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→Fluctuations within an isolated system in its own internal thermodynamic equilibrium
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，Eu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波动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。
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。
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.
当系统的宏观热观测值不再随着时间变化时，就会出现热平衡。例如，一种分布函数稳定到一个特定的麦克斯韦-波兹曼分布的理想气体即处于热平衡状态。这个结果可以将单一的温度和压力归因于整个系统。对于一个孤立的物体来说，在达到热平衡之前达到机械平衡是很有可能的，但是最终，所有方面的平衡，包括热平衡，对于热力学平衡来说都是必要的。
=== Thermal equilibrium ===
=== Thermal equilibrium ===