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。