75
个编辑
更改
跳到导航
跳到搜索
第213行：
第213行： − + − + 第227行：
第227行： − + 第239行：
第239行： − + 第245行：
第245行： − + 第253行：
第253行： − + 第261行：
第261行： − 热力学系综不能给出相同结果的重要情况包括:+ 第279行：
第279行： − 在这些情况下必须选择正确的热力学系综，因为这些系综之间不仅在涨落的大小方面有可观测的差异，而且在平均量方面都有可观察的差异，如粒子数的分布。正确的系综是对应于该系统的制备和表征的方式ーー换句话说，系综反映我们对系统的认知。+ 第499行：
第499行： − + 第505行：
第505行： − + 第529行：
第529行： − + − + 第543行：
第543行： − + 第569行：
第569行： − + 第613行：
第613行： − 所有这些过程都是以特征速率随时间发生的，这些速率对于工程来说非常重要。非平衡态统计力学研究领域关注的是在微观水平上理解这些非平衡过程。(统计热力学只能用来计算在外部不平衡被消除，整体回归到平衡状态之后的最终结果。)+ 第621行：
第621行： − + 第642行：
第642行： − − {{unordered list − − {{unordered list − − {无序列表 第653行：
第647行： − + 第669行：
第663行： − 一个与主题相关的量子技术是随机相位近似。+ 第683行：
第677行： − + 第717行：
第711行： − + 第797行：
第791行： − + 第817行：
第811行： − +
无编辑摘要
; [[Microcanonical ensemble]]
; [[Microcanonical ensemble]]
Microcanonical ensemble
Microcanonical Ensemble
<font color="#FF8000">微正则系综 Microcanonical ensemble</font>
<font color="#FF8000">微正则系综 Microcanonical Ensemble</font>
: describes a system with a precisely given energy and fixed composition (precise number of particles). The microcanonical ensemble contains with equal probability each possible state that is consistent with that energy and composition.
: describes a system with a precisely given energy and fixed composition (precise number of particles). The microcanonical ensemble contains with equal probability each possible state that is consistent with that energy and composition.
Canonical ensemble
Canonical ensemble
<font color="#FF8000">正则系综 Canonical ensemble</font>
<font color="#FF8000">正则系综 Canonical Ensemble</font>
: describes a system of fixed composition that is in [[thermal equilibrium]]{{NoteTag|The transitive thermal equilibrium (as in, "X is thermal equilibrium with Y") used here means that the ensemble for the first system is not perturbed when the system is allowed to weakly interact with the second system.}} with a [[heat bath]] of a precise [[thermodynamic temperature|temperature]]. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy.
: describes a system of fixed composition that is in [[thermal equilibrium]]{{NoteTag|The transitive thermal equilibrium (as in, "X is thermal equilibrium with Y") used here means that the ensemble for the first system is not perturbed when the system is allowed to weakly interact with the second system.}} with a [[heat bath]] of a precise [[thermodynamic temperature|temperature]]. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy.
Grand canonical ensemble
Grand canonical ensemble
<font color="#FF8000">巨正则系综 Grand canonical ensemble</font>
<font color="#FF8000">巨正则系综 Grand Canonical Ensemble</font>
: describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise [[chemical potential]]s for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers.
: describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise [[chemical potential]]s for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers.
describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise chemical potentials for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers.
describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise chemical potentials for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers.
描述了一个具有非固定成分(不确定粒子数)的系统，在热库中处于热力学和化学平衡。热库具有精确的温度，对各种类型的粒子具有精确的<font color="#FF8000">化学势 Chemical Potential</font>。巨正则系综包含不同能量和粒子数的状态; 根据总能量和粒子数的不同，系综中不同状态的概率也不同。
描述了一个具有非固定成分(不确定粒子数)在热库中处于热力学和化学平衡的系统。热库具有精确的温度，各种类型的粒子具有精确的<font color="#FF8000">化学势 Chemical Potential</font>。巨正则系综包含不同能量和状态的大量粒子; 根据总能量和粒子数的不同，系综中不同状态的概率也不同。
For systems containing many particles (the thermodynamic limit), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used. The Gibbs theorem about equivalence of ensembles was developed into the theory of concentration of measure phenomenon, which has applications in many areas of science, from functional analysis to methods of artificial intelligence and big data technology.
For systems containing many particles (the thermodynamic limit), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used. The Gibbs theorem about equivalence of ensembles was developed into the theory of concentration of measure phenomenon, which has applications in many areas of science, from functional analysis to methods of artificial intelligence and big data technology.
对于包含大量粒子的系统(<font color="#FF8000">热力学极限 Thermodynamic Limit</font>) ，上面列出的三种系综都倾向于给出相同的行为。因此，使用哪种系综只是一个简单的数学问题。关于系综等价的吉布斯定理被发展成为(<font color="#FF8000">测度现象集中理论 Concentration of Measure Phenomenon</font>，在从函数分析到人工智能和大数据技术等许多科学领域都有广泛的应用。
对于包含大量粒子的系统(<font color="#FF8000">热力学极限 Thermodynamic Limit</font>) ，上面列出的三种系综都倾向于体现出相同的行为。因此，使用哪种系综只是一个简单的数学问题。发展成为(<font color="#FF8000">测度现象集中理论 Concentration of Measure Phenomenon</font>系综等价的吉布斯定理，在从函数分析到人工智能和大数据技术等许多科学领域都有广泛的应用。
Important cases where the thermodynamic ensembles do not give identical results include:
Important cases where the thermodynamic ensembles do not give identical results include:
热力学系综不能给出相同结果的重要例子包括:
* Microscopic systems.
* Microscopic systems.
In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.
In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.
在这些情况下必须选择正确的热力学系综，因为这些系综之间不仅在涨落的大小方面有可观测的差异，而且在平均量方面都有可观察的差异，如粒子数的分布。正确的系综是对应于该系统的准备和表征的方式ーー换句话说，系综反映我们对系统的认知。
|}
|}
=== Calculation methods 计算方法===
=== Calculation Methods 计算方法===
Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.
Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.
Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.
Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.
一旦计算出一个系统的特征状态函数，该系统就被“解决”了(宏观观测量可以从特征状态函数中提取)。然而，计算热力学系综的特征状态函数并不一定是一项简单的工作，因为它涉及到考虑系统的每一种可能状态。虽然一些假设的系统已经被完全求解了，但是最一般的(和现实的)情况对于一个精确的解来说太复杂了。存在各种方法来近似真实的系综，并且计算平均量。
一旦计算出一个系统的特征状态函数，该系统就被“解决”了(宏观观测量可以从特征状态函数中提取)。然而，计算热力学系综的特征状态函数并不一定是一项简单的工作，因为它涉及到考虑系统的每一种可能状态。虽然一些假设的系统已经被完全求解了，但是对最一般的(和现实的)情况进行精确的求解实在是太复杂了。存在各种方法来近似真实的系综，并且计算平均量。
* 对于非常小的微观系统，可以通过简单地列举系统所有可能状态（利用量子力学中的严格对角化，或者经典力学中对所有相空间积分）来直接得到系综。
* 对于非常小的微观系统，可以通过简单地列举系统所有可能状态（利用量子力学中的严格对角化，或者经典力学中对所有相空间积分）来直接得到系综。
* 对于包含很多分离的微观系统的宏观系统，每个子系统可以单独分析。尤其是粒子间无相互作用的理想气体具有这种性质，从而可以精确地得到[[<font color="#FF8000">麦克斯韦–玻尔兹曼统计 Maxwell–Boltzmann statistics</font>]]， [[<font color="#FF8000">费米-狄拉克统计 Fermi–Dirac statistics</font>]]，和[[<font color="#FF8000">波色-爱因斯坦统计 Bose–Einstein statistics</font>]]。<ref name="tolman"/>
* 对于包含很多分离的微观系统的宏观系统，每个子系统可以单独分析。尤其是粒子间无相互作用的理想气体具有这种性质，从而可以精确地得到[[<font color="#FF8000">麦克斯韦–玻尔兹曼统计 Maxwell–Boltzmann Statistics</font>]]， [[<font color="#FF8000">费米-狄拉克统计 Fermi–Dirac Statistics</font>]]，和[[<font color="#FF8000">波色-爱因斯坦统计 Bose–Einstein Statistics</font>]]。<ref name="tolman"/>
* 某些存在相互作用的宏观系统也存在精确解。通过运用微妙的数学技巧，已经找到了几个[[玩具模型]]的精确解。<ref>{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc. | pages = }}</ref> 一些例子包括[[Bethe ansatz]]，零场下[[<font color="#FF8000"> 二维格点伊辛模型 square-lattice Ising model</font>]]，[[<font color="#FF8000">硬六边形模型 hard hexagon model</font>]]。
* 某些存在相互作用的宏观系统也存在精确解。通过运用微妙的数学技巧，已经找到了几个[[玩具模型]]的精确解。<ref>{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc. | pages = }}</ref> 一些例子包括[[Bethe ansatz]]，零场下的[[<font color="#FF8000"> 二维格点伊辛模型 Square-Lattice Ising Model</font>]]，[[<font color="#FF8000">硬六边形模型 Hard Hexagon Model</font>]]。
====Monte Carlo 蒙特卡罗方法====
====Monte Carlo 蒙特卡罗方法====
One approximate approach that is particularly well suited to computers is the Monte Carlo method, which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.
One approximate approach that is particularly well suited to computers is the Monte Carlo method, which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.
一个特别适合于计算机的近似方法是<font color="#FF8000">蒙特卡罗方法 Monte Carlo method</font>，它只随机选择(具有相当的权重)系统的几个可能状态进行检查。只要这些状态构成系统全部状态集的代表样本，就可以得到近似的特征函数。随着随机样本数量的增加，误差可以降低到任意低的水平。
一个特别适合于计算机的近似方法是<font color="#FF8000">蒙特卡罗方法 Monte Carlo Method</font>，它只随机选择(具有相当的权重)系统的几个可能状态进行检查。只要这些状态可构成系统全部状态集的代表样本，就可以得到近似的特征函数。随着随机样本数量的增加，误差可以降低到任意低的水平。
* 对于稠密流体，另一种近似方法是基于简化的分布函数，特别是[[径向分布函数 radial distribution function]]。<ref name="balescu"/>
* 对于稠密流体，另一种近似方法是基于简化的分布函数，特别是[[径向分布函数 radial distribution function]]。<ref name="balescu"/>
* [[分子动力学 Molecular dynamics]] 计算机模拟可以用来计算各态历经系统中的[[微正则系综]]平均。对于包含与随机热浴的连接的系统，他们还可以在正则和巨正则条件下建模。
* [[分子动力学 Molecular dynamics]] 计算机模拟可以用来计算各态历经系统中的[[微正则系综]]平均。对于包含与随机热浴有连接的系统，他们还可以在正则和巨正则条件下建模。
* 包含非平衡态统计力学结果（如下）的混合方法可能是很有用的。
* 包含非平衡态统计力学结果（如下）的混合方法可能是很有用的。
All of these processes occur over time with characteristic rates, and these rates are of importance for engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)
All of these processes occur over time with characteristic rates, and these rates are of importance for engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)
所有这些过程都是以特征速率随时间发生，这些速率对于工程来说非常重要。非平衡态统计力学研究领域关注的是在微观水平上理解这些非平衡过程。(统计热力学只能用来计算在外部不平衡被消除，整体回归到平衡状态之后的最终结果。)
In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as Liouville's equation or its quantum equivalent, the von Neumann equation. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. Unfortunately, these ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's Gibbs entropy is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics.
In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as Liouville's equation or its quantum equivalent, the von Neumann equation. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. Unfortunately, these ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's Gibbs entropy is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics.
原则上，非平衡态统计力学在数学上可以是精确的: 孤立系统的系综根据确定性方程随时间演化，如刘维尔方程或其量子等价、冯·诺依曼方程。这些方程是将运动力学方程独立应用于系综中每个状态的结果。不幸的是，这些系综演化方程继承了潜在动力学运动的大部分复杂性，因此很难得到精确解。此外，系综演化方程是完全可逆的，不会破坏信息(系综的<font color="#FF8000">吉布斯熵 Gibbs entropy</font>被保留)。为了在模拟不可逆过程中取得进展，除了概率和可逆力学外，还必须考虑其他因素。
原则上，非平衡态统计力学在数学上可以是精确的: 孤立系统的系综根据确定性方程随时间演化，如刘维尔方程或其量子等价、冯·诺依曼方程。这些方程是将运动力学方程独立应用于系综中每个状态的结果。不幸的是，这些系综演化方程继承了底层动力学运动的大部分复杂性，因此很难得到精确解。此外，系综演化方程是完全可逆的，不会破坏信息(系综的<font color="#FF8000">吉布斯熵 Gibbs entropy</font>被保留)。为了在模拟不可逆过程中取得进展，除了概率和可逆力学外，还必须考虑其他因素。
|1 = ''[[Boltzmann transport equation]]'': An early form of stochastic mechanics appeared even before the term "statistical mechanics" had been coined, in studies of [[kinetic theory of gases|kinetic theory]]. [[James Clerk Maxwell]] had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. [[Ludwig Boltzmann]] subsequently showed that, by taking this [[molecular chaos]] for granted as a complete randomization, the motions of particles in a gas would follow a simple [[Boltzmann transport equation]] that would rapidly restore a gas to an equilibrium state (see [[H-theorem]]).
|1 = ''[[Boltzmann transport equation]]'': An early form of stochastic mechanics appeared even before the term "statistical mechanics" had been coined, in studies of [[kinetic theory of gases|kinetic theory]]. [[James Clerk Maxwell]] had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. [[Ludwig Boltzmann]] subsequently showed that, by taking this [[molecular chaos]] for granted as a complete randomization, the motions of particles in a gas would follow a simple [[Boltzmann transport equation]] that would rapidly restore a gas to an equilibrium state (see [[H-theorem]]).
|1 = Boltzmann transport equation: An early form of stochastic mechanics appeared even before the term "statistical mechanics" had been coined, in studies of kinetic theory. James Clerk Maxwell had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. Ludwig Boltzmann subsequently showed that, by taking this molecular chaos for granted as a complete randomization, the motions of particles in a gas would follow a simple Boltzmann transport equation that would rapidly restore a gas to an equilibrium state (see H-theorem).
|1 = Boltzmann transport equation: An early form of stochastic mechanics appeared even before the term "statistical mechanics" had been coined, in studies of kinetic theory. James Clerk Maxwell had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. Ludwig Boltzmann subsequently showed that, by taking this molecular chaos for granted as a complete randomization, the motions of particles in a gas would follow a simple Boltzmann transport equation that would rapidly restore a gas to an equilibrium state (see H-theorem).
<font color="#FF8000">玻尔兹曼输运方程 Boltzmann transport equation</font>: 在动力学理论研究中，早期的随机力学形式甚至在“统计力学”一词被创造之前就已经出现了。詹姆斯·克拉克·麦克斯韦已经证明分子碰撞会导致气体内部明显的混沌运动。路德维希·玻尔兹曼随后证明，如果把这种分子混沌理所当然地看作是一种完全的随机化，那么气体中粒子的运动将遵循一个简单的玻尔兹曼输运方程，这个方程将使气体迅速恢复到平衡状态(见H-定理)。
<font color="#FF8000">玻尔兹曼输运方程 Boltzmann Transport Equation</font>: 在动力学理论研究中，早期的随机力学甚至在“统计力学”一词被创造之前就已经出现了。詹姆斯·克拉克·麦克斯韦已经证明分子碰撞会导致气体内部明显的混沌运动。路德维希·玻尔兹曼随后证明，如果把这种分子混沌理所当然地看作是一种完全的随机化，那么气体中粒子的运动将遵循一个简单的玻尔兹曼输运方程，这个方程将使气体迅速恢复到平衡状态(见H-定理)。
A quantum technique related in theme is the random phase approximation.
A quantum technique related in theme is the random phase approximation.
一个与主题相关的量子技术是<font color="#FF8000">随机相位近似 Random Phase Approximation</font>。
In liquids and dense gases, it is not valid to immediately discard the correlations between particles after one collision. The BBGKY hierarchy (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy) gives a method for deriving Boltzmann-type equations but also extending them beyond the dilute gas case, to include correlations after a few collisions.
In liquids and dense gases, it is not valid to immediately discard the correlations between particles after one collision. The BBGKY hierarchy (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy) gives a method for deriving Boltzmann-type equations but also extending them beyond the dilute gas case, to include correlations after a few collisions.
在液体和稠密气体中，不能在一次碰撞后立即丢掉粒子之间的关联。BBGKY 层级结构(Bogoliubov-Born-Green-Kirkwood-Yvon 层级结构)提供了一种推导玻尔兹曼型方程的方法，但也可以将它们扩展到稀薄气体情况之外，包括在几次碰撞之后的相关性。
在液体和稠密气体中，不能在一次碰撞后立即丢掉粒子之间的关联。BBGKY 层级结构(<font color="#FF8000">层级结构 Bogoliubov-Born-Green-Kirkwood-Yvon</font>)提供了一种推导玻尔兹曼型方程的方法，但也可以将它们扩展到稀薄气体情况之外，包括在几次碰撞之后的相关性。
Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in linear response theory. A remarkable result, as formalized by the fluctuation-dissipation theorem, is that the response of a system when near equilibrium is precisely related to the fluctuations that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or "know" how it came to be away from equilibrium.
Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in linear response theory. A remarkable result, as formalized by the fluctuation-dissipation theorem, is that the response of a system when near equilibrium is precisely related to the fluctuations that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or "know" how it came to be away from equilibrium.
非平衡态统计力学模型处理的另一类重要的系统，是对平衡态仅有非常轻微扰动的系统。在很小的扰动下，响应可以用<font color="#FF8000">线性响应理论 linear response theory</font>进行分析。<font color="#FF8000">涨落-耗散定理 fluctuation-dissipation theorem</font>是其中一个重要的结果，近平衡态系统的响应与系统总体平衡时的涨落准确相关。从本质上讲，一个系统如果稍微偏离平衡，无论是由于外力还是由于涨落，都会以同样的方式向平衡方向弛豫。因为这个系统无法区分偏离和回归，也“不知道”它是如何偏离平衡的。
非平衡态统计力学模型处理的另一类重要的系统，是对平衡态仅有非常轻微扰动的系统。在很小的扰动下，响应可以用<font color="#FF8000">线性响应理论 Linear Response Theory</font>进行分析。<font color="#FF8000">涨落-耗散定理 Fluctuation-Dissipation Theorem</font>是其中一个重要的结果，近平衡态系统的响应与系统总体平衡时的涨落准确相关。从本质上讲，一个系统如果稍微偏离平衡，无论是由于外力还是由于涨落，都会以同样的方式向平衡方向弛豫。因为这个系统无法区分偏离和回归，也“不知道”它是如何偏离平衡的。
In 1738, Swiss physicist and mathematician Daniel Bernoulli published Hydrodynamica which laid the basis for the kinetic theory of gases. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion.
In 1738, Swiss physicist and mathematician Daniel Bernoulli published Hydrodynamica which laid the basis for the kinetic theory of gases. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion.
1738年，瑞士的物理学家和数学家丹尼尔·伯努利发表了《水动力学》 ，这本书奠定了<font color="#FF8000">气体动力学理论 kinetic theory of gases</font>的基础。在这项工作中，伯努利假定气体是由大量向各个方向运动的分子组成的，它们对表面的影响导致了我们感觉到的气体压力，而我们感受到的热仅仅是它们运动的动能，这一点直到今天仍在沿用。
1738年，瑞士的物理学家和数学家丹尼尔·伯努利发表了《水动力学》 ，这本书奠定了<font color="#FF8000">气体动力学理论 Kinetic Theory of Gases</font>的基础。在这项工作中，伯努利假定气体是由大量向各个方向运动的分子组成的，它们对表面的影响导致了我们感觉到的气体压力，而我们感受到的热仅仅是它们运动的动能，这一点直到今天仍在沿用。
Statistical mechanics proper was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 Lectures on Gas Theory. Boltzmann's original papers on the statistical interpretation of thermodynamics, the H-theorem, transport theory, thermal equilibrium, the equation of state of gases, and similar subjects, occupy about 2,000 pages in the proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his H-theorem.
Statistical mechanics proper was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 Lectures on Gas Theory. Boltzmann's original papers on the statistical interpretation of thermodynamics, the H-theorem, transport theory, thermal equilibrium, the equation of state of gases, and similar subjects, occupy about 2,000 pages in the proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his H-theorem.
统计力学是19世纪70年代由玻尔兹曼的工作创立的，其中大部分在他1896年的气体理论演讲中集结出版。在维也纳学院和其他学会的会议记录中，玻尔兹曼关于热力学的统计解释、 <font color="#FF8000">H-定理H-theorem</font>、<font color="#FF8000">输运理论 transport theory</font>、<font color="#FF8000">热平衡thermal equilibrium</font>、气体的状态方程以及类似主题的原始论文占据了大约2000页。玻尔兹曼引入了平衡系综的概念，并用他的H-定理第一次研究了非平衡态统计力学。
统计力学是19世纪70年代由玻尔兹曼的工作创立的，其中大部分在他1896年的气体理论演讲中集结出版。在维也纳学院和其他学会的会议记录中，玻尔兹曼关于热力学的统计解释、 <font color="#FF8000">H-定理H-theorem</font>、<font color="#FF8000">输运理论 Transport Theory</font>、<font color="#FF8000">热平衡Thermal Equilibrium</font>、气体的状态方程以及类似主题的原始论文占据了大约2000页。玻尔兹曼引入了平衡系综的概念，并用他的H-定理第一次研究了非平衡态统计力学。