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→The force and diffusion terms “force”项与“diff”项
</math>
</math>
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{{NumBlk|2=<math>\begin{align}
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}
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]
& =\Delta f d^{3}\textbf{r}\, d^{3}\textbf{p}
& =\Delta f d^{3}\textbf{r}\, d^{3}\textbf{p}
\end{align}</math>|3={{EquationRef|1}}|：}}where Δ''f'' is the ''total'' change in ''f''. Dividing ({{EquationNote|1}}) by <math> d^3\bf{r}</math> <math> d^3\bf{p}</math> Δ''t'' and taking the limits Δ''t'' → 0 and Δ''f'' → 0, we have{{NumBlk|2=<math>\frac{d f}{d t} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}</math>|3={{EquationRef|2}}|：}}
\end{align}</math>|3={{EquationRef|1}}|：}}where Δ''f'' is the ''total'' change in ''f''. Dividing ({{EquationNote|1}}) by <math> d^3\bf{r}</math> <math> d^3\bf{p}</math> Δ''t'' and taking the limits Δ''t'' → 0 and Δ''f'' → 0, we have
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 a function|differential]] of ''f'' is:
<nowiki><math>f</math></nowiki>的全微分为：
{{NumBlk|:|<math>\begin{align}
{{NumBlk|:|<math>\begin{align}
d f & = \frac{\partial f}{\partial t} \, dt
d f & = \frac{\partial f}{\partial t} \, dt
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]]，
:<math>
:<math>
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===
是∇的动量类比的一个简写，'''ê'''<sub>''x''</sub>, '''ê'''<sub>''y''</sub>, '''ê'''<sub>''z''</sub> 为[[笛卡尔坐标系]]下的[[单位矢量]]。
===Final statement 最终形式===
Dividing ({{EquationNote|3}}) by ''dt'' and substituting into ({{EquationNote|2}}) gives:
Dividing ({{EquationNote|3}}) by ''dt'' and substituting into ({{EquationNote|2}}) gives:
对（{{EquationNote|3}}）两边同除以''dt'' 并代入（{{EquationNote|2}}）可得：
:<math>\frac{\partial f}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla f + \mathbf{F} \cdot \frac{\partial f}{\partial \mathbf{p}} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}</math>
:<math>\frac{\partial f}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla f + \mathbf{F} \cdot \frac{\partial f}{\partial \mathbf{p}} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}</math>
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}}。
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]]。
==The collision term (Stosszahlansatz) and molecular chaos 碰撞项（Stosszahlansatz）和分子混沌==
===Two-body collision term 双体碰撞项===
===Two-body collision term===
A key insight applied by [[Ludwig Boltzmann|Boltzmann]] was to determine the collision term resulting solely from two-body collisions between particles that are assumed to be uncorrelated prior to the collision. This assumption was referred to by Boltzmann as the "''Stosszahlansatz'' " and is also known as the "[[molecular chaos]] assumption". Under this assumption the collision term can be written as a momentum-space integral over the product of one-particle distribution functions:<ref name="Encyclopaediaof" />
A key insight applied by [[Ludwig Boltzmann|Boltzmann]] was to determine the collision term resulting solely from two-body collisions between particles that are assumed to be uncorrelated prior to the collision. This assumption was referred to by Boltzmann as the "''Stosszahlansatz'' " and is also known as the "[[molecular chaos]] assumption". Under this assumption the collision term can be written as a momentum-space integral over the product of one-particle distribution functions:<ref name="Encyclopaediaof" />
[[路德维希·玻尔兹曼|玻尔兹曼]]的一个关键见解就是对碰撞项的确定。他假设的碰撞项完全是由假定在碰撞前不相关的两个粒子的相互碰撞得到的。这个假设被波尔兹曼称为“Stosszahlansatz”，也叫做“{{le|分子混沌假设|Molecular chaos}}”。根据这一假设，碰撞项可以被写作单粒子分布函数的乘积在动量空间上的积分：<ref name="Encyclopaediaof"/>
:<math>
:<math>
where '''p'''<sub>''A''</sub> and '''p'''<sub>''B''</sub> are the momenta of any two particles (labeled as ''A'' and ''B'' for convenience) before a collision, '''p′'''<sub>''A''</sub> and '''p′'''<sub>''B''</sub> are the momenta after the collision,
where '''p'''<sub>''A''</sub> and '''p'''<sub>''B''</sub> are the momenta of any two particles (labeled as ''A'' and ''B'' for convenience) before a collision, '''p′'''<sub>''A''</sub> and '''p′'''<sub>''B''</sub> are the momenta after the collision,
其中 <math>\mathbf{p}_A</math> 和 <math>\mathbf{p}_B</math> 表示碰撞前任意两个粒子的动量（为了方便而标记为<math>A</math>和<math>B</math>）， <math>\mathbf{p}'_A</math> 和 <math>\mathbf{p}'_B</math> 表示碰撞后的动量，
:<math>
:<math>
is the magnitude of the relative momenta (see [[relative velocity]] for more on this concept), and ''I''(''g'', Ω) is the [[differential cross section]] of the collision, in which the relative momenta of the colliding particles turns through an angle θ into the element of the [[solid angle]] ''d''Ω, due to the collision.
is the magnitude of the relative momenta (see [[relative velocity]] for more on this concept), and ''I''(''g'', Ω) is the [[differential cross section]] of the collision, in which the relative momenta of the colliding particles turns through an angle θ into the element of the [[solid angle]] ''d''Ω, due to the collision.
指对应动量的大小（此概念参考[[相對速度|相对速度]]），<math>I(g, \Omega)</math> 是碰撞的[[截面 (物理)|微分散射截面]]，其中碰撞粒子的相对动量通过一个角θ变为[[实心角]]''d''Ω的元。
===Simplifications to the collision term===
===Simplifications to the collision term===