archive-date=|access-date=April 2, 2020}}<br />
</ref> <br />
<br />
<br />
在1996年，国际分级社团联盟 (IFCS)的成员在日本神户举行了两年一次的会议，在此，术语“数据科学”在由林知己夫（Chikio Hayashi）<br />
<ref name="Hayashi"><br />
{{Cite book<br />
|chapter-url=https://link.springer.com/chapter/10.1007/978-4-431-65950-1_3<br />
|url=https://www.springer.com/book/9784431702085<br />
|title=Data Science, Classification, and Related Methods<br />
|last=Hayashi<br />
|first=Chikio<br />
|date=1998-01-01<br />
|publisher=Springer Japan<br />
|isbn:9784431702085<br />
|editor-last=Hayashi<br />
|editor-first=Chikio<br />
|series=Studies in Classification, Data Analysis, and Knowledge Organization<br />
|location=<br />
|pages=40–51<br />
|language=en<br />
|chapter=What is Data Science? Fundamental Concepts and a Heuristic Example<br />
|doi:10.1007/978-4-431-65950-1_3<br />
|editor-last2=Yajima<br />
|editor-first2=Keiji<br />
|editor-last3=Bock<br />
|editor-first3=Hans-Hermann<br />
|editor-last4=Ohsumi<br />
|editor-first4=Noboru<br />
|editor-last5=Tanaka<br />
|editor-first5=Yutaka<br />
|editor-last6=Baba<br />
|editor-first6=Yasumasa<br />
}}</ref><br />
举办的圆桌讨论上经由介绍之后首次被纳入会议标题（“数据科学、分级、相关方法”）。<br />
<ref><br />
Press, Gil.<br />
[https://www.forbes.com/sites/gilpress/2013/05/28/a-very-short-history-of-data-science/ "A Very Short History Of Data Science"].<br />
</ref><br />
<br />
<br />
在1997年11月，吴建福（C.F. Jeff Wu）以[https://en.wikipedia.org/wiki/University_of_Michigan 密歇根大学]所授予的H.C Carver教授职位任命发表了题为“统计学=数据科学？”<br />
<ref name="cfjwutk"><br />
Wu, C. F. J. (1997).<br />
[http://www2.isye.gatech.edu/~jeffwu/presentations/datascience.pdf "Statistics = Data Science?"].<br />
Retrieved 9 October 2014.<br />
</ref><br />
的就职演讲<br />
<ref name="cfjwu01"><br />
[http://ur.umich.edu/9899/Nov09_98/4.htm "Identity of statistics in science examined"]<br />
.The University Records, 9 November 1997, The University of Michigan. Retrieved 12 August 2013.<br />
</ref><br />
，在演讲中他将统计学工作描述为数据收集、建模和分析、决策的三个流程。在结论中他首创了现代的、非计算机科学内的“数据科学”术语用法，并提倡统计学应更名为数据科学，统计学家应被称作为数据科学家。<br />
<ref name="cfjwutk"/>之后，他又在1998年纪念印度科学家和统计学家、[https://en.wikipedia.org/wiki/Indian_Statistical_Institute 印度统计学院]创立者[https://en.wikipedia.org/wiki/Prasanta_Chandra_Mahalanobis 马哈拉诺比斯（P.C. Mahalanobis）]的讲座上将同名的演讲作为其系列演讲<br />
<ref name="cfjwu02"><br />
[http://www.isical.ac.in/~statmath/html/pcm/pcm_recent.html "P.C. Mahalanobis Memorial Lectures, 7th series"]. <br />
P.C. Mahalanobis Memorial Lectures, Indian Statistical Institute.<br />
Archived from [https://web.archive.org/web/20131029191813/ the original]<br />
on 26 Feb 2017. Retrieved 18 Jul 2017.<br />
</ref><br />
的第一篇进而发表。<br />
<br />
<br />
1998年，林志雄主张数据科学是一个新的、跨学科的概念，包括数据设计、数据收集和数据分析三个方面。<ref><br />
{{Cite journal|last=Murtagh|first=Fionn|last2=Devlin|first2=Keith|date=2018|title=The Development of Data Science: Implications for Education, Employment, Research, and the Data Revolution for Sustainable Development|url=https://www.mdpi.com/2504-2289/2/2/14|journal=Big Data and Cognitive Computing|language=en|volume=2|issue=2|pages=14|doi=10.3390/bdcc2020014|via=|doi-access=free}}</ref> <br />
<br />
<br />
在20世纪90年代，在数据集中寻找模式的流行术语(数据集越来越大)包括“知识发现”和“数据挖掘”<ref name=":1">{{Cite web|url=https://www.forbes.com/sites/gilpress/2013/05/28/a-very-short-history-of-data-science/|title=A Very Short History Of Data Science|last=Press|first=Gil|website=Forbes|language=en|access-date=2020-04-03}}</ref><ref name=":0" /> <br />
<br />
<br />
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）下为旧版关于数据科学的词源演变由来的部分内容 可参考整合并进行填充<br />
<br />
=== Modern usage ===<br />
<br />
现代用法<br />
<br />
The modern conception of data science as an independent discipline is sometimes attributed to [[William S. Cleveland]].<ref>{{Cite web|url=https://www.stat.purdue.edu/~wsc/|title=William S Cleveland|last=Gupta|first=Shanti|date=December 11, 2015|website=|url-status=live|archive-url=|archive-date=|access-date=April 2, 2020}}</ref> In a 2001 paper, he advocated an expansion of statistics beyond theory into technical areas; because this would significantly change the field, it warranted a new name.<ref name=":1" /> "Data science" became more widely used in the next few years: in 2002, the [[Committee on Data for Science and Technology]] launched ''Data Science Journal.'' In 2003, Columbia University launched ''The Journal of Data Science''.<ref name=":1" /> In 2014, the [[American Statistical Association]]'s Section on Statistical Learning and Data Mining changed its name to the Section on Statistical Learning and Data Science, reflecting the ascendant popularity of data science.<ref>{{Cite news|last=Talley|first=Jill|url=https://magazine.amstat.org/blog/2016/06/01/datascience-2/|title=ASA Expands Scope, Outreach to Foster Growth, Collaboration in Data Science|date=June 1, 2016|work=Amstat News|access-date=|url-status=live|publisher=American Statistical Association}}</ref><br />
<br />
The modern conception of data science as an independent discipline is sometimes attributed to William S. Cleveland. In a 2001 paper, he advocated an expansion of statistics beyond theory into technical areas; because this would significantly change the field, it warranted a new name.<br />
<br />
<br />
<br />
The professional title of “data scientist” has been attributed to [[DJ Patil]] and [[Jeff Hammerbacher]] in 2008.<ref>{{Cite news|last=Davenport|first=Thomas H.|url=https://hbr.org/2012/10/data-scientist-the-sexiest-job-of-the-21st-century|title=Data Scientist: The Sexiest Job of the 21st Century|date=2012-10-01|work=Harvard Business Review|access-date=2020-04-03|last2=Patil|first2=D. J.|issue=October 2012|issn=0017-8012}}</ref> Though it was used by the [[National Science Board]] in their 2005 report, "Long-Lived Digital Data Collections: Enabling Research and Education in the 21st Century," it referred broadly to any key role in managing a digital data collection.<ref>{{Cite web|url=https://www.nsf.gov/pubs/2005/nsb0540/|title=US NSF - NSB-05-40, Long-Lived Digital Data Collections Enabling Research and Education in the 21st Century|website=www.nsf.gov|access-date=2020-04-03}}</ref><br />
<br />
The professional title of “data scientist” has been attributed to DJ Patil and Jeff Hammerbacher in 2008. Though it was used by the National Science Board in their 2005 report, "Long-Lived Digital Data Collections: Enabling Research and Education in the 21st Century," it referred broadly to any key role in managing a digital data collection.<br />
<br />
<br />
There is still no consensus on the definition of data science and it is considered by some to be a buzzword.<ref>{{Cite web|url=https://www.forbes.com/sites/gilpress/2013/08/19/data-science-whats-the-half-life-of-a-buzzword/|title=Data Science: What's The Half-Life Of A Buzzword?|last=Press|first=Gil|website=Forbes|language=en|access-date=2020-04-03}}</ref><br />
<br />
There is still no consensus on the definition of data science and it is considered by some to be a buzzword.<br />
<br />
<br />
<br />
<br />
<br />
数据科学作为一门独立学科的现代概念，一定程度上归功于威廉 · s · 克利夫兰 William S. Cleveland。<ref><br />
{{Cite web|url=https://www.stat.purdue.edu/~wsc/|title=William S Cleveland|last=Gupta|first=Shanti|date=December 11, 2015|website=|url-status=live|archive-url=|archive-date=|access-date=April 2, 2020}}<br />
</ref> <br />
在2001年，威廉·克利夫兰（William S.Cleveland）在他的文章《数据科学：一个用来扩大统计学领域技术范畴的行动计划》将数据科学作为一门独立学科引入，扩大了统计学的领域并使之包含“数据计算的前沿”，这篇文章发表在2001年4月版的《国际统计评论》（''International Statistical Review / Revue Internationale de Statistique''）的第69卷，第1篇上。<br />
<ref name="cleveland01"><br />
Cleveland, W. S. (2001). <br />
[https://pdfs.semanticscholar.org/915c/d8e2b39eb02723553913d592b2237d4d9960.pdf Data science: an action plan for expanding the technical areas of the field of statistics]. <br />
International Statistical Review / Revue Internationale de Statistique, 21–26.<br />
</ref> <br />
在他的报告中，他主张将统计学从理论扩展到技术领域; 因为这将大大改变这个领域，它需要一个新的名称。克利夫兰建立了他认为数据科学所关心的6个技术领域：多学科调查，数据模型和方法，数据计算，教学方法、工具评估及其理论。<br />
<br />
<br />
在2002年4月，国际科学委员会(ICSU)：数据科学与技术分会(CODATA)<br />
<ref name="ics12"><br />
International Council for Science : Committee on Data for Science and Technology. (2012, April). <br />
CODATA, The Committee on Data for Science and Technology. Retrieved from International Council for Science : Committee on Data for Science and Technology: http://www.codata.org/<br />
</ref><br />
创办了数据科学期刊（''Data Science Journal''）<br />
<ref name="dsj12"><br />
Data Science Journal. (2012, April). <br />
Available Volumes. <br />
Retrieved from Japan Science and Technology Information Aggregator, Electronic: http://www.jstage.jst.go.jp/browse/dsj/_vols<br />
</ref><br />
，这是一份聚焦于诸如数据系统描述及其网络出版物、应用和法律问题的出版物<br />
<ref name="dsj02"><br />
Data Science Journal. (2002, April). <br />
Contents of Volume 1, Issue 1, April 2002. <br />
Retrieved from Japan Science and Technology Information Aggregator, <br />
Electronic: http://www.jstage.jst.go.jp/browse/dsj/1/0/_contents<br />
</ref><br />
。之后不久，哥伦比亚大学在2003年1月开始出版数据科学期刊（''The Journal of Data Science''）<br />
<ref name="jds03"><br />
The Journal of Data Science. (2003, January).<br />
Contents of Volume 1, Issue 1, January 2003. <br />
Retrieved from http://www.jds-online.com/v1-1<br />
</ref> <br />
，为所有数据工作者提供了发表意见和交流想法的平台。这份期刊衷心致力于统计学方法应用和定量研究。在2005年，美国国家科学委员会出版了“长期数字数据收集：赋能21世纪的研究和教育”这一内容，注明数据科学家为“信息和计算机科学家、数据库和软件程序员、学科专家、管理者和注解专家、图书管理员、档案保管员，以及其它对数字化数据收集的成功管理起到关键性作用的人。”他们的首要活动是“进行创造性探究与分析。”<br />
<ref><br />
National Science Board.<br />
[http://www.nsf.gov/pubs/2005/nsb0540/ Long-Lived Digital Data Collections Enabling Research and Education in the 21st Century]<br />
. National Science Foundation<br />
. Retrieved 30 June 2013.<br />
</ref><br />
<br />
<br />
在2007年左右，<br />
<ref><br />
Citation needed<br />
</ref><br />
图灵奖得主[https://en.wikipedia.org/wiki/Jim_Gray_(computer_scientist) 吉姆·格雷]（Jim Gray）预见到使用大数据的分析计算作为主要科学方法的“数据驱动的科学”将成为科学的第四范式<br />
<ref name="TansleyTolle2009" /><br />
<ref name="BellHey2009" /><br />
，我们将迎来一个科学文献、科学数据全部链接且彼此运用的世界。<br />
<ref><br />
Markoff,John(2009-12-14).<br />
[https://www.nytimes.com/2009/12/15/science/15books.html "Essays Inspired by Microsoft’s Jim Gray, Who Saw Science Paradigm Shift"]. ''The New York Times''. Retrieved 2018-04-26.<br />
</ref><br />
<br />
<br />
2008年，DJ 帕蒂尔 DJ Patil和杰夫哈默巴赫尔 Jeff Hammerbacher被授予“数据科学家”的职称。<ref><br />
{{Cite news|last=Davenport|first=Thomas H.|url=https://hbr.org/2012/10/data-scientist-the-sexiest-job-of-the-21st-century|title=Data Scientist: The Sexiest Job of the 21st Century|date=2012-10-01|work=Harvard Business Review|access-date=2020-04-03|last2=Patil|first2=D. J.|issue=October 2012|issn=0017-8012}}<br />
</ref>尽管美国国家科学委员会(National Science Board)在其2005年的报告《长期数字数据收集: 21世纪的研究和教育成果》(Long-Lived Digital Data Collections: Enabling Research and Education in the 21st Century)中使用了这个词，但它泛指管理数字数据收集的任何关键角色。<ref>{{Cite web|url=https://www.nsf.gov/pubs/2005/nsb0540/|title=US NSF - NSB-05-40, Long-Lived Digital Data Collections Enabling Research and Education in the 21st Century|website=www.nsf.gov|access-date=2020-04-03}}</ref><br />
<br />
<br />
在2012年[https://en.wikipedia.org/wiki/Harvard_Business_Review 《哈佛商业评论》]的报道“数据科学家：21世纪最富有魅力的工作”中<br />
<ref name="Harvard"><br />
{{Cite journal<br />
|url=https://hbr.org/2012/10/data-scientist-the-sexiest-job-of-the-21st-century/<br />
|title=Data Scientist: The Sexiest Job of the 21st Century<br />
|publisher=Harvard Business Review<br />
|first=Thomas H.<br />
|last=Davenport<br />
|first2=DJ<br />
|last2=Patil<br />
|date=Oct 2012<br />
}}</ref><br />
，[https://en.wikipedia.org/wiki/DJ_Patil 帕蒂尔]（DJ Patil）声称其已于2008年和[https://en.wikipedia.org/wiki/Jeff_Hammerbacher 杰弗·哈梅巴赫]（Jeff Hammerbacher）共同创造了这一术语，用以标注他们在领英和脸书上的职业信息。他断言数据科学家将是一种全新的职业类型，并且数据科学家的短缺正成为某些领域的严重掣肘，但同时也将其描述为一个更加商业化导向的角色。<br />
<br />
<br />
2013年，IEEE数据科学和高等分析专门工作组<br />
<ref><br />
[http://www.dsaa.co "IEEE Task Force on Data Science and Advanced Analytics"]<br />
</ref> <br />
成立，同年第一届“欧洲数据分析大会(ECDA)”在卢森堡召开，会上成立了[http://euads.org/ 欧洲数据科学协会]（EuADS）。第一届国际会议——IEEE国际数据科学和高等分析会议也于2014年召开。<br />
<ref><br />
[http://datamining.it.uts.edu.au/conferences/dsaa14/ "2014 IEEE International Conference on Data Science and Advanced Analytics"]<br />
</ref><br />
同年，编程训练营始祖[https://en.wikipedia.org/wiki/General_Assembly_(school) General Assembly]开启了学生付费培训，[https://en.wikipedia.org/wiki/The_Data_Incubator 数据孵化器公司]成立了一个富有竞争力的自由数据科学团体。<br />
<ref><br />
[https://venturebeat.com/2014/04/15/ny-gets-new-bootcamp-for-data-scientists-its-free-but-harder-to-get-into-than-harvard/ "NY gets new bootcamp for data scientists: It’s free, but harder to get into than Harvard "]. <br />
''Venture Beat'' Retrieved 2016-02-22.<br />
</ref><br />
也是在2014年，[https://en.wikipedia.org/wiki/American_Statistical_Association 美国统计协会]的统计学习和数据挖掘部门将其期刊更名为“统计分析与数据挖掘：ASA数据科学期刊”，并在2016年将其部门更名为“统计学习与数据科学”。<br />
<ref name="ASA"><br />
Talley,Jill(2016-06-01)<br />
[http://magazine.amstat.org/blog/2016/06/01/datascience-2/ "ASA Expands Scope, Outreach to Foster Growth, Collaboration in Data Science"]<br />
. ''AMSTATNEWS''.<br />
American Statistical Association.<br />
Retrieved 2017-02-04<br />
</ref> <br />
<br />
<br />
2015年，Springer创办国际数据科学与分析杂志<br />
<ref><br />
[https://www.springer.com/41060 "Journal on Data Science and Analytics"]<br />
</ref><br />
，用来出版有关数据科学和大数据分析方面的原创性工作。2015年9月，[http://www.gfkl.org/welcome/ GfKI]在英国克彻斯特的[https://en.wikipedia.org/wiki/University_of_Essex 埃塞克斯大学]举办的第三届ECDA大会上增设“数据科学社团”。<br />
<br />
<br />
目前，对于数据科学的定义似乎还=依旧没有达成共识，有些人认为这是一个流行词。<ref>{{Cite web|url=https://www.forbes.com/sites/gilpress/2013/08/19/data-science-whats-the-half-life-of-a-buzzword/|title=Data Science: What's The Half-Life Of A Buzzword?|last=Press|first=Gil|website=Forbes|language=en|access-date=2020-04-03}}</ref><br />
<br />
==研究内容==<br />
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）并不限于所列出来的条目 可以根据研究内容进行自主填充<br />
===数据科学基础理论===<br />
数据科学中的新理念、理论、方法、技术及工具以及数据科学的研究目的、理论基础、研究内容、基本流程、主要原则、典型应用、人才培养、项目管理等。在此需要特别提醒的是--"基础理论"与"理论基础"是两个不同的概念。数据科学的"基础理论"在数据科学的研究边界之内，而其"理论基础"在数据科学的研究边界之外，是数据科学的理论依据和来源。<br />
<br />
===数据预处理===<br />
为了提升数据质量、降低数据计算的复杂度、减少数据计算量以及提升数据处理的准确性，数据科学中需要对原始数据进行预处理--进行数据审计、数据清洗、数据变换、数据集成、数据脱敏、数据规约和数据标注等。<br />
<br />
===数据计算===<br />
在数据科学中，计算模式发生了根本性的变化--从集中式计算、分布式计算、网格计算等传统计算过渡至云计算。有一定的代表性的是Google云计算3大技术、Hadoop MapReduce和YARN技术的出现。数据计算模式的变化意味着数据科学中所关注的数据计算的主要目标、瓶颈和矛盾发生了根本性变化。<br />
<br />
===数据管理===<br />
在完成"数据预处理"(或"数据计算")之后，我们需要对数据进行管理，以便进行(再次进行)"数据处理"以及数据的再利用和长久保管。在数据科学中，数据管理方法与技术发生了根本性的改变--不仅包括传统关系型数据库，而且还出现了一些新兴数据管理技术，例如NoSQL、NewSQL技术和关系云等。<br />
<br />
== Careers in data science 数据科学的相关职业==<br />
<br />
<br />
<br />
Data science is a growing field. A career as a data scientist is ranked at the third best job in America for 2020 by Glassdoor, and was ranked the number one best job from 2016-2019.<ref>{{Cite web|url=https://www.glassdoor.com/List/Best-Jobs-in-America-LST_KQ0,20.htm|title=Best Jobs in America|website=Glassdoor|language=en|access-date=2020-04-03}}</ref> Data scientists have a median salary of $118,370 per year or $56.91 per hour.<ref name=":2">{{Cite web|url=https://www.bls.gov/ooh/computer-and-information-technology/computer-and-information-research-scientists.htm|title=Computer and Information Research Scientists : Occupational Outlook Handbook: : U.S. Bureau of Labor Statistics|website=www.bls.gov|language=en-us|access-date=2020-04-03}}</ref> Job growth in this field is also above average, with a projected increase of 16% from 2018 to 2028.<ref name=":2" /> The largest employer of data scientists in the US is the federal government, employing 28% of the data science workforce.<ref name=":2" /> Other large employers of data scientists are computer system design services, research and development laboratories, and colleges and universities.<ref name=":2" /> Typically, data scientists work full time, and some work more than 40 hours a week.<ref name=":2" /><br />
<br />
Data science is a growing field. A career as a data scientist is ranked at the third best job in America for 2020 by Glassdoor, and was ranked the number one best job from 2016-2019. Data scientists have a median salary of $118,370 per year or $56.91 per hour. Job growth in this field is also above average, with a projected increase of 16% from 2018 to 2028. The largest employer of data scientists in the US is the federal government, employing 28% of the data science workforce. Other large employers of data scientists are computer system design services, research and development laboratories, and colleges and universities. Typically, data scientists work full time, and some work more than 40 hours a week.<br />
<br />
数据科学是一个不断发展的领域。数据科学家在Glassdoor(美国的一家做企业点评与职位搜索的职场社区)评价的2020年美国最佳职业中居第三名，在2016-2019年最佳职业中居首位。数据科学家的平均薪资是每年118,370美元或每小时56.91美元。该领域的就业增长率也高于平均水平，预计从2018年到2028年将增长16%。美国的数据科学家的最大雇佣方是美国联邦政府，占比28%；其他主要雇主包括计算机系统设计服务、研究和开发实验室以及高校。通常情况下，数据科学家全职工作，有些人每周工作超过40小时。<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
=== Educational path ===<br />
<br />
教育途径<br />
<br />
In order to become a data scientist, there is a significant amount of education and experience required. The first step in becoming a data scientist is to earn a bachelor's degree, typically in a field related to computing or mathematics.<ref name=":3">{{Cite web|url=https://www.mastersindatascience.org/careers/data-scientist/|title=What is a Data Scientist?|website=Master's in Data Science|language=en-US|access-date=2020-04-03}}</ref><ref name=":2" /> Coding bootcamps are also available and can be used as an alternate pre-qualification to supplement a bachelor's degree in another field.<ref name=":3" /> Most data scientists also complete a master’s degree or a PhD in data science.<ref name=":3" /> Once these qualifications are met, the next step to becoming a data scientist is to apply for an entry level job in the field.<ref name=":3" /> Some data scientists may later choose to specialize in a sub-field of data science.<ref name=":3" /><br />
<br />
In order to become a data scientist, there is a significant amount of education and experience required. The first step in becoming a data scientist is to earn a bachelor's degree, typically in a field related to computing or mathematics. Coding bootcamps are also available and can be used as an alternate pre-qualification to supplement a bachelor's degree in another field. Most data scientists also complete a master’s degree or a PhD in data science. Once these qualifications are met, the next step to becoming a data scientist is to apply for an entry level job in the field. Some data scientists may later choose to specialize in a sub-field of data science.<br />
<br />
要成为一名数据科学家，需要接受充分教育、积累大量经验。成为数据科学家的第一步通常是在与计算或数学相关的领域获得学士学位；其他领域的学士也可以通过参加编程训练营补充资格。大多数数据科学家也在该领域取得了硕士或博士学位。一旦符合这些条件，成为数据科学家的下一步就是申请该领域的入门级工作。一些数据科学家以后可能会选择专攻数据科学的一个分支领域。<br />
<br />
=== Specializations and associated careers ===<br />
<br />
专业细分和相关职业<br />
<br />
<br />
<br />
<br />
<br />
* 机器学习科学家: 机器学习科学家研究数据分析新方法，创立算法.<ref name=":4">{{Cite web|url=https://www.northeastern.edu/graduate/blog/data-science-careers-shaping-our-future/|title=11 Data Science Careers Shaping the Future|date=2018-11-23|website=Northeastern University Graduate Programs|language=en-US|access-date=2020-04-03}}</ref><br />
<br />
<br />
<br />
* 数据分析师: 数据分析师利用大型数据集来收集信息，以满足公司需求<ref name=":4" /><br />
<br />
<br />
<br />
* 数据顾问: 数据顾问与企业合作，以确定对数据分析产生的信息的最佳利用<ref name=":3" /><br />
<br />
<br />
<br />
* 数据架构师: 数据架构师提供针对性能和设计应用优化的数据解决方案<ref name=":4" /><br />
<br />
<br />
<br />
* 应用程序架构师: 应用程序架构师跟踪应用程序在整个业务中的使用情况，以及其与用户和其他应用程序的交互情况.<ref name=":4" /><br />
<br />
== Impacts of data science数据科学的影响 ==<br />
<br />
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）需要再进行补充 内容过少<br />
<br />
Big data is very quickly becoming a vital tool for businesses and companies of all sizes.<ref name=":5">{{Cite web|url=https://www.forbes.com/sites/peterpham/2015/08/28/the-impacts-of-big-data-that-you-may-not-have-heard-of/|title=The Impacts Of Big Data That You May Not Have Heard Of|last=Pham|first=Peter|website=Forbes|language=en|access-date=2020-04-03}}</ref> The availability and interpretation of big data has altered the business models of old industries and enabled the creation of new ones.<ref name=":5" /> Data-driven businesses are worth $1.2 trillion collectively in 2020, an increase from $333 billion in the year 2015.<ref name=":6">{{Cite web|url=https://towardsdatascience.com/how-data-science-will-impact-future-of-businesses-7f11f5699c4d|title=How Data Science will Impact Future of Businesses?|last=Martin|first=Sophia|date=2019-09-20|website=Medium|language=en|access-date=2020-04-03}}</ref> Data scientists are responsible for breaking down big data into usable information and creating software and algorithms that help companies and organizations determine optimal operations.<ref name=":6" /> As big data continues to have a major impact on the world, data science does as well due to the close relationship between the two.<ref name=":6" /> <br />
<br />
Big data is very quickly becoming a vital tool for businesses and companies of all sizes. The availability and interpretation of big data has altered the business models of old industries and enabled the creation of new ones. Data scientists are responsible for breaking down big data into usable information and creating software and algorithms that help companies and organizations determine optimal operations. As big data continues to have a major impact on the world, data science does as well due to the close relationship between the two. <br />
<br />
大数据正迅速成为各种规模的企业和公司的重要工具。大数据的可用性和解释改变了旧行业的商业模式，并促成了新行业的创建。数据科学家负责将大数据分解成可用的信息，并创建软件和算法，帮助企业和组织确定最佳操作。随着大数据继续对世界产生重大影响，数据科学也由于两者之间的密切关系而产生重大影响。<br />
<br />
<br />
大数据还引起了科技界对科学研究方法论的重新审视，正在引发科学研究思维与方法的一场革命。最早的科学研究只有实验科学，随后出现了以研究各种定律和定理为特征的理论科学。由于理论分析方法在许多问题上过于复杂，难以解决实际问题，人们开始寻求模拟的方法，导致计算科学的兴起。海量数据的出现催生了一种新的科研模式，即面对海量数据，科研人员只需从数据中直接查找或挖掘所需要的信息、知识和智慧，甚至无需直接接触需研究的对象。2007年，已故的图灵奖得主吉姆·格雷（Jim Gray）在他最后一次演讲中描绘了数据密集型科学研究的“第四范式”（TheFourth Paradigm）[5]，把数据密集型科学从计算科学中单独区分开来。格雷认为，要解决我们面临的某些最棘手的全球性挑战，“第四范式”可能是唯一具有系统性的方法。其实，“第四范式”不仅是<br />
科研方式的转变，也是人们思维方式的大变化。<br />
<br />
<br />
不仅如此，数据分析几乎涉及到了现代数学的所有分支。甚至于像表示论这样的极其抽象的分支，在数据科学领域也有其发挥作用的余地。所以数据科学对数学的要求和推动是全面的，而不是仅仅局限在几个领域。数据应该成为数、图形和方程之外数学研究的基本对象之一。<br />
<br />
对计算机学科的影响<br />
<br />
数据科学对计算机科学的发展也会带来很大的影响。图灵奖得主John Hopcroft曾经指出，在过去的几十年里，计算机科学的研究对象主要是计算机本身，包括硬件和软件。以后计算机科学的发展将主要围绕应用展开。而从计算机科学自身来看，这些应用领域提供的主要研究对象就是数据。虽然计算机科学一贯重视数据的研究，但数据在其中的地位将会得到更进一步的加强。<br />
<br />
对统计学的影响<br />
<br />
再看统计学。统计学一直就是一门研究数据的学科。所以它也是数据科学最核心的部分之一。但在数据科学的框架之下，统计学的发展也会受到很大的冲击。<br />
<br />
这种冲击至少表现在两个方面。一是关于数据的模型将会跳出传统的统计模型的框架。更一般的数学概念，如拓扑、几何和随机场的概念将会在数据分析中扮演重要的角色。二是算法和分布式计算将成为研究的中心课题之一。<br />
<br />
== Technologies and techniques 所涉及的技术和应用软件==<br />
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）所涉及的两个词 我查的时候都是技术的意思 在查看内容后也不知道怎么区别这两个词 cnki的查询结果：Technique：技术(59685)方法(19269)工艺(4963)；Technologies：技术(39330) 工艺(3185) 的技术(2197)；<br />
<br />
There are a variety of different technologies and techniques that are used for data science which depending on the application.<br />
<br />
There are a variety of different technologies and techniques that are used for data science which depending on the application.<br />
<br />
有各种不同的技术用于数据科学，这取决于在什么应用软件实现该技术。<br />
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=== Techniques技术 ===<br />
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* [[Cluster analysis|Clustering]] is a technique used to group data together.<br />
*[[聚类分析|聚类]]是一种用于将数据分组在一起的技术。<br />
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聚类，将相似的事物聚集在一起，将不相似的事物划分到不同的类别的过程。是将复杂数据简化为少数类别的一种手段。聚类分析的目标就是在相似的基础上收集数据来分类。聚类源于很多领域，包括数学，计算机科学，统计学，生物学和经济学。在不同的应用领域，很多聚类技术都得到了发展，这些技术方法被用作描述数据，衡量不同数据源间的相似性，以及把数据源分类到不同的簇中。<br />
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聚类的基本思想：<br />
#有大量的样本。<br />
#假定研究的样本之间存在程度不同的相似性，可以分为几类；相同类别的样本相似度高，不同类别的样本相似度差。<br />
#用一些数据指标来描述样本的若干属性，构成向量。<br />
#用某种方法度量样本之间或者类别 之间的相似性（或称距离），依据距离来进行分类。 <br />
#根据分类来研究各类样本的共性，找出规律。<br />
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* [[Dimensionality reduction]] is used to reduce the complexity of data computation so that it can be performed more quickly.<br />
*[[降维]]用于降低数据计算的复杂性，使其能够更快地执行。<br />
数据降维可以降低模型的计算量并减少模型运行时间、降低噪音变量信息对于模型结果的影响、便于通过可视化方式展示归约后的维度信息并减少数据存储空间。因此，大多数情况下，当我们面临高维数据时，都需要对数据做降维处理。<br />
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数据降维有两种方式：特征选择，维度转换。<br />
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# 特征选择<br />
特征选择指根据一定的规则和经验，直接在原有的维度中挑选一部分参与到计算和建模过程，用选择的特征代替所有特征，不改变原有特征，也不产生新的特征值。<br />
特征选择的降维方式好处是可以保留原有维度特征的基础上进行降维，既能满足后续数据处理和建模需求，又能保留维度原本的业务含义，以便于业务理解和应用。对于业务分析性的应用而言，模型的可理解性和可用性很多时候要有限于模型本身的准确率、效率等技术指标。例如，决策树得到的特征规则，可以作为选择用户样本的基础条件，而这些特征规则便是基于输入的维度产生。<br />
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#维度转换<br />
维度转换是按照一定数学变换方法，把给定的一组相关变量（维度）通过数学模型将高纬度空间的数据点映射到低纬度空间中，然后利用映射后变量的特征来表示原有变量的总体特征。这种方式是一种产生新维度的过程，转换后的维度并非原来特征，而是之前特征的转化后的表达式，新的特征丢失了原有数据的业务含义。 通过数据维度变换的降维方法是非常重要的降维方法，这种降维方法分为线性降维和非线性降维两种，其中常用的代表算法包括独立成分分析（ICA），主成分分析（PCA），因子分析（Factor Analysis，FA），线性判别分析（LDA），局部线性嵌入（LLE），核主成分分析（Kernel PCA）等。<br />
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* [[Machine learning]] is a technique used to perform tasks by inferencing patterns from data.<br />
*[[机器学习]]是一种通过从数据中推断模式来执行任务的技术。专门研究计算机怎样模拟或实现人类的学习行为，以获取新的知识或技能，重新组织已有的知识结构使之不断改善自身的性能。它是人工智能的核心，是使计算机具有智能的根本途径。机器学习也是用数据或以往的经验,以此优化计算机程序的性能标准。<br />
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--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）先将其意思译出来后再进行一些补充<br />
=== Technologies ===<br />
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技术<br />
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* [[Python (programming language)|Python]] is a programming language with simple syntax that is commonly used for data science.<ref>{{Cite web|url=https://sites.engineering.ucsb.edu/~shell/che210d/python.pdf|title=An introduction to Python for scientific computing|last=Shell|first=M Scott|date=September 24, 2019|website=|url-status=live|archive-url=|archive-date=|access-date=April 2, 2020}}</ref> There are a number of python libraries that are used in data science including numpy, pandas, and scipy.<br />
* [[Python(编程语言)|Python]]是一种语法简单的编程语言，通常用于数据科学。在数据科学中使用了大量的python库，包括numpy、panda和scipy。<br />
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* [[R (programming language)|R]] is a programming language that was designed for statisticians and data mining<ref>{{Cite web|url=https://cran.r-project.org/doc/FAQ/R-FAQ.html#What-is-R_003f|title=R FAQ|website=cran.r-project.org|access-date=2020-04-03}}</ref> and is optimized for computation.<br />
*[[R(程序设计语言)|R]]是一种为统计学家而设计程序语言，其可以用于数据挖掘和计算优化。<br />
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R作为一种统计分析软件，是集统计分析与图形显示于一体的。它可以运行于UNIX,Windows和Macintosh的操作系统上，而且嵌入了一个非常方便实用的帮助系统，相比于其他统计分析软件，R还有以下特点：<br />
#R是自由软件。这意味着它是完全免费,开放源代码的。可以在它的网站及其镜像中下载任何有关的安装程序、源代码、程序包及其源代码、文档资料。标准的安装文件身自身就带有许多模块和内嵌统计函数,安装好后可以直接实现许多常用的统计功能。<br />
#R是一种可编程的语言。作为一个开放的统计编程环境,语法通俗易懂,很容易学会和掌握语言的语法。而且学会之后,我们可以编制自己的函数来扩展现有的语言。这也就是为什么它的更新速度比一般统计软件,如,SPSS,SAS等快得多。大多数最新的统计方法和技术都可以在R中直接得到。 <br />
#所有R的函数和数据集是保存在程序包里面的。只有当一个包被载入时,它的内容才可以被访问。一些常用、基本的程序包已经被收入了标准安装文件中,随着新的统计分析方法的出现,标准安装文件中所包含的程序包也随着版本的更新而不断变化。在另外版安装文件中,已经包含的程序包有：base一R的基础模块、mle一极大似然估计模块、ts一时间序列分析模块、mva一多元统计分析模块、survival一生存分析模块等等。<br />
#R具有很强的互动性。除了图形输出是在另外的窗口处，它的输入输出窗口都是在同一个窗口进行的，输入语法中如果出现错误会马上在窗口口中得到提示,对以前输入过的命令有记忆功能,可以随时再现、编辑修改以满足用户的需要。输出的图形可以直接保存为JPG,BMP,PNG等图片格式,还可以直接保存为PDF文件。另外,和其他编程语言和数据库之间有很好的接口。<br />
#如果加入R的帮助邮件列表一,每天都可能会收到几十份关于R的邮件资讯。可以和全球一流的统计计算方面的专家讨论各种问题,可以说是全世界最大、最前沿的统计学家思维的聚集地。<br />
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* [[TensorFlow]] is a framework for creating machine learning models developed by Google.<br />
*[[TensorFlow]]是一个用于创建由谷歌开发的机器学习模型的框架，是一个端到端开源机器学习平台。<br />
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Tensorflow拥有一个全面而灵活的生态系统，其中包含各种工具、库和社区资源，可助力研究人员推动先进机器学习技术的发展，并使开发者能够轻松地构建和部署由机器学习提供支持的应用。TensorFlow也是一个基于'''数据流编程 Dataflow Programming''' 的符号数学系统，被广泛应用于各类'''机器学习 Machine Learning'''算法的编程实现，其前身是谷歌的神经网络算法库DistBelief 。<br />
Tensorflow拥有多层级结构，可部署于各类服务器、PC终端和网页并支持GPU和TPU高性能数值计算，被广泛应用于谷歌内部的产品开发和各领域的科学研究 。<br />
TensorFlow由谷歌人工智能团队'''谷歌大脑 Google Brain''' 开发和维护，拥有包括TensorFlow Hub、TensorFlow Lite、TensorFlow Research Cloud在内的多个项目以及各类'''应用程序接口 Application Programming Interface''' 。自2015年11月9日起，TensorFlow依据'''阿帕奇授权协议 Apache 2.0 open source license''' 开放源代码 。<br />
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* [[Pytorch]] is another framework for machine learning developed by Facebook.<br />
*[[Pytorch]]是Facebook开发的另一个机器学习框架。<br />
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PyTorch是一个开源的Python机器学习库，基于Torch，用于自然语言处理等应用程序。<br />
2017年1月，由Facebook人工智能研究院（FAIR）基于Torch推出了PyTorch。它是一个基于Python的可续计算包，提供两个高级功能：<br />
#具有强大的GPU加速的张量计算（如NumPy）。<br />
#包含自动求导系统的的深度神经网络。<br />
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* [[Jupyter Notebook]] is an interactive web interface for Python that allows faster experimentation.<br />
*[[Jupyter Notebook]]是一个用于Python的交互式web界面，允许进行更快的实验。<br />
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Jupyter Notebook（此前被称为 IPython notebook）是一个交互式笔记本，支持运行 40 多种编程语言。其本质是一个 Web 应用程序，便于创建和共享文学化程序文档，支持实时代码，数学方程，可视化。 用途包括：数据清理和转换，数值模拟，统计建模，机器学习等等 。<br />
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* [[Tableau Software|Tableau]] makes a variety of software that is used for data visualization<ref>{{Cite journal|url=https://www.wired.com/2014/07/a-drag-and-drop-toolkit-that-lets-anyone-create-interactive-maps/|journal=Wired|access-date=2020-04-03|title=A Dead-Simple Tool That Lets Anyone Create Interactive Maps|date=15 July 2014|last1=Rhodes|first1=Margaret}}</ref>.<br />
*[[Tableau软件|Tableau]]制作各种用于数据可视化的软件。<br />
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Tableau Software是桌面系统上的一种简单的商业智能工具软件。致力于帮助人们查看并理解数据。Tableau 帮助任何人快速分析、可视化并分享信息。超过 42,000 家客户通过使用 Tableau 在办公室或随时随地快速获得结果。数以万计的用户使用 Tableau Public 在博客与网站中分享数据。<br />
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* [[Apache Hadoop]] is a software framework that is used to process data over large distributed systems.<br />
*[[Apache Hadoop]]是一套用于在由通用硬件构建的大型集群上运行应用程序的框架。它实现了Map/Reduce编程范型，计算任务会被分割成小块（多次）运行在不同的节点上。除此之外，它还提供了一款分布式文件系统（HDFS），数据被存储在计算节点上以提供极高的跨数据中心聚合带宽。<br />
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==与机器学习、人工智能之间的异同==<br />
需要清楚的是，数据科学、机器学习、人工智能这三者之间的限定并不是那么绝对：并不是所有符合每个定义的东西都只属于这个领域。使用的是描述性的方法而不是说明性的方法来说明这三者的区别：<br />
*数据科学产生见解<br />
*机器学习产生预测<br />
*人工智能产生行为<br />
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===数据科学负责见解===<br />
数据科学与其他两个领域不同，因为它的目标跟人类的目标尤其接近：获得洞察力和理解力。'''Jeff Leek 杰夫韭菜''' 对数据科学所能达到的见解类型有一个很好的定义，包括描述性，探索性和因果关系。<br />
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）Jeff Leek 杰夫韭菜的cnki翻译意义感觉不太对<br />
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同样，并不是所有产生见解的东西都有资格成为数据科学(数据科学的经典定义是它包含了统计、软件工程和领域专家的结合)。但是我们可以用这个定义把数据科学同机器学习和人工智能区别出来。它们之间的主要区别在于，在数据科学中，总是有一个人在循环当中：某人理解这个观点，看到数字，或者从结论中获益。说“下棋算法使用数据科学来选择下一步行动”，或者“谷歌地图使用数据科学来推荐驾驶方向”都是毫无意义的。<br />
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因此，数据科学的定义强调：<br />
#统计推断<br />
#数据可视化<br />
#实验设计<br />
#领域知识<br />
#沟通<br />
数据科学家可能使用一些简单的工具：可以报告百分比，并根据SQL查询绘制线形图。还可以使用非常复杂的方法:可能与分布数据商店合作，分析数以万亿计的记录，开发前沿的统计技术，建立交互式可视化。不管使用什么，目的是为了更好地理解数据。<br />
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===机器学习负责预测===<br />
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机器学习是一个预测领域：“给定实例X具有特定的特征然后进行预测”。这些预测可能是关于未来的(“预测这个病人是否会进入脓毒症”)，但也可能是对于计算机来说并不是很轻易就识别出来的一些特质(“预测这个图像是否有鸟在里面”)。几乎所有的Kaggle竞赛都认为是机器学习的问题:提供一些训练数据，然后看看竞争对手能否对新的例子做出准确的预测。<br />
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数据科学和机器学习之间有很多重叠。例如，可以使用逻辑回归得出见解的关系(“富裕的用户更有可能会买我们的产品,所以我们应该改变我们的营销策略”),从而进行预测(“该用户有53%机会购买我们的产品,所以应该把产品推荐给他们”)。<br />
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像随机森林这种模型可解释性稍差，更适合“机器学习”的描述，而深度学习等方法则难以解释。如果你的目标是获取真知灼见，而不是做出预测，这可能会妨碍你。因此，我们可以假设有一个数据科学和机器学习的“光谱”，更倾向于数据科学的可解释模型和机器学习方面更多的“黑匣子”模型。<br />
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大多数的实践者会在两个任务之间来回切换非常轻松。在我的工作中使用了机器学习和数据科学：我可以在Stack Overflow上安装一个模型，以确定哪些用户可能正在寻找工作(机器学习)，然后构建总结和可视化来检验模型的工作原理(数据科学)。 这是发现模型中的缺陷，并与算法偏见作斗争的重要途径。这是数据科学家经常负责开发产品的机器学习组件的原因之一。<br />
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===人工智能负责行为===<br />
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人工智能是这三个名称当中迄今为止最古老以及最广为人知的，因此它的定义也是最具有挑战性的。感谢那些寻求金钱或关注的研究人员、记者和初创公司，使得人工智能这一术语被大量的传播和营销。<br />
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在“人工智能”的所有定义中，有一个共同点就是自主代理人执行或推荐操作(例如Poole, Mackworth和Goebel 1998, Russell和Norvig 2003)。一些我认为应该描述为AI的系统包括：<br />
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游戏算法(Deep Blue，AlphaGo)<br />
机器人技术与控制理论(运动规划，行走两足机器人)<br />
优化(谷歌地图选择路线)<br />
自然语言处理(bots)<br />
强化学习<br />
同样，我们可以看到很多与其他领域的重叠。深度学习既属于机器学习范畴，又属于AI领域，这是非常有趣的。典型的用例是对数据进行训练，然后生成预测，在像AlphaGo这样的博弈算法中已经取得了巨大的成功。<br />
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但也有区别。如果我分析一些销售数据并发现特定行业的客户比其他行业的客户更新更多，那么输出是一些数字和图形，而不是特定的操作。(高管们可能会用这些结论来调整销售策略，但这种做法并不是自动的)。<br />
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人工智能和机器学习之间的区别更加微妙，历史上的机器学习经常被认为是人工智能的一个分支(特别是计算机视觉，这是一个典型的人工智能问题)。但我认为，机器学习领域很大程度上已经从人工智能独立出来了，部分原因是上面描述的反弹案例:大多数从事预测问题的人不喜欢把自己描述为人工智能研究人员。<br />
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===案例研究：三项技术如何一起使用？===<br />
假设正在建造一辆无人驾驶车，现在卡在了停车标志这个具体问题上面。我们需要用到这三个领域之间的相关技术。<br />
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机器学习:汽车必须通过摄像头识别停车标志。我们构建了数以百万计的街边物体照片数据集，并训练算法来预测哪些街边是有停车标志。<br />
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人工智能:一旦汽车能够识别停车标志，它就需要决定何时采取刹车动作。我们需要它能够根据不同的路况进行判断(例如，在湿滑的路面上要能知道不能刹车刹的太快)，太早或太晚都是有危险的，这属于控制理论的范畴。<br />
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数据科学:街头测试的结果证明这辆车的性能还不够好，在某些场景下面，它可能沿着停车标志额右边行驶，但是这种情况漏报了。在分析了街道测试数据之后，我们得到了结论，漏报场景的概念根据一天当中不同的时间段有关:在日出之前或日落之后，更有可能错过一个停车标志。我们发现了大部分的数据集里面的数据只包含了白天的物体，因此又构造了一个更好的数据集，它包括了夜间图像，然后重新回到机器学习步骤。<br />
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来源：https://blog.csdn.net/dev_csdn/article/details/79127658 翻译原文http://varianceexplained.org/r/ds-ml-ai/ 原文作者：David Robinson<br />
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==Relationship to statistics与统计学的关系==<br />
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飞速增长的职位空缺表明“数据科学”的概念在商业界和学术界可谓一夜蹿红。<br />
<ref><br />
Darrow,Barb(May 21, 2015).<br />
[http://fortune.com/2015/05/21/data-science-white-hot/ "Data science is still white hot, but nothing lasts forever"]<br />
.''Fortune.'' Retrieved November 20, 2017.<br />
</ref><br />
然而许多持批判态度的学者和新闻记者并没有看出数据科学与[https://en.wikipedia.org/wiki/Statistics 统计学]的区别。吉尔·普莱斯（Gil Press）在[https://en.wikipedia.org/wiki/Forbes 福布斯杂志]上撰文主张数据科学只是一个缺乏清晰定义的[https://en.wikipedia.org/wiki/Buzzword 流行术语]，并且在诸如研究生的课程内容中成了“[https://en.wikipedia.org/wiki/Business_analytics 商业分析]”的简单替代。<br />
<ref name="GilPress"><br />
[https://www.forbes.com/sites/gilpress/2013/08/19/data-science-whats-the-half-life-of-a-buzzword/ Data Science: What's The Half-Life Of A Buzzword?].<br />
Forbes.2013-08-19.<br />
</ref> <br />
<br />
Many statisticians, including [[Nate Silver]], have argued that data science is not a new field, but rather another name for statistics.<ref>{{Cite web|url=https://www.statisticsviews.com/details/feature/5133141/Nate-Silver-What-I-need-from-statisticians.html|title=Nate Silver: What I need from statisticians - Statistics Views|website=www.statisticsviews.com|access-date=2020-04-03}}</ref> Others argue that data science is distinct from statistics because it focuses on problems and techniques unique to digital data.<ref>{{Cite web|url=http://priceonomics.com/whats-the-difference-between-data-science-and/|title=What's the Difference Between Data Science and Statistics?|website=Priceonomics|language=en|access-date=2020-04-03}}</ref> [[Vasant Dhar]] writes that statistics emphasizes quantitative data and description. In contrast, data science deals with quantitative and qualitative data (e.g. images) and emphasizes prediction and action.<ref>{{Cite journal|last=DharVasant|date=2013-12-01|title=Data science and prediction|journal=Communications of the ACM|volume=56|issue=12|pages=64–73|language=EN|doi=10.1145/2500499}}</ref> [[Andrew Gelman]] of Columbia University and data scientist Vincent Granville have described statistics as a nonessential part of data science.<ref>{{Cite web|url=https://statmodeling.stat.columbia.edu/2013/11/14/statistics-least-important-part-data-science/|title=Statistics is the least important part of data science « Statistical Modeling, Causal Inference, and Social Science|website=statmodeling.stat.columbia.edu|access-date=2020-04-03}}</ref><ref>{{Cite web|url=https://www.datasciencecentral.com/profiles/blogs/data-science-without-statistics-is-possible-even-desirable|title=Data science without statistics is possible, even desirable|last=Posted by Vincent Granville on December 8|first=2014 at 5:00pm|last2=Blog|first2=View|website=www.datasciencecentral.com|language=en|access-date=2020-04-03}}</ref><br />
<br />
Many statisticians, including Nate Silver, have argued that data science is not a new field, but rather another name for statistics. Others argue that data science is distinct from statistics because it focuses on problems and techniques unique to digital data. Vasant Dhar writes that statistics emphasizes quantitative data and description. In contrast, data science deals with quantitative and qualitative data (e.g. images) and emphasizes prediction and action. Andrew Gelman of Columbia University and data scientist Vincent Granville have described statistics as a nonessential part of data science.<br />
<br />
<br />
包括'''纳特•西尔弗 Nate Silver''' 在内的许多统计学家都认为，数据科学不是一个新领域，而是统计的另一个名称。其他人则认为，数据科学与统计学不同，因为它关注的是数字数据特有的问题和技术。'''瓦桑特·达尔 Vasant Dhar'''写道，统计学强调定量数据和描述。相比之下，数据科学处理定量和定性数据(如图像)，强调预测和行动。哥伦比亚大学的'''安德鲁·格尔曼 Andrew Gelman''' 和数据科学家'''文森特·格兰维尔 Vincent Granville'''将统计描述为数据科学中一个不重要的部分。<br />
<br />
<br />
Stanford professor [[David Donoho]] writes that data science is not distinguished from statistics by the size of datasets or use of computing, and that many graduate programs misleadingly advertise their analytics and statistics training as the essence of a data science program. He describes data science as an applied field growing out of traditional statistics.<ref name=":7" /> <br />
<br />
Stanford professor David Donoho writes that data science is not distinguished from statistics by the size of datasets or use of computing, and that many graduate programs misleadingly advertise their analytics and statistics training as the essence of a data science program. He describes data science as an applied field growing out of traditional statistics. <br />
<br />
斯坦福大学教授 '''大卫·林奇 David Donoho''' 写道，数据科学与统计学之间并不存在数据集的大小或计算机的使用，许多研究生课程错误地宣传他们的分析学和统计学训练是数据科学课程的本质。他把数据科学描述为从传统统计学中发展出来的一个应用领域。<br />
<br />
<br />
在[https://en.wikipedia.org/wiki/American_Statistical_Association 美国统计协会]的联合统计学会议上发表主旨演说后的问答部分，著名应用统计学家[https://en.wikipedia.org/wiki/Nate_Silver 内特·西尔弗]（Nate Silver）说道：“我认为数据科学家对于统计学家是一个富有魅力的词语…统计学是科学的一条分支。数据科学家在某种意义上略显多余，而且人们不应该痛斥统计学家这个词。”<br />
<ref name="NateSilver"><br />
[http://www.statisticsviews.com/details/feature/5133141/Nate-Silver-What-I-need-from-statisticians.html "Nate Silver: What I need from statisticians"]. 23 Aug 2013 <br />
</ref><br />
同样，在商业领域，各方研究者和分析师表示，仅仅有数据科学家远远不足以赋予公司真正的竞争优势，<br />
<ref><br />
{{Cite journal<br />
|last=Miller<br />
|first=Steven<br />
|date=2014-04-10<br />
|title=Collaborative Approaches Needed to Close the Big Data Skills Gap<br />
|url=http://www.jorgdesign.net/article/view/9823<br />
|journal=Journal of Organization Design<br />
|language=en<br />
|volume=3<br />
|issue=1<br />
|pages=26–30<br />
|doi:10.7146/jod.9823<br />
|issn:2245-408X<br />
}}</ref> <br />
而且，仅仅把数据科学家看作四项更伟大的工作种类之一，各公司需要为大数据进行有效的融资，亦即：数据分析师、数据科学家、大数据[https://en.wikipedia.org/wiki/Software_Developer 开发者]和大数据[https://en.wikipedia.org/wiki/Software_engineer 工程师]。<br />
<ref><br />
{{Cite journal<br />
|last=De Mauro<br />
|first=Andrea<br />
|last2=Greco<br />
|first2=Marco<br />
|last3=Grimaldi<br />
|first3=Michele<br />
|last4=Ritala<br />
|first4=Paavo<br />
|title=Human resources for Big Data professions: A systematic classification of job roles and required skill sets<br />
|url=http://linkinghub.elsevier.com/retrieve/pii/S0306457317300018<br />
|journal=Information Processing & Management<br />
|doi:10.1016/j.ipm.2017.05.004<br />
}}</ref><br />
<br />
另一方面，也有无数对批评的回应。在2014年一篇[https://en.wikipedia.org/wiki/The_Wall_Street_Journal 《华尔街日报》]的文章中，欧文·沃拉达斯凯-伯杰（Irving Wladawsky-Berger）比较了数据科学的狂热与[https://en.wikipedia.org/wiki/Computer_science 计算机科学]的黎明。他坚称，就像其他[https://en.wikipedia.org/wiki/Interdisciplinarity 交叉学科]领域一样，数据科学利用来自[https://en.wikipedia.org/wiki/Academy 学术界]和[https://en.wikipedia.org/wiki/Industry 工业界]的[https://en.wikipedia.org/wiki/Methodology 方法论]和实践，但之后会将它们变成一个新[https://en.wikipedia.org/wiki/Discipline_(academia) 学科]。他特别强调了现在一个广受认可的学术科目计算机科学曾面临的尖锐批评。<br />
<ref name=":1"><br />
Wladawsky-Berger,Irving (May 2, 2014).<br />
[https://blogs.wsj.com/cio/2014/05/02/why-do-we-need-data-science-when-weve-had-statistics-for-centuries/ "Why Do We Need Data Science When We’ve Had Statistics for Centuries?"].<br />
''The Wall Street Journal''. Retrieved November 20, 2017.<br />
</ref><br />
类似地，就像许多其他数据科学学界支持者一样，<br />
<ref name=":1" /><br />
[https://en.wikipedia.org/wiki/New_York_University 纽约大学][https://en.wikipedia.org/wiki/NYU_Stern_Center_for_Business_and_Human_Rights 斯特恩商学院]的瓦桑德·达尔（Vasant Dhar）在2013年12月更加明确地表示数据科学与现存的仅仅聚焦于解释[https://en.wikipedia.org/wiki/Data_set 数据集]的横跨所有[https://en.wikipedia.org/wiki/Discipline_(academia) 学科]的数据分析实践不同。数据科学为[https://en.wikipedia.org/wiki/Predictive_modelling 预测模型]寻求了可行和一致的[https://en.wikipedia.org/wiki/Pattern_recognition 模式]。<br />
<ref name=":0" /><br />
这项实际的工程目标采用了超越了传统[https://en.wikipedia.org/wiki/Analytics 数据分析]的数据科学。如今这些学科和[https://en.wikipedia.org/wiki/Applied_science 应用领域]的数据缺乏可靠[https://en.wikipedia.org/wiki/Theory 理论]以供形成有力的预测模型，就像[https://en.wikipedia.org/wiki/Health_science 健康科学]和[https://en.wikipedia.org/wiki/Social_science 社会科学]那样。<br />
<ref name=":0" /><br />
<br />
斯坦福大学教授[https://en.wikipedia.org/wiki/David_Donoho 大卫·多诺霍]（David Donoho）于2015年9月在一次与达尔类似的尝试中，通过抵制批评界对数据科学的三种过分简单化和误导性的定义，提出了更长远的主张。<br />
<ref name=":2"><br />
{{Cite journal<br />
|last=Donoho<br />
|first=David<br />
|date=September 2015<br />
|title=50 Years of Data Science<br />
|url=http://courses.csail.mit.edu/18.337/2015/docs/50YearsDataScience.pdf<br />
|journal=Based on a talk at Tukey Centennial workshop, Princeton NJ Sept 18 2015<br />
|volume=<br />
|pages=<br />
|via=<br />
}}</ref><br />
第一，对多诺霍而言，数据科学不等同于[https://en.wikipedia.org/wiki/Big_data 大数据]，因为数据集的规模不是区分数据科学和统计学的标准。<br />
<ref name=":2" /><br />
第二，数据科学不是由将大数据集分类整理的[https://en.wikipedia.org/wiki/Computing 计算]技能定义的，因为这些技能已经被广泛地用作所有学科的分析。<br />
<ref name=":2" /> <br />
第三，数据科学现在是一个[https://en.wikipedia.org/wiki/Graduate_school 学术项目]尚不足以给数据科学家日后的工作提供充足准备，而已然得到大量应用的领域，因为许多[https://en.wikipedia.org/wiki/Graduate_school 研究生项目]带有误导性地宣传他们的分析和统计学训练是一个数据科学项目的实质。<br />
<ref name=":2" /><br />
<ref><br />
{{Cite book<br />
|title=The Culture of Big Data<br />
|last=Barlow<br />
|first=Mike<br />
|publisher=O'Reilly Media, Inc.<br />
|year=2013<br />
|isbn=<br />
|location=<br />
|pages=<br />
}}</ref><br />
作为一名[https://en.wikipedia.org/wiki/Statistician 统计学家]，[https://en.wikipedia.org/wiki/David_Donoho 多诺霍]继承了学界诸多前辈的衣钵，拥护着数据科学研究范围的扩充，<br />
<ref name=":2" /> <br />
就像约翰·钱伯斯（John Chambers）极力主张统计学家采用一种包容的从数据中学习的概念、<br />
<ref><br />
{{Cite journal<br />
|last=Chambers<br />
|first=John M.<br />
|date=1993-12-01<br />
|title=Greater or lesser statistics: a choice for future research<br />
|url=https://link.springer.com/article/10.1007/BF00141776<br />
|journal=Statistics and Computing<br />
|language=en<br />
|volume=3<br />
|issue=4<br />
|pages=182–184<br />
|doi:10.1007/BF00141776<br />
|issn:0960-3174<br />
}}</ref><br />
威廉·克利夫兰（William Cleveland）强调把从数据中提取具有应用价值的[https://en.wikipedia.org/wiki/Predictive_modelling 预测工具]摆在比发掘[https://en.wikipedia.org/wiki/Explanatory_model 解释性理论]更高的优先级上一样。<br />
<ref name="cleveland01" /><br />
这些[https://en.wikipedia.org/wiki/Statistician 统计学家]们共同展望着一个日益包容、从传统的[https://en.wikipedia.org/wiki/Statistics 统计学]中生长出来并青出于蓝而胜于蓝的应用领域。<br />
<br />
为了数据科学的未来，多诺霍为[https://en.wikipedia.org/wiki/Open_science 开放性科学]规划了一个不断成长的环境，使所有研究者都可以访问用于[https://en.wikipedia.org/wiki/Academic_publishing 学术出版物]的数据集。<br />
<ref name=":2" /><br />
[https://en.wikipedia.org/wiki/National_Institutes_of_Health 美国国家卫生研究院]已经宣布了提高研究数据再现性和透明度的计划。<br />
<ref><br />
{{Cite journal<br />
|last=Collins<br />
|first=Francis S.<br />
|last2=Tabak<br />
|first2=Lawrence A.<br />
|date=2014-01-30<br />
|title=NIH plans to enhance reproducibility<br />
|journal=Nature<br />
|volume=505<br />
|issue=7485<br />
|pages=612–613<br />
|issn:0028-0836<br />
|pmc:4058759<br />
|pmid:24482835<br />
|doi:10.1038/505612a<br />
}}</ref><br />
其它的大型[https://en.wikipedia.org/wiki/Academic_journal 期刊]亦紧随其后。<br />
<ref><br />
{{Cite journal<br />
|last=McNutt<br />
|first=Marcia<br />
|date=2014-01-17<br />
|title=Reproducibility<br />
|url=http://science.sciencemag.org/content/343/6168/229<br />
|journal=Science<br />
|language=en<br />
|volume=343<br />
|issue=6168<br />
|pages=229–229<br />
|doi:10.1126/science.1250475<br />
|issn:0036-8075<br />
|pmid:24436391<br />
}}</ref><br />
<ref><br />
{{Cite journal<br />
|last=Peng<br />
|first=Roger D.<br />
|date=2009-07-01<br />
|title=Reproducible research and Biostatistics<br />
|url=https://academic.oup.com/biostatistics/article/10/3/405/293660<br />
|journal=Biostatistics<br />
|language=en<br />
|volume=10<br />
|issue=3<br />
|pages=405–408<br />
|doi:10.1093/biostatistics/kxp014<br />
|issn:1465-4644<br />
}}</ref><br />
这样，数据科学的未来不仅在规模和方法论上超越了[https://en.wikipedia.org/wiki/Statistical_theory 统计学理论]的界线，它还会彻底革新现在的学术和[https://en.wikipedia.org/wiki/Paradigm 研究范式]。<br />
<ref name=":2" /><br />
诚如多诺霍所言蔽之：“数据科学的范围和影响在今后数十年会继续扩充，科研数据和有关科学本身的数据将无处不在、俯拾即是。”<br />
<ref name=":2" /><br />
<br />
==References==<br />
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参考资料<br />
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{{Reflist|35em}}<br />
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[[Category:Information science]]<br />
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Category:Information science<br />
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类别: 信息科学<br />
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[[Category:Computer occupations]]<br />
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Category:Computer occupations<br />
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类别: 计算机职业<br />
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[[Category:Computational fields of study]]<br />
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Category:Computational fields of study<br />
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类别: 研究的计算领域<br />
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[[Category:Data analysis]]<br />
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Category:Data analysis<br />
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类别: 数据分析<br />
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<noinclude><br />
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<small>This page was moved from [[wikipedia:en:Data science]]. Its edit history can be viewed at [[数据科学/edithistory]]</small></noinclude><br />
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[[Category:待整理页面]]</div>Dorr