Historical Introduction

Statistical mechanics is a formalism that aims at explaining the physical properties of matter in bulk on the basis of the dynamical behavior of its microscopic constituents. The scope of the formalism is almost as unlimited as the very range of the natural phenomena, for in principle it is applicable to matter in any state whatsoever. It has, in fact, been applied, with considerable success, to the study of matter in the solid state, the liquid state, or the gaseous state, matter composed of several phases and/or several components, matter under extreme conditions of density and temperature, matter in equilibrium with radiation (as, for example, in astrophysics), matter in the form of a biological specimen, and so on. Furthermore, the formalism of statistical mechanics enables us to investigate the nonequilibrium states of matter as well as the equilibrium states; indeed, these investigations help us to understand the manner in which a physical system that happens to be “out of equilibrium” at a given time $t$ approaches a “state of equilibrium” as time passes.

统计力学是一种形式主义，旨在根据物质微观成分的动力学行为来解释大块物质的物理特性。 形式主义的范围几乎与自然现象的范围一样无限，因为原则上它适用于任何状态的物质。 事实上，它已被相当成功地应用于固态、液态或气态物质，由几相和/或几种成分组成的物质，密度和温度极端条件下的物质，与辐射平衡的物质（如天体物理学中的物质），生物标本形式的物质等的研究。 此外，统计力学的形式主义使我们能够研究物质的非平衡态以及平衡态；事实上，这些研究有助于我们理解在给定时间 $t$ 碰巧处于 “非平衡态” 的物理系统是如何随着时间的流逝而接近 “平衡态” 的。

In contrast with the present status of its development, the success of its applications, and the breadth of its scope, the beginnings of statistical mechanics were rather modest. Barring certain primitive references, such as those of Gassendi, Hooke, and so on, the real work on this subject started with the contemplations of Bernoulli (1738), Herapath (1821), and Joule (1851) who, in their own individual ways, attempted to lay a foundation for the so-called kinetic theory of gases — a discipline that finally turned out to be the forerunner of statistical mechanics. The pioneering work of these investigators established the fact that the pressure of a gas arose from the motion of its molecules and could, therefore, be computed by considering the dynamical influence of the molecular bombardment on the walls of the container. Thus, Bernoulli and Herapath could show that, if temperature remained constant, the pressure $P$ of an ordinary gas was inversely proportional to the volume $V$ of the container (Boyle’s law), and that it was essentially independent of the shape of the container. This, of course, involved the explicit assumption that, at a given temperature $T$, the (mean) speed of the molecules was independent of both pressure and volume. Bernoulli even attempted to determine the (first-order) correction to this law, arising from the finite size of the molecules, and showed that the volume $V$ appearing in the statement of the law should be replaced by $(V − b)$, where $b$ is the “actual” volume of the molecules.

与其发展现状、应用成功和范围广泛相比，统计力学的起步相当谦逊。 除了一些原始的参考文献，如 Gassendi、Hooke 等，这个主题的真正工作始于 Bernoulli（1738）、Herapath（1821）和 Joule（1851）的思考，他们各自以自己的方式试图为所谓的气体动力学理论奠定基础——这门学科最终成为统计力学的先驱。 这些调查人员的开创性工作确立了一个事实，即气体的压力来自其分子的运动，因此可以通过考虑分子轰击容器壁的动力学影响来计算。 因此，Bernoulli 和 Herapath 可以表明，如果温度保持恒定，普通气体的压力 $P$ 与容器的体积 $V$ 成反比（Boyle 定律），并且基本上与容器的形状无关。 当然，这涉及到明确的假设，即在给定温度 $T$ 下，分子的（平均）速度与压力和体积都无关。 Bernoulli 甚至试图确定由于分子的有限大小而产生的这个定律的（一阶）修正，并表明定律陈述中出现的体积 $V$ 应该被 $(V − b)$ 替换，其中 $b$ 是分子的 “实际” 体积。

Joule was the first to show that the pressure $P$ was directly proportional to the square of the molecular speed c, which he had initially assumed to be the same for all molecules. Kronig (1856) went a step further. Introducing the “quasistatistical” assumption that, at any time $t$, one-sixth of the molecules could be assumed to be flying in each of the six “independent” directions, namely $+x$,$−x$,$+y$,$−y$,$+z$, and $−z$, he derived the equation

$$ \begin{aligned} P = \frac{1}{3} nmc^{2},\tag{1} \end{aligned} $$

where $n$ is the number density of the molecules and m the molecular mass. Kronig, too, assumed the molecular speed $c$ to be the same for all molecules; so from (1), he inferred that the kinetic energy of the molecules should be directly proportional to the absolute temperature of the gas.

Joule 首先表明，压力 $P$ 与分子速度 $c$ 的平方成正比，他最初假设所有分子的速度相同。 Kronig（1856）更进一步。 引入 “准统计” 假设，即在任何时间 $t$，可以假设六分之一的分子在六个 “独立” 方向中的每一个中飞行，即 $+x$，$-x$，$+y$，$-y$，$+z$ 和 $-z$，他推导出方程

$$ \begin{aligned} P = \frac{1}{3} nmc^{2},\tag{1} \end{aligned} $$

其中 $n$ 是分子的数密度，$m$ 是分子质量。 Kronig 也假设分子速度 $c$ 对所有分子都是相同的；因此，从（1）中，他推断出分子的动能应该与气体的绝对温度成正比。

Kronig justified his method in these words: “The path of each molecule must be so irregular that it will defy all attempts at calculation. However, according to the laws of probability, one could assume a completely regular motion in place of a completely irregular one!” It must, however, be noted that it is only because of the special form of the summations appearing in the calculation of the pressure that Kronig’s argument leads to the same result as the one following from more refined models. In other problems, such as the ones involving diffusion, viscosity, or heat conduction, this is no longer the case.

Kronig 用这些话来证明他的方法：“每个分子的路径必须如此不规则，以至于它将使所有计算尝试都无法实现。 但是，根据概率定律，可以假设完全规则的运动代替完全不规则的运动！” 然而，必须注意的是，仅仅因为在计算压力时出现的求和的特殊形式，Kronig 的论点才会导致与从更精细的模型得出的结果相同。 在其他问题中，例如涉及扩散、粘度或传热的问题，情况就不再是这样了。

It was at this stage that Clausius entered the field. First of all, in 1857, he derived the ideal-gas law under assumptions far less stringent than Kronig’s. He discarded both leading ¨ assumptions of Kronig and showed that equation (1) was still true; of course, ¨ $c^{2}$ now became the mean square speed of the molecules. In a later paper (1859), Clausius introduced the concept of the mean free path and thus became the first to analyze transport phenomena. It was in these studies that he introduced the famous “Stosszahlansatz” — the hypothesis on the number of collisions (among the molecules) — which, later on, played a prominent role in the monumental work of Boltzmann.2 With Clausius, the introduction of the microscopic and statistical points of view into the physical theory was definitive, rather than speculative. Accordingly, Maxwell, in a popular article entitled “Molecules,” written for the Encyclopedia Britannica, referred to Clausius as the “principal founder of the kinetic theory of gases,” while Gibbs, in his Clausius obituary notice, called him the “father of statistical mechanics.”

正是在这个阶段，Clausius 进入了这个领域。 首先，在1857年，他在比 Kronig 更严格的假设下推导出了理想气体定律。 他抛弃了 Kronig 的两个主要假设，并表明方程（1）仍然成立；当然，$c^{2}$ 现在成为分子的平方平均速度。 在后来的一篇论文（1859年）中，Clausius引入了平均自由程的概念，从而成为第一个分析输运现象的人。 就是在这些研究中，他引入了著名的 “Stosszahlansatz”——关于碰撞次数（分子之间）的假设——这个假设后来在 Boltzmann 的重要工作中发挥了突出作用。 Clausius 引入了微观和统计观点到物理理论中，而不是推测性的。 因此，Maxwell 在为《大英百科全书》撰写的一篇名为 “分子” 的通俗文章中，将 Clausius 称为 “气体动力学理论的主要创始人”，而 Gibbs 在他对 Clausius 的讣告中称其为 “统计力学之父”。

The work of Clausius attracted Maxwell to the field. He made his first appearance with the memoir “Illustrations in the dynamical theory of gases” (1860), in which he went much farther than his predecessors by deriving his famous law of the “distribution of molecular speeds.” Maxwell’s derivation was based on elementary principles of probability and was clearly inspired by the Gaussian law of “distribution of random errors.” A derivation based on the requirement that “the equilibrium distribution of molecular speeds, once acquired, should remain invariant under molecular collisions” appeared in 1867. This led Maxwell to establish what is known as Maxwell’s transport equation which, if skilfully used, leads to the same results as one gets from the more fundamental equation due to Boltzmann.

Clausius 的工作吸引了 Maxwell 进入这个领域。 他首次亮相是在《气体动力学理论中的插图》（1860）一文中，他比他的前辈更进一步，推导出了他著名的 “分子速度分布定律”。 Maxwell 的推导是基于概率的基本原理，并明显受到 “随机误差分布” 的高斯定律的启发。 基于 “一旦获得分子速度的平衡分布，应该在分子碰撞下保持不变” 的要求，1867年出现了一种推导。 这使 Maxwell 建立了所谓的 Maxwell 输运方程，如果巧妙地使用，可以得到与由 Boltzmann 提出的更基本方程得到的结果相同。

Maxwell’s contributions to the subject diminished considerably after his appointment, in 1871, as the Cavendish Professor at Cambridge. By that time Boltzmann had already made his first strides. In the period 1868–1871 he generalized Maxwell’s distribution law to polyatomic gases, also taking into account the presence of external forces, if any; this gave rise to the famous Boltzmann factor $\text{exp}(−\beta\epsilon)$, where $\epsilon$ denotes the total energy of a molecule. These investigations also led to the equipartition theorem. Boltzmann further showed that, just like the original distribution of Maxwell, the generalized distribution (which we now call the Maxwell–Boltzmann distribution) is stationary with respect to molecular collisions.

Maxwell 在1871年被任命为剑桥大学 Cavendish 教授后，对这个主题的贡献大大减少。 那时，Boltzmann 已经迈出了他的第一步。 在1868年至1871年期间，他将 Maxwell 的分布定律推广到多原子气体，同时考虑了外部力的存在；这产生了著名的 Boltzmann 因子 $\text{exp}(−\beta\epsilon)$，其中 $\epsilon$ 表示分子的总能量。 这些研究还导致了能均分定理。 Boltzmann 进一步表明，就像 Maxwell 的原始分布一样，广义分布（我们现在称之为 Maxwell-Boltzmann 分布）在分子碰撞方面是稳定的。

In 1872 came the celebrated $H$-theorem, which provided a molecular basis for the natural tendency of physical systems to approach, and stay in, a state of equilibrium. This established a connection between the microscopic approach (which characterizes statistical mechanics) and the phenomenological approach (which characterized thermodynamics) much more transparently than ever before; it also provided a direct method for computing the entropy of a given physical system from purely microscopic considerations. As a corollary to the $H$-theorem, Boltzmann showed that the Maxwell–Boltzmann distribution is the only distribution that stays invariant under molecular collisions and that any other distribution, under the influence of molecular collisions, will ultimately go over to a Maxwell–Boltzmann distribution. In 1876 Boltzmann derived his famous transport equation, which, in the hands of Chapman and Enskog (1916–1917), has proved to be an extremely powerful tool for investigating macroscopic properties of systems in nonequilibrium states.

1872年出现了著名的 $H$ 定理，它为物理系统自然趋向于接近并保持在平衡状态提供了分子基础。 这比以往任何时候都更透明地建立了微观方法（表征统计力学）和现象学方法（表征热力学）之间的联系；它还提供了一种从纯粹的微观考虑中计算给定物理系统的熵的直接方法。 作为 $H$ 定理的推论，Boltzmann 表明 Maxwell-Boltzmann 分布是唯一在分子碰撞下保持不变的分布，任何其他分布在分子碰撞的影响下最终都会转变为 Maxwell-Boltzmann 分布。 1876年，Boltzmann 推导出了他著名的输运方程，由 Chapman 和 Enskog（1916-1917）使用，这已被证明是研究非平衡状态系统的宏观性质的极为强大的工具。

Things, however, proved quite harsh for Boltzmann. His $H$-theorem, and the consequent irreversible behavior of physical systems, came under heavy attack, mainly from Loschmidt (1876–1877) and Zermelo (1896). While Loschmidt wondered how the consequences of this theorem could be reconciled with the reversible character of the basic equations of motion of the molecules, Zermelo wondered how these consequences could be made to fit with the quasiperiodic behavior of closed systems (which arose in view of the so-called Poincare cycles). ´ Boltzmann defended himself against these attacks with all his might but, unfortunately, could not convince his opponents of the correctness of his viewpoint. At the same time, the energeticists, led by Mach and Ostwald, were criticizing the very (molecular) basis of the kinetic theory, while Kelvin was emphasizing the “nineteenth-century clouds hovering over the dynamical theory of light and heat.”

然而，对 Boltzmann 来说，事情变得非常艰难。 他的 $H$ 定理及其随之而来的物理系统的不可逆行为受到了严厉的攻击，主要来自 Loschmidt（1876-1877）和 Zermelo（1896）。 Loschmidt 怀疑这个定理的后果如何与分子的基本运动方程的可逆特性相协调，Zermelo 怀疑这些后果如何与封闭系统的准周期行为相吻合（这是由于所谓的 Poincare 循环）。 Boltzmann 全力以赴地抵御这些攻击，但不幸的是，他无法说服他的对手他的观点的正确性。 与此同时，由 Mach 和 Ostwald 领导的能量论者们批评了动力学理论的（分子）基础，而 Kelvin 强调了 “悬挂在光和热的动力学理论上的 19 世纪的云”。

All this left Boltzmann in a state of despair and induced in him a persecution complex. He wrote in the introduction to the second volume of his treatise Vorlesungen uber Gastheorie (1898):

I am convinced that the attacks (on the kinetic theory) rest on misunderstandings and that the role of the kinetic theory is not yet played out. In my opinion it would be a blow to science if contemporary opposition were to cause kinetic theory to sink into the oblivion which was the fate suffered by the wave theory of light through the authority of Newton. I am aware of the weakness of one individual against the prevailing currents of opinion. In order to insure that not too much will have to be rediscovered when people return to the study of kinetic theory I will present the most difficult and misunderstood parts of the subject in as clear a manner as I can.

所有这些让 Boltzmann 处于绝望状态，并在他身上引发了一种迫害情结。 他在他的著作 Vorlesungen uber Gastheorie（1898）第二卷的引言中写道：

我相信这些攻击（对动力学理论）是基于误解，动力学理论的作用尚未结束。 在我看来，如果当代的反对派导致动力学理论沉入牛顿权威所导致的遗忘中，这将对科学是一个打击。 我意识到一个人对主流舆论的逆流的弱点。 为了确保当人们重新研究动力学理论时不必重新发现太多，我将尽可能清楚地呈现这个主题中最困难和最被误解的部分。

We shall not dwell any further on the kinetic theory; we would rather move on to the development of the more sophisticated approach known as the ensemble theory, which may in fact be regarded as the statistical mechanics proper.9 In this approach, the dynamical state of a given system, as characterized by the generalized coordinates $q_{i}$ and the generalized momenta $p_{i}$, is represented by a phase point $G(q_{i},p_{i})$ in a phase space of appropriate dimensionality. The evolution of the dynamical state in time is depicted by the trajectory of the $G$-point in the phase space, the “geometry” of the trajectory being governed by the equations of motion of the system and by the nature of the physical constraints imposed on it. To develop an appropriate formalism, one considers the given system along with an infinitely large number of “mental copies” thereof; that is, an ensemble of similar systems under identical physical constraints (though, at any time $t$, the various systems in the ensemble would differ widely in respect of their dynamical states). In the phase space, then, one has a swarm of infinitely many $G$-points (which, at any time $t$, are widely dispersed and, with time, move along their respective trajectories). The fiction of a host of infinitely many, identical but independent, systems allows one to replace certain dubious assumptions of the kinetic theory of gases by readily acceptable statements of statistical mechanics. The explicit formulation of these statements was first given by Maxwell (1879) who on this occasion used the word “statistico-mechanical” to describe the study of ensembles (of gaseous systems) — though, eight years earlier, Boltzmann (1871) had already worked with essentially the same kind of ensembles.

我们将不再赘述动力学理论；我们将转向发展被称为系综理论的更复杂的方法，它实际上可以被视为统计力学的本体。在这种方法中，由广义坐标 $q_{i}$ 和广义矩量 $p_{i}$ 表征的给定系统的动力学状态由适当维度的相空间中的相点 $G(q_{i},p_{i})$ 表示。 $G$ 点在相空间中的轨迹描述了动力学状态在时间上的演化，轨迹的 “几何” 受系统运动方程和施加于系统的物理约束性质的制约。 为了建立一个适当的形式主义，我们需要考虑给定系统及其无限多的 “精神拷贝”；也就是说，在相同物理约束条件下的类似系统系综（不过，在任何时间 $t$，系综中的各种系统在其动力学状态方面会有很大差异）。 因此，在相空间中，存在着无限多的 $G$ 点群（在任何时间 $t$，这些点都非常分散，并随着时间的推移沿着各自的轨迹移动）。 虚构出无限多的相同但独立的系统，就可以用容易接受的统计力学陈述来取代气体动力学理论的某些可疑假设。 麦克斯韦（1879 年）首次明确提出了这些表述，他这次使用了 “统计力学” 一词来描述（气体系统）系综的研究–尽管早在八年前，玻尔兹曼（1871 年）就已经研究过基本相同的系综。

The most important quantity in the ensemble theory is the density function, $\rho(q_{i},p_{i};t)$, of the $G$-points in the phase space; a stationary distribution $(\partial\rho/\partial t = 0)$ characterizes a stationary ensemble, which in tum represents a system in equilibrium. Maxwell and Boltzmann confined their study to ensembles for which the function $\rho$ depended solely on the energy $E$ of the system. This included the special case of ergodic systems, which were so defined that “the undisturbed motion of such a system, if pursued for an unlimited time, would ultimately traverse (the neighborhood of) each and every phase point compatible with the fixed value $E$ of the energy.” Consequently, the ensemble average, $\langle f\rangle$, of a physical quantity $f$, taken at any given time $t$, would be the same as the long-time average, $\bar{f}$ , pertaining to any given member of the ensemble. Now, $\bar{f}$ is the value we expect to obtain for the quantity in question when we make an appropriate measurement on the system; the result of this measurement should, therefore, agree with the theoretical estimate $\langle f\rangle$. We thus acquire a recipe to bring about a direct contact between theory and experiment. At the same time, we lay down a rational basis for a microscopic theory of matter as an alternative to the empirical approach of thermodynamics!

系综理论中最重要的量是相空间中 $G$ 点的密度函数 $\rho(q_{i},p_{i};t)$；一个稳态分布 $(\partial\rho/\partial t = 0)$ 表征一个稳态系综，这反过来又代表一个处于平衡状态的系统。 麦克斯韦和玻尔兹曼将他们的研究限制在密度函数 $\rho$ 仅依赖于系统的能量 $E$ 的系综上。 这包括了能动系统的特殊情况，这些系统被定义为 “(这样定义的)，即 ‘如果这样的系统的不受干扰的运动持续无限时间，最终将历经（附近的）与能量 $E$ 的固定值相兼容的每一个相点。” 因此，物理量 $f$ 的系综平均值 $\langle f\rangle$，在任何给定时间 $t$，将与任何给定系综成员相关的长时平均值 $\bar{f}$ 相同。 现在，$\bar{f}$ 是我们期望在对系统进行适当测量时获得的量的值；因此，这个测量的结果应该与理论估计 $\langle f\rangle$ 一致。 因此，我们获得了一种方法，使理论与实验直接接触。 同时，我们为物质的微观理论奠定了一个理性基础，作为热力学经验方法的替代！

A significant advance in this direction was made by Gibbs who, with his Elementary Principles of Statistical Mechanics (1902), turned ensemble theory into a most efficient tool for the theorist. He emphasized the use of “generalized” ensembles and developed schemes which, in principle, enabled one to compute a complete set of thermodynamic quantities of a given system from purely mechanical properties of its microscopic constituents. In its methods and results, the work of Gibbs turned out to be much more general than any preceding treatment of the subject; it applied to any physical system that met the simple-minded requirements that (i) it was mechanical in structure and (ii) it obeyed Lagrange’s and Hamilton’s equations of motion. In this respect, Gibbs’s work may be considered to have accomplished for thermodynamics as much as Maxwell’s had accomplished for electrodynamics.

Gibbs 在他的《统计力学基本原理》（1902）中取得了这方面的重大进展，将系综理论转变为理论家最有效的工具。 他强调了 “广义” 系综的使用，并开发了方案，原则上使人能够从系统微观成分的纯机械性质计算给定系统的完整热力学量。 在方法和结果上，Gibbs 的工作比以往任何对待这个主题的处理都更为普遍；它适用于任何满足简单要求的物理系统，即 (i) 它的结构是机械的，(ii) 它遵循拉格朗日和哈密顿的运动方程。 在这方面，Gibbs 的工作可以被认为已经为热力学所做的事情与 Maxwell 为电动力学所做的事情一样多。

These developments almost coincided with the great revolution that Planck’s work of 1900 brought into physics. As is well known, Planck’s quantum hypothesis successfully resolved the essential mysteries of the black-body radiation — a subject where the three best-established disciplines of the nineteenth century, namely mechanics, electrodynamics, and thermodynamics, were all focused. At the same time, it uncovered both the strengths and the weaknesses of these disciplines. It would have been surprising if statistical mechanics, which linked thermodynamics with mechanics, could have escaped the repercussions of this revolution.

这些发展几乎与普朗克 1900 年的工作带来的物理学的巨大革命相吻合。 众所周知，普朗克的量子假设成功地解决了黑体辐射的基本谜团——一个 19 世纪三个最成熟的学科，即力学、电动力学和热力学，都集中在这个主题上。 同时，它揭示了这些学科的优点和缺点。 如果将热学与力学联系起来的统计力学能够逃脱这场革命的影响，那将是令人惊讶的。

The subsequent work of Einstein (1905a) on the photoelectric effect and of Compton (1923a,b) on the scattering of x-rays established, so to say, the “existence” of the quantum of radiation, or the photon as we now call it. It was then natural for someone to derive Planck’s radiation formula by treating black-body radiation as a gas of photons in the same way as Maxwell had derived his law of distribution of molecular speeds for a gas of conventional molecules. But, then, does a gas of photons differ so radically from a gas of conventional molecules that the two laws of distribution should be so different from one another?

爱因斯坦（1905a）关于光电效应的工作和康普顿（1923a，b）关于 X 射线散射的工作，可以说，确立了辐射量子或我们现在称之为光子的 “存在”。 于是，有人自然而然地通过将黑体辐射视为光子气体来推导普朗克的辐射公式，就像麦克斯韦通过将分子速度分布定律推导为传统分子气体一样。 但是，光子气体是否与传统分子气体有如此根本的不同，以至于这两个分布定律应该如此不同？

The answer to this question was provided by the manner in which Planck’s formula was derived by Bose. In his historic paper of 1924, Bose treated black-body radiation as a gas of photons; however, instead of considering the allocation of the “individual” photons to the various energy states of the system, he fixed his attention on the number of states that contained “a particular number” of photons. Einstein, who seems to have translated Bose’s paper into German from an English manuscript sent to him by the author, at once recognized the importance of this approach and added the following note to his translation: “Bose’s derivation of Planck’s formula is in my opinion an important step forward. The method employed here would also yield the quantum theory of an ideal gas, which I propose to demonstrate elsewhere.”

对这个问题的回答是由玻色提出普朗克公式的推导方式提供的。 在他 1924 年的历史性论文中，玻色将黑体辐射视为光子气体；然而，他没有考虑将 “个别” 光子分配给系统的各种能量状态，而是将注意力集中在包含 “特定数量” 光子的状态的数量上。 爱因斯坦似乎已经将玻色的论文从作者发给他的英文手稿翻译成德文，立即认识到了这种方法的重要性，并在他的翻译中添加了以下注释：“玻色对普朗克公式的推导在我看来是一个重要的进步。 这里使用的方法也将产生理想气体的量子理论，我打算在其他地方进行演示。”

Implicit in Bose’s approach was the fact that in the case of photons what really mattered was “the set of numbers of photons in various energy states of the system” and not the specification as to “which photon was in which state”; in other words, photons were mutually indistinguishable. Einstein argued that what Bose had implied for photons should be true for material particles as well (for the property of indistinguishability arose essentially from the wave character of these entities and, according to de Broglie, material particles also possessed that character). In two papers, which appeared soon after, Einstein (1924, 1925) applied Bose’s method to the study of an ideal gas and thereby developed what we now call Bose–Einstein statistics. In the second of these papers, the fundamental difference between the new statistics and the classical Maxwell–Boltzmann statistics comes out so transparently in terms of the indistinguishability of the molecules. In the same paper, Einstein discovered the phenomenon of Bose–Einstein condensation which, 13 years later, was adopted by London (1938a,b) as the basis for a microscopic understanding of the curious properties of liquid $\text{He}^{4}$ at low temperatures.

玻色的方法隐含的事实是，在光子的情况下，真正重要的是 “系统的各种能量状态中的光子数量集合”，而不是 “哪个光子在哪个状态” 的规定；换句话说，光子是相互不可区分的。 爱因斯坦认为，玻色对光子所暗示的对物质粒子也应该是真实的（因为不可区分性质基本上是由这些实体的波特性引起的，根据德布罗意，物质粒子也具有这种特性）。 在两篇随后出现的论文中，爱因斯坦（1924，1925）将玻色的方法应用于理想气体的研究，从而发展了我们现在称之为玻色-爱因斯坦统计的统计学。 在这两篇论文中的第二篇中，新统计学与经典的 Maxwell-Boltzmann 统计学之间的根本区别在于分子的不可区分性方面变得如此透明。 在同一篇论文中，爱因斯坦发现了玻色-爱因斯坦凝聚现象，13年后，伦敦（1938a，b）将其作为液体 $\text{He}^{4}$ 在低温下的奇特性质的微观理解的基础。

Following the enunciation of Pauli’s exclusion principle (1925), Fermi (1926) showed that certain physical systems would obey a different kind of statistics, namely the Fermi–Dirac statistics, in which not more than one particle could occupy the same energy state $(n_{i} = 0, 1)$. It seems important to mention here that Bose’s method of 1924 leads to the Fermi–Dirac distribution as well, provided that one limits the occupancy of an energy state to at most one particle.

在宣布了 Pauli 不相容原理（1925）之后，费米（1926）表明，某些物理系统将遵循一种不同的统计学，即费米-狄拉克统计学，其中不超过一个粒子可以占据相同的能量状态 $(n_{i} = 0, 1)$。 在这里似乎很重要提到，玻色 1924 年的方法也会导致费米-狄拉克分布，只要将能量状态的占用限制在最多一个粒子。

Soon after its appearance, the Fermi–Dirac statistics were applied by Fowler (1926) to discuss the equilibrium states of white dwarf stars and by Pauli (1927) to explain the weak, temperature-independent paramagnetism of alkali metals; in each case, one had to deal with a “highly degenerate” gas of electrons that obey Fermi–Dirac statistics. In the wake of this, Sommerfeld produced his monumental work of 1928 that not only put the electron theory of metals on a physically secure foundation but also gave it a fresh start in the right direction. Thus, Sommerfeld could explain practically all the major properties of metals that arose from conduction electrons and, in each case, obtained results that showed much better agreement with experiment than the ones following from the classical theories of Riecke (1898), Drude (1900), and Lorentz (1904–1905). Around the same time, Thomas (1927) and Fermi (1928) investigated the electron distribution in heavier atoms and obtained theoretical estimates for the relevant binding energies; these investigations led to the development of the so-called Thomas–Fermi model of the atom, which was later extended so that it could be applied to molecules, solids, and nuclei as well.

费米-狄拉克统计学一出现，费勒（1926）就将其应用于讨论白矮星的平衡状态，并由 Pauli（1927）解释了碱金属的弱、与温度无关的顺磁性；在每种情况下，都必须处理遵循费米-狄拉克统计学的 “高度简并” 电子气体。 在此之后，Sommerfeld 在 1928 年发表了他的重要工作，不仅使金属的电子理论在物理上得到了牢固的基础，而且使其朝着正确的方向重新开始。 因此，Sommerfeld 可以解释金属的所有主要性质，这些性质都来自导电电子，并且在每种情况下，得到的结果与从 Riecke（1898）、Drude（1900）和 Lorentz（1904-1905）的经典理论得到的结果相比，与实验的符合度要好得多。 大约在同一时间，Thomas（1927）和 Fermi（1928）研究了较重原子中的电子分布，并为相关的结合能得到了理论估计；这些研究导致了所谓的 Thomas-Fermi 原子模型的发展，后来这个模型被扩展，以便也可以应用于分子、固体和核。

Thus, the whole structure of statistical mechanics was overhauled by the introduction of the concept of indistinguishability of (identical) particles.16 The statistical aspect of the problem, which was already there in view of the large number of particles present, was now augmented by another statistical aspect that arose from the probabilistic nature of the wave mechanical description. One had, therefore, to carry out a two-fold averaging of the dynamical variables over the states of the given system in order to obtain the relevant expectation values. That sort of a situation was bound to necessitate a reformulation of the ensemble theory itself, which was carried out step by step. First, Landau (1927) and von Neumann (1927) introduced the so-called density matrix, which was the quantum-mechanical analogue of the density function of the classical phase space; this was elaborated, both from statistical and quantummechanical points of view, by Dirac (1929–1931). Guided by the classical ensemble theory, these authors considered both microcanonical and canonical ensembles; the introduction of grand canonical ensembles in quantum statistics was made by Pauli (1927).

因此，通过引入（相同）粒子不可区分性的概念，统计力学的整个结构得到了彻底的改造。问题的统计方面，由于存在大量粒子，已经存在，现在又增加了另一个统计方面，这是由于波动力学描述的概率性质而产生的。 因此，为了获得相关的期望值，必须对给定系统的状态进行动力学变量的双重平均。 这种情况必然需要对系综理论本身进行重新表述，这是逐步进行的。 首先，Landau（1927）和 von Neumann（1927）引入了所谓的密度矩阵，这是经典相空间密度函数的量子力学类比；这是由 Dirac（1929-1931）从统计和量子力学的角度详细阐述的。 在经典系综理论的指导下，这些作者考虑了微正则系综和正则系综；Pauli（1927）引入了量子统计中的大正则系综。

The important question as to which particles would obey Bose–Einstein statistics and which Fermi–Dirac remained theoretically unsettled until Belinfante (1939) and Pauli (1940) discovered the vital connection between spin and statistics.18 It turns out that those particles whose spin is an integral multiple of $\hbar$ obey Bose–Einstein statistics while those whose spin is a half-odd integral multiple of $\hbar$ obey Fermi–Dirac statistics. To date, no third category of particles has been discovered.

关于哪些粒子将遵循玻色-爱因斯坦统计学，哪些将遵循费米-狄拉克统计学的重要问题，在 Belinfante（1939）和 Pauli（1940）发现自旋与统计之间的重要联系之前，一直没有理论上解决。 结果表明，那些自旋是 $\hbar$ 的整数倍的粒子遵循玻色-爱因斯坦统计学，而那些自旋是 $\hbar$ 的半奇整数倍的粒子遵循费米-狄拉克统计学。 到目前为止，还没有发现第三类粒子。

Apart from the foregoing milestones, several notable contributions toward the development of statistical mechanics have been made from time to time; however, most of those contributions were concerned with the development or perfection of mathematical techniques that make application of the basic formalism to actual physical problems more fruitful. A review of these developments is out of place here; they will be discussed at their appropriate place in the text.

除了前述的里程碑，还有一些显著的贡献促进了统计力学的发展；然而，这些贡献大多涉及开发或完善数学技术，使基本形式主义的应用对实际物理问题更有成效。 这些发展的回顾在这里是不合适的；它们将在文本中适当的位置进行讨论。