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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.<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。

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">强度量 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。

 

 

 

 

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。

这种由外力引起的平衡不均匀性，对于强度量'''<font color="#ff8000">温度 Temperature</font>'''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。

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.

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，它必须处于它自己的内部热平衡状态是必要不充分的; 一个系统在到达内部热平衡之前到达内部机械平衡是可能的。

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。

 

 

===Number of real variables needed for specification===

===Number of real variables needed for specification===