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A thermodynamic system consisting of a single phase in the absence of external forces, in its own internal thermodynamic equilibrium, is homogeneous. 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." 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.

A thermodynamic system consisting of a single phase in the absence of external forces, in its own internal thermodynamic equilibrium, is homogeneous. 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." 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.

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。普朗克在论文中简要介绍了热量、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，它们在整个物质中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。虽然普朗克提到了温度，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。

在内部热力学平衡中，没有外力的情况下由单相组成的热力学系统是均匀的。Planck在论文中简要介绍了热量、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，它们在整个物质中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。

===Homogeneity in the absence of external forces===

===Homogeneity in the absence of external forces===

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.

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.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。

为了使一个系统处于它自己的内部热力学平衡状态，它处于自己热平衡的内部状态当然是必要的，但不是充分的;系统在达到内部热平衡之前，可以达到内部机械平衡。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.

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>

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。

为了使一个系统处于它自己的内部热力学平衡状态，它必须处于它自己的内部热平衡状态当然是必要不充分的; 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.

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21热力学平衡用于研究系统的实验就会遇到严重的困难。

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，研究热力学平衡系统的实验就会遇到巨大的困难。