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===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:

求解波尔兹曼方程时，许多挑战都来自于其复杂的碰撞项；因此我们会做一些对碰撞项“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近似中假设分子的碰撞会迫使一个物理空间中的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，波尔兹曼方程可被写作以下的BGK形式：

 

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：

:<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>

where <math>\nu</math> is the molecular collision frequency, and <math>f_0</math> is the local Maxwellian distribution function given the gas temperature at this point in space.

where <math>\nu</math> is the molecular collision frequency, and <math>f_0</math> is the local Maxwellian distribution function given the gas temperature at this point in space.

其中 <math>\nu</math> 是分子碰撞频率，和[[驰豫时间]] <math>\tau</math> 具有倒数关系：<math>\nu = 1/\tau</math>。<math>f_0</math>是此处局域的麦克斯韦分布函数，由空间中这一点的气体温度给定。

其中 <math>\nu</math> 是分子碰撞频率，和驰豫时间 <math>\tau</math> 具有倒数关系：<math>\nu = 1/\tau</math>。<math>f_0</math>是局域麦克斯韦分布函数，由空间中这一点的气体温度给定。

==通用方程（对于混合物）==

==通用方程（对于混合物）==