更改

Consider particles described by <math> f</math> , each experiencing an ''external'' force '''<math> F</math>''' not due to other particles (see the collision term for the latter treatment).

Consider particles described by <math> f</math> , each experiencing an ''external'' force '''<math> F</math>''' not due to other particles (see the collision term for the latter treatment).

考虑一群以<math>f</math>分布的粒子。每个粒子均受到外力<math>\mathbf{F}</math>的作用（不包括粒子间作用力。粒子间的作用见后面对“coll”项的处理)。

考虑一群以 <math>f</math> 分布的粒子。每个粒子均受到外力<math>\mathbf{F}</math>的作用（不包括粒子间作用力。粒子间的作用见后面对“coll”项的处理)。

Suppose at time <math>t</math> some number of particles all have position '''<math>r</math>''' within element <math> d^3\bf{r}</math> and momentum '''<math> p</math>''' within <math> d^3\bf{p}</math>. If a force '''<math> F</math>''' instantly acts on each particle, then at time <math> t+\Delta t</math> their position will be <math> \mathbf{r}+\Delta \mathbf{r}= \textbf{r}+\frac{\textbf{p}}{m}\Delta t</math> and momentum <math> \mathbf{p}+\Delta \mathbf{p}= \mathbf{p}+\mathbf{F}\Delta t</math>. Then, in the absence of collisions, <math> f</math> must satisfy

Suppose at time <math>t</math> some number of particles all have position '''<math>r</math>''' within element <math> d^3\bf{r}</math> and momentum '''<math> p</math>''' within <math> d^3\bf{p}</math>. If a force '''<math> F</math>''' instantly acts on each particle, then at time <math> t+\Delta t</math> their position will be <math> \mathbf{r}+\Delta \mathbf{r}= \textbf{r}+\frac{\textbf{p}}{m}\Delta t</math> and momentum <math> \mathbf{p}+\Delta \mathbf{p}= \mathbf{p}+\mathbf{F}\Delta t</math>. Then, in the absence of collisions, <math> f</math> must satisfy

假设在时间 <nowiki><math>t</math></nowiki>，一定数量的粒子都有位置 <nowiki><math>\mathbf{r}</math></nowiki>（于微元 <nowiki><math> d^3\mathbf{r}</math></nowiki> 内），和动量 <nowiki><math>\mathbf{p}</math></nowiki>（于微元 <nowiki><math> d^3\mathbf{p}</math></nowiki> 内）。如果此时有一个力<nowiki><math>\mathbf{F}</math></nowiki>在这一瞬作用在每个颗粒上，那么在时间 <nowiki><math>t + \Delta\,t</math></nowiki>，它们的位置将会是<nowiki><math>\mathbf{r} + \Delta\,\mathbf{r} = \mathbf{r} + \mathbf{p} \Delta\,t/m</math></nowiki>，动量将变成 <nowiki><math>\mathbf{p} + \Delta\,\mathbf{p} = \mathbf{p} + \mathbf{F}\Delta\,t</math></nowiki>。在没有碰撞的情况下，<nowiki><math>f</math></nowiki>必须满足

假设在时间 <math>t</math>，一定数量的粒子都有位置 <math>\mathbf{r}</math>（于微元 <math> d^3\mathbf{r}</math> 内），和动量 <math>\mathbf{p}</math>（于微元 <math> d^3\mathbf{p}</math> 内）。如果此时有一个力<math>\mathbf{F}</math>在这一瞬作用在每个颗粒上，那么在时间 <math>t + \Delta\,t</math>，它们的位置将会是<math>\mathbf{r} + \Delta\,\mathbf{r} = \mathbf{r} + \mathbf{p} \Delta\,t/m</math>，动量将变成 <math>\mathbf{p} + \Delta\,\mathbf{p} = \mathbf{p} + \mathbf{F}\Delta\,t</math>。在没有碰撞的情况下，<math>f</math>必须满足

<math>

<math>

==应用与推广==

==应用与推广==

===Conservation equations ===

===Conservation equations===

The Boltzmann equation can be used to derive the fluid dynamic conservation laws for mass, charge, momentum, and energy.<ref name="dG1984">{{cite book|last1=de Groot|first1=S. R.|last2=Mazur|first2=P.|title=Non-Equilibrium Thermodynamics|year=1984|publisher=Dover Publications Inc.|location=New York|isbn=978-0-486-64741-8}}</ref>{{rp|p 163}} For a fluid consisting of only one kind of particle, the number density ''n'' is given by

The Boltzmann equation can be used to derive the fluid dynamic conservation laws for mass, charge, momentum, and energy.<ref name="dG1984">{{cite book|last1=de Groot|first1=S. R.|last2=Mazur|first2=P.|title=Non-Equilibrium Thermodynamics|year=1984|publisher=Dover Publications Inc.|location=New York|isbn=978-0-486-64741-8}}</ref>{{rp|p 163}} For a fluid consisting of only one kind of particle, the number density ''n'' is given by

<references />{{reflist|40em}}

<references />{{reflist|40em}}

== 参考文献==

==参考文献 ==

*{{cite book|last1=Harris|first1=Stewart|title=An introduction to the theory of the Boltzmann equation|publisher=Dover Books|pages=221|year=1971|isbn=978-0-486-43831-3|url=https://books.google.com/books?id=KfYK1lyq3VYC}}. Very inexpensive introduction to the modern framework (starting from a formal deduction from Liouville and the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy (BBGKY) in which the Boltzmann equation is placed). Most statistical mechanics textbooks like Huang still treat the topic using Boltzmann's original arguments. To derive the equation, these books use a heuristic explanation that does not bring out the range of validity and the characteristic assumptions that distinguish Boltzmann's from other transport equations like [[wikipedia:Fokker–Planck_equation|Fokker–Planck]] or [[wikipedia:Landau_equation|Landau equations]].

*{{cite book|last1=Harris|first1=Stewart|title=An introduction to the theory of the Boltzmann equation|publisher=Dover Books|pages=221|year=1971|isbn=978-0-486-43831-3|url=https://books.google.com/books?id=KfYK1lyq3VYC}}. Very inexpensive introduction to the modern framework (starting from a formal deduction from Liouville and the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy (BBGKY) in which the Boltzmann equation is placed). Most statistical mechanics textbooks like Huang still treat the topic using Boltzmann's original arguments. To derive the equation, these books use a heuristic explanation that does not bring out the range of validity and the characteristic assumptions that distinguish Boltzmann's from other transport equations like [[wikipedia:Fokker–Planck_equation|Fokker–Planck]] or [[wikipedia:Landau_equation|Landau equations]].

*{{cite journal|last1=DiPerna|first1=R. J.|last2=Lions|first2=P.-L.|title=On the Cauchy problem for Boltzmann equations: global existence and weak stability|journal=Ann. of Math.|series=2|volume=130|issue=2|pages=321–366|year=1989|doi=10.2307/1971423|jstor=1971423}}

*{{cite journal|last1=DiPerna|first1=R. J.|last2=Lions|first2=P.-L.|title=On the Cauchy problem for Boltzmann equations: global existence and weak stability|journal=Ann. of Math.|series=2|volume=130|issue=2|pages=321–366|year=1989|doi=10.2307/1971423|jstor=1971423}}

== 外部链接==

==外部链接==

*[http://homepage.univie.ac.at/franz.vesely/sp_english/sp/node7.html The Boltzmann Transport Equation by Franz Vesely]

*[http://homepage.univie.ac.at/franz.vesely/sp_english/sp/node7.html The Boltzmann Transport Equation by Franz Vesely]