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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联系是由玻尔兹曼在他的气体动力学理论中首次提出的”。也就是著名的熵公式：<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，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”。也就是著名的熵公式：<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，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.

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.

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.