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第125行： − − | last1= Harris − − 1 = Harris − − | first1= Stewart − − 1 = Stewart − − |author1-link= − − 1-link = − − | 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 Fokker–Planck or Landau equations. − − | title = 玻尔兹曼方程理论导论 | 出版商 = Dover Books | pages = 221 | year = 1971 | isbn = 978-0-486-43831-3 | url = https://Books.google.com/Books?id=kfyk1lyq3vyc }。非常便宜的现代框架介绍(从一个正式的推论从 Liouville 和 Bogoliubov-Born-Green-Kirkwood-伊冯等级(BBGKY)的玻尔兹曼方程是放置)。大多数统计力学的教科书，比如 Huang，仍然使用 Boltzmann 的原始论点来处理这个话题。为了推导这个方程，这些书使用了一种启发式的解释，这种解释没有提出波尔兹曼方程与其他传输方程(如福克-普朗克方程或兰道方程)区别开来的有效性范围和特征性假设。 − − − − | last1= Arkeryd − − | last1= Arkeryd 第292行：
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→Final statement
Dividing ({{EquationNote|3}}) by ''dt'' and substituting into ({{EquationNote|2}}) gives:
Dividing ({{EquationNote|3}}) by ''dt'' and substituting into ({{EquationNote|2}}) gives:
:<math>\frac{\partial f}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla f + \mathbf{F} \cdot \frac{\partial f}{\partial \mathbf{p}} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}</math>
:<math>\frac{\partial f}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla f + \mathbf{F} \cdot \frac{\partial f}{\partial \mathbf{p}} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}</math>
In this context, '''F'''('''r''', ''t'') is the [[Force field (chemistry)|force field]] acting on the particles in the fluid, and ''m'' is the [[mass]] of the particles. The term on the right hand side is added to describe the effect of collisions between particles; if it is zero then the particles do not collide. The collisionless Boltzmann equation, where individual collisions are replaced with long-range aggregated interactions, e.g. [[Coulomb interaction]]s, is often called the [[Vlasov equation]].
In this context, '''F'''('''r''', ''t'') is the [[Force field (chemistry)|force field]] acting on the particles in the fluid, and ''m'' is the [[mass]] of the particles. The term on the right hand side is added to describe the effect of collisions between particles; if it is zero then the particles do not collide. The collisionless Boltzmann equation, where individual collisions are replaced with long-range aggregated interactions, e.g. [[Coulomb interaction]]s, is often called the [[Vlasov equation]].
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.
==The collision term (Stosszahlansatz) and molecular chaos==
==The collision term (Stosszahlansatz) and molecular chaos==
==参考文献==
==参考文献==
| last1= Harris | first1= Stewart |author1-link=| 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 Fokker–Planck or Landau equations.
| last1= Arkeryd
| last1= Arkeryd
[[分类:偏微分方程]]
[[分类:偏微分方程]]
[[分类:统计力学]]
[[分类:统计力学]]