统计力学

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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.^[1]^[2]^[3]模板:NoteTag

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.

统计力学 Statistical mechanics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。

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，被称为统计热力学 statistical thermodynamics或平衡态统计力学 equilibrium statistical mechanics。

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.^[1]

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(比如热容 heat capacity)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。

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。非平衡统计力学 non-equilibrium statistical mechanics(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。涨落-耗散定理 fluctuation–dissipation theorem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。

Principles: mechanics and ensembles 原理：力学和系综 ensembles

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: 经典力学 classical mechanics和量子力学 quantum mechanics。对于这两种类型的力学，标准的数学方法是考虑两个概念

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).

力学系统在给定时间内的完整状态，用数学表示为相空间 phase space中的点(经典力学)或纯量子态矢量(量子力学)。

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: 哈密尔顿方程 Hamilton's equations(经典力学)或含时薛定谔方程 time-dependent Schrödinger equation(量子力学)

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

使用这两个概念，系统在任何时间的状态，无论过去或未来，原则上都可以计算出来。

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，。

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,模板:NoteTag 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，而统计力学引入了统计系综 statistical ensemble，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概率分布 probability distribution。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) ，通常表现为正则坐标 canonical coordinates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，可以简单地概括为密度矩阵 density matrix。

As is usual for probabilities, the ensemble can be interpreted in different ways:^[1]

As is usual for probabilities, the ensemble can be interpreted in different ways:

与通常的概率一样，系综可以用不同的方式来解释:

an ensemble can be taken to represent the various possible states that a single system could be in (epistemic probability, a form of knowledge), or

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 empirical probability）。

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

这两种意义在很多情况下是等价的，在本文中可以互换使用。

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。系综演化由刘维尔方程 Liouville equation(经典力学)或冯·诺依曼方程 von Neumann equation(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.模板:NoteTag 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。这样的系综称为平衡系综 equilibrium ensembles，它们的状态称为统计平衡 statistical equilibrium。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 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.

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 classical thermodynamics。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。

Fundamental postulate 基本假设

A 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.).^[1]

A 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.).

孤立系统统计平衡的一个充分(但不是必要)条件是，其概率分布只是某些守恒量(总能量、总粒子数等)的函数。

There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics.^[1] 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.

有许多不同的平衡系综可以考虑，只有一些适用于热力学。为了说明为什么给定系统的系综具有这样或那样的形式，还需要一些额外的假设。

A common approach found in many textbooks is to take the equal a priori probability postulate.^[2] 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先验概率相等假设 equal a priori probability postulate。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 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。

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 microcanonical ensemble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.

Principle of indifference: In the absence of any further information, we can only assign equal probabilities to each compatible situation.

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).^[4]

[[各态历经假设 Ergodic hypothesis]]: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。

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。^[5]

Other fundamental postulates for statistical mechanics have also been proposed.^[6]

Other fundamental postulates for statistical mechanics have also been proposed.

其他关于统计力学的基本假设也有被提出。

Three thermodynamic ensembles 三种热力学系综

There are three equilibrium ensembles with a simple form that can be defined for any isolated system bounded inside a finite volume.^[1] 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 isolated system，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

微正则系综 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

正则系综 Canonical ensemble

describes a system of fixed composition that is in thermal equilibrium 模板:NoteTag 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。

Grand canonical ensemble

Grand canonical ensemble

巨正则系综 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，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 chemical potential。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.^[7] The Gibbs theorem about equivalence of ensembles^[8] was developed into the theory of concentration of measure phenomenon,^[9] which has applications in many areas of science, from functional analysis to methods of artificial intelligence and big data technology.^[10]

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 thermodynamic limit) ，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:

热力学系综不能给出相同结果的重要情况包括:

Microscopic systems.

Large systems at a phase transition.

Large systems with long-range interactions.

微观系统

相变处的宏观系统

长程关联的宏观系统

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.^[2]

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.

在这些情况下必须选择正确的热力学系综，因为这些系综之间不仅在涨落的大小方面有可观测的差异，而且在平均量方面都有可观察的差异，如粒子数的分布。正确的系综是对应于该系统的制备和表征的方式ーー换句话说，系综反映我们对系统的认知。

{ | class“ wikable sortable mw-collable mw-collapse”


			Thermodynamic ensembles^[1]			Thermodynamic ensembles			热力学系数
		Microcanonical	Microcanonical	微正则化	Canonical	Canonical	典范	Grand canonical	Grand canonical	巨正典
Fixed variables	Fixed variables	固定变量	$N, E, V$	中心，中心	$N, T, V$	中心，中心	$μ, T, V$	中心，中心
Microscopic features	Microscopic features	微观特征		“ div class”“ plainlist” Number of microstates [math]\displaystyle{ W }[/math] / div		“ div class”“ plainlist” Canonical partition function [math]\displaystyle{ Z = \sum_k e^{- E_k / k_B T} }[/math] / div		“ div class”“ plainlist” Grand partition function [math]\displaystyle{ \mathcal Z = \sum_k e^{ -(E_k - \mu N_k) /k_B T} }[/math] / div
Macroscopic function	Macroscopic function	宏观功能		“ div class”“ plainlist” Boltzmann entropy [math]\displaystyle{ S = k_B \log W }[/math] / div		“ div class”“ plainlist” Helmholtz free energy [math]\displaystyle{ F = - k_B T \log Z }[/math] / div		“ div class”“ plainlist” Grand potential [math]\displaystyle{ \Omega =- k_B T \log \mathcal Z }[/math] / div

|}

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.

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。

Exact 精确解

There are some cases which allow exact solutions.

有些情况可以得到精确解。

For very small microscopic systems, the ensembles can be directly computed by simply enumerating over all possible states of the system (using exact diagonalization in quantum mechanics, or integral over all phase space in classical mechanics).

Some large systems consist of many separable microscopic systems, and each of the subsystems can be analysed independently. Notably, idealized gases of non-interacting particles have this property, allowing exact derivations of Maxwell–Boltzmann statistics, Fermi–Dirac statistics, and Bose–Einstein statistics.^[2]

A few large systems with interaction have been solved. By the use of subtle mathematical techniques, exact solutions have been found for a few toy models.^[11] Some examples include the Bethe ansatz, square-lattice Ising model in zero field, hard hexagon model.

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 Maxwell–Boltzmann statistics]]， [[费米-狄拉克统计 Fermi–Dirac statistics]]，和[[波色-爱因斯坦统计 Bose–Einstein statistics]]。^[2]

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。^[12] 一些例子包括Bethe ansatz，零场下[[ 二维格点伊辛模型 square-lattice Ising model]]，[[硬六边形模型 hard hexagon model]]。

Monte Carlo 蒙特卡罗方法

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 Monte Carlo method，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。

The Metropolis–Hastings algorithm is a classic Monte Carlo method which was initially used to sample the canonical ensemble.

Path integral Monte Carlo, also used to sample the canonical ensemble.

Metropolis–Hastings 算法是一种经典的蒙特卡罗算法，最初用于正则系综的采样。

路径积分蒙特卡罗方法也可以用于正则系综的采样。

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.^[3]

For dense fluids, another approximate approach is based on reduced distribution functions, in particular the radial distribution function.^[3]

Molecular dynamics computer simulations can be used to calculate microcanonical ensemble averages, in ergodic systems. With the inclusion of a connection to a stochastic heat bath, they can also model canonical and grand canonical conditions.

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 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。^[3]

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。^[3]

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。

Non-equilibrium statistical mechanics 非平衡态统计力学

There are many physical phenomena of interest that involve quasi-thermodynamic processes out of equilibrium, for example:

有许多令人感兴趣的物理现象都涉及到失去平衡的准热力学过程，例如:

heat transport by the internal motions in a material, driven by a temperature imbalance,

electric currents carried by the motion of charges in a conductor, driven by a voltage imbalance,

spontaneous chemical reactions driven by a decrease in free energy,

friction, dissipation, quantum decoherence,

systems being pumped by external forces (optical pumping, etc.),

and irreversible processes in general.

材料的内部运动来传导热量，由热度不平衡来驱动，

导体内电荷的运动产生电流，由电压不平衡来驱动，

自发的化学反应，由自由能的下降驱动，

摩擦，耗散，[[量子退相干]，

由外力泵送的系统([光泵]]等)

以及一般意义的可逆过程

All of these processes occur over time with characteristic rates, and these rates are of importance for engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)

所有这些过程都是以特征速率随时间发生的，这些速率对于工程来说非常重要。非平衡态统计力学研究领域关注的是在微观水平上理解这些非平衡过程。(统计热力学只能用来计算在外部不平衡被消除，整体回归到平衡状态之后的最终结果。)

In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as Liouville's equation or its quantum equivalent, the von Neumann equation. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. Unfortunately, these ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's Gibbs entropy is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics.

原则上，非平衡态统计力学在数学上可以是精确的: 孤立系统的系综根据确定性方程随时间演化，如刘维尔方程或其量子等价、冯·诺依曼方程。这些方程是将运动力学方程独立应用于系综中每个状态的结果。不幸的是，这些系综演化方程继承了潜在动力学运动的大部分复杂性，因此很难得到精确解。此外，系综演化方程是完全可逆的，不会破坏信息(系综的吉布斯熵 Gibbs entropy被保留)。为了在模拟不可逆过程中取得进展，除了概率和可逆力学外，还必须考虑其他因素。

Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections.

因此，非平衡力学是一个活跃的理论研究领域，因为这些额外假设的有效范围仍将继续探索。在下面的小节中描述了一些方法。

Stochastic methods 随机方法

One approach to non-equilibrium statistical mechanics is to incorporate stochastic (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from hypothetical situations involving black holes, a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as chaotic or pseudorandom influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.

处理非平衡态统计力学的一个方法是将随机行为引入系统。随机行为可以破坏系综中包含的信息。虽然这在技术上是不准确的(除了涉及黑洞的假设情况外，黑洞系统本身不会导致信息丢失) ，但这种随机性是为了反映出，随着时间的推移，感兴趣的信息会在系统内部转化为微妙的相关性，或者系统与环境之间的相关性。这些关联表现为对感兴趣的变量的混沌 Chaos或伪随机的影响。用适当的随机性取代这些相关性，计算可以变得容易得多。

{{unordered list

{无序列表

|1 = Boltzmann transport equation: An early form of stochastic mechanics appeared even before the term "statistical mechanics" had been coined, in studies of kinetic theory. James Clerk Maxwell had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. Ludwig Boltzmann subsequently showed that, by taking this molecular chaos for granted as a complete randomization, the motions of particles in a gas would follow a simple Boltzmann transport equation that would rapidly restore a gas to an equilibrium state (see H-theorem).

|1 = Boltzmann transport equation: An early form of stochastic mechanics appeared even before the term "statistical mechanics" had been coined, in studies of kinetic theory. James Clerk Maxwell had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. Ludwig Boltzmann subsequently showed that, by taking this molecular chaos for granted as a complete randomization, the motions of particles in a gas would follow a simple Boltzmann transport equation that would rapidly restore a gas to an equilibrium state (see H-theorem).

玻尔兹曼输运方程 Boltzmann transport equation: 在动力学理论研究中，早期的随机力学形式甚至在“统计力学”一词被创造之前就已经出现了。詹姆斯·克拉克·麦克斯韦已经证明分子碰撞会导致气体内部明显的混沌运动。路德维希·玻尔兹曼随后证明，如果把这种分子混沌理所当然地看作是一种完全的随机化，那么气体中粒子的运动将遵循一个简单的玻尔兹曼输运方程，这个方程将使气体迅速恢复到平衡状态(见H-定理)。

The Boltzmann transport equation and related approaches are important tools in non-equilibrium statistical mechanics due to their extreme simplicity. These approximations work well in systems where the "interesting" information is immediately (after just one collision) scrambled up into subtle correlations, which essentially restricts them to rarefied gases. The Boltzmann transport equation has been found to be very useful in simulations of electron transport in lightly doped semiconductors (in transistors), where the electrons are indeed analogous to a rarefied gas.

玻耳兹曼输运方程及其相关方法是非平衡态统计力学的重要工具，因为它们极其简单。这些近似方法在“感兴趣的”信息立即(在一次碰撞之后)变成微妙关联的系统中非常有效，这种关联本质上限制它们为稀薄气体。玻耳兹曼输运方程被发现在模拟轻掺杂半导体(晶体管)的电子输运中非常有用，其中的电子确实类似于稀薄气体。

A quantum technique related in theme is the random phase approximation.

一个与主题相关的量子技术是随机相位近似。

|2 = BBGKY hierarchy:

|2 = BBGKY hierarchy:

2 BBGKY 层级:

In liquids and dense gases, it is not valid to immediately discard the correlations between particles after one collision. The BBGKY hierarchy (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy) gives a method for deriving Boltzmann-type equations but also extending them beyond the dilute gas case, to include correlations after a few collisions.

在液体和稠密气体中，不能在一次碰撞后立即丢掉粒子之间的关联。BBGKY 层级结构(Bogoliubov-Born-Green-Kirkwood-Yvon 层级结构)提供了一种推导玻尔兹曼型方程的方法，但也可以将它们扩展到稀薄气体情况之外，包括在几次碰撞之后的相关性。

|3 = Keldysh formalism (a.k.a. NEGF—non-equilibrium Green functions):

|3 = Keldysh formalism (a.k.a. NEGF—non-equilibrium Green functions):

| 3 Keldysh 公式。NEGF—非平衡态格林函数):

A quantum approach to including stochastic dynamics is found in the Keldysh formalism. This approach often used in electronic quantum transport calculations.

人们在 Keldysh 公式中发明了包含随机动力学的量子方法，这种方法常用于电子量子输运计算。

|4 = Stochastic Liouville equation

| 4随机刘维尔方程

}}

Near-equilibrium methods 近平衡态方法

Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in linear response theory. A remarkable result, as formalized by the fluctuation-dissipation theorem, is that the response of a system when near equilibrium is precisely related to the fluctuations that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or "know" how it came to be away from equilibrium.^[3]^:664

Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in linear response theory. A remarkable result, as formalized by the fluctuation-dissipation theorem, is that the response of a system when near equilibrium is precisely related to the fluctuations that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or "know" how it came to be away from equilibrium.

非平衡态统计力学模型处理的另一类重要的系统，是对平衡态仅有非常轻微扰动的系统。在很小的扰动下，响应可以用线性响应理论 linear response theory进行分析。涨落-耗散定理 fluctuation-dissipation theorem是其中一个重要的结果，近平衡态系统的响应与系统总体平衡时的涨落准确相关。从本质上讲，一个系统如果稍微偏离平衡，无论是由于外力还是由于涨落，都会以同样的方式向平衡方向弛豫。因为这个系统无法区分偏离和回归，也“不知道”它是如何偏离平衡的。

This provides an indirect avenue for obtaining numbers such as ohmic conductivity and thermal conductivity by extracting results from equilibrium statistical mechanics. Since equilibrium statistical mechanics is mathematically well defined and (in some cases) more amenable for calculations, the fluctuation-dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical mechanics.

这提供了一个间接的方法，通过从平衡态统计力学中提取结果来获得诸如欧姆电导率和热导率之类的物理量。由于平衡态统计力学在数学上有很好的定义，而且(在某些情况下)更易于计算，因此在近平衡态统计力学中，涨落-耗散关联可以成为一种方便的计算捷径。

A few of the theoretical tools used to make this connection include:

用于建立这种关联的一些理论工具包括:

Fluctuation–dissipation theorem

Onsager reciprocal relations

Green–Kubo relations

Landauer–Büttiker formalism

Mori–Zwanzig formalism

涨落-耗散定理

昂萨格倒易关系

格林-库伯关系

Landauer–Büttiker 公式

Mori–Zwanzig 公式

Hybrid methods 组合方法

An advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects (weak localization, conductance fluctuations) in the conductance of an electronic system is the use of the Green-Kubo relations, with the inclusion of stochastic dephasing by interactions between various electrons by use of the Keldysh method.^[13]^[14]

An advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects (weak localization, conductance fluctuations) in the conductance of an electronic system is the use of the Green-Kubo relations, with the inclusion of stochastic dephasing by interactions between various electrons by use of the Keldysh method.

一种先进的方法使用了随机方法和线性响应理论的结合。例如，计算电子系统电导中的量子相干效应(弱局域化，电导涨落)的一种方法是使用 Green-Kubo 关系，包括随机退相的各种电子之间的相互作用，使用Keldysh 方法。

Applications outside thermodynamics

The ensemble formalism also can be used to analyze general mechanical systems with uncertainty in knowledge about the state of a system. Ensembles are also used in:

系综也可以用来分析系统状态有不确定性的一般机械系统。在下列情况中也会使用系综:

propagation of uncertainty over time,^[1]

regression analysis of gravitational orbits,

ensemble forecasting of weather,

dynamics of neural networks,

bounded-rational potential games in game theory and economics.

不确定性随时间传播，

引力轨道的回归分析，

天气的系综预报，

神经网络动力学，

博弈论和经济学中的有界理性潜在对策。

History 历史

In 1738, Swiss physicist and mathematician Daniel Bernoulli published Hydrodynamica which laid the basis for the kinetic theory of gases. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion.^[6]

In 1738, Swiss physicist and mathematician Daniel Bernoulli published Hydrodynamica which laid the basis for the kinetic theory of gases. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion.

1738年，瑞士的物理学家和数学家丹尼尔·伯努利发表了《水动力学》，这本书奠定了气体动力学理论 kinetic theory of gases的基础。在这项工作中，伯努利假定气体是由大量向各个方向运动的分子组成的，它们对表面的影响导致了我们感觉到的气体压力，而我们感受到的热仅仅是它们运动的动能，这一点直到今天仍在沿用。

In 1859, after reading a paper on the diffusion of molecules by Rudolf Clausius, Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range.引用错误：没有找到与</ref>对应的<ref>标签 This was the first-ever statistical law in physics.^[15] Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium.^[16] Five years later, in 1864, Ludwig Boltzmann, a young student in Vienna, came across Maxwell's paper and spent much of his life developing the subject further.

Statistical mechanics proper was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 Lectures on Gas Theory.^[17] Boltzmann's original papers on the statistical interpretation of thermodynamics, the H-theorem, transport theory, thermal equilibrium, the equation of state of gases, and similar subjects, occupy about 2,000 pages in the proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his H-theorem.

Statistical mechanics proper was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 Lectures on Gas Theory. Boltzmann's original papers on the statistical interpretation of thermodynamics, the H-theorem, transport theory, thermal equilibrium, the equation of state of gases, and similar subjects, occupy about 2,000 pages in the proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his H-theorem.

统计力学是19世纪70年代由玻尔兹曼的工作创立的，其中大部分在他1896年的气体理论演讲中集结出版。在维也纳学院和其他学会的会议记录中，玻尔兹曼关于热力学的统计解释、 H-定理H-theorem、输运理论 transport theory、热平衡thermal equilibrium、气体的状态方程以及类似主题的原始论文占据了大约2000页。玻尔兹曼引入了平衡系综的概念，并用他的H-定理第一次研究了非平衡态统计力学。

The term "statistical mechanics" was coined by the American mathematical physicist J. Willard Gibbs in 1884.^[18]模板:NoteTag "Probabilistic mechanics" might today seem a more appropriate term, but "statistical mechanics" is firmly entrenched.^[19] Shortly before his death, Gibbs published in 1902 Elementary Principles in Statistical Mechanics, a book which formalized statistical mechanics as a fully general approach to address all mechanical systems—macroscopic or microscopic, gaseous or non-gaseous.^[1] Gibbs' methods were initially derived in the framework classical mechanics, however they were of such generality that they were found to adapt easily to the later quantum mechanics, and still form the foundation of statistical mechanics to this day.^[2]

The term "statistical mechanics" was coined by the American mathematical physicist J. Willard Gibbs in 1884. "Probabilistic mechanics" might today seem a more appropriate term, but "statistical mechanics" is firmly entrenched. Shortly before his death, Gibbs published in 1902 Elementary Principles in Statistical Mechanics, a book which formalized statistical mechanics as a fully general approach to address all mechanical systems—macroscopic or microscopic, gaseous or non-gaseous. Gibbs' methods were initially derived in the framework classical mechanics, however they were of such generality that they were found to adapt easily to the later quantum mechanics, and still form the foundation of statistical mechanics to this day.

1884年，美国数学物理学家约西亚·威拉德·吉布斯首创了“统计力学”一词。在今天看来，“概率力学”似乎是一个更合适的术语，但“统计力学”却根深蒂固。在吉布斯去世前不久，他于1902年在《统计力学》出版了《基本原理》一书，这本书正式确定了统计力学是解决所有力学系统——宏观的或微观的、气态的或非气态的——的一种完全通用的方法。吉布斯的方法最初是在经典力学的框架下产生的，然而它们是如此的普遍，以至于人们发现它们很容易适应后来的量子力学，直到今天仍然是统计力学的基础。

Notes

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References

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参考文献

^[1]

^[2]

^[3]

^[6]

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External links

模板:Commons category

Philosophy of Statistical Mechanics article by Lawrence Sklar for the Stanford Encyclopedia of Philosophy.

Sklogwiki - Thermodynamics, statistical mechanics, and the computer simulation of materials. SklogWiki is particularly orientated towards liquids and soft condensed matter.

Statistical Thermodynamics - Historical Timeline

Thermodynamics and Statistical Mechanics by Richard Fitzpatrick

Lecture Notes in Statistical Mechanics and Mesoscopics by Doron Cohen

Videos of lecture series in statistical mechanics on YouTube taught by Leonard Susskind.

Vu-Quoc, L., Configuration integral (statistical mechanics), 2008. this wiki site is down; see this article in the web archive on 2012 April 28.

模板:-

模板:Statistical mechanics topics

模板:Physics-footer

Category:Concepts in physics

分类: 物理概念

Category:Physics

类别: 物理学

Category:Thermodynamics

分类: 热力学

This page was moved from wikipedia:en:Statistical mechanics. Its edit history can be viewed at 统计力学/edithistory

↑ ^1.0 ^1.1 ^1.2 ^1.3 ^1.4 ^1.5 ^1.6 ^1.7 ^1.8 ^1.9 Gibbs, Josiah Willard (1902). Elementary Principles in Statistical Mechanics. New York: Charles Scribner's Sons.
↑ ^2.0 ^2.1 ^2.2 ^2.3 ^2.4 ^2.5 ^2.6 Tolman, R. C. (1938). The Principles of Statistical Mechanics. Dover Publications. ISBN 9780486638966.
↑ ^3.0 ^3.1 ^3.2 ^3.3 ^3.4 ^3.5 ^3.6 Balescu, Radu (1975). Equilibrium and Non-Equilibrium Statistical Mechanics. John Wiley & Sons. ISBN 9780471046004.
↑ Jaynes, E. (1957). "Information Theory and Statistical Mechanics". Physical Review. 106 (4): 620–630. Bibcode:1957PhRv..106..620J. doi:10.1103/PhysRev.106.620.
↑ Jaynes, E. (1957). "Information Theory and Statistical Mechanics". Physical Review. 106 (4): 620–630. Bibcode:1957PhRv..106..620J. doi:10.1103/PhysRev.106.620.
↑ ^6.0 ^6.1 ^6.2 J. Uffink, "Compendium of the foundations of classical statistical physics." (2006)
↑ Reif, F. (1965). Fundamentals of Statistical and Thermal Physics. McGraw–Hill. p. 227. ISBN 9780070518001. https://archive.org/details/fundamentalsofst00fred/page/227.
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↑ Baxter, Rodney J. (1982). Exactly solved models in statistical mechanics. Academic Press Inc.. ISBN 9780120831807.
↑ Baxter, Rodney J. (1982). Exactly solved models in statistical mechanics. Academic Press Inc.. ISBN 9780120831807.
↑ Altshuler, B. L.; Aronov, A. G.; Khmelnitsky, D. E. (1982). "Effects of electron-electron collisions with small energy transfers on quantum localisation". Journal of Physics C: Solid State Physics. 15 (36): 7367. Bibcode:1982JPhC...15.7367A. doi:10.1088/0022-3719/15/36/018.
↑ Aleiner, I.; Blanter, Y. (2002). "Inelastic scattering time for conductance fluctuations". Physical Review B. 65 (11): 115317. arXiv:cond-mat/0105436. Bibcode:2002PhRvB..65k5317A. doi:10.1103/PhysRevB.65.115317.
↑ Mahon, Basil (2003). The Man Who Changed Everything – the Life of James Clerk Maxwell. Hoboken, NJ: Wiley. ISBN 978-0-470-86171-4. OCLC 52358254.
↑ Gyenis, Balazs (2017). "Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium". Studies in History and Philosophy of Modern Physics. 57: 53–65. arXiv:1702.01411. Bibcode:2017SHPMP..57...53G. doi:10.1016/j.shpsb.2017.01.001.
↑ Ebeling, Werner; Sokolov, Igor M. (2005). Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems. Series on Advances in Statistical Mechanics. 8. pp. 3–12. Bibcode 2005stst.book.....E. doi:10.1142/2012. ISBN 978-90-277-1674-3. https://books.google.com/books?id=KUjFHbid8A0C. (section 1.2)
↑ J. W. Gibbs, "On the Fundamental Formula of Statistical Mechanics, with Applications to Astronomy and Thermodynamics." Proceedings of the American Association for the Advancement of Science, 33, 57-58 (1884). Reproduced in The Scientific Papers of J. Willard Gibbs, Vol II (1906), pp. 16.
↑ Mayants, Lazar (1984). The enigma of probability and physics. Springer. p. 174. ISBN 978-90-277-1674-3. https://books.google.com/books?id=zmwEfXUdBJ8C&pg=PA174.

[gibbs-1] 1.0 ^1.1 ^1.2 ^1.3 ^1.4 ^1.5 ^1.6 ^1.7 ^1.8 ^1.9 Gibbs, Josiah Willard (1902). Elementary Principles in Statistical Mechanics. New York: Charles Scribner's Sons.

[tolman-2] 2.0 ^2.1 ^2.2 ^2.3 ^2.4 ^2.5 ^2.6 Tolman, R. C. (1938). The Principles of Statistical Mechanics. Dover Publications. ISBN 9780486638966.

[balescu-3] 3.0 ^3.1 ^3.2 ^3.3 ^3.4 ^3.5 ^3.6 Balescu, Radu (1975). Equilibrium and Non-Equilibrium Statistical Mechanics. John Wiley & Sons. ISBN 9780471046004.

[4] Jaynes, E. (1957). "Information Theory and Statistical Mechanics". Physical Review. 106 (4): 620–630. Bibcode:1957PhRv..106..620J. doi:10.1103/PhysRev.106.620.

[5] Jaynes, E. (1957). "Information Theory and Statistical Mechanics". Physical Review. 106 (4): 620–630. Bibcode:1957PhRv..106..620J. doi:10.1103/PhysRev.106.620.

[uffink-6] 6.0 ^6.1 ^6.2 J. Uffink, "Compendium of the foundations of classical statistical physics." (2006)

[Reif-7] Reif, F. (1965). Fundamentals of Statistical and Thermal Physics. McGraw–Hill. p. 227. ISBN 9780070518001. https://archive.org/details/fundamentalsofst00fred/page/227.

[8] Touchette, Hugo (2015). "Equivalence and Nonequivalence of Ensembles: Thermodynamic, Macrostate, and Measure Levels". Journal of Statistical Physics. 159 (5): 987–1016. arXiv:1403.6608. doi:10.1007/s10955-015-1212-2.

[9] Ledoux, Michel (2005). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs. 89. doi:10.1090/surv/089. ISBN 9780821837924. .

[10] Gorban, A. N.; Tyukin, I. Y. (2018). "Blessing of dimensionality: Mathematical foundations of the statistical physics of data". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 376 (2118): 20170237. doi:10.1098/rsta.2017.0237. PMC 5869543.

[11] Baxter, Rodney J. (1982). Exactly solved models in statistical mechanics. Academic Press Inc.. ISBN 9780120831807.

[12] Baxter, Rodney J. (1982). Exactly solved models in statistical mechanics. Academic Press Inc.. ISBN 9780120831807.

[13] Altshuler, B. L.; Aronov, A. G.; Khmelnitsky, D. E. (1982). "Effects of electron-electron collisions with small energy transfers on quantum localisation". Journal of Physics C: Solid State Physics. 15 (36): 7367. Bibcode:1982JPhC...15.7367A. doi:10.1088/0022-3719/15/36/018.

[14] Aleiner, I.; Blanter, Y. (2002). "Inelastic scattering time for conductance fluctuations". Physical Review B. 65 (11): 115317. arXiv:cond-mat/0105436. Bibcode:2002PhRvB..65k5317A. doi:10.1103/PhysRevB.65.115317.

[15] Mahon, Basil (2003). The Man Who Changed Everything – the Life of James Clerk Maxwell. Hoboken, NJ: Wiley. ISBN 978-0-470-86171-4. OCLC 52358254.

[16] Gyenis, Balazs (2017). "Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium". Studies in History and Philosophy of Modern Physics. 57: 53–65. arXiv:1702.01411. Bibcode:2017SHPMP..57...53G. doi:10.1016/j.shpsb.2017.01.001.

[17] Ebeling, Werner; Sokolov, Igor M. (2005). Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems. Series on Advances in Statistical Mechanics. 8. pp. 3–12. Bibcode 2005stst.book.....E. doi:10.1142/2012. ISBN 978-90-277-1674-3. https://books.google.com/books?id=KUjFHbid8A0C. (section 1.2)

[18] J. W. Gibbs, "On the Fundamental Formula of Statistical Mechanics, with Applications to Astronomy and Thermodynamics." Proceedings of the American Association for the Advancement of Science, 33, 57-58 (1884). Reproduced in The Scientific Papers of J. Willard Gibbs, Vol II (1906), pp. 16.

[19] Mayants, Lazar (1984). The enigma of probability and physics. Springer. p. 174. ISBN 978-90-277-1674-3. https://books.google.com/books?id=zmwEfXUdBJ8C&pg=PA174.

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