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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]]的一个关键见解就是对碰撞项的确定。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“{{le|分子混沌假设|Molecular chaos}}”。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|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“[[分子混沌假设 Molecular chaos]]”。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]]''d''Ω的元。

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]]''d''Ω的元。

===Simplifications to the collision term===

===Simplifications to the collision term===

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.

==通用方程（对于混合物）==

==通用方程（对于混合物） ==

For a mixture of chemical species labelled by indices ''i'' = 1, 2, 3, ..., ''n'' the equation for species ''i'' is<ref name="Encyclopaediaof" />  

For a mixture of chemical species labelled by indices ''i'' = 1, 2, 3, ..., ''n'' the equation for species ''i'' is<ref name="Encyclopaediaof" />  

where <math>u = \tfrac{1}{2} \rho \langle (w_i-V_i) (w_i-V_i) \rangle</math> is the kinetic thermal energy density, and <math>J_{qi} = \tfrac{1}{2} \rho \langle(w_i - V_i)(w_k - V_k)(w_k - V_k)\rangle</math> is the heat flux vector.

where <math>u = \tfrac{1}{2} \rho \langle (w_i-V_i) (w_i-V_i) \rangle</math> is the kinetic thermal energy density, and <math>J_{qi} = \tfrac{1}{2} \rho \langle(w_i - V_i)(w_k - V_k)(w_k - V_k)\rangle</math> is the heat flux vector.

===Hamiltonian mechanics===

===Hamiltonian mechanics ===

In [[wikipedia:Hamiltonian_mechanics|Hamiltonian mechanics]], the Boltzmann equation is often written more generally as

In [[wikipedia:Hamiltonian_mechanics|Hamiltonian mechanics]], the Boltzmann equation is often written more generally as

:<math>\hat{\mathbf{L}}_\mathrm{NR} = \frac{\partial}{\partial t} + \frac{\mathbf{p}}{m} \cdot \nabla + \mathbf{F}\cdot\frac{\partial}{\partial \mathbf{p}}\,.</math>

:<math>\hat{\mathbf{L}}_\mathrm{NR} = \frac{\partial}{\partial t} + \frac{\mathbf{p}}{m} \cdot \nabla + \mathbf{F}\cdot\frac{\partial}{\partial \mathbf{p}}\,.</math>

===Quantum theory and violation of particle number conservation===

=== Quantum theory and violation of particle number conservation ===

It is possible to write down relativistic [[wikipedia:Quantum_Boltzmann_equation|quantum Boltzmann equations]] for [[wikipedia:Quantum_field_theory|relativistic]] quantum systems in which the number of particles is not conserved in collisions. This has several applications in [[wikipedia:Physical_cosmology|physical cosmology]],<ref name="KolbTurner">{{cite book|author1=Edward Kolb|author2=Michael Turner|name-list-style=amp|title=The Early Universe|year=1990|publisher=Westview Press|isbn=9780201626742}}</ref> including the formation of the light elements in [[wikipedia:Big_Bang_nucleosynthesis|Big Bang nucleosynthesis]], the production of [[wikipedia:Dark_matter|dark matter]] and [[wikipedia:Baryogenesis|baryogenesis]]. It is not a priori clear that the state of a quantum system can be characterized by a classical phase space density ''f''. However, for a wide class of applications a well-defined generalization of ''f'' exists which is the solution of an effective Boltzmann equation that can be derived from first principles of [[wikipedia:Quantum_field_theory|quantum field theory]].<ref name="BEfromQFT">{{cite journal|author1=M. Drewes|author2=C. Weniger|author3=S. Mendizabal|journal=Phys. Lett. B|date=8 January 2013|volume=718|issue=3|pages=1119–1124|doi=10.1016/j.physletb.2012.11.046|arxiv=1202.1301|bibcode=2013PhLB..718.1119D|title=The Boltzmann equation from quantum field theory|s2cid=119253828}}</ref>

It is possible to write down relativistic [[wikipedia:Quantum_Boltzmann_equation|quantum Boltzmann equations]] for [[wikipedia:Quantum_field_theory|relativistic]] quantum systems in which the number of particles is not conserved in collisions. This has several applications in [[wikipedia:Physical_cosmology|physical cosmology]],<ref name="KolbTurner">{{cite book|author1=Edward Kolb|author2=Michael Turner|name-list-style=amp|title=The Early Universe|year=1990|publisher=Westview Press|isbn=9780201626742}}</ref> including the formation of the light elements in [[wikipedia:Big_Bang_nucleosynthesis|Big Bang nucleosynthesis]], the production of [[wikipedia:Dark_matter|dark matter]] and [[wikipedia:Baryogenesis|baryogenesis]]. It is not a priori clear that the state of a quantum system can be characterized by a classical phase space density ''f''. However, for a wide class of applications a well-defined generalization of ''f'' exists which is the solution of an effective Boltzmann equation that can be derived from first principles of [[wikipedia:Quantum_field_theory|quantum field theory]].<ref name="BEfromQFT">{{cite journal|author1=M. Drewes|author2=C. Weniger|author3=S. Mendizabal|journal=Phys. Lett. B|date=8 January 2013|volume=718|issue=3|pages=1119–1124|doi=10.1016/j.physletb.2012.11.046|arxiv=1202.1301|bibcode=2013PhLB..718.1119D|title=The Boltzmann equation from quantum field theory|s2cid=119253828}}</ref>

===General relativity and astronomy===

=== General relativity and astronomy===

The Boltzmann equation is of use in galactic dynamics. A galaxy, under certain assumptions, may be approximated as a continuous fluid; its mass distribution is then represented by ''f''; in galaxies, physical collisions between the stars are very rare, and the effect of ''gravitational collisions'' can be neglected for times far longer than the [[wikipedia:Age_of_the_universe|age of the universe]].

The Boltzmann equation is of use in galactic dynamics. A galaxy, under certain assumptions, may be approximated as a continuous fluid; its mass distribution is then represented by ''f''; in galaxies, physical collisions between the stars are very rare, and the effect of ''gravitational collisions'' can be neglected for times far longer than the [[wikipedia:Age_of_the_universe|age of the universe]].

在接近局部均衡的情况下，玻尔兹曼方程的解可以用一个克努森数的渐近展开表示(Chapman-Enskog 展开式)。这个展开式的前两项给出了欧拉方程和纳维－斯托克斯方程。较高的项有奇点。从原子观点(以玻尔兹曼方程为代表)到连续统运动定律的极限过程的数学发展问题，是希尔伯特第六个问题的重要组成部分。

在接近局部均衡的情况下，玻尔兹曼方程的解可以用一个克努森数的渐近展开表示(Chapman-Enskog 展开式)。这个展开式的前两项给出了欧拉方程和纳维－斯托克斯方程。较高的项有奇点。从原子观点(以玻尔兹曼方程为代表)到连续统运动定律的极限过程的数学发展问题，是希尔伯特第六个问题的重要组成部分。

==另见==

==另见 ==

{{Div col|colwidth=20em}}

{{Div col|colwidth=20em}}

* [[Vlasov equation]]

* [[Vlasov equation]]

<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]].