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}

这里注意我们知道相空间元 <math> d^3\mathbf{r}\,d^3\mathbf{p}</math> 是恒定的，这一事实可以从[[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]

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[[wikipedia:Maxwell–Boltzmann distribution|'''麦克斯韦-玻尔兹曼分布 Maxwell–Boltzmann Distribution''']]，[[wikipedia:Fermi–Dirac distribution|'''费米-狄拉克分布 Fermi–Dirac Distribution''']]或[[wikipedia:Bose–Einstein distribution|'''玻色–爱因斯坦分布 Bose–Einstein Distribution''']]。

这个方程比上一节中的一般形式更加有用，然而它依旧是不完整的：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轻而易举地得到——这一项是表示粒子碰撞的统计项，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-狄拉克分布 Fermi–Dirac Distribution''']]或[[wikipedia:Bose–Einstein distribution|'''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==

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 Ludwig Edward Boltzmann|玻尔兹曼]]在确定碰撞项时所应用到的关键见解就是：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“'''[[wikipedia:molecular chaos|分子混沌 Molecular Chaos]]假设'''”。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" />

[[路德维希·玻尔兹曼 Ludwig Edward Boltzmann|玻尔兹曼]]在确定碰撞项时所应用到的关键见解是：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。这个假设被玻尔兹曼称为“Stosszahlansatz”，也叫做“'''[[wikipedia:molecular chaos|分子混沌 Molecular Chaos]]'''假设”。根据这一假设，碰撞项可以被写作单粒子分布函数乘积在动量空间上的积分：<ref name="Encyclopaediaof" />

:<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[[wikipedia:relative velocity|'''相对速度 Relative Velocity''']]），<math>I(g, \Omega)</math> 是碰撞的[[wikipedia:differential cross section|'''微分散射截面 Differential Cross Section''']]，其中碰撞粒子的相对动量通过一个角θ变为[[wikipedia:solid angle|'''实心角 Solid Angle''']]dΩ的元。

指对应动量的大小（此概念参考[[wikipedia:relative velocity|'''相对速度 Relative Velocity''']]），<math>I(g, \Omega)</math> 是碰撞的[[wikipedia:differential cross section|'''微分散射截面 Differential Cross Section''']]，其中碰撞粒子的相对动量通过一个角θ变为[[wikipedia:solid angle|'''实心角 Solid Angle''']] dΩ的元。

===Simplifications to the collision term 对碰撞项的简化===

===Simplifications to the collision term 对碰撞项的简化===

Since much of the challenge in solving the Boltzmann equation originates with the complex collision term, attempts have been made to "model" and simplify the collision term.  The best known model equation is due to Bhatnagar, Gross and Krook.<ref name=":4">{{Cite journal|title=A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems|journal=Physical Review|date=1954-05-01|pages=511–525|volume=94|issue=3|doi=10.1103/PhysRev.94.511|first1=P. L.|last1=Bhatnagar|first2=E. P.|last2=Gross|first3=M.|last3=Krook|bibcode=1954PhRv...94..511B}}</ref>  The assumption in the BGK approximation is that the effect of molecular collisions is to force a non-equilibrium distribution function at a point in physical space back to a Maxwellian equilibrium distribution function and that the rate at which this occurs is proportional to the molecular collision frequency.  The Boltzmann equation is therefore modified to the BGK form:

Since much of the challenge in solving the Boltzmann equation originates with the complex collision term, attempts have been made to "model" and simplify the collision term.  The best known model equation is due to Bhatnagar, Gross and Krook.<ref name=":4">{{Cite journal|title=A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems|journal=Physical Review|date=1954-05-01|pages=511–525|volume=94|issue=3|doi=10.1103/PhysRev.94.511|first1=P. L.|last1=Bhatnagar|first2=E. P.|last2=Gross|first3=M.|last3=Krook|bibcode=1954PhRv...94..511B}}</ref>  The assumption in the BGK approximation is that the effect of molecular collisions is to force a non-equilibrium distribution function at a point in physical space back to a Maxwellian equilibrium distribution function and that the rate at which this occurs is proportional to the molecular collision frequency.  The Boltzmann equation is therefore modified to the BGK form:

由于求解波尔兹曼方程时，许多挑战都来自于其复杂的碰撞项；因此人们会对碰撞项做一些建模和简化的尝试。现知最好的模型是由Bhatnagar，Gross和Krook作出的BGK近似<ref name=":4" />。BGK近似假设分子碰撞的影响会迫使物理空间中某一点的非平衡分布函数回到麦克斯韦平衡分布函数，且其发生率正比于分子碰撞频率。于是，波尔兹曼方程可写作以下的BGK形式：

由于求解玻尔兹曼方程时，许多挑战都来自于其复杂的碰撞项；因此人们会对碰撞项做一些建模和简化的尝试。现知最好的模型是由Bhatnagar，Gross和Krook作出的BGK近似<ref name=":4" />。BGK近似假设分子碰撞的影响会迫使物理空间中某一点的非平衡分布函数回到麦克斯韦平衡分布函数，且其发生率正比于分子碰撞频率。于是，玻尔兹曼方程可写作以下的BGK形式：

:<math>\frac{\partial f}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla f + \mathbf{F} \cdot \frac{\partial f}{\partial \mathbf{p}} = \nu (f_0 - f),</math>

:<math>\frac{\partial f}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla f + \mathbf{F} \cdot \frac{\partial f}{\partial \mathbf{p}} = \nu (f_0 - f),</math>

Exact solutions to the Boltzmann equations have been proven to exist in some cases;<ref name=":9">Philip T. Gressman, Robert M. Strain (2011). "Global Classical Solutions of the Boltzmann Equation without Angular Cut-off". ''Journal of the American Mathematical Society''. '''24''' (3): 771. arXiv:1011.5441. doi:10.1090/S0894-0347-2011-00697-8. S2CID&nbsp;115167686.</ref> this analytical approach provides insight, but is not generally usable in practical problems.

Exact solutions to the Boltzmann equations have been proven to exist in some cases;<ref name=":9">Philip T. Gressman, Robert M. Strain (2011). "Global Classical Solutions of the Boltzmann Equation without Angular Cut-off". ''Journal of the American Mathematical Society''. '''24''' (3): 771. arXiv:1011.5441. doi:10.1090/S0894-0347-2011-00697-8. S2CID&nbsp;115167686.</ref> this analytical approach provides insight, but is not generally usable in practical problems.

在某些情况下，可以证明玻尔兹曼方程存在精确解<ref name=":9" />。这意味着，如果对服从波尔兹曼方程的系统施加一个微扰，此系统最终将回到平衡状态，而不是发散到无穷，或表现出其他的行为。然而，这种存在性证明无助于我们在现实问题中求解该方程。

在某些情况下，可以证明玻尔兹曼方程存在精确解<ref name=":9" />。这意味着，如果对服从玻尔兹曼方程的系统施加一个微扰，此系统最终将回到平衡状态，而不是发散到无穷，或表现出其他的行为。然而，这种存在性证明无助于我们在现实问题中求解该方程。

Instead, [[wikipedia:Numerical_methods_in_fluid_mechanics|numerical methods]] (including [[wikipedia:Finite_elements|finite elements]] and [[wikipedia:Lattice_Boltzmann_methods|lattice Boltzmann methods]]) are generally used to find approximate solutions to the various forms of the Boltzmann equation. Example applications range from [[wikipedia:Hypersonic_speed|hypersonic aerodynamics]] in rarefied gas flows<ref name=":10">{{Cite journal|title=A discontinuous finite element solution of the Boltzmann kinetic equation in collisionless and BGK forms for macroscopic gas flows|url=https://cronfa.swan.ac.uk/Record/cronfa6256|journal=Applied Mathematical Modelling|date=2011-03-01|pages=996–1015|volume=35|issue=3|doi=10.1016/j.apm.2010.07.027|first1=Ben|last1=Evans|first2=Ken|last2=Morgan|first3=Oubay|last3=Hassan}}</ref><ref name=":11">{{Cite journal|last1=Evans|first1=B.|last2=Walton|first2=S.P.|date=December 2017|title=Aerodynamic optimisation of a hypersonic reentry vehicle based on solution of the Boltzmann–BGK equation and evolutionary optimisation|journal=Applied Mathematical Modelling|volume=52|pages=215–240|doi=10.1016/j.apm.2017.07.024|issn=0307-904X|url=https://cronfa.swan.ac.uk/Record/cronfa34688}}</ref> to plasma flows.<ref name=":12">{{Cite journal|title=Numerical Solution of the Boltzmann Equation I: Spectrally Accurate Approximation of the Collision Operator|journal=SIAM Journal on Numerical Analysis|date=2000-01-01|issn=0036-1429|pages=1217–1245|volume=37|issue=4|doi=10.1137/S0036142998343300|first1=L.|last1=Pareschi|first2=G.|last2=Russo|citeseerx=10.1.1.46.2853}}</ref> An application of the Boltzmann equation in electrodynamics is the calculation of the electrical conductivity - the result is in leading order identical with the semiclassical result.<ref name=":13">H.J.W. Müller-Kirsten, Basics of Statistical Mechanics, Chapter 13, 2nd ed., World Scientific (2013), <nowiki>ISBN 978-981-4449-53-3</nowiki>. </ref>

Instead, [[wikipedia:Numerical_methods_in_fluid_mechanics|numerical methods]] (including [[wikipedia:Finite_elements|finite elements]] and [[wikipedia:Lattice_Boltzmann_methods|lattice Boltzmann methods]]) are generally used to find approximate solutions to the various forms of the Boltzmann equation. Example applications range from [[wikipedia:Hypersonic_speed|hypersonic aerodynamics]] in rarefied gas flows<ref name=":10">{{Cite journal|title=A discontinuous finite element solution of the Boltzmann kinetic equation in collisionless and BGK forms for macroscopic gas flows|url=https://cronfa.swan.ac.uk/Record/cronfa6256|journal=Applied Mathematical Modelling|date=2011-03-01|pages=996–1015|volume=35|issue=3|doi=10.1016/j.apm.2010.07.027|first1=Ben|last1=Evans|first2=Ken|last2=Morgan|first3=Oubay|last3=Hassan}}</ref><ref name=":11">{{Cite journal|last1=Evans|first1=B.|last2=Walton|first2=S.P.|date=December 2017|title=Aerodynamic optimisation of a hypersonic reentry vehicle based on solution of the Boltzmann–BGK equation and evolutionary optimisation|journal=Applied Mathematical Modelling|volume=52|pages=215–240|doi=10.1016/j.apm.2017.07.024|issn=0307-904X|url=https://cronfa.swan.ac.uk/Record/cronfa34688}}</ref> to plasma flows.<ref name=":12">{{Cite journal|title=Numerical Solution of the Boltzmann Equation I: Spectrally Accurate Approximation of the Collision Operator|journal=SIAM Journal on Numerical Analysis|date=2000-01-01|issn=0036-1429|pages=1217–1245|volume=37|issue=4|doi=10.1137/S0036142998343300|first1=L.|last1=Pareschi|first2=G.|last2=Russo|citeseerx=10.1.1.46.2853}}</ref> An application of the Boltzmann equation in electrodynamics is the calculation of the electrical conductivity - the result is in leading order identical with the semiclassical result.<ref name=":13">H.J.W. Müller-Kirsten, Basics of Statistical Mechanics, Chapter 13, 2nd ed., World Scientific (2013), <nowiki>ISBN 978-981-4449-53-3</nowiki>. </ref>