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第1行： − + + + − It can be used to explain the [[thermodynamics|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. − 统计力学一个分支可以用来解释大系统的热力学行为。该分支完善和扩展了经典热力学，被称为<font color="#FF8000">统计热力学 Statistical Thermodynamics</font>或<font color="#FF8000">平衡态统计力学 Equilibrium Statistical Mechanics</font>。 + + − Statistical mechanics describes how macroscopic observations (such as temperature and pressure) are related to microscopic parameters that fluctuate around an average. It connects thermodynamic quantities (such as [[heat capacity]]) to microscopic behavior, whereas, in [[classical thermodynamics]], the only available option would be to measure and tabulate such quantities for various materials.<ref name="gibbs"/>+ − Statistical mechanics describes how macroscopic observations (such as temperature and pressure) are related to microscopic parameters that fluctuate around an average. It connects thermodynamic quantities (such as heat capacity) to microscopic behavior, whereas, in classical thermodynamics, the only available option would be to measure and tabulate such quantities for various materials.+ − − 统计力学描述了宏观观测量(如温度和压强)与围绕平均值波动的微观参数的关系。它将热力学量(比如<font color="#FF8000">热容 Heat Capacity</font>)与微观行为联系起来。而在<font color="#FF8000">经典热力学 Classical Thermodynamics</font>中，唯一可行的选择就是测量和列出各种材料的热力学量。 − − − − Statistical mechanics can also be used to study systems that are out of equilibrium. An important subbranch known as '''non-equilibrium statistical mechanics''' (sometimes called '''statistical dynamics''') deals with the issue of microscopically modelling the speed of [[irreversible process]]es that are driven by imbalances. Examples of such processes include chemical reactions or flows of particles and heat. The [[fluctuation–dissipation theorem]] is the basic knowledge obtained from applying [[non-equilibrium statistical mechanics]] to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles. − − Statistical mechanics can also be used to study systems that are out of equilibrium. An important subbranch known as non-equilibrium statistical mechanics (sometimes called statistical dynamics) deals with the issue of microscopically modelling the speed of irreversible processes that are driven by imbalances. Examples of such processes include chemical reactions or flows of particles and heat. The fluctuation–dissipation theorem is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles. − − 统计力学也可以用来研究非平衡的系统。<font color="#FF8000">非平衡统计力学 Non-equilibrium Statistical Mechanics</font>(有时称为统计动力学)是统计力学的重要分支，它涉及的问题是对由非平衡导致的不可逆过程的速度进行微观模拟。例如化学反应或粒子流和热流。<font color="#FF8000">涨落-耗散定理 Fluctuation–Dissipation Theorem</font>是人们从非平衡态统计力学中获得的基本知识，它是在应用非平衡态统计力学来研究多粒子系统中稳态电流流动这样的最简单的非平衡态情况下所发现的。 − − − − == Principles: mechanics and ensembles 原理：力学和<font color="#FF8000">系综 Ensembles</font>== − − {{main|Mechanics|Statistical ensemble (mathematical physics)|l2=Statistical ensemble}} − − − − In physics, two types of mechanics are usually examined: [[classical mechanics]] and [[quantum mechanics]]. For both types of mechanics, the standard mathematical approach is to consider two concepts: − − In physics, two types of mechanics are usually examined: classical mechanics and quantum mechanics. For both types of mechanics, the standard mathematical approach is to consider two concepts: − − 在物理学中，有两种力学被广泛研究: <font color="#FF8000">经典力学 Classical Mechanics</font>和<font color="#FF8000">量子力学 Quantum Mechanics</font>。对于这两种力学，标准的数学方法与两个概念有关 − − − # The complete state of the mechanical system at a given time, mathematically encoded as a [[phase space|phase point]] (classical mechanics) or a pure [[quantum state vector]] (quantum mechanics). − − The complete state of the mechanical system at a given time, mathematically encoded as a phase 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<font color="#FF8000">相空间 Phase Space</font>中的点(经典力学)或<font color="#FF8000">纯量子态矢量 Pure Quantum State Vector</font>(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) − − − − − Using these two concepts, the state at any other time, past or future, can in principle be calculated. − − Using these two concepts, the state at any other time, past or future, can in principle be calculated. − − There is however a disconnection between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in. − − There is however a disconnection between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in. − + − Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the [[Statistical ensemble (mathematical physics)|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,{{NoteTag|The probabilities in quantum statistical mechanics should not be confused with [[quantum superposition]]. While a quantum ensemble can contain states with quantum superpositions, a single quantum state cannot be used to represent an ensemble.}} 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. − − − − − As is usual for probabilities, the ensemble can be interpreted in different ways:<ref name="gibbs" /> − − As is usual for probabilities, the ensemble can be interpreted in different ways: − * 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. − − − − * 系综的元素可以理解为在无限次试验的极限下，在类似但不完全受控的独立系统中，重复进行实验得到的系统的状态（<font color="#FF8000">经验概率 Empirical Probability</font>）。 − − These two meanings are equivalent for many purposes, and will be used interchangeably in this article. − These two meanings are equivalent for many purposes, and will be used interchangeably in this article.+ + + − However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the [[Liouville's theorem (Hamiltonian)|Liouville equation]] (classical mechanics) or the [[von Neumann equation]] (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state.+ − − However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the Liouville equation (classical mechanics) or the von Neumann equation (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state. − − 然而，这种概率是可被解释的，系综中的每个随时间演化的状态都可以由运动方程给出。因此，系综本身(状态的概率分布)也在随时间演化，因为系综中的虚拟系统不断地离开一个状态进入另一个状态。系综演化由<font color="#FF8000">刘维尔方程 Liouville Equation</font>(经典力学)或<font color="#FF8000">冯·诺依曼方程 Von Neumann Equation</font>(量子力学)给出。这些方程是简单地通过应用力学运动方程到系综中的每个虚拟系统而导出的，虚拟系统的概率随时间演化过程中是守恒的。 − − − One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as ''equilibrium ensembles'' and their condition is known as ''statistical equilibrium''. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state.{{NoteTag|Statistical equilibrium should not be confused with ''[[mechanical equilibrium]]''. The latter occurs when a mechanical system has completely ceased to evolve even on a microscopic scale, due to being in a state with a perfect balancing of forces. Statistical equilibrium generally involves states that are very far from mechanical equilibrium.}} 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. − − 有一种特殊的系综是不随时间演化的。这样的系综称为<font color="#FF8000">平衡系综 Equilibrium Ensembles</font>，它们的状态称为<font color="#FF8000">统计平衡 Statistical Equilibrium</font>。如果对于每个状态，无论是未来还是过去，该系综都包含在内，并且其概率等于处于该状态的概率，则出现统计平衡的情况。孤立系统的平衡系综是统计热力学研究的重点。非平衡统计力学研究更一般情况下的可以随时间演化的系综，以及（或）非孤立系统的系综。 − − == Statistical thermodynamics 统计热力学== − − The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the [[classical thermodynamics]] of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in [[thermodynamic equilibrium]], and the microscopic behaviours and motions occurring inside the material. − − The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the classical thermodynamics of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in thermodynamic equilibrium, and the microscopic behaviours and motions occurring inside the material. − − 统计热力学(也称为平衡态统计力学)的主要目标是根据组成某材料的粒子的性质和它们之间的相互作用，推导出材料的<font color="#FF8000">经典热力学 Classical Thermodynamics</font>。换句话说，统计热力学提供了热力学平衡态中物质的宏观性质与物质内部微观行为和运动之间的联系。 − − − − 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.+ − + − − − − A [[sufficient condition|sufficient]] (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.).<ref name="gibbs" /> − − A sufficient (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.). − − There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics.<ref name="gibbs" /> Additional postulates are necessary to motivate why the ensemble for a given system should have one form or another. − − There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics. Additional postulates are necessary to motivate why the ensemble for a given system should have one form or another. − + − − A common approach found in many textbooks is to take the ''equal a priori probability postulate''.<ref name="tolman"/> This postulate states that − − A common approach found in many textbooks is to take the equal a priori probability postulate. This postulate states that − − − − ''For an isolated system with an exactly known energy and exactly known composition, the system can be found with ''equal probability'' in any [[microstate (statistical mechanics)|microstate]] consistent with that knowledge.'' − − For an isolated system with an exactly known energy and exactly known composition, the system can be found with equal probability in any microstate consistent with that knowledge. − The equal a priori probability postulate therefore provides a motivation for the [[microcanonical ensemble]] described below. There are various arguments in favour of the equal a priori probability postulate:+ − The equal a priori probability postulate therefore provides a motivation for the microcanonical ensemble described below. There are various arguments in favour of the equal a priori probability postulate:+ − 因此，先验概率相等假设为下面描述的<font color="#FF8000">微正则系综 Microcanonical Ensemble</font>提供了一个动力。有各种各样的论据支持先验概率相等假设:+ − + − * [[Principle of indifference]]: In the absence of any further information, we can only assign equal probabilities to each compatible situation.+ − * [[Maximum entropy thermodynamics|Maximum information entropy]]: A more elaborate version of the principle of indifference states that the correct ensemble is the ensemble that is compatible with the known information and that has the largest [[Gibbs entropy]] ([[information entropy]]).<ref>{{cite journal | last = Jaynes | first = E.| author-link = Edwin Thompson Jaynes | title = Information Theory and Statistical Mechanics | doi = 10.1103/PhysRev.106.620 | journal = Physical Review | volume = 106 | issue = 4 | pages = 620–630 | year = 1957 | pmid = | pmc = |bibcode = 1957PhRv..106..620J }}</ref>+ − * [[<font color="#FF8000">各态历经假设 Ergodic hypothesis</font>]]:各态历经系统是一种随着时间的演化而探索“所有可到达”状态的系统:所有具有相同能量和组成的状态。在各态历经系统中，微正则系综是唯一可能的具有固定能量的平衡系综。这种方法的适用性有限，因为大多数系统不是各态历经的。+ − * [[<font color="#FF8000">无差别原则 Principle of Indifference</font>]]: 在没有更多信息的情况下，我们只能对每一个相容的情况分配相等的概率。+ − − * [[<font color="#FF8000">最大热力学熵|最大信息熵 Maximum Entropy Thermodynamics|Maximum Information Entropy</font>]]: 无差异原则的一个更详细的版本表明，正确的系综是与已知信息兼容且具有最大[[吉布斯熵]] ([[信息熵]])的系综。<ref>{{cite journal | last = Jaynes | first = E.| author-link = Edwin Thompson Jaynes | title = Information Theory and Statistical Mechanics | doi = 10.1103/PhysRev.106.620 | journal = Physical Review | volume = 106 | issue = 4 | pages = 620–630 | year = 1957 | pmid = | pmc = |bibcode = 1957PhRv..106..620J }}</ref> − − Other fundamental postulates for statistical mechanics have also been proposed.<ref name="uffink"/> − − Other fundamental postulates for statistical mechanics have also been proposed. − − 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=== − − {{main|Microcanonical ensemble|Canonical ensemble|Grand canonical ensemble}} − − − − There are three equilibrium ensembles with a simple form that can be defined for any [[isolated system]] bounded inside a finite volume.<ref name="gibbs"/> These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics. − − There are three equilibrium ensembles with a simple form that can be defined for any isolated system bounded inside a finite volume. These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics. − − 对于任何有限体积的<font color="#FF8000">孤立系统 Isolated System</font>，可以定义三种简单形式的平衡系综。这些是统计热力学中最经常讨论的系综。在宏观极限(定义如下) ，它们都与经典热力学有对应。 − − ; [[Microcanonical ensemble]] − − 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. − − ; [[Canonical ensemble]] − − Canonical ensemble − − <font color="#FF8000">正则系综 Canonical Ensemble</font> − − : describes a system of fixed composition that is in [[thermal equilibrium]]{{NoteTag|The transitive thermal equilibrium (as in, "X is thermal equilibrium with Y") used here means that the ensemble for the first system is not perturbed when the system is allowed to weakly interact with the second system.}} with a [[heat bath]] of a precise [[thermodynamic temperature|temperature]]. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy. − − describes a system of fixed composition that is in thermal equilibrium 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. − ; [[Grand canonical ensemble]]+ − − 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 potential]]s for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers. − − describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise chemical potentials for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers. − + − − 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.<ref name="Reif">{{cite book | last = Reif | first = F. | title = Fundamentals of Statistical and Thermal Physics | publisher = McGraw–Hill | year = 1965 | isbn = 9780070518001 | page = [https://archive.org/details/fundamentalsofst00fred/page/227 227] | url-access = registration | url = https://archive.org/details/fundamentalsofst00fred/page/227 }}</ref> The Gibbs theorem about equivalence of ensembles<ref>{{cite journal |doi=10.1007/s10955-015-1212-2|title=Equivalence and Nonequivalence of Ensembles: Thermodynamic, Macrostate, and Measure Levels|journal=Journal of Statistical Physics|volume=159|issue=5|pages=987–1016|year=2015|last1=Touchette|first1=Hugo|arxiv=1403.6608}}</ref> was developed into the theory of [[concentration of measure]] phenomenon,<ref>{{cite book |doi=10.1090/surv/089|title=The Concentration of Measure Phenomenon|volume=89|series=Mathematical Surveys and Monographs|year=2005|isbn=9780821837924|last1=Ledoux|first1=Michel}}.</ref> which has applications in many areas of science, from functional analysis to methods of [[artificial intelligence]] and [[big data]] technology.<ref>{{cite journal |doi=10.1098/rsta.2017.0237|pmc=5869543|title=Blessing of dimensionality: Mathematical foundations of the statistical physics of data|journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=376|issue=2118|pages=20170237|year=2018|last1=Gorban|first1=A. N.|last2=Tyukin|first2=I. Y.}}</ref> − − For systems containing many particles (the thermodynamic limit), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used. The Gibbs theorem about equivalence of ensembles was developed into the theory of concentration of measure phenomenon, which has applications in many areas of science, from functional analysis to methods of artificial intelligence and big data technology. − − 对于包含大量粒子的系统(<font color="#FF8000">热力学极限 Thermodynamic Limit</font>) ，上面列出的三种系综都倾向于体现出相同的行为。因此，使用哪种系综只是一个简单的数学问题。发展成为<font color="#FF8000">测度现象集中理论 Concentration of Measure Phenomenon</font>系综等价的吉布斯定理，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: − − * Microscopic systems. − − * Large systems at a phase transition. − − * Large systems with long-range interactions. 第255行：
第85行： − 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.<ref name="tolman" />+ − − 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准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。 − − 第479行：
第303行： − + − − 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. − + − − − − − − There are some cases which allow exact solutions. − − There are some cases which allow exact solutions. − − − − * 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]].<ref name="tolman"/> − − * 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 model]]s.<ref>{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc. | pages = }}</ref> Some examples include the [[Bethe ansatz]], [[square-lattice Ising model]] in zero field, [[hard hexagon model]]. − + − + − − {{main|Monte Carlo method}} − − − One approximate approach that is particularly well suited to computers is the [[Monte Carlo method]], which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.+ − One approximate approach that is particularly well suited to computers is the Monte Carlo method, which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.+ − 一个特别适合于计算机的近似方法是<font color="#FF8000">蒙特卡罗方法 Monte Carlo Method</font>，它只随机选择(具有相当的权重)系统的几个可能状态进行检查。只要这些状态可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。+ + + − + − + − − * [[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]].<ref name="balescu" /> − − * For dense fluids, another approximate approach is based on reduced distribution functions, in particular the [[radial distribution function]].<ref name="balescu"/> − − * [[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]]。<ref name="balescu" /> − − * 对于稠密流体，另一种近似方法是基于简化的分布函数，特别是[[径向分布函数 radial distribution function]]。<ref name="balescu"/> − − * [[分子动力学 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有连接的系统，他们还可以在正则和巨正则条件下建模。 − + − − {{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: − + − − * [[Electrical conduction|electric currents carried by the motion of charges in a conductor]], driven by a voltage imbalance, − − * spontaneous [[chemical reaction]]s 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. − − * [[热传导|材料内部的物质运动来传导热量]]，由热度不平衡来驱动， − + − + − + − + − − 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.) − + − − 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 theorem (Hamiltonian)|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. − − − − − − 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. + + − === 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 [[Black hole information paradox|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 [[Chaos theory|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<font color="#FF8000">混沌 Chaos</font>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。 − − − − |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 of gases|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). − − <font color="#FF8000">玻尔兹曼输运方程 Boltzmann Transport Equation</font>: 在动力学理论研究中，早期的随机力学甚至在“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 [[semiconductor]]s (in [[transistor]]s), 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. + + − 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<font color="#FF8000">随机相位近似 Random Phase Approximation</font>。 − − − − |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，不能在一次碰撞后立即丢掉粒子之间的关联。BBGKY 层级结构(<font color="#FF8000">层级结构 Bogoliubov-Born-Green-Kirkwood-Yvon</font>)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): − − 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. − − |4 = Stochastic [[Liouville's theorem (Hamiltonian)|Liouville equation]] − − |4 = Stochastic Liouville equation − }} − − }} − − }} − === 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 [[Statistical fluctuations|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.<ref name="balescu"/>{{rp|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<font color="#FF8000">线性响应理论 Linear Response Theory</font>进行分析。<font color="#FF8000">涨落-耗散定理 Fluctuation-Dissipation Theorem</font>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 [[Ohm's law|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. − − − − A few of the theoretical tools used to make this connection include: − − A few of the theoretical tools used to make this connection include: − + − − * [[Onsager reciprocal relations]] − − * [[Green–Kubo relations]] − − * [[Ballistic conduction#Landauer-Büttiker formalism|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]] − − * [[Ballistic conduction#Landauer-Büttiker formalism|Landauer–Büttiker 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.<ref>{{Cite journal | last1 = Altshuler | first1 = B. L. | last2 = Aronov | first2 = A. G. | last3 = Khmelnitsky | first3 = D. E. | doi = 10.1088/0022-3719/15/36/018 | title = Effects of electron-electron collisions with small energy transfers on quantum localisation | journal = Journal of Physics C: Solid State Physics | volume = 15 | issue = 36 | pages = 7367 | year = 1982 | pmid = | pmc = |bibcode = 1982JPhC...15.7367A }}</ref><ref>{{Cite journal | last1 = Aleiner | first1 = I. | last2 = Blanter | first2 = Y. | doi = 10.1103/PhysRevB.65.115317 | title = Inelastic scattering time for conductance fluctuations | journal = Physical Review B | volume = 65 | issue = 11 | pages = 115317 | year = 2002 | pmid = | pmc = |arxiv = cond-mat/0105436 |bibcode = 2002PhRvB..65k5317A | url = http://resolver.tudelft.nl/uuid:e7736134-6c36-47f4-803f-0fdee5074b5a }}</ref>+ − 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.+ − 一种先进的方法结合了随机方法和线性响应理论。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。+ − + − 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:+ − + − − * [[regression analysis]] of gravitational [[orbit]]s, − − * [[ensemble forecasting]] of weather, − − * dynamics of [[neural networks]], − − * bounded-rational [[potential game]]s 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.<ref name="uffink"/> − 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<font color='red'><s>《水动力学》</s></font> <font color='blue'>《流体动力学》</font> ，这本书奠定了<font color="#FF8000">气体动力学理论 Kinetic Theory of Gases</font>的基础。在这<font color='red'><s>项工作</s></font><font color='blue'>部著作</font>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.<ref>See:+ − 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.<ref>See:+ − * Maxwell, J.C. (1860) [https://books.google.com/books?id=-YU7AQAAMAAJ&pg=PA19#v=onepage&q&f=false "Illustrations of the dynamical theory of gases. Part I. On the motions and collisions of perfectly elastic spheres,"] ''Philosophical Magazine'', 4th series, '''19''' : 19–32. + − − * Maxwell, J.C. (1860) [https://books.google.com/books?id=DIc7AQAAMAAJ&pg=PA21#v=onepage&q&f=false "Illustrations of the dynamical theory of gases. Part II. On the process of diffusion of two or more kinds of moving particles among one another,"] ''Philosophical Magazine'', 4th series, '''20''' : 21–37.</ref> This was the first-ever statistical law in physics.<ref>{{cite book |last = Mahon |first = Basil |title=The Man Who Changed Everything – the Life of James Clerk Maxwell |location=Hoboken, NJ |publisher=Wiley |year=2003 |isbn=978-0-470-86171-4 |oclc=52358254}}</ref> Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium.<ref>{{cite journal | last = Gyenis | first = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G }}</ref> 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''.<ref>{{cite book |title = Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems |journal=Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems. Edited by Ebeling Werner & Sokolov Igor M. Published by World Scientific Press |volume=8 |last1=Ebeling |first1=Werner |last2=Sokolov |first2=Igor M. |year=2005 |isbn=978-90-277-1674-3 |pages=3–12 |url = https://books.google.com/books?id=KUjFHbid8A0C|bibcode=2005stst.book.....E |doi=10.1142/2012 |series = Series on Advances in Statistical Mechanics }} (section 1.2)</ref> Boltzmann's original papers on the statistical interpretation of thermodynamics, the [[H-theorem]], [[transport theory (statistical physics)|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|''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、 <font color="#FF8000">H-定理H-theorem</font>、<font color="#FF8000">输运理论 Transport Theory</font>、<font color="#FF8000">热平衡Thermal Equilibrium</font>、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 [[Josiah Willard Gibbs|J. Willard Gibbs]] in 1884.<ref>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), [https://archive.org/stream/scientificpapers02gibbuoft#page/16/mode/2up pp. 16].</ref>{{NoteTag|1 = According to Gibbs, the term "statistical", in the context of mechanics, i.e. statistical mechanics, was first used by the Scottish physicist [[James Clerk Maxwell]] in 1871. From: J. Clerk Maxwell, ''Theory of Heat'' (London, England: Longmans, Green, and Co., 1871), [https://books.google.com/books?id=DqAAAAAAMAAJ&pg=PA309 p. 309]: "In dealing with masses of matter, while we do not perceive the individual molecules, we are compelled to adopt what I have described as the statistical method of calculation, and to abandon the strict dynamical method, in which we follow every motion by the calculus."}} "Probabilistic mechanics" might today seem a more appropriate term, but "statistical mechanics" is firmly entrenched.<ref>{{cite book |title = The enigma of probability and physics |last=Mayants |first=Lazar |year=1984 |publisher=Springer |isbn=978-90-277-1674-3 |page=174 |url = https://books.google.com/books?id=zmwEfXUdBJ8C&pg=PA174 }}</ref> 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.<ref name="gibbs" /> 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.<ref name="tolman" /> − 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<font color='red'><s>在《统计力学》出版了《基本原理》</s></font><font color='blue'>出版了《统计力学的基本原理Elementary Principles in Statistical Mechanics》</font>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<font color='red'><s>它们</s></font><font color='blue'>这些方法</font>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。+ − + − {{Books-inline|Fundamentals of Statistical Mechanics}}+ − − − + − + − + − + − + − + − + − {{Div col}}+ − − {{Books-inline|统计力学基础}} − − * [[热力学]]: [[非平衡态热力学]], [[化学热力学]] − − * [[力学]]: [[经典力学]], [[量子力学]] − − * [[概率]], [[统计系综（数学物理）]] − − * 数值方法: [[蒙特卡洛方法]], [[分子动力学]] − − * [[统计物理]] − − * [[量子统计力学]] − − * [[统计力学著名教科书列表]] − − * [[统计力学重要文献列表]] − − {{Div col end}} − − − − {{NoteFoot}} − − − − == References == − − {{Reflist − − {{Reflist − − {通货再膨胀 − − |refs = − − |refs = − − 参考文献 − − <ref name="gibbs">{{cite book |last=Gibbs |first=Josiah Willard |author-link=Josiah Willard Gibbs |title=Elementary Principles in Statistical Mechanics |year=1902 |publisher=[[Charles Scribner's Sons]] |location=New York |title-link=Elementary Principles in Statistical Mechanics }}</ref> − − <ref name="tolman">{{cite book | last=Tolman |first=R. C. | author-link = Richard C. Tolman | year=1938 | title = The Principles of Statistical Mechanics | publisher=[[Dover Publications]] | isbn = 9780486638966 }}</ref> − − <ref name="balescu">{{cite book | isbn = 9780471046004 | title = Equilibrium and Non-Equilibrium Statistical Mechanics |last = Balescu |first = Radu | author-link = Radu Balescu | year = 1975 | publisher = John Wiley & Sons }}</ref> − − <ref name="uffink">J. Uffink, "[http://philsci-archive.pitt.edu/2691/1/UffinkFinal.pdf Compendium of the foundations of classical statistical physics.]" (2006)</ref> − − }} − − }} − − }} − − − − == External links == − − {{Commons category|Statistical mechanics}}
无编辑摘要
<font color="#FF8000">统计力学 Statistical Mechanics</font>是现代物理学的支柱之一。对于具有多个<font color="#FF8000">自由度 Degrees of Freedom</font>的物理系统的基础研究，统计力学是不可或缺的。统计力学的方法是基于<font color="#FF8000">统计学方法 Statistical Methods</font>、<font color="#FF8000">概率论 Probability Theory</font>和<font color="#FF8000">微观物理定律 Microscopic Physical Laws</font>。
<font color="#FF8000">统计力学 </font>是现代物理学的支柱之一。对于具有多个<font color="#FF8000">自由度</font>的物理系统的基础研究，统计力学是不可或缺的。统计力学的方法是基于<font color="#FF8000">统计学方法</font>、<font color="#FF8000">概率论</font>和<font color="#FF8000">微观物理定律</font>。
统计力学一个分支可以用来解释大系统的热力学行为。该分支完善和扩展了经典热力学，被称为<font color="#FF8000">统计热力学</font>或<font color="#FF8000">平衡态统计力学</font>。
统计力学描述了宏观观测量(如温度和压强)与围绕平均值波动的微观参数的关系。它将热力学量(比如<font color="#FF8000">热容 Heat Capacity</font>)与微观行为联系起来。而在<font color="#FF8000">经典热力学 </font>中，唯一可行的选择就是测量和列出各种材料的热力学量。
统计力学也可以用来研究非平衡的系统。<font color="#FF8000">非平衡统计力学 </font>(有时称为统计动力学)是统计力学的重要分支，它涉及的问题是对由非平衡导致的不可逆过程的速度进行微观模拟。例如化学反应或粒子流和热流。<font color="#FF8000">涨落-耗散定理 </font>是人们从非平衡态统计力学中获得的基本知识，它是在应用非平衡态统计力学来研究多粒子系统中稳态电流流动这样的最简单的非平衡态情况下所发现的。
== 原理：力学和<font color="#FF8000">系综 </font>==
在物理学中，有两种力学被广泛研究: <font color="#FF8000">经典力学 </font>和<font color="#FF8000">量子力学 </font>。对于这两种力学，标准的数学方法与两个概念有关
力学系统在给定时间内的完整状态，用数学表示为<font color="#FF8000">相空间</font>中的点(经典力学)或<font color="#FF8000">纯量子态矢量</font>(量子力学)。
一个运动方程描述状态在时间上的演化: <font color="#FF8000">哈密尔顿方程</font>(经典力学)或<font color="#FF8000">含时薛定谔方程</font>(量子力学)
一个运动方程描述状态在时间上的演化: <font color="#FF8000">哈密尔顿方程 Hamilton's Equations</font>(经典力学)或<font color="#FF8000">含时薛定谔方程 Time-dependent Schrödinger Equation</font>(量子力学)
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。
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。
然而，这些定律与日常生活经验之间存在着脱节。因为对于在人类尺度上进行的过程(例如化学反应)，我们没有必要(甚至在理论上也不可能)在微观层面上准确地知道每个分子所在的位置及其速度。统计力学通过增加一些对于系统状态的不确定性，填补了力学定律和人类不完全知识的实践经验之间的脱节。
然而，这些定律与日常生活经验之间存在着脱节。因为对于在人类尺度上进行的过程(例如化学反应)，我们没有必要(甚至在理论上也不可能)在微观层面上准确地知道每个分子所在的位置及其速度。统计力学通过增加一些对于系统状态的不确定性，填补了力学定律和人类不完全知识的实践经验之间的脱节。
普通力学只考虑单一状态的行为，而统计力学引入了<font color="#FF8000">统计系综 </font>，它是系统在各种状态下的大量虚拟、独立的拷贝的集合。系综是一个覆盖系统所有可能状态的<font color="#FF8000">概率分布 </font>。在经典的统计力学中，系综是相点上的概率分布(与普通力学中的单相点相反) ，通常表现为<font color="#FF8000">正则坐标</font>下相空间中的分布。在量子统计力学中，系综是纯态上的概率分布，可以简单地概括为<font color="#FF8000">密度矩阵</font>。
普通力学只考虑单一状态的行为，而统计力学引入了<font color="#FF8000">统计系综 Statistical Ensemble</font>，它是系统在各种状态下的大量虚拟、独立的拷贝的集合。系综是一个覆盖系统所有可能状态的<font color="#FF8000">概率分布 Probability Distribution</font>。在经典的统计力学中，系综是相点上的概率分布(与普通力学中的单相点相反) ，通常表现为<font color="#FF8000">正则坐标 Canonical Coordinates</font>下相空间中的分布。在量子统计力学中，系综是纯态上的概率分布，可以简单地概括为<font color="#FF8000">密度矩阵 Density Matrix</font>。
与通常的概率一样，系综可以用不同的方式来解释:
与通常的概率一样，系综可以用不同的方式来解释:
* 系综可以表示"单个系统"的所有可能状态（<font color="#FF8000">认识概率</font>，知识的一种形式），或者
* 系综可以表示"单个系统"的所有可能状态（<font color="#FF8000">认识概率 Epistemic Probability</font>，知识的一种形式），或者
* 系综的元素可以理解为在无限次试验的极限下，在类似但不完全受控的独立系统中，重复进行实验得到的系统的状态（<font color="#FF8000">经验概率</font>）。
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。
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。
然而，这种概率是可被解释的，系综中的每个随时间演化的状态都可以由运动方程给出。因此，系综本身(状态的概率分布)也在随时间演化，因为系综中的虚拟系统不断地离开一个状态进入另一个状态。系综演化由<font color="#FF8000">刘维尔方程</font>(经典力学)或<font color="#FF8000">冯·诺依曼方程</font>(量子力学)给出。这些方程是简单地通过应用力学运动方程到系综中的每个虚拟系统而导出的，虚拟系统的概率随时间演化过程中是守恒的。
有一种特殊的系综是不随时间演化的。这样的系综称为<font color="#FF8000">平衡系综 </font>，它们的状态称为<font color="#FF8000">统计平衡 </font>。如果对于每个状态，无论是未来还是过去，该系综都包含在内，并且其概率等于处于该状态的概率，则出现统计平衡的情况。孤立系统的平衡系综是统计热力学研究的重点。非平衡统计力学研究更一般情况下的可以随时间演化的系综，以及（或）非孤立系统的系综。
==统计热力学==
统计热力学(也称为平衡态统计力学)的主要目标是根据组成某材料的粒子的性质和它们之间的相互作用，推导出材料的<font color="#FF8000">经典热力学 </font>。换句话说，统计热力学提供了热力学平衡态中物质的宏观性质与物质内部微观行为和运动之间的联系。
然而统计力学本身就涉及到动态变化，此时的关注点集中在统计平衡(稳态)上。统计平衡并不意味着粒子已经停止运动(力学平衡) ，相反，只是系综没有进化。
然而统计力学本身就涉及到动态变化，此时的关注点集中在统计平衡(稳态)上。统计平衡并不意味着粒子已经停止运动(力学平衡) ，相反，只是系综没有进化。
===基本假设===
=== Fundamental postulate 基本假设===
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。
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。
有许多不同的平衡系综可以考虑，但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。
有许多不同的平衡系综可以考虑，但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。
在许多教科书中常见的一种方法是采用<font color="#FF8000">先验概率相等假设</font>。这个假设表明
在许多教科书中常见的一种方法是采用<font color="#FF8000">先验概率相等假设 Equal A Priori Probability Postulate</font>。这个假设表明
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等概率的找到该系统。
对于一个已知精确能量和组成的孤立系统，可以在任何符合条件的微观状态下等概率的找到该系统。
因此，先验概率相等假设为下面描述的<font color="#FF8000">微正则系综 </font>提供了一个动力。有各种各样的论据支持先验概率相等假设:
* <font color="#FF8000">各态历经假设 </font>:各态历经系统是一种随着时间的演化而探索“所有可到达”状态的系统:所有具有相同能量和组成的状态。在各态历经系统中，微正则系综是唯一可能的具有固定能量的平衡系综。这种方法的适用性有限，因为大多数系统不是各态历经的。
* <font color="#FF8000">无差别原则 </font>: 在没有更多信息的情况下，我们只能对每一个相容的情况分配相等的概率。
* [[Ergodic hypothesis]]: An ergodic system is one that evolves over time to explore "all accessible" states: all those with the same energy and composition. In an ergodic system, the microcanonical ensemble is the only possible equilibrium ensemble with fixed energy. This approach has limited applicability, since most systems are not ergodic.
* <font color="#FF8000">最大热力学熵|最大信息熵</font>: 无差异原则的一个更详细的版本表明，正确的系综是与已知信息兼容且具有最大[[吉布斯熵]] ([[信息熵]])的系综。<ref>{{cite journal | last = Jaynes | first = E.| author-link = Edwin Thompson Jaynes | title = Information Theory and Statistical Mechanics | doi = 10.1103/PhysRev.106.620 | journal = Physical Review | volume = 106 | issue = 4 | pages = 620–630 | year = 1957 | pmid = | pmc = |bibcode = 1957PhRv..106..620J }}</ref>
其他关于统计力学的基本假设也有被提出。<ref name="uffink"/>
===三种热力学系综===
对于任何有限体积的<font color="#FF8000">孤立系统 </font>，可以定义三种简单形式的平衡系综。<ref name="gibbs"/>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。
<font color="#FF8000">微正则系综 </font>
<font color="#FF8000">微正则系综 Microcanonical Ensemble</font>
描述了一个具有精确给定能量和固定成分(精确数量的粒子)的系统。微正则系综中，与能量和组成相一致的每个可能状态的概率是相等的。
描述了一个具有精确给定能量和固定成分(精确数量的粒子)的系统。微正则系综中，与能量和组成相一致的每个可能状态的概率是相等的。
描述了一个固定成分的系统，这个系统与一个精确温度的热浴形成热平衡。正则系综包含能量不同但组成完全相同的状态; 根据总能量的不同，系综中不同的状态被赋予不同的概率。
描述了一个固定成分的系统，这个系统与一个精确温度的热浴形成热平衡。正则系综包含能量不同但组成完全相同的状态; 根据总能量的不同，系综中不同的状态被赋予不同的概率。
<font color="#FF8000">巨正则系综</font>
<font color="#FF8000">巨正则系综 Grand Canonical Ensemble</font>
描述了一个具有非固定成分(不确定粒子数)在热库中处于热力学和化学平衡的系统。热库具有精确的温度，各种类型的粒子具有精确的<font color="#FF8000">化学势 Chemical Potential</font>。巨正则系综包含不同能量和状态的大量粒子; 根据总能量和粒子数的不同，系综中不同状态的概率也不同。
描述了一个具有非固定成分(不确定粒子数)在热库中处于热力学和化学平衡的系统。热库具有精确的温度，各种类型的粒子具有精确的<font color="#FF8000">化学势 Chemical Potential</font>。巨正则系综包含不同能量和状态的大量粒子; 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。
对于包含大量粒子的系统(<font color="#FF8000">热力学极限 </font>) ，上面列出的三种系综都倾向于体现出相同的行为。<ref name="Reif">{{cite book | last = Reif | first = F. | title = Fundamentals of Statistical and Thermal Physics | publisher = McGraw–Hill | year = 1965 | isbn = 9780070518001 | page = [https://archive.org/details/fundamentalsofst00fred/page/227 227] | url-access = registration | url = https://archive.org/details/fundamentalsofst00fred/page/227 }}</ref> 因此，使用哪种系综只是一个简单的数学问题。<ref>{{cite journal |doi=10.1007/s10955-015-1212-2|title=Equivalence and Nonequivalence of Ensembles: Thermodynamic, Macrostate, and Measure Levels|journal=Journal of Statistical Physics|volume=159|issue=5|pages=987–1016|year=2015|last1=Touchette|first1=Hugo|arxiv=1403.6608}}</ref>发展成为<font color="#FF8000">测度现象<ref>{{cite book |doi=10.1090/surv/089|title=The Concentration of Measure Phenomenon|volume=89|series=Mathematical Surveys and Monographs|year=2005|isbn=9780821837924|last1=Ledoux|first1=Michel}}.</ref> 集中理论</font>系综等价的吉布斯定理，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。<ref>{{cite journal |doi=10.1098/rsta.2017.0237|pmc=5869543|title=Blessing of dimensionality: Mathematical foundations of the statistical physics of data|journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=376|issue=2118|pages=20170237|year=2018|last1=Gorban|first1=A. N.|last2=Tyukin|first2=I. Y.}}</ref>
热力学系综不能给出相同结果的重要例子包括:
热力学系综不能给出相同结果的重要例子包括:
* 微观系统
* 微观系统
* 长程关联的宏观系统
* 长程关联的宏观系统
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准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。<ref name="tolman" />
:{| class="wikitable sortable mw-collapsible mw-collapsed"
:{| class="wikitable sortable mw-collapsible mw-collapsed"
|}
|}
=== Calculation Methods 计算方法===
===计算方法===
一旦计算出一个系统的特征状态函数，该系统就被“解决”了(宏观观测量可以从特征状态函数中提取)。然而，计算热力学系综的特征状态函数并不一定是一项简单的工作，因为它涉及到考虑系统的每一种可能状态。虽然一些假设的系统已经被完全求解了，但是对最一般的(和现实的)情况进行精确的求解实在是太复杂了。存在各种方法来近似真实的系综，并且计算平均量。
一旦计算出一个系统的特征状态函数，该系统就被“解决”了(宏观观测量可以从特征状态函数中提取)。然而，计算热力学系综的特征状态函数并不一定是一项简单的工作，因为它涉及到考虑系统的每一种可能状态。虽然一些假设的系统已经被完全求解了，但是对最一般的(和现实的)情况进行精确的求解实在是太复杂了。存在各种方法来近似真实的系综，并且计算平均量。
====精确解====
====Exact 精确解====
有些情况可以得到精确解。
有些情况可以得到精确解。
* 对于非常小的微观系统，可以通过简单地列举系统所有可能状态（利用量子力学中的严格对角化，或者经典力学中对所有相空间积分）来直接得到系综。
* 对于非常小的微观系统，可以通过简单地列举系统所有可能状态（利用量子力学中的严格对角化，或者经典力学中对所有相空间积分）来直接得到系综。
* 对于包含很多分离的微观系统的宏观系统，每个子系统可以单独分析。尤其是粒子间无相互作用的理想气体具有这种性质，从而可以精确地得到[[<font color="#FF8000">麦克斯韦–玻尔兹曼统计 Maxwell–Boltzmann Statistics</font>]]， [[<font color="#FF8000">费米-狄拉克统计 Fermi–Dirac Statistics</font>]]，和[[<font color="#FF8000">波色-爱因斯坦统计 Bose–Einstein Statistics</font>]]。<ref name="tolman"/>
* 对于包含很多分离的微观系统的宏观系统，每个子系统可以单独分析。尤其是粒子间无相互作用的理想气体具有这种性质，从而可以精确地得到麦克斯韦–玻尔兹曼统计，费米-狄拉克统计，和波色-爱因斯坦统计。<ref name="tolman"/>
* 某些存在相互作用的宏观系统也存在精确解。通过运用微妙的数学技巧，已经找到了几个[[玩具模型]]的精确解。<ref>{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc. | pages = }}</ref> 一些例子包括[[Bethe ansatz]]，零场下的[[<font color="#FF8000"> 二维格点伊辛模型 Square-Lattice Ising Model</font>]]，[[<font color="#FF8000">硬六边形模型 Hard Hexagon Model</font>]]。
* 某些存在相互作用的宏观系统也存在精确解。通过运用微妙的数学技巧，已经找到了几个[[玩具模型]]的精确解。<ref>{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc. | pages = }}</ref> 一些例子包括[[Bethe ansatz]]，零场下的[[<font color="#FF8000"> 二维格点伊辛模型 Square-Lattice Ising Model</font>]]，[[<font color="#FF8000">硬六边形模型 Hard Hexagon Model</font>]]。
====Monte Carlo 蒙特卡罗方法====
====蒙特卡罗方法====
一个特别适合于计算机的近似方法是蒙特卡罗方法 ，它只随机选择(具有相当的权重)系统的几个可能状态进行检查。只要这些状态可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。
* 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。
==== 其他====
* 对于稀薄的非理想气体，团簇膨胀等方法使用微扰理论来涵盖弱相互作用的影响，得到维里展开。<ref name="balescu" />
* The [[Metropolis–Hastings algorithm]] is a classic Monte Carlo method which was initially used to sample the canonical ensemble.
* 对于稠密流体，另一种近似方法是基于简化的分布函数，特别是径向分布函数。<ref name="balescu"/>
* [[Path integral Monte Carlo]], also used to sample the canonical ensemble.
* 分子动力学计算机模拟可以用来计算各态历经系统中的[[微正则系综]]平均。对于包含与随机热浴有连接的系统，他们还可以在正则和巨正则条件下建模。
* 包含非平衡态统计力学结果（如下）的混合方法可能是很有用的。
* 包含非平衡态统计力学结果（如下）的混合方法可能是很有用的。
== Non-equilibrium statistical mechanics 非平衡态统计力学==
==非平衡态统计力学==
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:
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:
* [[Thermal conduction|heat transport by the internal motions in a material]], driven by a temperature imbalance,
* 热传导|材料内部的物质运动来传导热量，由热度不平衡来驱动，
* [[导电|导体内电荷的运动产生电流]]，由电压不平衡来驱动，
* 导电|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，
* 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]，
* 摩擦，耗散，量子退相干，
* 由外力泵送的系统([光泵]]等)
* 由外力泵送的系统光泵等
* 以及一般意义的可逆过程
* 以及一般意义的可逆过程
所有这些过程都是以特征速率随时间发生，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。)
所有这些过程都是以特征速率随时间发生，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。)
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底层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。
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底层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<font color="#FF8000">吉布斯熵 Gibbs entropy</font>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。
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。
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。
===随机方法===
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。
玻尔兹曼输运方程: 在动力学理论研究中，早期的随机力学甚至在“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)。
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。
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。
一个与主题相关的量子技术是随机相位近似。
层级:
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。层级结构(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 公式。NEGF—非平衡态格林函数):
| 3 Keldysh 公式。NEGF—非平衡态格林函数):
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。
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随机 刘维尔方程
| 4随机 刘维尔方程
===近平衡态方法 ===
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进行分析。涨落-耗散定理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
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。
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。
用于建立这种关联的一些理论工具包括:
用于建立这种关联的一些理论工具包括:
* [[Fluctuation–dissipation theorem]]
* 涨落-耗散定理
* [[Mori–Zwanzig 公式]]
* 昂萨格倒易关系
* 格林-库伯关系
* Ballistic conduction#Landauer-Büttiker formalism|Landauer–Büttiker 公式
* Mori–Zwanzig 公式
===组合方法===
==Applications outside thermodynamics==
一种先进的方法结合了随机方法和线性响应理论。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。<ref>{{Cite journal | last1 = Altshuler | first1 = B. L. | last2 = Aronov | first2 = A. G. | last3 = Khmelnitsky | first3 = D. E. | doi = 10.1088/0022-3719/15/36/018 | title = Effects of electron-electron collisions with small energy transfers on quantum localisation | journal = Journal of Physics C: Solid State Physics | volume = 15 | issue = 36 | pages = 7367 | year = 1982 | pmid = | pmc = |bibcode = 1982JPhC...15.7367A }}</ref><ref>{{Cite journal | last1 = Aleiner | first1 = I. | last2 = Blanter | first2 = Y. | doi = 10.1103/PhysRevB.65.115317 | title = Inelastic scattering time for conductance fluctuations | journal = Physical Review B | volume = 65 | issue = 11 | pages = 115317 | year = 2002 | pmid = | pmc = |arxiv = cond-mat/0105436 |bibcode = 2002PhRvB..65k5317A | url = http://resolver.tudelft.nl/uuid:e7736134-6c36-47f4-803f-0fdee5074b5a }}</ref>
==热力学以外的应用==
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:
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,<ref name="gibbs"/>
* 不确定性随时间传播，<ref name="gibbs"/>
* 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。
==历史==
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》《流体动力学》 ，这本书奠定了气体动力学理论的基础。在这项工作部著作中，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。
1859年，在阅读了 Rudolf Clausius 的一篇关于分子扩散的论文之后，苏格兰物理学家詹姆斯·克拉克·麦克斯韦提出了分子速度的麦克斯韦-玻尔兹曼分布，它给出了在一个特定范围内具有某种速度的分子的比例。这是物理学里第一个统计定律。麦克斯韦还提出了第一个力学论点，即分子碰撞必然导致温度的平衡，从而趋向平衡。五年之后，也就是1864年，维也纳的年轻学生路德维希·玻尔兹曼(Ludwig Boltzmann)偶然发现了麦克斯韦尔的论文，并花了大半辈子的时间来进一步研究这一课题。
1859年，在阅读了 Rudolf Clausius 的一篇关于分子扩散的论文之后，苏格兰物理学家詹姆斯·克拉克·麦克斯韦提出了分子速度的麦克斯韦-玻尔兹曼分布，它给出了在一个特定范围内具有某种速度的分子的比例。这是物理学里第一个统计定律。麦克斯韦还提出了第一个力学论点，即分子碰撞必然导致温度的平衡，从而趋向平衡。五年之后，也就是1864年，维也纳的年轻学生路德维希·玻尔兹曼(Ludwig Boltzmann)偶然发现了麦克斯韦尔的论文，并花了大半辈子的时间来进一步研究这一课题。
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。<ref>{{cite book |title = Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems |journal=Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems. Edited by Ebeling Werner & Sokolov Igor M. Published by World Scientific Press |volume=8 |last1=Ebeling |first1=Werner |last2=Sokolov |first2=Igor M. |year=2005 |isbn=978-90-277-1674-3 |pages=3–12 |url = https://books.google.com/books?id=KUjFHbid8A0C|bibcode=2005stst.book.....E |doi=10.1142/2012 |series = Series on Advances in Statistical Mechanics }} (section 1.2)</ref> 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、输运理论、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。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。
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《统计力学的基本原理》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，然而它们这些方法是如此的普遍，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 其他可见==
==其他可见==
{{Div col}}
{{Div col}}
* 热力学: 非平衡态热力学, 化学热力学
* [[Thermodynamics]]: [[Non-equilibrium thermodynamics|non-equilibrium]], [[Chemical thermodynamics|chemical]]
* [[Mechanics]]: [[Classical mechanics|classical]], [[Quantum mechanics|quantum]]
* 力学: 经典力学, 量子力学
* [[Probability]], [[Statistical ensemble (mathematical physics)|statistical ensemble]]
* 概率, 统计系综（数学物理）
* Numerical methods: [[Monte Carlo method]], [[molecular dynamics]]
* 数值方法: 蒙特卡洛方法, 分子动力学
* [[Statistical physics]]
* 统计物理
* [[Quantum statistical mechanics]]
* 量子统计力学
* [[List of notable textbooks in statistical mechanics]]
* 统计力学著名教科书列表
* [[List of publications in physics#Statistical mechanics|List of important publications in statistical mechanics]]
* 统计力学重要文献列表
{{Div col end}}
{{Div col end}}
==外链==
== Notes ==
* [http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics] article by Lawrence Sklar for the [[Stanford Encyclopedia of Philosophy]].
* [http://plato.stanford.edu/entries/statphys-statmech/ Philosophy of Statistical Mechanics] article by Lawrence Sklar for the [[Stanford Encyclopedia of Philosophy]].