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第503行： + + − In his exposition of his scheme of closed system equilibrium thermodynamics, C. Carathéodory initially postulates that experiment reveals that a definite number of real variables define the states that are the points of the manifold of equilibria.<ref name="Caratheodory" /> In the words of Prigogine and Defay (1945): "It is a matter of experience that when we have specified a certain number of macroscopic properties of a system, then all the other properties are fixed."<ref>Prigogine, I., Defay, R. (1950/1954), p. 1.</ref><ref>Silbey, R.J., [[Robert A. Alberty|Alberty, R.A.]], Bawendi, M.G. (1955/2005), p. 4.</ref> As noted above, according to A. Münster, the number of variables needed to define a thermodynamic equilibrium is the least for any state of a given isolated system. As noted above, J.G. Kirkwood and I. Oppenheim point out that a state of thermodynamic equilibrium may be defined by a special subclass of intensive variables, with a definite number of members in that subclass.+ 第508行：
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第560行： − + − 如果介观系统进一步重复分裂，最终产生一个微观系统。物质的分子性质和动量传递的量子性质在涨落过程中起着重要作用。一个人已经离开了经典热力学或宏观热力学的领域，他需要量子统计力学。波动可以变得相对占主导地位，测量问题变得重要。+ + 第566行：
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→Characteristics of a state of internal thermodynamic equilibrium
==Characteristics of a state of internal thermodynamic equilibrium==
==Characteristics of a state of internal thermodynamic equilibrium==
内部热力学平衡状态的特征
===Homogeneity in the absence of external forces===
===Homogeneity in the absence of external forces===
在没有外力的情况下的同质性
A thermodynamic system consisting of a single phase in the absence of external forces, in its own internal thermodynamic equilibrium, is homogeneous.<ref name="Planck 1903 3"/> This means that the material in any small volume element of the system can be interchanged with the material of any other geometrically congruent volume element of the system, and the effect is to leave the system thermodynamically unchanged. In general, a strong external force field makes a system of a single phase in its own internal thermodynamic equilibrium inhomogeneous with respect to some [[Intensive and extensive properties|intensive variables]]. For example, a relatively dense component of a mixture can be concentrated by centrifugation.
A thermodynamic system consisting of a single phase in the absence of external forces, in its own internal thermodynamic equilibrium, is homogeneous.<ref name="Planck 1903 3"/> This means that the material in any small volume element of the system can be interchanged with the material of any other geometrically congruent volume element of the system, and the effect is to leave the system thermodynamically unchanged. In general, a strong external force field makes a system of a single phase in its own internal thermodynamic equilibrium inhomogeneous with respect to some [[Intensive and extensive properties|intensive variables]]. For example, a relatively dense component of a mixture can be concentrated by centrifugation.
在没有外力的情况下，由单一相组成的热力学系统，在其自身的内部热力学平衡中是均匀的。这意味着系统中任何小体积单元中的材料可以与系统中任何其他几何相等的体积单元中的材料互换，其效果是使系统在热力学上保持不变。一般来说，一个强外力场使得一个单相系统在其自身的内部热力学平衡中对于一些'''<font color="#ff8000">强变量 Intensive Variable</font>'''是不均匀的。例如，可以通过离心来浓缩混合物中相对密度较大的组分。
The temperature within a system in thermodynamic equilibrium is uniform in space as well as in time. In a system in its own state of internal thermodynamic equilibrium, there are no net internal macroscopic flows. In particular, this means that all local parts of the system are in mutual radiative exchange equilibrium. This means that the temperature of the system is spatially uniform. Considerations of kinetic theory or statistical mechanics also support this statement.
The temperature within a system in thermodynamic equilibrium is uniform in space as well as in time. In a system in its own state of internal thermodynamic equilibrium, there are no net internal macroscopic flows. In particular, this means that all local parts of the system are in mutual radiative exchange equilibrium. This means that the temperature of the system is spatially uniform. Considerations of kinetic theory or statistical mechanics also support this statement.
===Uniform temperature===
===Uniform temperature===
均匀温度
In order that a system may be in its own internal state of thermodynamic equilibrium, it is of course necessary, but not sufficient, that it be in its own internal state of thermal equilibrium; it is possible for a system to reach internal mechanical equilibrium before it reaches internal thermal equilibrium. As noted above, according to A. Münster, the number of variables needed to define a thermodynamic equilibrium is the least for any state of a given isolated system. As noted above, J.G. Kirkwood and I. Oppenheim point out that a state of thermodynamic equilibrium may be defined by a special subclass of intensive variables, with a definite number of members in that subclass.
In order that a system may be in its own internal state of thermodynamic equilibrium, it is of course necessary, but not sufficient, that it be in its own internal state of thermal equilibrium; it is possible for a system to reach internal mechanical equilibrium before it reaches internal thermal equilibrium. As noted above, according to A. Münster, the number of variables needed to define a thermodynamic equilibrium is the least for any state of a given isolated system. As noted above, J.G. Kirkwood and I. Oppenheim point out that a state of thermodynamic equilibrium may be defined by a special subclass of intensive variables, with a definite number of members in that subclass.
为了使一个系统处于它自己的内部热力学平衡状态，它当然是必要的，但不是充分的，它必须处于它自己的内部热平衡状态; 一个系统在到达内部热平衡之前到达内部力学平衡是可能的。如上所述，根据 a. m ü nster 的说法，对于给定的孤立系统的任何状态，定义一个热力学平衡所需的变量数是最少的。如上所述，j.g。Kirkwood 和 i. Oppenheim 指出，热力学平衡状态可以由一个特殊的子类的强变量定义，该子类中有一定数量的成员。
为了使一个系统处于它自己的内部热力学平衡状态，它当然是必要的，但不是充分的，它必须处于它自己的内部热平衡状态; 一个系统在到达内部热平衡之前到达内部力学平衡是可能的。如上所述，根据 A.Münster的说法，对于给定的孤立系统的任何状态，定义一个热力学平衡所需的变量数是最少的。如上所述，J.G.Kirkwood 和 I.Oppenheim 指出，热力学平衡状态可以由一个特殊的子类的强变量定义，该子类中有一定数量的成员。
Such equilibrium inhomogeneity, induced by external forces, does not occur for the intensive variable [[temperature]]. According to [[Edward A. Guggenheim|E.A. Guggenheim]], "The most important conception of thermodynamics is temperature."<ref>[[Edward A. Guggenheim|Guggenheim, E.A.]] (1949/1967), p.5.</ref> Planck introduces his treatise with a brief account of heat and temperature and thermal equilibrium, and then announces: "In the following we shall deal chiefly with homogeneous, isotropic bodies of any form, possessing throughout their substance the same temperature and density, and subject to a uniform pressure acting everywhere perpendicular to the surface."<ref name="Planck 1903 3">[[Max Planck|Planck, M.]] (1897/1927), p.3.</ref> As did Carathéodory, Planck was setting aside surface effects and external fields and anisotropic crystals. Though referring to temperature, Planck did not there explicitly refer to the concept of thermodynamic equilibrium. In contrast, Carathéodory's scheme of presentation of classical thermodynamics for closed systems postulates the concept of an "equilibrium state" following Gibbs (Gibbs speaks routinely of a "thermodynamic state"), though not explicitly using the phrase 'thermodynamic equilibrium', nor explicitly postulating the existence of a temperature to define it.
Such equilibrium inhomogeneity, induced by external forces, does not occur for the intensive variable [[temperature]]. According to [[Edward A. Guggenheim|E.A. Guggenheim]], "The most important conception of thermodynamics is temperature."<ref>[[Edward A. Guggenheim|Guggenheim, E.A.]] (1949/1967), p.5.</ref> Planck introduces his treatise with a brief account of heat and temperature and thermal equilibrium, and then announces: "In the following we shall deal chiefly with homogeneous, isotropic bodies of any form, possessing throughout their substance the same temperature and density, and subject to a uniform pressure acting everywhere perpendicular to the surface."<ref name="Planck 1903 3">[[Max Planck|Planck, M.]] (1897/1927), p.3.</ref> As did Carathéodory, Planck was setting aside surface effects and external fields and anisotropic crystals. Though referring to temperature, Planck did not there explicitly refer to the concept of thermodynamic equilibrium. In contrast, Carathéodory's scheme of presentation of classical thermodynamics for closed systems postulates the concept of an "equilibrium state" following Gibbs (Gibbs speaks routinely of a "thermodynamic state"), though not explicitly using the phrase 'thermodynamic equilibrium', nor explicitly postulating the existence of a temperature to define it.
这种由外力引起的平衡不均匀性，在强烈变化的'''<font color="#ff8000">温度 Temperature</font>'''下不会发生。'''<font color="#ff8000">E.A.古根海姆 E.A. Guggenheim</font>'''认为，“热力学最重要的概念是温度。“Planck在介绍他的论文时，简要叙述了热、温度和热平衡，然后宣布: ”在下文中，我们将主要讨论任何形式的均匀、各向同性的物体，它们的物质具有相同的温度和密度，并受到到处垂直于表面的均匀压力的作用和Carathéodory 一样，Planck将表面效应、外场和各向异性晶体排除在外。虽然Planck提到了温度，但并没有明确提到热力学平衡的概念。相比之下，Carathéodory关于封闭系统的经典热力学演示方案假设了一个遵循 Gibbs 的“平衡态”的概念(Gibbs 经常提到一个“热力学状态”) ，虽然没有明确地使用短语‘热力学平衡’ ，也没有明确地假设存在一个温度来定义它。
The temperature within a system in thermodynamic equilibrium is uniform in space as well as in time. In a system in its own state of internal thermodynamic equilibrium, there are no net internal macroscopic flows. In particular, this means that all local parts of the system are in mutual radiative exchange equilibrium. This means that the temperature of the system is spatially uniform.<ref name="Planck 1914 40"/> This is so in all cases, including those of non-uniform external force fields. For an externally imposed gravitational field, this may be proved in macroscopic thermodynamic terms, by the calculus of variations, using the method of Langrangian multipliers.<ref>Gibbs, J.W. (1876/1878), pp. 144-150.</ref><ref>[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966), pp. 127–130.</ref><ref>Münster, A. (1970), pp. 309–310.</ref><ref>Bailyn, M. (1994), pp. 254-256.</ref><ref>{{cite journal | last1 = Verkley | first1 = W.T.M. | last2 = Gerkema | first2 = T. | year = 2004 | title = On maximum entropy profiles | journal = J. Atmos. Sci. | volume = 61 | issue = 8| pages = 931–936 | doi=10.1175/1520-0469(2004)061<0931:omep>2.0.co;2| bibcode = 2004JAtS...61..931V | doi-access = free }}</ref><ref>{{cite journal | last1 = Akmaev | first1 = R.A. | year = 2008 | title = On the energetics of maximum-entropy temperature profiles | url = | journal = Q. J. R. Meteorol. Soc. | volume = 134 | issue = 630| pages = 187–197 | doi=10.1002/qj.209| bibcode = 2008QJRMS.134..187A }}</ref> Considerations of kinetic theory or statistical mechanics also support this statement.<ref>Maxwell, J.C. (1867).</ref><ref>Boltzmann, L. (1896/1964), p. 143.</ref><ref>Chapman, S., Cowling, T.G. (1939/1970), Section 4.14, pp. 75–78.</ref><ref>[[J. R. Partington|Partington, J.R.]] (1949), pp. 275–278.</ref><ref>{{cite journal | last1 = Coombes | first1 = C.A. | last2 = Laue | first2 = H. | year = 1985 | title = A paradox concerning the temperature distribution of a gas in a gravitational field | url = | journal = Am. J. Phys. | volume = 53 | issue = 3| pages = 272–273 | doi=10.1119/1.14138| bibcode = 1985AmJPh..53..272C }}</ref><ref>{{cite journal | last1 = Román | first1 = F.L. | last2 = White | first2 = J.A. | last3 = Velasco | first3 = S. | year = 1995 | title = Microcanonical single-particle distributions for an ideal gas in a gravitational field | url = | journal = Eur. J. Phys. | volume = 16 | issue = 2| pages = 83–90 | doi=10.1088/0143-0807/16/2/008| bibcode = 1995EJPh...16...83R }}</ref><ref>{{cite journal | last1 = Velasco | first1 = S. | last2 = Román | first2 = F.L. | last3 = White | first3 = J.A. | year = 1996 | title = On a paradox concerning the temperature distribution of an ideal gas in a gravitational field | url = | journal = Eur. J. Phys. | volume = 17 | issue = | pages = 43–44 | doi=10.1088/0143-0807/17/1/008}}</ref>
The temperature within a system in thermodynamic equilibrium is uniform in space as well as in time. In a system in its own state of internal thermodynamic equilibrium, there are no net internal macroscopic flows. In particular, this means that all local parts of the system are in mutual radiative exchange equilibrium. This means that the temperature of the system is spatially uniform.<ref name="Planck 1914 40"/> This is so in all cases, including those of non-uniform external force fields. For an externally imposed gravitational field, this may be proved in macroscopic thermodynamic terms, by the calculus of variations, using the method of Langrangian multipliers.<ref>Gibbs, J.W. (1876/1878), pp. 144-150.</ref><ref>[[Dirk ter Haar|ter Haar, D.]], [[Harald Wergeland|Wergeland, H.]] (1966), pp. 127–130.</ref><ref>Münster, A. (1970), pp. 309–310.</ref><ref>Bailyn, M. (1994), pp. 254-256.</ref><ref>{{cite journal | last1 = Verkley | first1 = W.T.M. | last2 = Gerkema | first2 = T. | year = 2004 | title = On maximum entropy profiles | journal = J. Atmos. Sci. | volume = 61 | issue = 8| pages = 931–936 | doi=10.1175/1520-0469(2004)061<0931:omep>2.0.co;2| bibcode = 2004JAtS...61..931V | doi-access = free }}</ref><ref>{{cite journal | last1 = Akmaev | first1 = R.A. | year = 2008 | title = On the energetics of maximum-entropy temperature profiles | url = | journal = Q. J. R. Meteorol. Soc. | volume = 134 | issue = 630| pages = 187–197 | doi=10.1002/qj.209| bibcode = 2008QJRMS.134..187A }}</ref> Considerations of kinetic theory or statistical mechanics also support this statement.<ref>Maxwell, J.C. (1867).</ref><ref>Boltzmann, L. (1896/1964), p. 143.</ref><ref>Chapman, S., Cowling, T.G. (1939/1970), Section 4.14, pp. 75–78.</ref><ref>[[J. R. Partington|Partington, J.R.]] (1949), pp. 275–278.</ref><ref>{{cite journal | last1 = Coombes | first1 = C.A. | last2 = Laue | first2 = H. | year = 1985 | title = A paradox concerning the temperature distribution of a gas in a gravitational field | url = | journal = Am. J. Phys. | volume = 53 | issue = 3| pages = 272–273 | doi=10.1119/1.14138| bibcode = 1985AmJPh..53..272C }}</ref><ref>{{cite journal | last1 = Román | first1 = F.L. | last2 = White | first2 = J.A. | last3 = Velasco | first3 = S. | year = 1995 | title = Microcanonical single-particle distributions for an ideal gas in a gravitational field | url = | journal = Eur. J. Phys. | volume = 16 | issue = 2| pages = 83–90 | doi=10.1088/0143-0807/16/2/008| bibcode = 1995EJPh...16...83R }}</ref><ref>{{cite journal | last1 = Velasco | first1 = S. | last2 = Román | first2 = F.L. | last3 = White | first3 = J.A. | year = 1996 | title = On a paradox concerning the temperature distribution of an ideal gas in a gravitational field | url = | journal = Eur. J. Phys. | volume = 17 | issue = | pages = 43–44 | doi=10.1088/0143-0807/17/1/008}}</ref>
热力学平衡系统内的温度在时间和空间上都是均匀的。在一个处于内部热力学平衡的系统中，不存在净的内部宏观流动。特别是，这意味着系统的所有局部都处于相互辐射交换平衡。这意味着系统的温度在空间上是均匀的。这在所有情况下都是如此，包括那些非均匀外力场。对于外加的引力场，这可以用宏观热力学条件来证明，用变分法乘数法可以证明。对动力学理论或统计力学的考虑也支持这种说法。
In order that a system may be in its own internal state of thermodynamic equilibrium, it is of course necessary, but not sufficient, that it be in its own internal state of thermal equilibrium; it is possible for a system to reach internal mechanical equilibrium before it reaches internal thermal equilibrium.<ref name="Fitts 43"/>
In order that a system may be in its own internal state of thermodynamic equilibrium, it is of course necessary, but not sufficient, that it be in its own internal state of thermal equilibrium; it is possible for a system to reach internal mechanical equilibrium before it reaches internal thermal equilibrium.<ref name="Fitts 43"/>
为了使一个系统处于它自己的内部热力学平衡状态，它当然是必要的，但不是充分的，它必须处于它自己的内部热平衡状态; 一个系统在到达内部热平衡之前到达内部力学平衡是可能的。
As noted above, J.R. Partington points out that a state of thermodynamic equilibrium is stable against small transient perturbations. Without this condition, in general, experiments intended to study systems in thermodynamic equilibrium are in severe difficulties.
As noted above, J.R. Partington points out that a state of thermodynamic equilibrium is stable against small transient perturbations. Without this condition, in general, experiments intended to study systems in thermodynamic equilibrium are in severe difficulties.
===Number of real variables needed for specification===
===Number of real variables needed for specification===
规范所需的实变量数目
In his exposition of his scheme of closed system equilibrium thermodynamics, C. Carathéodory initially postulates that experiment reveals that a definite number of real variables define the states that are the points of the manifold of equilibria.<ref name="Caratheodory" /> In the words of Prigogine and Defay (1945): "It is a matter of experience that when we have specified a certain number of macroscopic properties of a system, then all the other properties are fixed."<ref>Prigogine, I., Defay, R. (1950/1954), p. 1.</ref><ref>Silbey, R.J., [[Robert A. Alberty|Alberty, R.A.]], Bawendi, M.G. (1955/2005), p. 4.</ref> As noted above, according to A. Münster, the number of variables needed to define a thermodynamic equilibrium is the least for any state of a given isolated system. As noted above, J.G. Kirkwood and I. Oppenheim point out that a state of thermodynamic equilibrium may be defined by a special subclass of intensive variables, with a definite number of members in that subclass.
在他关于封闭系统平衡态热力学方案的论述中，C.Carathéodory 最初假定实验揭示了一定数量的实变量定义了作为平衡态流形点的状态。用 Prigogine 和 Defay (1945)的话说: “这是一个经验问题，当我们确定了一个系统的一定数量的宏观属性，那么所有其他属性都是固定的。如上所述，A. Münster认为，对于给定孤立系统的任何状态来说，定义热力学平衡所需的变量数量是最少的。如上所述，J.G. Kirkwood 和 I. Oppenheim 指出，热力学平衡状态可以由一个特殊的子类的集约变量来定义，该子类中有一定数量的成员。
When a body of material starts from a non-equilibrium state of inhomogeneity or chemical non-equilibrium, and is then isolated, it spontaneously evolves towards its own internal state of thermodynamic equilibrium. It is not necessary that all aspects of internal thermodynamic equilibrium be reached simultaneously; some can be established before others. For example, in many cases of such evolution, internal mechanical equilibrium is established much more rapidly than the other aspects of the eventual thermodynamic equilibrium. Another example is that, in many cases of such evolution, thermal equilibrium is reached much more rapidly than chemical equilibrium.
When a body of material starts from a non-equilibrium state of inhomogeneity or chemical non-equilibrium, and is then isolated, it spontaneously evolves towards its own internal state of thermodynamic equilibrium. It is not necessary that all aspects of internal thermodynamic equilibrium be reached simultaneously; some can be established before others. For example, in many cases of such evolution, internal mechanical equilibrium is established much more rapidly than the other aspects of the eventual thermodynamic equilibrium. Another example is that, in many cases of such evolution, thermal equilibrium is reached much more rapidly than chemical equilibrium.
If the thermodynamic equilibrium lies in an external force field, it is only the temperature that can in general be expected to be spatially uniform. Intensive variables other than temperature will in general be non-uniform if the external force field is non-zero. In such a case, in general, additional variables are needed to describe the spatial non-uniformity.
If the thermodynamic equilibrium lies in an external force field, it is only the temperature that can in general be expected to be spatially uniform. Intensive variables other than temperature will in general be non-uniform if the external force field is non-zero. In such a case, in general, additional variables are needed to describe the spatial non-uniformity.
如果热力学平衡位于一个外力场中，那么通常只有温度在空间上是均匀的。如果外力场非零，温度以外的强度变量通常是不均匀的。在这种情况下，一般需要附加变量来描述空间非均匀性。
===Stability against small perturbations===
===Stability against small perturbations===
对小扰动的稳定性
In an isolated system, thermodynamic equilibrium by definition persists over an indefinitely long time. In classical physics it is often convenient to ignore the effects of measurement and this is assumed in the present account.
In an isolated system, thermodynamic equilibrium by definition persists over an indefinitely long time. In classical physics it is often convenient to ignore the effects of measurement and this is assumed in the present account.
As noted above, J.R. Partington points out that a state of thermodynamic equilibrium is stable against small transient perturbations. Without this condition, in general, experiments intended to study systems in thermodynamic equilibrium are in severe difficulties.
As noted above, J.R. Partington points out that a state of thermodynamic equilibrium is stable against small transient perturbations. Without this condition, in general, experiments intended to study systems in thermodynamic equilibrium are in severe difficulties.
如上所述，J.R. Partington 指出热力学平衡状态在小的瞬态扰动下是稳定的。如果没有这个条件，一般来说，在热力学平衡用于研究系统的实验就会遇到严重的困难。
===Approach to thermodynamic equilibrium within an isolated system===
===Approach to thermodynamic equilibrium within an isolated system===
孤立系统中的热力学平衡
When a body of material starts from a non-equilibrium state of inhomogeneity or chemical non-equilibrium, and is then isolated, it spontaneously evolves towards its own internal state of thermodynamic equilibrium. It is not necessary that all aspects of internal thermodynamic equilibrium be reached simultaneously; some can be established before others. For example, in many cases of such evolution, internal mechanical equilibrium is established much more rapidly than the other aspects of the eventual thermodynamic equilibrium.<ref name="Fitts 43">Fitts, D.D. (1962), p. 43.</ref> Another example is that, in many cases of such evolution, thermal equilibrium is reached much more rapidly than chemical equilibrium.<ref>Denbigh, K.G. (1951), p. 42.</ref>
When a body of material starts from a non-equilibrium state of inhomogeneity or chemical non-equilibrium, and is then isolated, it spontaneously evolves towards its own internal state of thermodynamic equilibrium. It is not necessary that all aspects of internal thermodynamic equilibrium be reached simultaneously; some can be established before others. For example, in many cases of such evolution, internal mechanical equilibrium is established much more rapidly than the other aspects of the eventual thermodynamic equilibrium.<ref name="Fitts 43">Fitts, D.D. (1962), p. 43.</ref> Another example is that, in many cases of such evolution, thermal equilibrium is reached much more rapidly than chemical equilibrium.<ref>Denbigh, K.G. (1951), p. 42.</ref>
当一个物质体从不均匀的非平衡状态或化学非平衡状态开始，然后被孤立，它自发地朝着自己的内部热力学平衡状态演化。没有必要同时达到内部热力学平衡的所有方面; 有些方面可以先于其他方面建立起来。例如，在这种演变的许多情况下，内部力学平衡的建立比最终热力学平衡的其他方面要快得多。另一个例子是，在这种演变的许多情况下，热平衡的发展要比化学平衡快得多
If the system is truly macroscopic as postulated by classical thermodynamics, then the fluctuations are too small to detect macroscopically. This is called the thermodynamic limit. In effect, the molecular nature of matter and the quantal nature of momentum transfer have vanished from sight, too small to see. According to Buchdahl: "... there is no place within the strictly phenomenological theory for the idea of fluctuations about equilibrium (see, however, Section 76)."
If the system is truly macroscopic as postulated by classical thermodynamics, then the fluctuations are too small to detect macroscopically. This is called the thermodynamic limit. In effect, the molecular nature of matter and the quantal nature of momentum transfer have vanished from sight, too small to see. According to Buchdahl: "... there is no place within the strictly phenomenological theory for the idea of fluctuations about equilibrium (see, however, Section 76)."
===Fluctuations within an isolated system in its own internal thermodynamic equilibrium===
===Fluctuations within an isolated system in its own internal thermodynamic equilibrium===
孤立系统内部热力学平衡的波动
In an isolated system, thermodynamic equilibrium by definition persists over an indefinitely long time. In classical physics it is often convenient to ignore the effects of measurement and this is assumed in the present account.
In an isolated system, thermodynamic equilibrium by definition persists over an indefinitely long time. In classical physics it is often convenient to ignore the effects of measurement and this is assumed in the present account.
在一个孤立的系统中，根据定义，热力学平衡可以持续无限长的时间。在经典物理学中，忽略测量的影响通常是很方便的，现在我们假设这一点。
If the mesoscopic system is further repeatedly divided, eventually a microscopic system is produced. Then the molecular character of matter and the quantal nature of momentum transfer become important in the processes of fluctuation. One has left the realm of classical or macroscopic thermodynamics, and one needs quantum statistical mechanics. The fluctuations can become relatively dominant, and questions of measurement become important.
If the mesoscopic system is further repeatedly divided, eventually a microscopic system is produced. Then the molecular character of matter and the quantal nature of momentum transfer become important in the processes of fluctuation. One has left the realm of classical or macroscopic thermodynamics, and one needs quantum statistical mechanics. The fluctuations can become relatively dominant, and questions of measurement become important.
如果介观系统进一步重复分裂，最终产生一个微观系统。物质的分子性质和动量传递的量子性质在涨落过程中起着重要作用。这已经离开了经典热力学或宏观热力学的领域，即需要量子统计力学。波动可以变得相对占主导地位，测量问题变得重要。
To consider the notion of fluctuations in an isolated thermodynamic system, a convenient example is a system specified by its extensive state variables, internal energy, volume, and mass composition. By definition they are time-invariant. By definition, they combine with time-invariant nominal values of their conjugate intensive functions of state, inverse temperature, pressure divided by temperature, and the chemical potentials divided by temperature, so as to exactly obey the laws of thermodynamics.<ref>Tschoegl, N.W. (2000). ''Fundamentals of Equilibrium and Steady-State Thermodynamics'', Elsevier, Amsterdam, {{ISBN|0-444-50426-5}}, p. 21.</ref> But the laws of thermodynamics, combined with the values of the specifying extensive variables of state, are not sufficient to provide knowledge of those nominal values. Further information is needed, namely, of the constitutive properties of the system.
To consider the notion of fluctuations in an isolated thermodynamic system, a convenient example is a system specified by its extensive state variables, internal energy, volume, and mass composition. By definition they are time-invariant. By definition, they combine with time-invariant nominal values of their conjugate intensive functions of state, inverse temperature, pressure divided by temperature, and the chemical potentials divided by temperature, so as to exactly obey the laws of thermodynamics.<ref>Tschoegl, N.W. (2000). ''Fundamentals of Equilibrium and Steady-State Thermodynamics'', Elsevier, Amsterdam, {{ISBN|0-444-50426-5}}, p. 21.</ref> But the laws of thermodynamics, combined with the values of the specifying extensive variables of state, are not sufficient to provide knowledge of those nominal values. Further information is needed, namely, of the constitutive properties of the system.
考虑孤立热力学系统中的涨落概念，一个方便的例子是由其广泛的状态变量、内能、体积和质量组成指定的系统。根据定义，它们是时不变的。根据定义，它们与它们的共轭强度函数的时不变名义值相结合，即状态函数、反温度函数、压力除以温度函数、化学势除以温度函数，从而精确服从热力学定律。但是热力学定律，加上指定广泛的状态变量的值，不足以提供这些名义值的知识。需要进一步的信息，即系统的构成性质。
It may be admitted that on repeated measurement of those conjugate intensive functions of state, they are found to have slightly different values from time to time. Such variability is regarded as due to internal fluctuations. The different measured values average to their nominal values.
It may be admitted that on repeated measurement of those conjugate intensive functions of state, they are found to have slightly different values from time to time. Such variability is regarded as due to internal fluctuations. The different measured values average to their nominal values.
可以承认，在重复测量这些共轭密集态函数时，发现它们的值随时间略有不同。这种可变性被认为是由于内部波动。不同测量值与其名义值的平均值。
If the system is truly macroscopic as postulated by classical thermodynamics, then the fluctuations are too small to detect macroscopically. This is called the thermodynamic limit. In effect, the molecular nature of matter and the quantal nature of momentum transfer have vanished from sight, too small to see. According to Buchdahl: "... there is no place within the strictly phenomenological theory for the idea of fluctuations about equilibrium (see, however, Section 76)."<ref>Buchdahl, H.A. (1966), p. 16.</ref>
If the system is truly macroscopic as postulated by classical thermodynamics, then the fluctuations are too small to detect macroscopically. This is called the thermodynamic limit. In effect, the molecular nature of matter and the quantal nature of momentum transfer have vanished from sight, too small to see. According to Buchdahl: "... there is no place within the strictly phenomenological theory for the idea of fluctuations about equilibrium (see, however, Section 76)."<ref>Buchdahl, H.A. (1966), p. 16.</ref>
如果这个系统真的像经典热力学所假定的那样是宏观的，那么这个系统的涨落太小了，宏观上无法检测到。这就是所谓的热力学极限。实际上，物质的分子性质和动量转移的量子性质已经从我们的视线中消失，因为它们太小而看不见。根据Buchdahl: “ ... ... 在严格的现象学理论中，平衡的涨落概念是没有位置的
If the system is repeatedly subdivided, eventually a system is produced that is small enough to exhibit obvious fluctuations. This is a mesoscopic level of investigation. The fluctuations are then directly dependent on the natures of the various walls of the system. The precise choice of independent state variables is then important. At this stage, statistical features of the laws of thermodynamics become apparent.
If the system is repeatedly subdivided, eventually a system is produced that is small enough to exhibit obvious fluctuations. This is a mesoscopic level of investigation. The fluctuations are then directly dependent on the natures of the various walls of the system. The precise choice of independent state variables is then important. At this stage, statistical features of the laws of thermodynamics become apparent.
如果系统被重复细分，最终会产生一个足够小以显示明显波动的系统。这是一个介观层面的研究。波动则直接取决于系统各墙的性质。因此，精确地选择独立状态变量是很重要的。在这个阶段，热力学定律的统计特征变得明显。
An explicit distinction between 'thermal equilibrium' and 'thermodynamic equilibrium' is made by B. C. Eu. He considers two systems in thermal contact, one a thermometer, the other a system in which there are occurring several irreversible processes, entailing non-zero fluxes; the two systems are separated by a wall permeable only to heat. He considers the case in which, over the time scale of interest, it happens that both the thermometer reading and the irreversible processes are steady. Then there is thermal equilibrium without thermodynamic equilibrium. Eu proposes consequently that the zeroth law of thermodynamics can be considered to apply even when thermodynamic equilibrium is not present; also he proposes that if changes are occurring so fast that a steady temperature cannot be defined, then "it is no longer possible to describe the process by means of a thermodynamic formalism. In other words, thermodynamics has no meaning for such a process." This illustrates the importance for thermodynamics of the concept of temperature.
An explicit distinction between 'thermal equilibrium' and 'thermodynamic equilibrium' is made by B. C. Eu. He considers two systems in thermal contact, one a thermometer, the other a system in which there are occurring several irreversible processes, entailing non-zero fluxes; the two systems are separated by a wall permeable only to heat. He considers the case in which, over the time scale of interest, it happens that both the thermometer reading and the irreversible processes are steady. Then there is thermal equilibrium without thermodynamic equilibrium. Eu proposes consequently that the zeroth law of thermodynamics can be considered to apply even when thermodynamic equilibrium is not present; also he proposes that if changes are occurring so fast that a steady temperature cannot be defined, then "it is no longer possible to describe the process by means of a thermodynamic formalism. In other words, thermodynamics has no meaning for such a process." This illustrates the importance for thermodynamics of the concept of temperature.
热平衡和热力学平衡之间的明确区分是由 B.c. Eu 提出的。他认为两个系统在热接触，一个是温度计，另一个是一个系统，其中有几个不可逆过程，产生非零通量; 这两个系统被一个只透热的壁隔开。他考虑了这样一种情况，在有兴趣的时间尺度上，温度计读数和不可逆过程都是稳定的。然后是没有热平衡的热力学平衡。因此，欧盟提出，即使在没有热力学第零定律的情况下，也可以考虑应用热力学平衡; 他还提出，如果变化发生得太快，以至于无法确定一个稳定的温度，那么“用热力学形式主义来描述这一过程就不再可能了。换句话说，热力学对这样一个过程没有意义。”这说明了温度概念对热力学的重要性。
热平衡和热力学平衡之间的明确区分是由 B.C.Eu 提出的。他认为两个系统在热接触，一个是温度计，另一个是一个系统，其中有几个不可逆过程，产生非零通量; 这两个系统被一个只透热的壁隔开。他考虑了这样一种情况，在有兴趣的时间尺度上，温度计读数和不可逆过程都是稳定的。然后是没有热平衡的热力学平衡。因此，欧盟提出，即使在没有热力学第零定律的情况下，也可以考虑应用热力学平衡; 他还提出，如果变化发生得太快，以至于无法确定一个稳定的温度，那么“用热力学形式主义来描述这一过程就不再可能了。换句话说，热力学对这样一个过程没有意义。”这说明了温度概念对热力学的重要性。
If the mesoscopic system is further repeatedly divided, eventually a microscopic system is produced. Then the molecular character of matter and the quantal nature of momentum transfer become important in the processes of fluctuation. One has left the realm of classical or macroscopic thermodynamics, and one needs quantum statistical mechanics. The fluctuations can become relatively dominant, and questions of measurement become important.
If the mesoscopic system is further repeatedly divided, eventually a microscopic system is produced. Then the molecular character of matter and the quantal nature of momentum transfer become important in the processes of fluctuation. One has left the realm of classical or macroscopic thermodynamics, and one needs quantum statistical mechanics. The fluctuations can become relatively dominant, and questions of measurement become important.
如果介观系统进一步重复分裂，最终产生一个微观系统。物质的分子性质和动量传递的量子性质在涨落过程中起着重要作用。一个人已经离开了经典热力学或宏观热力学的领域，他需要量子统计力学。波动可以变得相对占主导地位，测量问题变得重要。
Thermal equilibrium is achieved when two systems in thermal contact with each other cease to have a net exchange of energy. It follows that if two systems are in thermal equilibrium, then their temperatures are the same.
Thermal equilibrium is achieved when two systems in thermal contact with each other cease to have a net exchange of energy. It follows that if two systems are in thermal equilibrium, then their temperatures are the same.
The statement that 'the system is its own internal thermodynamic equilibrium' may be taken to mean that 'indefinitely many such measurements have been taken from time to time, with no trend in time in the various measured values'. Thus the statement, that 'a system is in its own internal thermodynamic equilibrium, with stated nominal values of its functions of state conjugate to its specifying state variables', is far far more informative than a statement that 'a set of single simultaneous measurements of those functions of state have those same values'. This is because the single measurements might have been made during a slight fluctuation, away from another set of nominal values of those conjugate intensive functions of state, that is due to unknown and different constitutive properties. A single measurement cannot tell whether that might be so, unless there is also knowledge of the nominal values that belong to the equilibrium state.
The statement that 'the system is its own internal thermodynamic equilibrium' may be taken to mean that 'indefinitely many such measurements have been taken from time to time, with no trend in time in the various measured values'. Thus the statement, that 'a system is in its own internal thermodynamic equilibrium, with stated nominal values of its functions of state conjugate to its specifying state variables', is far far more informative than a statement that 'a set of single simultaneous measurements of those functions of state have those same values'. This is because the single measurements might have been made during a slight fluctuation, away from another set of nominal values of those conjugate intensive functions of state, that is due to unknown and different constitutive properties. A single measurement cannot tell whether that might be so, unless there is also knowledge of the nominal values that belong to the equilibrium state.
系统是它自己的内部热力学平衡的说法可能被理解为无限期地多次这样的测量已经不时地被采取，在不同的测量值中没有时间的趋势。因此，一个系统处于它自己的内部热力学平衡，它的状态变量与它的状态变量共轭的函数的标称值相对应，这种说法远比一个状态函数的一组单一的同时测量值具有相同的值的说法信息量大得多。这是因为单次测量可能是由于未知和不同的组成性质，在轻微的波动中，远离那些共轭的状态密集函数的另一组名义值。一个单一的测量不能告诉是否可能是这样，除非还有属于平衡状态的名义值的知识。
Thermal equilibrium occurs when a system's macroscopic thermal observables have ceased to change with time. For example, an ideal gas whose distribution function has stabilised to a specific Maxwell–Boltzmann distribution would be in thermal equilibrium. This outcome allows a single temperature and pressure to be attributed to the whole system. For an isolated body, it is quite possible for mechanical equilibrium to be reached before thermal equilibrium is reached, but eventually, all aspects of equilibrium, including thermal equilibrium, are necessary for thermodynamic equilibrium.
Thermal equilibrium occurs when a system's macroscopic thermal observables have ceased to change with time. For example, an ideal gas whose distribution function has stabilised to a specific Maxwell–Boltzmann distribution would be in thermal equilibrium. This outcome allows a single temperature and pressure to be attributed to the whole system. For an isolated body, it is quite possible for mechanical equilibrium to be reached before thermal equilibrium is reached, but eventually, all aspects of equilibrium, including thermal equilibrium, are necessary for thermodynamic equilibrium.
=== Thermal equilibrium ===
=== Thermal equilibrium ===
热平衡
{{Main|Thermal equilibrium}}
{{Main|Thermal equilibrium}}
管理系统的定律远离平衡也是有争议的。这些系统的指导原则之一就是最大产生熵原则。它指出，非平衡系统的演化，如最大化其产生熵。
管理系统的定律远离平衡也是有争议的。这些系统的指导原则之一就是最大产生熵原则。它指出，非平衡系统的演化，如最大化其产生熵。
==Non-equilibrium==
==Non-equilibrium==