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The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the classical thermodynamics of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in thermodynamic equilibrium, and the microscopic behaviours and motions occurring inside the material.

The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the classical thermodynamics of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in thermodynamic equilibrium, and the microscopic behaviours and motions occurring inside the material.

统计热力学(也称为平衡态统计力学)的主要目标是根据组成某材料的粒子的性质和它们之间的相互作用，推导出材料的经典热力学。换句话说，统计热力学提供了热力学平衡态中物质的宏观性质与物质内部微观行为和运动之间的联系。

统计热力学(也称为平衡态统计力学)的主要目标是根据组成某材料的粒子的性质和它们之间的相互作用，推导出材料的<font color="#FF8000">经典热力学 classical thermodynamics</font>。换句话说，统计热力学提供了热力学平衡态中物质的宏观性质与物质内部微观行为和运动之间的联系。

=== Fundamental postulate ===

=== Fundamental postulate 基本假设===

 

A [[sufficient condition|sufficient]] (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.).<ref name="gibbs" />

A [[sufficient condition|sufficient]] (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.).<ref name="gibbs" />

A common approach found in many textbooks is to take the equal a priori probability postulate. This postulate states that

A common approach found in many textbooks is to take the equal a priori probability postulate. This postulate states that

: ''For an isolated system with an exactly known energy and exactly known composition, the system can be found with ''equal probability'' in any [[microstate (statistical mechanics)|microstate]] consistent with that knowledge.''

: ''For an isolated system with an exactly known energy and exactly known composition, the system can be found with ''equal probability'' in any [[microstate (statistical mechanics)|microstate]] consistent with that knowledge.''

The equal a priori probability postulate therefore provides a motivation for the microcanonical ensemble described below. There are various arguments in favour of the equal a priori probability postulate:

The equal a priori probability postulate therefore provides a motivation for the microcanonical ensemble described below. There are various arguments in favour of the equal a priori probability postulate:

因此，先验概率相等假设为下面描述的微正则系综提供了一个动机。有各种各样的论据支持先验概率相等假设:

因此，先验概率相等假设为下面描述的<font color="#FF8000">微正则系综 microcanonical ensemble</font>提供了一个动机。有各种各样的论据支持先验概率相等假设:

* [[Ergodic hypothesis]]: An ergodic system is one that evolves over time to explore "all accessible" states: all those with the same energy and composition. In an ergodic system, the microcanonical ensemble is the only possible equilibrium ensemble with fixed energy. This approach has limited applicability, since most systems are not ergodic.

* [[Ergodic hypothesis]]: An ergodic system is one that evolves over time to explore "all accessible" states: all those with the same energy and composition. In an ergodic system, the microcanonical ensemble is the only possible equilibrium ensemble with fixed energy. This approach has limited applicability, since most systems are not ergodic.

* [[Maximum entropy thermodynamics|Maximum information entropy]]: A more elaborate version of the principle of indifference states that the correct ensemble is the ensemble that is compatible with the known information and that has the largest [[Gibbs entropy]] ([[information entropy]]).<ref>{{cite journal | last = Jaynes | first = E.| author-link = Edwin Thompson Jaynes | title = Information Theory and Statistical Mechanics | doi = 10.1103/PhysRev.106.620 | journal = Physical Review | volume = 106 | issue = 4 | pages = 620–630 | year = 1957 | pmid =  | pmc = |bibcode = 1957PhRv..106..620J }}</ref>

* [[Maximum entropy thermodynamics|Maximum information entropy]]: A more elaborate version of the principle of indifference states that the correct ensemble is the ensemble that is compatible with the known information and that has the largest [[Gibbs entropy]] ([[information entropy]]).<ref>{{cite journal | last = Jaynes | first = E.| author-link = Edwin Thompson Jaynes | title = Information Theory and Statistical Mechanics | doi = 10.1103/PhysRev.106.620 | journal = Physical Review | volume = 106 | issue = 4 | pages = 620–630 | year = 1957 | pmid =  | pmc = |bibcode = 1957PhRv..106..620J }}</ref>

* [[各态历经假设]]:各态历经系统是一种随着时间的演化而探索“所有可到达”状态的系统:所有具有相同能量和组成的状态。在各态历经系统中，微正则系综是唯一可能的具有固定能量的平衡系综。这种方法的适用性有限，因为大多数系统不是各态历经的。

* [[<font color="#FF8000">各态历经假设 Ergodic hypothesis</font>]]:各态历经系统是一种随着时间的演化而探索“所有可到达”状态的系统:所有具有相同能量和组成的状态。在各态历经系统中，微正则系综是唯一可能的具有固定能量的平衡系综。这种方法的适用性有限，因为大多数系统不是各态历经的。

* [[无差别原则]]: 在没有更多信息的情况下，我们只能对每一个相容的情况分配相等的概率。

* [[无差别原则]]: 在没有更多信息的情况下，我们只能对每一个相容的情况分配相等的概率。

其他关于统计力学的基本假设也有被提出。

其他关于统计力学的基本假设也有被提出。

===Three thermodynamic ensembles===

===Three thermodynamic ensembles 三种热力学系综===

{{main|Microcanonical ensemble|Canonical ensemble|Grand canonical ensemble}}

{{main|Microcanonical ensemble|Canonical ensemble|Grand canonical ensemble}}

 

There are three equilibrium ensembles with a simple form that can be defined for any [[isolated system]] bounded inside a finite volume.<ref name="gibbs"/> 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.<ref name="gibbs"/> 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.

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。

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

微正则系综

<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.

  Canonical ensemble

  Canonical ensemble

正则系综

<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.

  Grand canonical ensemble

  Grand canonical ensemble

巨正则系综

<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。

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(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。

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。

|}

|}

=== Calculation methods ===

=== Calculation methods 计算方法===

 

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.

====Exact====

====Exact 精确解====

 

There are some cases which allow exact solutions.

There are some cases which allow exact solutions.

* 对于非常小的微观系统，可以通过简单地列举系统所有可能状态（利用量子力学中的严格对角化，或者经典力学中对所有相空间积分）来直接得到系综。

* 对于非常小的微观系统，可以通过简单地列举系统所有可能状态（利用量子力学中的严格对角化，或者经典力学中对所有相空间积分）来直接得到系综。

* 对于包含很多分离的微观系统的宏观系统，每个子系统可以单独分析。尤其是粒子间无相互作用的理想气体具有这种性质，从而可以精确地得到[麦克斯韦–玻尔兹曼统计]]， [[费米-狄拉克统计]]，和[[波色-爱因斯坦统计]]。<ref name="tolman"/>

* 对于包含很多分离的微观系统的宏观系统，每个子系统可以单独分析。尤其是粒子间无相互作用的理想气体具有这种性质，从而可以精确地得到[[<font color="#FF8000">麦克斯韦–玻尔兹曼统计 Maxwell–Boltzmann statistics</font>]]， [[<font color="#FF8000">费米-狄拉克统计 Fermi–Dirac statistics</font>]]，和[[<font color="#FF8000">波色-爱因斯坦统计 Bose–Einstein statistics</font>]]。<ref name="tolman"/>

* 某些存在相互作用的宏观系统也存在精确解。通过运用微妙的数学技巧，已经找到了几个[[玩具模型]]的精确解。<ref>{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc.  | pages =  }}</ref> 一些例子包括[[Bethe ansatz]]，零场下[[二维格点伊辛模型]]，[[硬六边形模型]]。

* 某些存在相互作用的宏观系统也存在精确解。通过运用微妙的数学技巧，已经找到了几个[[玩具模型]]的精确解。<ref>{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc.  | pages =  }}</ref> 一些例子包括[[Bethe ansatz]]，零场下[[<font color="#FF8000"> 二维格点伊辛模型 square-lattice Ising model</font>]]，[[<font color="#FF8000">硬六边形模型 hard hexagon model</font>]]。

====Monte Carlo====

====Monte Carlo 蒙特卡罗方法====

{{main|Monte Carlo method}}

{{main|Monte Carlo method}}

 

One approximate approach that is particularly well suited to computers is the [[Monte Carlo method]], which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.

One approximate approach that is particularly well suited to computers is the [[Monte Carlo method]], which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.

One approximate approach that is particularly well suited to computers is the Monte Carlo method, which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.

One approximate approach that is particularly well suited to computers is the Monte Carlo method, which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.

一个特别适合于计算机的近似方法是蒙特卡罗方法，它只随机选择(具有相当的权重)系统的几个可能状态进行检查。只要这些状态构成系统全部状态集的代表样本，就可以得到近似的特征函数。随着随机样本数量的增加，误差可以降低到任意低的水平。

一个特别适合于计算机的近似方法是<font color="#FF8000">蒙特卡罗方法 Monte Carlo method</font>，它只随机选择(具有相当的权重)系统的几个可能状态进行检查。只要这些状态构成系统全部状态集的代表样本，就可以得到近似的特征函数。随着随机样本数量的增加，误差可以降低到任意低的水平。

==== Other ====

==== Other 其他====

 

* For rarefied non-ideal gases, approaches such as the [[cluster expansion]] use [[perturbation theory]] to include the effect of weak interactions, leading to a [[virial expansion]].<ref name="balescu" />

* For rarefied non-ideal gases, approaches such as the [[cluster expansion]] use [[perturbation theory]] to include the effect of weak interactions, leading to a [[virial expansion]].<ref name="balescu" />

* Mixed methods involving non-equilibrium statistical mechanical results (see below) may be useful.

* Mixed methods involving non-equilibrium statistical mechanical results (see below) may be useful.

* 对于稀薄的非理想气体，[[团簇膨胀]]等方法使用[[微扰理论]]来涵盖弱相互作用的影响，得到[[维里展开]]。<ref name="balescu" />

* 对于稀薄的非理想气体，[[团簇膨胀 cluster expansion]]等方法使用[[微扰理论 perturbation theory]]来涵盖弱相互作用的影响，得到[[维里展开 virial expansion]]。<ref name="balescu" />

* 对于稠密流体，另一种近似方法是基于简化的分布函数，特别是[[径向分布函数]]。<ref name="balescu"/>

* 对于稠密流体，另一种近似方法是基于简化的分布函数，特别是[[径向分布函数 radial distribution function]]。<ref name="balescu"/>

* [[分子动力学]] 计算机模拟可以用来计算各态历经系统中的[[微正则系综]]平均。对于包含与随机热浴的连接的系统，他们还可以在正则和巨正则条件下建模。

* [[分子动力学 Molecular dynamics]] 计算机模拟可以用来计算各态历经系统中的[[微正则系综]]平均。对于包含与随机热浴的连接的系统，他们还可以在正则和巨正则条件下建模。

* 包含非平衡态统计力学结果（如下）的混合方法可能是很有用的。

* 包含非平衡态统计力学结果（如下）的混合方法可能是很有用的。