本科毕业设计（论文）
外文翻译
统计力学简史
作者：R.K.Pathria, Paul D.Beale
国籍：美国
出处：《统计力学》
统计力学是一种理论体系，旨在根据其微观成分的动力学行为来解释大块物质的物理特性。 这一理论体系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下的物质。 实际上，它已被成功地用于研究固态，液态或气态的物质，该物质由几种相或几种成分组成，在极端条件下均如此。密度和温度，与辐射保持平衡的物质（例如，在天体物理学中），生物样本形式的物质等。 此外，统计力学的这一理论体系使我们能够研究物质的非平衡态以及平衡态事实上，这些研究帮助我们理解，一个物理系统在给定时间t处于“不平衡”状态时，随着时间的推移，它是如何接近“平衡状态”的。
与它的发展现状，其应用的成功以及其范围的广度相比，统计力学的起步还比较缓慢。与它的发展现状，其应用的成功以及其范围的广度相比，统计力学的起步还比较缓慢。这些研究人员的开创性工作建立了这样一个事实，即气体的压力源自其分子的运动，因此可以通过考虑分子轰击对容器壁的动态影响来计算因此，Bernoulli和Herapath可以证明，如果温度保持恒定，则普通气体的压力P与容器的体积V成反比（鲍尔定律），并且基本上与容器的形状无关。当然，这涉及明确的假设，即在给定的温度T下，分子的（平均）速度与压力和体积无关。伯努利甚至试图确定由于分子的有限大小而引起的对该法则的（一阶）修正，并表明应将法则中出现的体积V替换为（V-b），其中b是分子的“实际”体积。焦耳最早显示出压力P与分子速度c的平方成正比，他最初假设所有分子均相同。克罗尼格（Kronig，1856年）研究更加深入。引入“准统计”假设，即在任何时候t都可以假设六分之一的分子在六个“独立”方向（即 x，-x， y，-y， z，-z）中的每个方向飞行，他推导出了方程。
其中n是分子的数量密度，m是分子质量。克罗尼格也假设所有分子的分子速度c是相同的。因此，他推断分子的动能应与气体的绝对温度成正比。
克罗尼格用以下这些话来证明他的方法是正确的：“每个分子的路径必须如此不规则，以至于无法进行所有的计算尝试。但是，根据概率定律，可以假设一个完全规则的运动来代替一个完全不规则的运动！”但是，必须指出的是，这仅是因为求和的特殊形式出现在计算中。克罗尼格的论证所导致的压力与从更精细的模型得出的结果相同，而在其他问题（例如涉及扩散，粘度或热传导的问题）中，情况已不再如此。
在这个阶段，克劳修斯进入了领域。首先。1857年，他根据远不如克罗尼格（Kronig）严格的假设推导出理想气体定律。他抛弃了罗尼格(Kronig)的两个主要假设，并证明等式仍然成立；当然，c2 现在变成了分子的均方速度。在后来的论文（1859年）中，克劳修斯（Clausius）引入了平均自由程的概念，从而成为分析运输现象的第一人。正是在这些研究中，他介绍了著名的“ Stosszahlansatz”（关于分子间碰撞数的假设），后来在玻尔兹曼的单方面思想工作中发挥了重要作用。^与克劳修斯一起，将微观和统计观点引入物理理论是确定的，而不是推测性的。因此，麦克斯韦（Maxwell）在为《大不列颠百科全书》撰写的标题为“分子”的热门文章中，称克劳修斯为“气体动力学理论的主要奠基人”，而吉布斯在克劳修斯的讣告中称其为“气体之父”
麦克斯韦于1871年被任命为剑桥卡文迪许教授后，对该学科的贡献大大减少。到那时，玻尔兹曼已经迈出了第一步。在1868-1871年间，他将麦克斯韦的分布定律推广到了多原子气体，同时也考虑了外力（如果有）的存在。这产生了著名的玻尔兹曼因子exp（-beta;ε），其中ε表示分子的总能量。这些研究也导致了等分定理。Boltzmann进一步证明，就像Maxwell的原始分布一样，广义分布（我们现在将其称为Maxwell-Boltzmann分布）对于分子碰撞而言是平稳的。
1872年出现了著名的H定理，它为物理系统趋近并保持平衡的自然趋势提供了分子基础。这在微观方法（表征统计力学）和现象学方法（表征热力学）之间建立了联系，比以往任何时候都更加透明；它还提供了一种直接的计算方法（纯物理上给定物理系统的熵）作为H定理的推论，玻尔兹曼证明了麦克斯韦-玻尔兹曼分布是唯一在分子碰撞下保持不变的分布，并且在分子碰撞的影响下，任何其他分布最终都将延续到1876年，玻尔兹曼推导了他著名的输运方程，在查普曼和恩斯科格（Chapman and Enskog，1916-1917）的手中，它被证明是研究非平衡态系统宏观性质的极其有力的工具。
然而，事实证明，对于玻尔兹曼来说，这是非常严酷的。他的H定理以及随之而来的物理系统不可逆行为受到了严重的攻击，主要来自Loschmidt（1876-1877）和Zermelo（1896）。而洛施密特（Loschmidt）则想知道，该定理的结果如何与分子基本运动方程的可逆性相一致。策尔梅洛想知道如何才能使这些结果与封闭系统的准周期性行为相适应（这是由于所谓的庞加莱循环而产生的）。玻尔兹曼竭尽全力为自己抵御这些攻击，但不幸的是，他无法说服反对者自己观点的正确性。同时，以马赫（Mach）和奥斯特瓦尔德（Ostwald）为首的高能主义者批评了动力学理论的（分子）基础，而开尔文则强调了“十九世纪的云团笼罩着光和热的动力学理论”。
这一切使玻尔兹曼陷入了绝望的境地，并给他造成了迫害。7他在他的专着《伏尔松根·伊伯·加泰罗理》（1898年）第二卷的序言中写道：
我相信，对动力学理论的攻击是基于误解，而动力学理论的作用还没有发挥出来。在我看来，如果当代的反对意见导致运动理论被遗忘，这将是对科学的一个打击，这是通过牛顿权威的光的波动理论所遭受的命运。我意识到一个人在主流观点面前的弱点。为了确保当人们回到运动理论的研究时，不会有太多的东西被重新发现，我将以尽可能清楚的方式呈现这个主题中最困难和被误解的部分。
我们将不再进一步讨论动力学理论：我们将继续发展称为系综理论的更复杂的方法，该方法实际上可以被视为统计力学。在这种方法中，动力学状态由广义坐标qj和广义矩Pi表征的给定系统的相位差由具有适当维数的相空间中的相位点G{qi,pi}表示。动态状态的演化通过相空间中G点的轨迹来描述，轨迹的“几何形状”由系统的运动方程式以及施加于其上的物理约束的性质控制。发展适当的理论体系。人们会考虑给定的系统及其无限多的“心理副本”；也就是说，在相同的物理约束下，类似系统的集合（尽管在任何时候t，集合中的各种系统的动态状态都将有很大差异）。这样，在相空间中，就有无数个G点（在任何时间t上，它们广泛分散并且随着时间沿着它们各自的轨迹移动）而组成的一群。大量无数相同，但独立的系统的构想允许人们用容易接受的统计力学陈述来代替气体动力学理论的某些可疑假设。这些陈述的明确表述首先是由麦克斯韦（Maxwell，1879年）提出的，在这种情况下，他使用“统计力学”一词来描述（气态系统）研究，尽管八年前，玻尔兹曼（1871）已经基本上使用了方法。
在系综理论中，最重要的量是相空间中G点的密度函数p（qi，pi； t）。固定分布（part;p/part;t= 0）表征了一个固定的集合体，该集合体又表示一个处于平衡状态的系统。Maxwell和Boltzmann将他们的研究仅限于集合，其功能完全取决于系统的能量E。这包括遍历系统的特殊情况，其定义如下：“如果对系统进行不受干扰的运动，并进行无限制的时间，则最终将遍历（固定）与E的固定值E相匹配的每个相点。因此，在任何给定时间t所获得的物理量f的集合平均lt;fgt;将与与该集合的任何给定成员有关的长期平均f相同。现在，f是在系统上进行适当测量时我们期望获得的有关量的值；因此，测量结果应与理论估计值lt;fgt;一致，因此我们获得了可以直接获得收益的方法理论与实验之间的联系；同时，我们为物质微观理论奠定了合理的基础，以替代热力学的经验方法。
吉布斯在这个方向上取得了重大进展，他以统计力学的基本原理（1902年）将系综理论变成了理论家最有效的工具。他强调使用“广义”合奏并开发了方案，从原理上讲，它使人们可以从其微观组成部分的纯机械性质计算给定系统的完整热力学量。在其方法和结果中，事实证明，吉布斯比以前对该受试者的治疗普遍得多。它适用于满足以下简单要求的任何物理系统：（i）在结构上是机械的；（ii）遵循拉格朗日和汉密尔顿的运动方程。在这方面，可以认为吉布斯的工作在热力学上已经完成，就象麦克斯韦在电动力学上所完成的一样。
Historical Introduction of the Statistical Mechanics
作者：R.K.Pathria and Paul D.Beale
国籍：America
出处：《Statistic Mechanics》
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.
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
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Historical Introduction of the Statistical Mechanics
作者：R.K.Pathria and Paul D.Beale
国籍：America
出处：《Statistic Mechanics》
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.
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 (Boyles 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.1
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
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.
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 Kronigs 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.
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 Kronigs. He discarded both leading assumptions of Kronig and showed that equation (1) was still true; of course, c2 now became the mean square speed of the molecules. In a later paper (1859), Clausius ihentroduced t 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) 一 which, later on, played a prominent role in the monumental work of Boltzmann.^ 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.'3
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.' Maxwells 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 Maxwells transport equation which, if sk
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