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第19行： − 玻尔兹曼方程是[[wikipedia:Nonlinear system integro-differential equation|非线性积分微分方程 Nonlinear Integro-Differential Equation]]，方程中的未知函数是位置和动量在六维空间中的概率密度函数。方程解的存在唯一性仍然是未完全解决的问题，但是一些研究显示解决这一问题是很有希望的。<ref name=":0" /><ref name=":1" />+ 第49行：
第49行： − 虽然f与一群粒子有关，但相空间是针对单一粒子进行讨论(对于所有粒子的分析通常是确定性多体系统的情况)，因为只有一个r和p是需要考虑的。使用r1, p1代表粒子1,r2, p2代表粒子2，......，直到rN, pN代表粒子N，都不在考虑范围之内。+ 第97行：
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</ref><ref name=":1">Philip T. Gressman & Robert M. Strain (2010). "Global classical solutions of the Boltzmann equation with long-range interactions". ''Proceedings of the National Academy of Sciences''. '''107''' (13): 5744–5749. arXiv:1002.3639. Bibcode:2010PNAS..107.5744G. doi:10.1073/pnas.1001185107. PMC 2851887. <nowiki>PMID 20231489</nowiki>.</ref>
</ref><ref name=":1">Philip T. Gressman & Robert M. Strain (2010). "Global classical solutions of the Boltzmann equation with long-range interactions". ''Proceedings of the National Academy of Sciences''. '''107''' (13): 5744–5749. arXiv:1002.3639. Bibcode:2010PNAS..107.5744G. doi:10.1073/pnas.1001185107. PMC 2851887. <nowiki>PMID 20231489</nowiki>.</ref>
玻尔兹曼方程是非线性积分微分方程 Nonlinear Integro-Differential Equation，方程中的未知函数是位置和动量在六维空间中的概率密度函数。方程解的存在唯一性仍然是未完全解决的问题，但是一些研究显示解决这一问题是很有希望的。<ref name=":0" /><ref name=":1" />
==Overview 概述==
==Overview 概述==
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''.
这是一个六重积分。虽然f与一群粒子有关，但相空间是针对单一粒子进行讨论(对于所有粒子的分析通常是确定性[[wikipedia:many body problem|多体系统 Many-Body]]的情况)，因为只有一个'''r'''和'''p'''是需要考虑的。使用'''r'''<sub>1</sub>, '''p'''<sub>1</sub>代表粒子1,'''r'''<sub>2</sub>, '''p'''<sub>2</sub>代表粒子2，......，直到'''r'''<sub>''N''</sub>, '''p'''<sub>''N''</sub>代表粒子''N''，都不在考虑范围之内。
It is assumed the particles in the system are identical (so each has an identical [[mass]] ''m''). For a mixture of more than one [[chemical species]], one distribution is needed for each, see below.
It is assumed the particles in the system are identical (so each has an identical [[mass]] ''m''). For a mixture of more than one [[chemical species]], one distribution is needed for each, see below.
Note that we have used the fact that the phase space volume element <math> d^3\bf{r}</math> <math> d^3\bf{p}</math> is constant, which can be shown using [[Hamilton's equations]] (see the discussion under [[Liouville's theorem (Hamiltonian)|Liouville's theorem]]). However, since collisions do occur, the particle density in the phase-space volume <math> d^3\bf{r}</math> '<math> d^3\bf{p}</math> changes, so
Note that we have used the fact that the phase space volume element <math> d^3\bf{r}</math> <math> d^3\bf{p}</math> is constant, which can be shown using [[Hamilton's equations]] (see the discussion under [[Liouville's theorem (Hamiltonian)|Liouville's theorem]]). However, since collisions do occur, the particle density in the phase-space volume <math> d^3\bf{r}</math> '<math> d^3\bf{p}</math> changes, so
这里，注意到相空间元 <math> d^3\mathbf{r}\,d^3\mathbf{p}</math> 是恒定的这个事实可以从[[哈密顿方程]]（见[[刘维尔定理 (哈密顿力学)|刘维尔定理]]）得知。然而，由于存在碰撞，相空间元 <math> d^3\mathbf{r}\,d^3\mathbf{p}</math> 中的粒子密度是可变的，所以有：{{NumBlk|2=<math>\begin{align}
这里，注意到相空间元 <math> d^3\mathbf{r}\,d^3\mathbf{p}</math> 是恒定的这个事实可以从[[wikipedia:Hamilton's equations|哈密顿方程 Hamilton's Equations]]（见[[wikipedia:Liouville's theorem (Hamiltonian)|刘维尔定理 Liouville's Theorem（哈密顿力学）]]）得知。然而，由于存在碰撞，相空间元 <math> d^3\mathbf{r}\,d^3\mathbf{p}</math> 中的粒子密度是可变的，所以有：{{NumBlk|2=<math>\begin{align}
dN_{coll} &= \left ( \frac{\partial f}{\partial t} \right )_{coll}\Delta td^{3}\textbf{r}\, d^{3}\textbf{p}\\[5pt]
dN_{coll} &= \left ( \frac{\partial f}{\partial t} \right )_{coll}\Delta td^{3}\textbf{r}\, d^{3}\textbf{p}\\[5pt]
& = f\left ( \textbf{r}+\frac{\textbf{p}}{m}\Delta t,\textbf{p}+\textbf{F}\Delta t,t+\Delta t \right )\, d^{3}\textbf{r}\, d^{3}\textbf{p}- f(\textbf{r},\textbf{p},t)\, d^{3}\textbf{r}\, d^{3}\textbf{p}\\[5pt]
& = f\left ( \textbf{r}+\frac{\textbf{p}}{m}\Delta t,\textbf{p}+\textbf{F}\Delta t,t+\Delta t \right )\, d^{3}\textbf{r}\, d^{3}\textbf{p}- f(\textbf{r},\textbf{p},t)\, d^{3}\textbf{r}\, d^{3}\textbf{p}\\[5pt]