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第457行： − + − 规范所需的实变量数目+ − + − − 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. − − 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，它自发地演化到自己的内部热力学平衡状态。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，内部机械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。 第475行：
第471行： − + − 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. − 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。 第487行：
第480行： − 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. 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波动概念，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，它们与它们的共轭状态密集函数的时不变名义值相结合，反向温度，压力除以温度，化学势除以温度，以便准确地服从热力学定律。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。我们需要进一步的信息，即关于该系统的构成特性的信息。 − − ===Approach to thermodynamic equilibrium within an isolated system=== − 孤立系统中的热力学平衡 + − 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强度函数时，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。不同测量值平均到其名义值。 − 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，它自发地演化到自己的内部热力学平衡状态。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，内部机械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)." − − 如果这个系统真的像经典热力学所假定的那样是宏观的，那么这个系统的波动太小了，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由于它们太小而看不见，已经从我们的视线中消失。根据Buchdahl: “ ... 在严格的现象学理论中，平衡的波动概念是没有位置的。” 第519行：
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→Characteristics of a state of internal thermodynamic equilibrium 内部热力学平衡状态的特征
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。
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 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.
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一定数量的宏观属性时，那么所有其他属性都是固定的。如上所述，A. Münster认为，定义热力学平衡所需的变量数量对于给定孤立系统的任何状态来说都是最少的。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子类的强度变量来定义，该子类中有一定数量的成员。
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一定数量的宏观属性时，那么所有其他属性都是固定的”。如上所述，A. Münster认为，定义热力学平衡所需的变量数量相对于给定孤立系统的任何状态来说都是最少的。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子类的强度变量来定义，该子类中有一定数量的成员。
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。
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===
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.
===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. 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.<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，它会自发地演化到自己的内部热力学平衡状态。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，在这种演化的许多情况下，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，在这种演化的许多情况下，热平衡的发展要比化学平衡快得多。
===Fluctuations within an isolated system in its own internal thermodynamic equilibrium===
===Fluctuations within an isolated system in its own internal thermodynamic equilibrium===
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。
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。
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.
考虑孤立热力学系统中的波动概念，一个方便的例子是由其众多的状态变量，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，它们与它们的共轭状态密集函数的时不变名义值相结合，反向温度，压力除以温度，化学势除以温度，以便准确地服从热力学定律。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。我们需要进一步的信息，即关于该系统的构成特性的信息。
考虑孤立热力学系统中的波动概念，一个方便的例子是由其众多的状态变量，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，它们与它们的共轭状态密集函数的时不变名义值相结合，反向温度，压力除以温度，化学势除以温度，以便准确地服从热力学定律。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。我们需要进一步的信息，即关于该系统的构成特性的信息。
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. 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波动概念，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，它们与它们的共轭状态密集函数的时不变名义值相结合，反向温度，压力除以温度，化学势除以温度，以便准确地服从热力学定律。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。我们需要进一步的信息，即关于该系统的构成特性的信息。
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>
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)."
如果这个系统真的像经典热力学所假定的那样是宏观的，那么这个系统的波动太小了，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由于它们太小而看不见，已经从我们的视线中消失。根据Buchdahl: “ ... 在严格的现象学理论中，平衡的波动概念是没有位置的。”
如果这个系统真的像经典热力学所假定的那样是宏观的，那么这个系统的波动太小了，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由于它们太小而看不见，已经从我们的视线中消失。根据Buchdahl: “ ... 在严格的现象学理论中，平衡的波动概念是没有位置的。”
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.