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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.

对于任何有限体积的<font color="#FF8000">孤立系统 isolated system</font>，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。

对于任何有限体积的<font color="#FF8000">孤立系统 Isolated System</font>，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]]

Microcanonical ensemble

Microcanonical ensemble

<font color="#FF8000">微正则系综 Microcanonical ensemble</font>

<font color="#FF8000">微正则系综 Microcanonical ensemble</font>

: 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.

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。微正则系综中，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]]

Canonical ensemble

Canonical ensemble

<font color="#FF8000">正则系综 Canonical ensemble</font>

<font color="#FF8000">正则系综 Canonical ensemble</font>

: describes a system of fixed composition that is in [[thermal equilibrium]]{{NoteTag|The transitive thermal equilibrium (as in, "X is thermal equilibrium with Y") used here means that the ensemble for the first system is not perturbed when the system is allowed to weakly interact with the second system.}} with a [[heat bath]] of a precise [[thermodynamic temperature|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]]{{NoteTag|The transitive thermal equilibrium (as in, "X is thermal equilibrium with Y") used here means that the ensemble for the first system is not perturbed when the system is allowed to weakly interact with the second system.}} with a [[heat bath]] of a precise [[thermodynamic temperature|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.

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。

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]]

Grand canonical ensemble

Grand canonical ensemble

<font color="#FF8000">巨正则系综 Grand canonical ensemble</font>

<font color="#FF8000">巨正则系综 Grand canonical ensemble</font>

: 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 potential]]s 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 potential]]s 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.

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，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<font color="#FF8000">化学势 chemical potential</font>。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。

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<font color="#FF8000">化学势 Chemical Potential</font>。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.

对于包含大量粒子的系统(<font color="#FF8000">热力学极限 thermodynamic limit</font>) ，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。

对于包含大量粒子的系统(<font color="#FF8000">热力学极限 Thermodynamic Limit</font>) ，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问题。关于系综等价的吉布斯定理被发展成为(<font color="#FF8000">测度现象集中理论 Concentration of Measure Phenomenon</font>，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。

* 微观系统

* 微观系统

* 相变处的宏观系统

* 处于相变的宏观系统

* 长程关联的宏观系统

* 长程关联的宏观系统

Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.

Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.

一旦计算出一个系统的特征状态函数，该系统就“解决”了(宏观观测量可以从特征状态函数中提取)。然而，计算热力学系综的特征状态函数并不一定是一项简单的工作，因为它涉及到考虑系统的每一种可能状态。虽然一些假设的系统已经被完全求解了，但是最一般的(和现实的)情况对于一个精确的解来说太复杂了。存在各种方法来近似真实的系综，并且计算平均量。

一旦计算出一个系统的特征状态函数，该系统就被“解决”了(宏观观测量可以从特征状态函数中提取)。然而，计算热力学系综的特征状态函数并不一定是一项简单的工作，因为它涉及到考虑系统的每一种可能状态。虽然一些假设的系统已经被完全求解了，但是最一般的(和现实的)情况对于一个精确的解来说太复杂了。存在各种方法来近似真实的系综，并且计算平均量。