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* the members of the ensemble can be understood as the states of the systems in experiments repeated on independent systems which have been prepared in a similar but imperfectly controlled manner ([[empirical probability]]), in the limit of an infinite number of trials.

* the members of the ensemble can be understood as the states of the systems in experiments repeated on independent systems which have been prepared in a similar but imperfectly controlled manner ([[empirical probability]]), in the limit of an infinite number of trials.

* 系综可以表示"单个系统"的所有可能状态([[epistemic probability]], a form of knowledge)，或者

* 系综可以表示"单个系统"的所有可能状态<font color="#32CD32">([[epistemic probability]], a form of knowledge)</font>，或者

* 系综的元素可以理解为在无限次试验的极限下，在类似但不完全受控的独立系统中，重复进行实验得到的系统的状态（[经验概率]]）。

* 系综的元素可以理解为在无限次试验的极限下，在类似但不完全受控的独立系统中，重复进行实验得到的系统的状态（<font color="#FF8000">经验概率 empirical probability</font>）。

These two meanings are equivalent for many purposes, and will be used interchangeably in this article.

These two meanings are equivalent for many purposes, and will be used interchangeably in this article.

However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the Liouville equation (classical mechanics) or the von Neumann equation (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state.

However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the Liouville equation (classical mechanics) or the von Neumann equation (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state.

然而，这种概率是被解释的，系综中的每个状态随时间的演化都可以由运动方程给出。因此，系综本身(状态的概率分布概率分布)也在随时间演化，因为系综中的虚拟系统不断地离开一个状态进入另一个状态。系综演化由<font color="#FFD700">刘维尔方程</font>(经典力学)或<font color="#FFD700">冯·诺依曼方程</font>(量子力学)给出。这些方程是简单地通过分别应用力学运动方程到系综中的每个虚拟系统而导出的，虚拟系统随时间演化过程中概率是守恒的。

然而，这种概率是被解释的，系综中的每个状态随时间的演化都可以由运动方程给出。因此，系综本身(状态的概率分布概率分布)也在随时间演化，因为系综中的虚拟系统不断地离开一个状态进入另一个状态。系综演化由<font color="#FF8000">刘维尔方程 Liouville equation</font>(经典力学)或<font color="#FF8000">冯·诺依曼方程 von Neumann equation</font>(量子力学)给出。这些方程是简单地通过分别应用力学运动方程到系综中的每个虚拟系统而导出的，虚拟系统随时间演化过程中概率是守恒的。

One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as equilibrium ensembles and their condition is known as statistical equilibrium. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state. The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems.

One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as equilibrium ensembles and their condition is known as statistical equilibrium. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state. The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems.

系综的一种特殊情况是不随时间演化的系综。这样的系综称为<font color="#FFD700">平衡系综</font>，它们的状态称为<font color="#FFD700">统计平衡</font>。如果对于系综中的每个状态，系综也包含其所有的未来和过去的状态，并且其概率等于处于该状态的概率，则出现统计平衡。孤立系统的平衡系综是统计热力学研究的重点。非平衡统计力学研究更一般的情况下的可以随时间演化的系综，以及（或）非孤立系统的系综。

系综的一种特殊情况是不随时间演化的系综。这样的系综称为<font color="#FF8000">平衡系综 equilibrium ensembles</font>，它们的状态称为<font color="#FF8000">统计平衡 statistical equilibrium</font>。如果对于系综中的每个状态，系综也包含其所有的未来和过去的状态，并且其概率等于处于该状态的概率，则出现统计平衡。孤立系统的平衡系综是统计热力学研究的重点。非平衡统计力学研究更一般的情况下的可以随时间演化的系综，以及（或）非孤立系统的系综。

== Statistical thermodynamics ==

== Statistical thermodynamics ==