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dN是在t时刻，关于（r，p）的微体积元<math> d^3\bf{r}</math>和微动量元<math> \mathrm{d}^3\bf{p}</math>内的分子数目。在位置空间和动量空间的一个区域上积分，得出在该区域中具有位置和动量的粒子总数：
dN是在t时刻，关于（r，p）的微体积元<math> d^3\bf{r}</math>和微动量元<math> \mathrm{d}^3\bf{p}</math>内的分子数目。在位置空间和动量空间的一个区域上积分，得出在该区域中具有位置和动量的粒子总数：
<math>N=\int d^{3}\mathbf{p}\int d^{3}\mathbf{r}f(\mathbf{r},\mathbf{p},t)=\iiint\iiint f(x,y,z,p_{x},p_{y},p_{z},t) \text{d}x\,\text{d}y\,\text{d}z\,\text{d}p_x\,\text{d}p_y\,\text{d}p_z</math>
which is a [[multiple integral|6-fold integral]]. While ''f'' is associated with a number of particles, the phase space is for one-particle (not all of them, which is usually the case with [[deterministic]] [[many body problem|many-body]] systems), since only one '''r''' and '''p''' is in question. It is not part of the analysis to use '''r'''<sub>1</sub>, '''p'''<sub>1</sub> for particle 1, '''r'''<sub>2</sub>, '''p'''<sub>2</sub> for particle 2, etc. up to '''r'''<sub>''N''</sub>, '''p'''<sub>''N''</sub> for particle ''N''.
which is a [[multiple integral|6-fold integral]]. While ''f'' is associated with a number of particles, the phase space is for one-particle (not all of them, which is usually the case with [[deterministic]] [[many body problem|many-body]] systems), since only one '''r''' and '''p''' is in question. It is not part of the analysis to use '''r'''<sub>1</sub>, '''p'''<sub>1</sub> for particle 1, '''r'''<sub>2</sub>, '''p'''<sub>2</sub> for particle 2, etc. up to '''r'''<sub>''N''</sub>, '''p'''<sub>''N''</sub> for particle ''N''.
is a shorthand for the momentum analogue of ∇, and '''ê'''<sub>''x''</sub>, '''ê'''<sub>''y''</sub>, '''ê'''<sub>''z''</sub> are [[cartesian coordinates|Cartesian]] [[unit vector]]s.
is a shorthand for the momentum analogue of ∇, and '''ê'''<sub>''x''</sub>, '''ê'''<sub>''y''</sub>, '''ê'''<sub>''z''</sub> are [[cartesian coordinates|Cartesian]] [[unit vector]]s.
===Final statement===
=== Final statement===
Dividing ({{EquationNote|3}}) by ''dt'' and substituting into ({{EquationNote|2}}) gives:
Dividing ({{EquationNote|3}}) by ''dt'' and substituting into ({{EquationNote|2}}) gives:
1-link = Leif Arkeryd
1-link = Leif Arkeryd
=== Two-body collision term ===
===Two-body collision term ===
| title= On the Boltzmann equation part II: The full initial value problem | journal= Arch. Rational Mech. Anal. | volume= 45 | issue= 1
| title= On the Boltzmann equation part II: The full initial value problem | journal= Arch. Rational Mech. Anal. | volume= 45 | issue= 1
类别: 统计力学
类别: 统计力学
=== Simplifications to the collision term ===
===Simplifications to the collision term===
where <math>\nu</math> is the molecular collision frequency, and <math>f_0</math> is the local Maxwellian distribution function given the gas temperature at this point in space.
where <math>\nu</math> is the molecular collision frequency, and <math>f_0</math> is the local Maxwellian distribution function given the gas temperature at this point in space.
<small>This page was moved from [[wikipedia:en:Boltzmann equation]]. Its edit history can be viewed at [[玻尔兹曼方程/edithistory]]</small></noinclude>
<small>This page was moved from [[wikipedia:en:Boltzmann equation]]. Its edit history can be viewed at [[玻尔兹曼方程/edithistory]]</small></noinclude>
== 通用方程（对于混合物） ==
==通用方程（对于混合物）==
For a mixture of chemical species labelled by indices i = 1, 2, 3, ..., n the equation for species i is
For a mixture of chemical species labelled by indices i = 1, 2, 3, ..., n the equation for species i is
== 应用与推广 ==
==应用与推广==
守恒方程
守恒方程
20世纪90年代物理宇宙学，完全协变方法被用于研究宇宙微波背景辐射。更一般地说，对早期宇宙过程的研究往往试图考虑量子力学和广义相对论的影响。
20世纪90年代物理宇宙学，完全协变方法被用于研究宇宙微波背景辐射。更一般地说，对早期宇宙过程的研究往往试图考虑量子力学和广义相对论的影响。
== 方程求解 ==
==方程求解==
this analytical approach provides insight, but is not generally usable in practical problems.
this analytical approach provides insight, but is not generally usable in practical problems.
在接近局部均衡的情况下，玻尔兹曼方程的解可以用一个克努森数的渐近展开表示(Chapman-Enskog 展开式)。这个展开式的前两项给出了欧拉方程和纳维－斯托克斯方程。较高的项有奇点。从原子观点(以玻尔兹曼方程为代表)到连续统运动定律的极限过程的数学发展问题，是希尔伯特第六个问题的重要组成部分。
在接近局部均衡的情况下，玻尔兹曼方程的解可以用一个克努森数的渐近展开表示(Chapman-Enskog 展开式)。这个展开式的前两项给出了欧拉方程和纳维－斯托克斯方程。较高的项有奇点。从原子观点(以玻尔兹曼方程为代表)到连续统运动定律的极限过程的数学发展问题，是希尔伯特第六个问题的重要组成部分。
== 另见 ==
==另见==
== 注释 ==
==注释==
<references />
<references />
== 参考文献 ==
==参考文献==
[[分类:偏微分方程]]
[[分类:偏微分方程]]
[[分类:统计力学]]
[[分类:统计力学]]