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第92行： − − <math> S = k_B \ln W </math> − − [ math ] s = k _ b ln w [ math ] − − where k<sub>B</sub> is Boltzmann's constant, and ln is the natural logarithm. W is Wahrscheinlichkeit, a German word meaning the probability of occurrence of a macrostate or, more precisely, the number of possible microstates corresponding to the macroscopic state of a system — the number of (unobservable) "ways" in the (observable) thermodynamic state of a system that can be realized by assigning different positions and momenta to the various molecules. Boltzmann's paradigm was an ideal gas of N identical particles, of which N<sub>i</sub> are in the ith microscopic condition (range) of position and momentum. W can be counted using the formula for permutations − − 其中 k < sub > b </sub > 是 Boltzmann 常数，ln 是自然对数。W 是 Wahrscheinlichkeit，一个德语单词，意思是发生宏观状态的可能性，或者更准确地说，是对应于系统宏观状态的可能的微观状态的数量---- 一个系统的可观测的热力学状态中的“方式”的数量，可以通过给各种分子分配不同的位置和动量来实现。玻耳兹曼的范式是全同粒子的理想气体，其中 n < sub > i </sub > 处于位置和动量的微观条件。W 可以用排列的公式来计算 − − − − <math> W = N! \prod_i \frac{1}{N_i!} </math> − − W = n！1} n i数学 − − − − where i ranges over all possible molecular conditions, and where <math>!</math> denotes factorial. The "correction" in the denominator account for indistinguishable particles in the same condition. − − 这里 i 可以覆盖所有可能的分子条件，这里 < math > ! </math > 表示阶乘。在相同条件下，分母中的“修正”解释了不可区分的粒子。 − − − − Boltzmann could also be considered one of the forerunners of quantum mechanics due to his suggestion in 1877 that the energy levels of a physical system could be discrete. − − 由于他在1877年提出物理系统的能级可以是离散的，Boltzmann 也可以被认为是量子力学的先驱之一。 − − − − − Boltzmann's bust in the courtyard arcade of the main building, University of Vienna. − − 在维也纳大学主楼的庭院拱廊中的玻尔兹曼的半身像。 − − − − − The Boltzmann equation was developed to describe the dynamics of an ideal gas. − − 玻尔兹曼方程是用来描述理想气体的动力学的。 − − − − <math> \frac{\partial f}{\partial t}+ v \frac{\partial f}{\partial x}+ \frac{F}{m} \frac{\partial f}{\partial v} = \frac{\partial f}{\partial t}\left.{\!\!\frac{}{}}\right|_\mathrm{collision} </math> − − { partial t } + v frac { partial f }{ partial x } + frac { f }{ m } frac { partial f }{{ partial v } = frac { partial f }{ partial t }.{ ! ! frac {}{}}右 | _ mathrm { collision } </math > − − − − where ƒ represents the distribution function of single-particle position and momentum at a given time (see the Maxwell–Boltzmann distribution), F is a force, m is the mass of a particle, t is the time and v is an average velocity of particles. − − F 表示一定时间内单个粒子的位置和动量的分布函数(见麦克斯韦-波兹曼分布) ，f 表示一个力，m 表示一个粒子的质量，t 表示时间，v 表示粒子的平均速度。 − − − − This equation describes the temporal and spatial variation of the probability distribution for the position and momentum of a density distribution of a cloud of points in single-particle phase space. (See Hamiltonian mechanics.) The first term on the left-hand side represents the explicit time variation of the distribution function, while the second term gives the spatial variation, and the third term describes the effect of any force acting on the particles. The right-hand side of the equation represents the effect of collisions. − − 这个方程描述了单粒子相空间中点云密度分布的位置和动量的概率分布的时间和空间变化。(参见哈密顿力学。)左边的第一项表示分布函数的显式时间变化，第二项表示空间变化，第三项描述作用在粒子上的任何力的效应。方程的右边表示碰撞的影响。 − − − − In principle, the above equation completely describes the dynamics of an ensemble of gas particles, given appropriate boundary conditions. This first-order differential equation has a deceptively simple appearance, since ƒ can represent an arbitrary single-particle distribution function. Also, the force acting on the particles depends directly on the velocity distribution function ƒ. The Boltzmann equation is notoriously difficult to integrate. David Hilbert spent years trying to solve it without any real success. − − 原则上，上述方程在给定适当的边界条件下，完全描述了气体粒子系综的动力学行为。这个一阶微分方程的外观看似简单，因为它可以表示任意的单粒子分布函数。作用在粒子上的力的大小直接取决于它的速度分布函数。众所周知，玻尔兹曼方程是难以整合的。大卫 · 希尔伯特花了数年时间试图解决这个问题，但没有取得任何真正的成功。 − − − − The form of the collision term assumed by Boltzmann was approximate. However, for an ideal gas the standard Chapman–Enskog solution of the Boltzmann equation is highly accurate. It is expected to lead to incorrect results for an ideal gas only under shock wave conditions. − − Boltzmann 采用的碰撞术语的形式是近似的。然而，对于理想气体，玻尔兹曼方程的标准 Chapman-Enskog 解决方案是高度准确的。只有在冲击波条件下，理想气体才有可能得到不正确的结果。 − − − − Boltzmann tried for many years to "prove" the second law of thermodynamics using his gas-dynamical equation — his famous H-theorem. However the key assumption he made in formulating the collision term was "molecular chaos", an assumption which breaks time-reversal symmetry as is necessary for anything which could imply the second law. It was from the probabilistic assumption alone that Boltzmann's apparent success emanated, so his long dispute with Loschmidt and others over Loschmidt's paradox ultimately ended in his failure. − − 多年来，Boltzmann 一直试图用他的气体动力学方程——著名的 h 定理——来“证明”热力学第二定律。然而，他在构造碰撞术语时所作的关键假设是“分子混沌” ，这个假设破坏了时间反转对称性，这对任何可能暗示第二定律的事物都是必要的。波尔兹曼表面上的成功仅仅来自于概率假设，因此他与洛施密特和其他人就洛施密特悖论的长期争论最终以他的失败而告终。 − −
无编辑摘要
| year = 1949
| year = 1949
| page = 300}}</ref>
| page = 300}}</ref>
路德维希·玻尔兹曼是奥地利物理学家和哲学家，1844年2月20日至1906年9月5日。他最伟大的成就是统计力学的发展，以及热力学第二定律的统计学解释。1877年，他提出了当前熵的定义，s = kblnω，被解释为一个系统的统计无序度量。马克斯 · 普朗克把这个常数命名为 kB，即波兹曼常数。[2]
Statistical mechanics is one of the pillars of modern [[physics]]. It 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">{{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>
Statistical mechanics is one of the pillars of modern [[physics]]. It 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">{{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>
==Biography==
==Biography==
===Childhood and education===
===Childhood and education===
Boltzmann was born in Erdberg, a suburb of [[Vienna]]. His father, Ludwig Georg Boltzmann, was a revenue official. His grandfather, who had moved to Vienna from Berlin, was a clock manufacturer, and Boltzmann's mother, Katharina Pauernfeind, was originally from [[Salzburg]]. He received his primary education at the home of his parents.<ref>{{cite book
Boltzmann was born in Erdberg, a suburb of [[Vienna]]. His father, Ludwig Georg Boltzmann, was a revenue official. His grandfather, who had moved to Vienna from Berlin, was a clock manufacturer, and Boltzmann's mother, Katharina Pauernfeind, was originally from [[Salzburg]]. He received his primary education at the home of his parents.<ref>{{cite book
|title=The Scientific 100
|title=The Scientific 100
|first1=John
|first1=John
|last1=Simmons
|last1=Simmons
|first2=Lynda
|first2=Lynda
|last2=Simmons
|last2=Simmons
|isbn=9780806536781
|isbn=9780806536781
|page=123
|page=123
|publisher=Kensington Publishing Corp.
|publisher=Kensington Publishing Corp.
|year=2000
|year=2000
}}</ref> Boltzmann attended high school in [[Linz]], [[Upper Austria]]. When Boltzmann was 15, his father died.<ref name=james2004>{{cite book
}}</ref> Boltzmann attended high school in [[Linz]], [[Upper Austria]]. When Boltzmann was 15, his father died.<ref name=james2004>{{cite book
|title=Remarkable Physicists: From Galileo to Yukawa
|title=Remarkable Physicists: From Galileo to Yukawa
|url=https://archive.org/details/remarkablephysic00jame
|url=https://archive.org/details/remarkablephysic00jame
|url-access=limited
|url-access=limited
|first1=Ioan
|first1=Ioan
|last1=James
|last1=James
|isbn=9780521017060
|isbn=9780521017060
|page=[https://archive.org/details/remarkablephysic00jame/page/n185 169]
|page=[https://archive.org/details/remarkablephysic00jame/page/n185 169]
|publisher=Cambridge University Press
|publisher=Cambridge University Press
|year=2004
|year=2004
}}</ref>
}}</ref>
Starting in 1863, Boltzmann studied [[mathematics]] and [[physics]] at the [[University of Vienna]]. He received his doctorate in 1866 and his [[venia legendi]] in 1869. Boltzmann worked closely with [[Josef Stefan]], director of the institute of physics. It was Stefan who introduced Boltzmann to [[James Clerk Maxwell|Maxwell's]] work.<ref name=james2004 />
Starting in 1863, Boltzmann studied [[mathematics]] and [[physics]] at the [[University of Vienna]]. He received his doctorate in 1866 and his [[venia legendi]] in 1869. Boltzmann worked closely with [[Josef Stefan]], director of the institute of physics. It was Stefan who introduced Boltzmann to [[James Clerk Maxwell|Maxwell's]] work.<ref name=james2004 />
===Academic career===
===Academic career===