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The Boltzmann equation can be used to determine how physical quantities change, such as [[heat]] energy and [[momentum]], when a fluid is in transport. One may also derive other properties characteristic to fluids such as [[viscosity]], [[thermal conductivity]], and [[electrical conductivity]] (by treating the charge carriers in a material as a gas).<ref name="Encyclopaediaof" /> See also [[convection–diffusion equation]].

The Boltzmann equation can be used to determine how physical quantities change, such as [[heat]] energy and [[momentum]], when a fluid is in transport. One may also derive other properties characteristic to fluids such as [[viscosity]], [[thermal conductivity]], and [[electrical conductivity]] (by treating the charge carriers in a material as a gas).<ref name="Encyclopaediaof" /> See also [[convection–diffusion equation]].

在流体运输过程中，玻尔兹曼方程可以用来确定物理量如何变化，比如热能和动量。人们还可以推导出流体的其他特性，如粘度、热导率和电导率(通过将材料中的载流子当作气体来处理)。<ref name="Encyclopaediaof" /> 参见[[wikipedia:convection–diffusion equation|对流扩散方程 Convection–Diffusion Equation]]。

在流体运输过程中，玻尔兹曼方程可以用来确定物理量如何变化，比如热能和动量。人们还可以推导出流体的其他特性，如粘度、热导率和电导率(通过将材料中的载流子当作气体来处理)。<ref name="Encyclopaediaof" /> 参见[[wikipedia:convection–diffusion equation|'''对流扩散方程 Convection–Diffusion Equation''']]。

The equation is a [[Nonlinear system|nonlinear]] [[integro-differential equation]], and the unknown function in the equation is a probability density function in six-dimensional space of a particle position and momentum. The problem of existence and uniqueness of solutions is still not fully resolved, but some recent results are quite promising.<ref name=":0">DiPerna, R. J.; Lions, P.-L. (1989). "On the Cauchy problem for Boltzmann equations: global existence and weak stability". ''Ann. of Math''. 2. '''130''' (2): 321–366. doi:10.2307/1971423. JSTOR 1971423.

The equation is a [[Nonlinear system|nonlinear]] [[integro-differential equation]], and the unknown function in the equation is a probability density function in six-dimensional space of a particle position and momentum. The problem of existence and uniqueness of solutions is still not fully resolved, but some recent results are quite promising.<ref name=":0">DiPerna, R. J.; Lions, P.-L. (1989). "On the Cauchy problem for Boltzmann equations: global existence and weak stability". ''Ann. of Math''. 2. '''130''' (2): 321–366. doi:10.2307/1971423. JSTOR 1971423.

</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" />

玻尔兹曼方程是非线性积分微分方程，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'''₁, '''p'''₁ for particle 1, '''r'''₂, '''p'''₂ for particle 2, etc. up to '''r'''_''N'', '''p'''_''N'' 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'''₁, '''p'''₁ for particle 1, '''r'''₂, '''p'''₂ for particle 2, etc. up to '''r'''_''N'', '''p'''_''N'' 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'''₁, '''p'''₁代表粒子1,'''r'''₂, '''p'''₂代表粒子2，......，直到'''r'''_''N'', '''p'''_''N''代表粒子''N''，都不在考虑范围之内。

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'''₁, '''p'''₁代表粒子1,'''r'''₂, '''p'''₂代表粒子2，......，直到'''r'''_''N'', '''p'''_''N''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.

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]]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“[[Index.php?title=分子混沌假设|分子混沌假设 ]]”。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" />

:<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]]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: