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There are three equilibrium ensembles with a simple form that can be defined for any isolated system bounded inside a finite volume. These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.

There are three equilibrium ensembles with a simple form that can be defined for any isolated system bounded inside a finite volume. These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.

有三个简单形式的平衡系综，可以定义为任何有限体积内的孤立系统。这些是统计热力学中最经常讨论的集合。在宏观极限(定义如下) ，它们都对应于经典热力学。

对于任何有限体积的孤立系统，可以定义三种简单形式的平衡系综。这些是统计热力学中最经常讨论的系综。在宏观极限(定义如下) ，它们都与经典热力学有对应。

; [[Microcanonical ensemble]]

; [[Microcanonical ensemble]]

  describes a system with a precisely given energy and fixed composition (precise number of particles). The microcanonical ensemble contains with equal probability each possible state that is consistent with that energy and composition.

  describes a system with a precisely given energy and fixed composition (precise number of particles). The microcanonical ensemble contains with equal probability each possible state that is consistent with that energy and composition.

描述了一个具有精确给定的能量和固定成分(精确数量的粒子)的系统。微正则系综包含与能量和组成相一致的每个可能状态的概率是相等的。

描述了一个具有精确给定的能量和固定成分(精确数量的粒子)的系统。微正则系综中，与能量和组成相一致的每个可能状态的概率是相等的。

; [[Canonical ensemble]]

; [[Canonical ensemble]]

  describes a system of fixed composition that is in thermal equilibrium with a heat bath of a precise temperature. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy.

  describes a system of fixed composition that is in thermal equilibrium with a heat bath of a precise temperature. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy.

描述了一个固定成分的系统，这个系统的热平衡是一个精确温度的热浴。正则系综包含能量不同但组成完全相同的状态; 根据总能量的不同，总体中不同的状态被赋予不同的概率。

描述了一个固定成分的系统，这个系统与一个精确温度的热浴形成热平衡。正则系综包含能量不同但组成完全相同的状态; 根据总能量的不同，系综中不同的状态被赋予不同的概率。

; [[Grand canonical ensemble]]

; [[Grand canonical ensemble]]

  describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise chemical potentials for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers.

  describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise chemical potentials for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers.

描述了一个具有非固定成分(不确定粒子数)的系统，这个系统处于热力和化学平衡中，有一个热力储存器。储层具有精确的温度，对各种类型的颗粒具有精确的化学势。巨正则系综包含不同能量和粒子数量的状态; 根据粒子总能量和粒子数量的不同，系综中不同状态的概率也不同。

描述了一个具有非固定成分(不确定粒子数)的系统，在热库中处于热力学和化学平衡。热库具有精确的温度，对各种类型的粒子具有精确的化学势。巨正则系综包含不同能量和粒子数的状态; 根据总能量和粒子数的不同，系综中不同状态的概率也不同。

For systems containing many particles (the thermodynamic limit), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used. The Gibbs theorem about equivalence of ensembles was developed into the theory of concentration of measure phenomenon, which has applications in many areas of science, from functional analysis to methods of artificial intelligence and big data technology.

For systems containing many particles (the thermodynamic limit), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used. The Gibbs theorem about equivalence of ensembles was developed into the theory of concentration of measure phenomenon, which has applications in many areas of science, from functional analysis to methods of artificial intelligence and big data technology.

对于包含许多粒子的系统(热力学极限) ，上面列出的所有3个系综都倾向于给出相同的行为。因此，使用集合只是一个简单的数学方便问题。集合等价吉布斯定理被发展成为测度现象集中理论，在从函数分析到人工智能和大数据技术等许多科学领域都有广泛的应用。

对于包含大量粒子的系统(热力学极限) ，上面列出的三种系综都倾向于给出相同的行为。因此，使用哪种系综只是一个简单的数学方便问题。关于系综等价的吉布斯定理被发展成为测度现象集中理论，在从函数分析到人工智能和大数据技术等许多科学领域都有广泛的应用。

Important cases where the thermodynamic ensembles do not give identical results include:

Important cases where the thermodynamic ensembles do not give identical results include:

热力学系数不能给出相同结果的重要情况包括:

热力学系综不能给出相同结果的重要情况包括:

* Microscopic systems.

* Microscopic systems.

* Large systems with long-range interactions.

* Large systems with long-range interactions.

In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.<ref name="tolman" />

In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.<ref name="tolman" />

In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.

In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.

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=== Calculation methods ===

=== Calculation methods ===