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第562行： − 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。+ − 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。+ + − 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+ + − 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。+ +
→Fluctuations within an isolated system in its own internal thermodynamic equilibrium
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"的说法可能意味着"无限期地，许多这样的测量是不时进行的，在各种测量值中没有时间趋势"。因此，一个系统处于它自己的内部热力学平衡，它的状态变量与它状态变量共轭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”的说法丰富得多。这是因为单个测量可能是在轻微波动期间进行的，而不是由于未知和不同的构成属性而导致的，即远离那些共轭的状态密集函数的另一组名义值。非已知属于平衡状态的标称值，否则根据单一的度量无法进行判断。
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。
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.
如果系统被重复细分，最终产生的系统足够小，可以表现出明显的波动。这是一个介观层面的研究。波动则直接取决于系统各壁的性质。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.