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where ''i'' ranges over all possible molecular conditions, and where <math>!</math> denotes [[factorial]]. The "correction" in the denominator account for [[Identical particles|indistinguishable]] particles in the same condition.

where ''i'' ranges over all possible molecular conditions, and where <math>!</math> denotes [[factorial]]. The "correction" in the denominator account for [[Identical particles|indistinguishable]] particles in the same condition.

普朗克曾说：“熵和概率之间的对数关系是由玻尔兹曼在他的气体动力学理论中首次提出的”。<ref>Max Planck, p. 119.</ref>也就是著名的熵公式：<ref>The concept of [[entropy]] was introduced by [[Rudolf Clausius]] in 1865. He was the first to enunciate the [[second law of thermodynamics]] by saying that "entropy always increases".</ref><ref>An alternative is the [[Information entropy#Formal definitions|information entropy]] definition introduced in 1948 by [[Claude Elwood Shannon|Claude Shannon]].[https://archive.is/20070503225307/http://cm.bell-labs.com/cm/ms/what/shannonday/paper.html] It was intended for use in communication theory, but is applicable in all areas. It reduces to Boltzmann's expression when all the probabilities are equal, but can, of course, be used when they are not. Its virtue is that it yields immediate results without resorting to [[factorial]]s or [[Stirling's approximation]]. Similar formulas are found, however, as far back as the work of Boltzmann, and explicitly in [[H-theorem#Quantum mechanical H-theorem|Gibbs]] (see reference).</ref><math> S = k_B \ln W </math>，其中''k_B'' 是玻尔兹曼常数，''W'' 代表德文中宏观状态出现的概率，<ref>{{cite book|last=Pauli| first=Wolfgang| title=Statistical Mechanics|publisher=MIT Press|location=Cambridge|year=1973|isbn=978-0-262-66035-8}}, p. 21</ref>更准确一些来说，是对应于系统宏观状态的可能微观状态的数量——在一个系统的(可观察的)热力学状态下的(不可观测的)“方式”的数量，可以通过分配不同的位置和动量给不同的分子来实现。玻尔兹曼的范式是N个相同粒子的理想气体，其中Ni处于第i个微观位置和动量条件(范围)。''W''  可以用排列公式计算：<math> W = N! \prod_i \frac{1}{N_i!} </math>，其中''i'' 的范围包含所有可能的分子状态，<math>!</math>代表阶乘。分母中的“修正”解释了相同条件下难以区分的粒子。

普朗克曾说：“熵和概率之间的对数关系是由玻尔兹曼在他的气体动力学理论中首次提出的”。<ref>Max Planck, p. 119.</ref> 也就是著名的熵公式：<ref>The concept of [[entropy]] was introduced by [[Rudolf Clausius]] in 1865. He was the first to enunciate the [[second law of thermodynamics]] by saying that "entropy always increases".</ref><ref>An alternative is the [[Information entropy#Formal definitions|information entropy]] definition introduced in 1948 by [[Claude Elwood Shannon|Claude Shannon]].[https://archive.is/20070503225307/http://cm.bell-labs.com/cm/ms/what/shannonday/paper.html] It was intended for use in communication theory, but is applicable in all areas. It reduces to Boltzmann's expression when all the probabilities are equal, but can, of course, be used when they are not. Its virtue is that it yields immediate results without resorting to [[factorial]]s or [[Stirling's approximation]]. Similar formulas are found, however, as far back as the work of Boltzmann, and explicitly in [[H-theorem#Quantum mechanical H-theorem|Gibbs]] (see reference).</ref><math> S = k_B \ln W </math>，其中''k_B'' 是玻尔兹曼常数，''W'' 代表德文中宏观状态出现的概率，<ref>{{cite book|last=Pauli| first=Wolfgang| title=Statistical Mechanics|publisher=MIT Press|location=Cambridge|year=1973|isbn=978-0-262-66035-8}}, p. 21</ref>更准确一些来说，是对应于系统宏观状态的可能微观状态的数量——在一个系统的(可观察的)热力学状态下的(不可观测的)“方式”的数量，可以通过分配不同的位置和动量给不同的分子来实现。玻尔兹曼的范式是N个相同粒子的理想气体，其中Ni处于第i个微观位置和动量条件(范围)。''W''  可以用排列公式计算：<math> W = N! \prod_i \frac{1}{N_i!} </math>，其中''i'' 的范围包含所有可能的分子状态，<math>!</math>代表阶乘。分母中的“修正”解释了相同条件下难以区分的粒子。

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.

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.

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.

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.

其中 ''ƒ''  代表某一时刻单个粒子位置和动量的函数分布（见[[麦克斯韦-玻尔兹曼方程]]），''F'' 代表力，''m'' 代表粒子的质量，''t'' 是时间，''v'' 是粒子的平均速度。

其中 ''ƒ''  代表某一时刻单个粒子位置和动量的函数分布（见[[麦克斯韦-玻尔兹曼方程|麦克斯韦-玻尔兹曼分布]]），''F'' 代表力，''m'' 代表粒子的质量，''t'' 是时间，''v'' 是粒子的平均速度。

This equation describes the [[time|temporal]] and [[space|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.

This equation describes the [[time|temporal]] and [[space|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.

该方程描述了单粒子相空间中一团点密度分布的位置和动量概率分布的时空变化(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，而第二项给出空间变化，第三项描述作用于粒子的任何力响。方程右边表示碰撞的影响。

该方程描述了单粒子相空间中一团点密度分布的位置和动量概率分布的时空变化(见'''[[哈密顿力学]] [[Hamiltonian mechanics]]''')。左边的第一项表示分布函数的显式时间变化，而第二项给出空间变化，第三项描述作用于粒子的任何力响。方程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&nbsp;''ƒ''. The Boltzmann equation is notoriously difficult to [[Integral|integrate]]. [[David Hilbert]] spent years trying to solve it without any real success.

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&nbsp;''ƒ''. The Boltzmann equation is notoriously difficult to [[Integral|integrate]]. [[David Hilbert]] spent years trying to solve it without any real success.

The form of the collision term assumed by Boltzmann was approximate. However, for an ideal gas the standard [[Chapman–Enskog theory|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.

The form of the collision term assumed by Boltzmann was approximate. However, for an ideal gas the standard [[Chapman–Enskog theory|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 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 [[CPT symmetry|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 [[Johann Josef Loschmidt|Loschmidt]] and others over [[Loschmidt's paradox]] ultimately ended in his failure.

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 [[CPT symmetry|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 [[Johann Josef Loschmidt|Loschmidt]] and others over [[Loschmidt's paradox]] ultimately ended in his failure.