更改

Note that we have used the fact that the phase space volume element <math> d^3\bf{r}</math>&nbsp;<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>&nbsp;'<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>&nbsp;<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>&nbsp;'<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[[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]]）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''']]）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]

其中 <math>\Delta f</math> 指的是<math>f</math>的总变化量。（{{EquationNote|1}}）式除以 <math> d^3\mathbf{r}\,d^3\mathbf{p}\,\Delta t</math> 并取极限 <math> \Delta t\,\rightarrow 0</math> 和 <math> \Delta f\,\rightarrow 0</math> 可得：{{NumBlk|2=<math>\frac{d f}{d t} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}</math>|3={{EquationRef|2}}|：}}

其中 <math>\Delta f</math> 指的是<math>f</math>的总变化量。（{{EquationNote|1}}）式除以 <math> d^3\mathbf{r}\,d^3\mathbf{p}\,\Delta t</math> 并取极限 <math> \Delta t\,\rightarrow 0</math> 和 <math> \Delta f\,\rightarrow 0</math> 可得：{{NumBlk|2=<math>\frac{d f}{d t} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}</math>|3={{EquationRef|2}}|：}}

The total [[differential of a function|differential]] of ''f'' is:

The total differential of ''f'' is:

<nowiki><math>f</math></nowiki>的全微分为：

<nowiki><math>f</math></nowiki>的全微分为：

where ∇ is the [[gradient]] operator, '''·''' is the [[dot product]],

where ∇ is the [[gradient]] operator, '''·''' is the [[dot product]],

其中''' ∇''' 为[[梯度]]算符，'''·''' 为[[点积]]，

其中''' ∇''' 为梯度算符，'''·''' 为点积，

:<math>

:<math>

is a shorthand for the momentum analogue of ∇, and '''ê'''_''x'', '''ê'''_''y'', '''ê'''_''z'' are [[cartesian coordinates|Cartesian]] [[unit vector]]s.

is a shorthand for the momentum analogue of ∇, and '''ê'''_''x'', '''ê'''_''y'', '''ê'''_''z'' are [[cartesian coordinates|Cartesian]] [[unit vector]]s.

是∇的动量类比的一个简写，'''ê'''_''x'', '''ê'''_''y'', '''ê'''_''z'' 为[[笛卡尔坐标系]]下的[[单位矢量]]。

是∇的动量类比的一个简写，'''ê'''_''x'', '''ê'''_''y'', '''ê'''_''z'' 为笛卡尔坐标系下的单位矢量。

===Final statement 最终形式===

===Final statement 最终形式===

In this context, '''F'''('''r''', ''t'') is the [[Force field (chemistry)|force field]] acting on the particles in the fluid, and ''m'' is the [[mass]] of the particles. The term on the right hand side is added to describe the effect of collisions between particles; if it is zero then the particles do not collide. The collisionless Boltzmann equation, where individual collisions are replaced with long-range aggregated interactions, e.g. [[Coulomb interaction]]s, is often called the [[Vlasov equation]].

In this context, '''F'''('''r''', ''t'') is the [[Force field (chemistry)|force field]] acting on the particles in the fluid, and ''m'' is the [[mass]] of the particles. The term on the right hand side is added to describe the effect of collisions between particles; if it is zero then the particles do not collide. The collisionless Boltzmann equation, where individual collisions are replaced with long-range aggregated interactions, e.g. [[Coulomb interaction]]s, is often called the [[Vlasov equation]].

这里，<math>F (\mathbf{r}, t)</math> 为流体中作用在粒子上的[[分子力场|力场]]，<math>m</math>为粒子[[质量]]。 右边的一项用于描述粒子间相互碰撞产生的影响；如果此项为零，则说明粒子之间没有碰撞。无碰撞情况下的玻尔兹曼方程常被称为{{le|弗拉索夫方程式|Vlasov equation}}。

这里，<math>F (\mathbf{r}, t)</math> 为流体中作用在粒子上的力场，<math>m</math>为粒子质量。 右边的一项用于描述粒子间相互碰撞产生的影响；如果此项为零，则说明粒子之间没有碰撞。无碰撞情况下的玻尔兹曼方程常被称为[[wikipedia:Vlasov equation|'''弗拉索夫方程式 Vlasov Equation''']]。

This equation is more useful than the principal one above, yet still incomplete, since ''f'' cannot be solved unless the collision term in ''f'' is known. This term cannot be found as easily or generally as the others – it is a statistical term representing the particle collisions, and requires knowledge of the statistics the particles obey, like the [[Maxwell–Boltzmann distribution|Maxwell–Boltzmann]], [[Fermi–Dirac distribution|Fermi–Dirac]] or [[Bose–Einstein distribution|Bose–Einstein]] distributions.

This equation is more useful than the principal one above, yet still incomplete, since ''f'' cannot be solved unless the collision term in ''f'' is known. This term cannot be found as easily or generally as the others – it is a statistical term representing the particle collisions, and requires knowledge of the statistics the particles obey, like the [[Maxwell–Boltzmann distribution|Maxwell–Boltzmann]], [[Fermi–Dirac distribution|Fermi–Dirac]] or [[Bose–Einstein distribution|Bose–Einstein]] distributions.

这个方程比上一节“主要论述”中的一般形式更加有用。然而这个方程依旧是不完整的：除非已知<math>f</math>中的碰撞项，否则<math>f</math>是解不出来的。这一项并不像其他项一样可以简单地或一般地得到——这一项是表示粒子的碰撞的'''统计项'''，需要知道粒子遵守怎样的统计规律，例如[[麦克斯韦-玻尔兹曼分布]]，[[费米-狄拉克统计|费米-狄拉克分布]]或[[玻色–爱因斯坦统计|玻色–爱因斯坦分布]]。

这个方程比上一节“主要论述”中的一般形式更加有用。然而这个方程依旧是不完整的：除非已知<math>f</math>中的碰撞项，否则<math>f</math>是解不出来的。这一项并不像其他项一样可以简单地或一般地得到——这一项是表示粒子的碰撞的'''统计项'''，需要知道粒子遵守怎样的统计规律，例如[[wikipedia:Maxwell–Boltzmann distribution|'''麦克斯韦-玻尔兹曼分布 Maxwell–Boltzmann Distribution''']]，[[wikipedia:Fermi–Dirac distribution|费米-狄拉克统计|'''费米-狄拉克分布 Fermi–Dirac Distribution''']]或[[wikipedia:Bose–Einstein distribution|玻色–爱因斯坦统计|玻色–爱因斯坦分布 '''Bose–Einstein Distribution''']]。

 

==The collision term (Stosszahlansatz) and molecular chaos 碰撞项（Stosszahlansatz）和分子混沌==

==The collision term (Stosszahlansatz) and molecular chaos 碰撞项（Stosszahlansatz）和分子混沌==