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→Fluctuations within an isolated system in its own internal thermodynamic equilibrium
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。
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.
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。
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.
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。
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.<ref>Tschoegl, N.W. (2000). ''Fundamentals of Equilibrium and Steady-State Thermodynamics'', Elsevier, Amsterdam, {{ISBN|0-444-50426-5}}, p. 21.</ref> 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.
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.<ref>Tschoegl, N.W. (2000). ''Fundamentals of Equilibrium and Steady-State Thermodynamics'', Elsevier, Amsterdam, {{ISBN|0-444-50426-5}}, p. 21.</ref> 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.
考虑隔离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。
当一个系统的宏观热观测量不再随时间变化时，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将在热平衡。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，在达到力学平衡之前达到热平衡是很有可能的，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。
当一个系统的宏观热观测量不再随时间变化时，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将在热平衡。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，在达到力学平衡之前达到热平衡是很有可能的，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 ===