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此词条由栗子CUGB翻译整理。[[File:StairsOfReduction.svg|thumb|The place of the Boltzmann kinetic equation on the stairs of model reduction from microscopic dynamics to macroscopic continuum dynamics (illustration to the content of the book<ref>

此词条由栗子CUGB翻译整理。[[File:StairsOfReduction.svg|thumb|The place of the Boltzmann kinetic equation on the stairs of model reduction from microscopic dynamics to macroscopic continuum dynamics (illustration to the content of the book<ref name=":2">

The place of the Boltzmann kinetic equation on the stairs of model reduction from microscopic dynamics to macroscopic continuum dynamics (illustration to the content of the book)

{{cite book |last1=Gorban |first1= Alexander N.|last2= Karlin |first2= Ilya V. |date=2005 |title= Invariant Manifolds for Physical and Chemical Kinetics|url= https://www.academia.edu/17378865|url-access=| location= Berlin, Heidelberg |publisher= Springer|series= Lecture Notes in Physics (LNP, vol. 660)| isbn= 978-3-540-22684-0|doi= 10.1007/b98103|via= |quote=}} [https://archive.org/details/gorban-karlin-lnp-2005 Alt URL]</ref>)

玻耳兹曼动力学方程在从微观动力学到宏观连续动力学的模型简化阶梯上的位置(本书内容的说明)

玻耳兹曼动力学方程在从微观动力学到宏观连续动力学的模型简化阶梯上的位置(本书内容的说明)<ref name=":2" />|链接=Special:FilePath/StairsOfReduction.svg]]

 

{{{cite book |last1=Gorban |first1= Alexander N.|last2= Karlin |first2= Ilya V. |date=2005 |title= Invariant Manifolds for Physical and Chemical Kinetics|url= https://www.academia.edu/17378865|url-access=| location= Berlin, Heidelberg |publisher= Springer|series= Lecture Notes in Physics (LNP, vol. 660)| isbn= 978-3-540-22684-0|doi= 10.1007/b98103|via= |quote=}} [https://archive.org/details/gorban-karlin-lnp-2005 Alt URL]</ref>)|链接=Special:FilePath/StairsOfReduction.svg]]

<math> \text{d}^3\mathbf{r}\,\text{d}^3\mathbf{p} = \text{d}x\,\text{d}y\,\text{d}z\,\text{d}p_x\,\text{d}p_y\,\text{d}p_z. </math>.

<math> \text{d}^3\mathbf{r}\,\text{d}^3\mathbf{p} = \text{d}x\,\text{d}y\,\text{d}z\,\text{d}p_x\,\text{d}p_y\,\text{d}p_z. </math>.

Since the probability of ''N'' molecules which ''all'' have '''r''' and '''p''' within <math> \mathrm{d}^3\bf{r}</math>&nbsp;<math> \mathrm{d}^3\bf{p}</math> is in question, at the heart of the equation is a quantity ''f'' which gives this probability per unit phase-space volume, or probability per unit length cubed per unit momentum cubed, at an instant of time ''t''. This is a [[probability density function]]: ''f''('''r''', '''p''', ''t''), defined so that,

Since the probability of ''N'' molecules which ''all'' have '''r''' and '''p''' within <math> \mathrm{d}^3\bf{r}</math>&nbsp;<math> \mathrm{d}^3\bf{p}</math> is in question, at the heart of the equation is a quantity ''f'' which gives this probability per unit phase-space volume, or probability per unit length cubed per unit momentum cubed, at an instant of time ''t''. This is a [[probability density function]]: ''f'' ('''r''', '''p''', ''t''), defined so that,

<math>\text{d}N = f (\mathbf{r},\mathbf{p},t)\,\text{d}^3\mathbf{r}\,\text{d}^3\mathbf{p}</math>

<math>\text{d}N = f (\mathbf{r},\mathbf{p},t)\,\text{d}^3\mathbf{r}\,\text{d}^3\mathbf{p}</math>

<math>\text{d}N = f (\mathbf{r},\mathbf{p},t)\,\text{d}^3\mathbf{r}\,\text{d}^3\mathbf{p}</math>

<math>\text{d}N = f (\mathbf{r},\mathbf{p},t)\,\text{d}^3\mathbf{r}\,\text{d}^3\mathbf{p}</math>

is the number of molecules which all have positions lying within a volume element <math> d^3\bf{r}</math> about r and momenta lying within a momentum space element <math> \mathrm{d}^3\bf{p}</math> about p, at time t. Integrating over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region:

is the number of molecules which all have positions lying within a volume element <math> d^3\bf{r}</math> about r and momenta lying within a momentum space element <math> \mathrm{d}^3\bf{p}</math> about p, at time t<ref>Huang, Kerson (1987). ''Statistical Mechanics'' (Second ed.). New York: Wiley. p. 53. ISBN <bdi>978-0-471-81518-1</bdi>.</ref>. Integrating over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region:

dN是在t时刻，关于（r，p）的微体积元<math> d^3\bf{r}</math>和微动量元<math> \mathrm{d}^3\bf{p}</math>内的分子数目。在位置空间和动量空间的一个区域上积分，得出在该区域中具有位置和动量的粒子总数：

dN是在t时刻，关于（r，p）的微体积元<math> d^3\bf{r}</math>和微动量元<math> \mathrm{d}^3\bf{p}</math>内的分子数目。在位置空间和动量空间的一个区域上积分，得出在该区域中具有位置和动量的粒子总数：

其中“force”一词指外界对粒子施加的力（而不是粒子间的作用），“diff”表示粒子扩散，“coll”表示粒子碰撞，指碰撞中粒子间相互的作用力。上述三项的具体形式将会在下文给出。注意，一些作者会使用 '''v''' 表示粒子的速度，而不是动量 '''p。'''这两个物理量可以通过动量的定义'''p''' = m'''v'''联系起来。

其中“force”一词指外界对粒子施加的力（而不是粒子间的作用），“diff”表示粒子扩散，“coll”表示粒子碰撞，指碰撞中粒子间相互的作用力。上述三项的具体形式将会在下文给出。注意，一些作者会使用 '''v''' 表示粒子的速度，而不是动量 '''p。'''这两个物理量可以通过动量的定义'''p''' = m'''v'''联系起来。

where the "force" term corresponds to the forces exerted on the particles by an external influence (not by the particles themselves), the "diff" term represents the diffusion of particles, and "coll" is the collision term – accounting for the forces acting between particles in collisions. Expressions for each term on the right side are provided below.  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:

where the "force" term corresponds to the forces exerted on the particles by an external influence (not by the particles themselves), the "diff" term represents the [[diffusion]] of particles, and "coll" is the [[collision]] term – accounting for the forces acting between particles in collisions. Expressions for each term on the right side are provided below<ref>McGraw Hill Encyclopaedia of Physics (2nd Edition), C. B. Parker, 1994, <nowiki>ISBN 0-07-051400-3</nowiki>.</ref>.

 

如果“力”这个术语对应于外部影响(而不是粒子本身)施加在粒子上的力，“ diff”这个术语代表粒子的扩散，“ coll”是碰撞术语——解释粒子之间在碰撞中的作用力。下面提供了右边每个术语的表达式。BGK 近似中的假设是，分子碰撞的效果是迫使物理空间中某一点的非平衡分布函数回到马克斯韦尔平衡分布函数，而这种情况发生的速率与分子碰撞频率成正比。因此，《玻尔兹曼方程修改为 BGK 格式:

 

\left(\frac{\partial f}{\partial t}\right)_\text{coll},

 

</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 the "force" term corresponds to the forces exerted on the particles by an external influence (not by the particles themselves), the "diff" term represents the [[diffusion]] of particles, and "coll" is the [[collision]] term – accounting for the forces acting between particles in collisions. Expressions for each term on the right side are provided below.<ref name="McGrawHill" />

Note that some authors use the particle velocity '''v''' instead of momentum '''p'''; they are related in the definition of momentum by '''p''' = ''m'''''v'''.

Note that some authors use the particle velocity '''v''' instead of momentum '''p'''; they are related in the definition of momentum by '''p''' = ''m'''''v'''.

==The force and diffusion terms==

==The force and diffusion terms “force”项与“diff”项==

For a mixture of chemical species labelled by indices i = 1, 2, 3, ..., n the equation for species i is For a fluid consisting of only one kind of particle, the number density n is given by

For a mixture of chemical species labelled by indices i = 1, 2, 3, ..., n the equation for species i is For a fluid consisting of only one kind of particle, the number density n is given by

对于以指数 i = 1,2,3，... ，n 标记的化学物种混合物，物种 i 的方程是: 对于只包含一种粒子的流体，数密度 n 由

对于以指数 i = 1,2,3，... ，n 标记的化学物种混合物，物种 i 的方程是: 对于只包含一种粒子的流体，数密度 n 由

<math>n = \int f \,d^3p.</math>

<math>n = \int f \,d^3p.</math>