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.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，被称为统计热力学或平衡态统计力学。

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(比如热容)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(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

Main articles: Mechanics and 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: 经典力学和量子力学。对于这两种类型的力学，标准的数学方法是考虑两个概念:

1. 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中的点(经典力学)或纯量子态矢量(量子力学)。

1. 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(经典力学)或含时薛定谔方程(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.

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.

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，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。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。

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

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.

• 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.

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.

这两个意思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.

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，这种可能性是被解释的，根据运动方程，这种合奏中的每个状态都随着时间而演化。因此，整体本身(概率分布)也在发展，因为整体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 Liouville 方程(经典力学)或 von Neumann 方程(量子力学)给出。这些方程是简单地通过应用力学运动方程分别包含在系综中的每个虚拟系统，随着时间的推移虚拟系统从一个状态演化到另一个状态的概率而导出的。

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.

合奏的一个特殊类别是那些不随时间演变的合奏。这些系综称为平衡系综，它们的状态称为统计平衡。如果对于集合中的每个状态，集合也包含其所有的未来和过去状态，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。非平衡统计力学解决了更一般的情况下的系综，随着时间的推移而改变，和 / 或非孤立系统的系综。

Statistical thermodynamics

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(也称为平衡统计力学)的主要目标是根据其组成粒子的性质和它们之间的相互作用推导出经典的材料热力学。换句话说，统计热力学提供了热力学平衡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，只是整体没有进化。

Fundamental postulate

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.).

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)条件是概率分布只是守恒性质(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. 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，只有一些对应于热力学。为了说明为什么给定系统的整体应该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. 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的假设。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.

对于一个具有精确已知能量和精确已知组成的孤立系统，可以在任何与该知识一致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，等量的先验概率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相等的先验概率假设:

• 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).

Other fundamental postulates for statistical mechanics have also been proposed.

Other fundamental postulates for statistical mechanics have also been proposed.

其他关于统计力学的基本假设也被提出。

Three thermodynamic ensembles

Main articles: Microcanonical ensemble, Canonical ensemble, and 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. These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.

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

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

Microcanonical ensemble

Microcanonical ensemble

微正则系综

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

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

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

Canonical ensemble

Canonical ensemble

正则系综

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

Grand canonical ensemble

Grand canonical ensemble

巨正则系综

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

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

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

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

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

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

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

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

热力学系数不能给出相同结果的重要情况包括:

• Microscopic systems.

• Large systems at a phase transition.

• Large systems with long-range interactions.

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

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，而且在平均数量方面，如粒子的分布方面，都有可观察的差异。正确的集成是对应于该系统的制备和特征的方式ーー换句话说，反映该系统知识的集成。

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

			Thermodynamic ensembles			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

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解决了，但是最一般的(和现实的)情况对于一个精确的解决方案来说太复杂了。存在各种方法来近似真实的总体，并允许计算平均量。

Exact

There are some cases which allow exact solutions.

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.

• 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. Some examples include the Bethe ansatz, square-lattice Ising model in zero field, hard hexagon model.

Monte Carlo

Main article: 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.

一个特别适合于计算机的近似方法是蒙特卡罗方法分析法，它只检查系统的几个可能状态，状态是随机选择的(具有相当的权重)。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，误差降低到了一个任意低的水平。

• 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.

Other

• 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.

• For dense fluids, another approximate approach is based on reduced distribution functions, in particular the radial distribution function.

• 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.

Non-equilibrium statistical mechanics

See also: Non-equilibrium thermodynamics

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

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.

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.)

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最终结果，在外部不平衡被消除，整体回归到平衡状态之后。)

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.

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，非平衡态统计力学可以在数学上是精确的: 孤立系统的整体随时间演化，根据确定性方程，如刘维尔方程或其量子等价物冯 · 诺依曼方程。这些方程是将机械运动方程独立应用于整体中每个状态的结果。不幸的是，这些集合演化方程继承了潜在机械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，不破坏信息(系综的吉布斯熵被保留)。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.

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，因为这些额外的假设的有效范围继续探索。在下面的小节中描述了一些方法。

Stochastic methods

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.

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.

处理非平衡统计力学的一个方法是将随机行为引入系统。随机行为破坏了集合中包含的信息。虽然这在技术上是不准确的(除了涉及黑洞的假设情况外，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。这些关联表现为混沌或伪随机对感兴趣的变量的影响。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

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|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: 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.

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统计力学的重要工具，因为它们极其简单。这些近似方法在“有趣的”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了它们在稀薄气体中的应用。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，其中电子确实类似于稀薄气体。

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

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.

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，在一次碰撞后立即抛弃粒子之间的关联是无效的。Bbgky 层次结构(Bogoliubov-Born-Green-Kirkwood-Yvon 层次结构)提供了一种推导 boltzmann 型方程的方法，但也将它们扩展到稀释气体情况之外，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 形式主义。NEGF—non-equilibrium Green functions):

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

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

在 Keldysh 公式中发现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 = Stochastic Liouville equation

| 4随机 Liouville 方程

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Near-equilibrium methods

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.

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.

另一类重要的非平衡统计力学模型处理的系统，只有非常轻微的扰动从平衡。在很小的扰动下，响应可以用线性响应理论进行分析。一个显著的结果，正如涨落耗散定理的形式化，是系统的反应时，接近平衡恰恰相关的涨落，发生时，系统是在总平衡。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波动，都会以同样的方式向平衡方向放松，因为这个系统既不能说出差别，也不能“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.

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，通过从平衡电导率和热导率统计力学中提取结果来获得诸如欧姆电导率和电导率之类的数字。由于平衡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，因此在近平衡统计力学中，涨落-耗散连接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:

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

用于建立这种联系的一些理论工具包括:

• Fluctuation–dissipation theorem

• Onsager reciprocal relations

• Green–Kubo relations

• Landauer–Büttiker formalism

• Mori–Zwanzig formalism

Hybrid methods

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.

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。作为一个例子，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，使用凯尔迪什方法。

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:

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:

集成形式主义也可以用来分析一般机械系统的不确定性知识的状态系统。在下列情况中也会使用便服:

• propagation of uncertainty over time,

• regression analysis of gravitational orbits,

• ensemble forecasting of weather,

• dynamics of neural networks,

• bounded-rational potential games in game theory and economics.

History

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.

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》 ，这本书奠定了分子运动论的基础。在这项工作中，伯努利假定了，直到今天仍然沿用的论点，即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。

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. This was the first-ever statistical law in physics. Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium. 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. 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年代由玻尔兹曼的著作发起的，其中大部分在他1896年的气体理论演讲中集体出版。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、传输理论、热平衡、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。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统计力学。

The term "statistical mechanics" was coined by the American mathematical physicist J. Willard Gibbs in 1884.模板:NoteTag "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.

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 j. Willard Gibbs 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机械系统ー宏观的或微观的、气态的或非气态的ー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。

See also

• Thermodynamics: non-equilibrium, chemical

• Mechanics: classical, quantum

• Probability, statistical ensemble

• Numerical methods: Monte Carlo method, molecular dynamics

• Statistical physics

• Quantum statistical mechanics

• List of notable textbooks in statistical mechanics

• List of important publications in statistical mechanics

Notes

References

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{{Reflist

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参考文献

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External links

• 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.

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