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Statistical mechanics is one of the pillars of modern physics. It is necessary for the fundamental study of any physical system that has many degrees of freedom. The approach is based on statistical methods, probability theory and the microscopic physical laws.

Statistical mechanics is one of the pillars of modern physics. It is necessary for the fundamental study of any physical system that has many degrees of freedom. The approach is based on statistical methods, probability theory and the microscopic physical laws.

<font color="#FF8000">统计力学 Statistical mechanics</font>是现代物理学的支柱之一，对于任何具有多个自由度的物理系统的基础研究都很必要。统计力学的基础是统计学方法、概率论和微观物理定律。

<font color="#FF8000">统计力学 Statistical mechanics</font>是现代物理学的支柱之一。对于具有多个<font color="#FF8000">自由度 Degrees of Freedom</font>的物理系统的基础研究，统计力学是不可或缺的。统计力学的方法是基于<font color="#FF8000">统计学方法 Statistical Methods</font>、<font color="#FF8000">概率论 Probability Theory</font>和<font color="#FF8000">微观物理定律 Microscopic Physical Laws</font>。

It can be used to explain the thermodynamic behaviour of large systems. This branch of statistical mechanics, which treats and extends classical thermodynamics, is known as statistical thermodynamics or equilibrium statistical mechanics.

It can be used to explain the thermodynamic behaviour of large systems. This branch of statistical mechanics, which treats and extends classical thermodynamics, is known as statistical thermodynamics or equilibrium statistical mechanics.

统计力学可以用来解释大系统的热力学行为，其中一个分支处理和扩展了经典热力学，被称为<font color="#FF8000">统计热力学 statistical thermodynamics</font>或<font color="#FF8000">平衡态统计力学 equilibrium statistical mechanics</font>。

统计力学一个分支可以用来解释大系统的热力学行为。该分支完善和扩展了经典热力学，被称为<font color="#FF8000">统计热力学 Statistical Thermodynamics</font>或<font color="#FF8000">平衡态统计力学 Equilibrium Statistical Mechanics</font>。

Statistical mechanics describes how macroscopic observations (such as temperature and pressure) are related to microscopic parameters that fluctuate around an average. It connects thermodynamic quantities (such as heat capacity) to microscopic behavior, whereas, in classical thermodynamics, the only available option would be to measure and tabulate such quantities for various materials.

Statistical mechanics describes how macroscopic observations (such as temperature and pressure) are related to microscopic parameters that fluctuate around an average. It connects thermodynamic quantities (such as heat capacity) to microscopic behavior, whereas, in classical thermodynamics, the only available option would be to measure and tabulate such quantities for various materials.

统计力学描述了宏观观测量(如温度和压强)与围绕平均值波动的微观参数的关系。它将热力学量(比如<font color="#FF8000">热容 heat capacity</font>)与微观行为联系起来，而在经典热力学中，唯一可行的选择就是测量和列出各种材料的热力学量。

统计力学描述了宏观观测量(如温度和压强)与围绕平均值波动的微观参数的关系。它将热力学量(比如<font color="#FF8000">热容 Heat Capacity</font>)与微观行为联系起来。而在<font color="#FF8000">经典热力学 Classical Thermodynamics</font>中，唯一可行的选择就是测量和列出各种材料的热力学量。

Statistical mechanics can also be used to study systems that are out of equilibrium. An important subbranch known as non-equilibrium statistical mechanics (sometimes called statistical dynamics) deals with the issue of microscopically modelling the speed of irreversible processes that are driven by imbalances. Examples of such processes include chemical reactions or flows of particles and heat. The fluctuation–dissipation theorem is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles.

Statistical mechanics can also be used to study systems that are out of equilibrium. An important subbranch known as non-equilibrium statistical mechanics (sometimes called statistical dynamics) deals with the issue of microscopically modelling the speed of irreversible processes that are driven by imbalances. Examples of such processes include chemical reactions or flows of particles and heat. The fluctuation–dissipation theorem is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles.

统计力学也可以用来研究非平衡的系统。<font color="#FF8000">非平衡统计力学 non-equilibrium statistical mechanics</font>(有时称为统计动力学)是统计力学的重要分支，它涉及的问题是对由非平衡导致的不可逆过程的速度进行微观模拟。例如化学反应或粒子流和热流。<font color="#FF8000">涨落-耗散定理 fluctuation–dissipation theorem</font>是人们从非平衡态统计力学中获得的基本知识，这是在应用非平衡态统计力学来研究多粒子系统中稳态电流流动这样的最简单的非平衡态情况下所发现的。

统计力学也可以用来研究非平衡的系统。<font color="#FF8000">非平衡统计力学 non-equilibrium statistical mechanics</font>(有时称为统计动力学)是统计力学的重要分支，它涉及的问题是对由非平衡导致的不可逆过程的速度进行微观模拟。例如化学反应或粒子流和热流。<font color="#FF8000">涨落-耗散定理 Fluctuation–Dissipation Theorem</font>是人们从非平衡态统计力学中获得的基本知识，它是在应用非平衡态统计力学来研究多粒子系统中稳态电流流动这样的最简单的非平衡态情况下所发现的。

== Principles: mechanics and ensembles 原理：力学和<font color="#FF8000">系综 ensembles</font>==

== Principles: mechanics and ensembles 原理：力学和<font color="#FF8000">系综 Ensembles</font>==

{{main|Mechanics|Statistical ensemble (mathematical physics)|l2=Statistical ensemble}}

{{main|Mechanics|Statistical ensemble (mathematical physics)|l2=Statistical ensemble}}

In physics, two types of mechanics are usually examined: classical mechanics and quantum mechanics. For both types of mechanics, the standard mathematical approach is to consider two concepts:

In physics, two types of mechanics are usually examined: classical mechanics and quantum mechanics. For both types of mechanics, the standard mathematical approach is to consider two concepts:

在物理学中，通常有两种力学被研究: <font color="#FF8000">经典力学 classical mechanics</font>和<font color="#FF8000">量子力学 quantum mechanics</font>。对于这两种类型的力学，标准的数学方法是考虑两个概念

在物理学中，有两种力学被广泛研究: <font color="#FF8000">经典力学 Classical Mechanics</font>和<font color="#FF8000">量子力学 Quantum Mechanics</font>。对于这两种力学，标准的数学方法与两个概念有关

 

# The complete state of the mechanical system at a given time, mathematically encoded as a [[phase space|phase point]] (classical mechanics) or a pure [[quantum state vector]] (quantum mechanics).

# The complete state of the mechanical system at a given time, mathematically encoded as a [[phase space|phase point]] (classical mechanics) or a pure [[quantum state vector]] (quantum mechanics).

The complete state of the mechanical system at a given time, mathematically encoded as a phase point (classical mechanics) or a pure quantum state vector (quantum mechanics).

The complete state of the mechanical system at a given time, mathematically encoded as a phase point (classical mechanics) or a pure quantum state vector (quantum mechanics).

 

# An equation of motion which carries the state forward in time: [[Hamilton's equations]] (classical mechanics) or the [[time-dependent Schrödinger equation]] (quantum mechanics)

# An equation of motion which carries the state forward in time: [[Hamilton's equations]] (classical mechanics) or the [[time-dependent Schrödinger equation]] (quantum mechanics)

An equation of motion which carries the state forward in time: Hamilton's equations (classical mechanics) or the time-dependent Schrödinger equation (quantum mechanics)

An equation of motion which carries the state forward in time: Hamilton's equations (classical mechanics) or the time-dependent Schrödinger equation (quantum mechanics)

 

一个运动方程描述状态在时间上的演化: <font color="#FF8000">哈密尔顿方程 Hamilton's Equations</font>(经典力学)或<font color="#FF8000">含时薛定谔方程 Time-dependent Schrödinger Equation</font>(量子力学)

Using these two concepts, the state at any other time, past or future, can in principle be calculated.

Using these two concepts, the state at any other time, past or future, can in principle be calculated.

There is however a disconnection between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.

There is however a disconnection between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.

然而，这些定律与日常生活经验之间存在脱节。因为对于在人类尺度上进行过程(例如化学反应)，我们没有必要(甚至在理论上也不可能)在微观层面上准确地知道每个分子所在的位置及其速度。统计力学通过增加一些对于系统状态的不确定性，填补了力学定律和不完全知识的实践经验之间的这种脱节，。

然而，这些定律与日常生活经验之间存在着脱节。因为对于在人类尺度上进行过程(例如化学反应)，我们没有必要(甚至在理论上也不可能)在微观层面上准确地知道每个分子所在的位置及其速度。统计力学通过增加一些对于系统状态的不确定性，填补了力学定律和人类不完全知识的实践经验之间的脱节。

Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the statistical ensemble, which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a probability distribution over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a phase space with canonical coordinates. In quantum statistical mechanics, the ensemble is a probability distribution over pure states, and can be compactly summarized as a density matrix.

Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the statistical ensemble, which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a probability distribution over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a phase space with canonical coordinates. In quantum statistical mechanics, the ensemble is a probability distribution over pure states, and can be compactly summarized as a density matrix.

普通力学只考虑单一状态的行为，而统计力学引入了<font color="#FF8000">统计系综 statistical ensemble</font>，它是系统在各种状态下的大量虚拟、独立副本的集合。系综是一个覆盖系统所有可能状态的<font color="#FF8000">概率分布 probability distribution</font>。在经典的统计力学中，系综是相点上的概率分布(与普通力学中的单相点相反) ，通常表现为<font color="#FF8000">正则坐标 canonical coordinates</font>下相空间中的分布。在量子统计力学中，系综是纯态上的概率分布，可以简单地概括为<font color="#FF8000">密度矩阵 density matrix</font>。

普通力学只考虑单一状态的行为，而统计力学引入了<font color="#FF8000">统计系综 Statistical Ensemble</font>，它是系统在各种状态下的大量虚拟、独立的拷贝的集合。系综是一个覆盖系统所有可能状态的<font color="#FF8000">概率分布 Probability Distribution</font>。在经典的统计力学中，系综是相点上的概率分布(与普通力学中的单相点相反) ，通常表现为<font color="#FF8000">正则坐标 Canonical Coordinates</font>下相空间中的分布。在量子统计力学中，系综是纯态上的概率分布，可以简单地概括为<font color="#FF8000">密度矩阵 Density Matrix</font>。  

 

* 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<font color="#32CD32">([[epistemic probability]], a form of knowledge)</font>，或者

* 系综可以表示"单个系统"的所有可能状态（<font color="#FF8000">认识概率 Epistemic Probability</font>，知识的一种形式），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（<font color="#FF8000">经验概率 empirical probability</font>）。

* 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="#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。

然而，这种概率是可被解释的，系综中的每个随时间演化的状态都可以由运动方程给出。因此，系综本身(状态的概率分布)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，虚拟系统的概率随时间演化过程中是守恒的。

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="#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。

有一种特殊的系综是不随时间演化的。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>。如果对于每个状态，无论是未来还是过去，该系综都包含在内，并且其概率等于处于该状态的概率，则出现统计平衡的情况。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 统计热力学==

== Statistical thermodynamics 统计热力学==

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<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。

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。

Whereas statistical mechanics proper involves dynamics, here the attention is focussed on statistical equilibrium (steady state). Statistical equilibrium does not mean that the particles have stopped moving (mechanical equilibrium), rather, only that the ensemble is not evolving.

Whereas statistical mechanics proper involves dynamics, here the attention is focussed on statistical equilibrium (steady state). Statistical equilibrium does not mean that the particles have stopped moving (mechanical equilibrium), rather, only that the ensemble is not evolving.

然而统计力学本身涉及到动态，这里的注意力集中在统计平衡(稳态)上。统计平衡并不意味着粒子已经停止运动(力学平衡) ，相反，只是系综没有进化。

然而统计力学本身就涉及到动态变化，此时的关注点集中在统计平衡(稳态)上。统计平衡并不意味着粒子已经停止运动(力学平衡) ，相反，只是系综没有进化。

There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics. Additional postulates are necessary to motivate why the ensemble for a given system should have one form or another.

There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics. Additional postulates are necessary to motivate why the ensemble for a given system should have one form or another.

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

在许多教科书中常见的一种方法是采用<font color="#FF8000">先验概率相等假设 equal a priori probability postulate</font>。这个假设表明

在许多教科书中常见的一种方法是采用<font color="#FF8000">先验概率相等假设 Equal A Priori Probability Postulate</font>。这个假设表明

: ''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.''

For an isolated system with an exactly known energy and exactly known composition, the system can be found with equal probability in any 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 consistent with that knowledge.

对于一个已知精确能量和组成的孤立系统，可以在任何符合条件的微观状态下以等概率找到该系统。

对于一个已知精确能量和组成的孤立系统，可以在任何符合条件的微观状态下以等概率找到该系统。

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<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:

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>提供了一个动力。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.

* [[<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。

* [[<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。

* [[<font color="#FF8000">无差别原则 Principle of Indifference</font>]]: 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。

* [[最大热力学熵|最大信息熵]]: 无差异原则的一个更详细的版本表明，正确的系综是与已知信息兼容且具有最大[[吉布斯熵]] ([[信息熵]])的系综。<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>

* [[<font color="#FF8000">最大热力学熵|最大信息熵 Maximum Entropy Thermodynamics|Maximum Information Entropy</font>]]: 无差异原则的一个更详细的版本表明，正确的系综是与已知信息兼容且具有最大[[吉布斯熵]] ([[信息熵]])的系综。<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>

Other fundamental postulates for statistical mechanics have also been proposed.<ref name="uffink"/>

Other fundamental postulates for statistical mechanics have also been proposed.<ref name="uffink"/>