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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]].

这里，<math>F (\mathbf{r}, t)</math> 为流体中作用在粒子上的力场，<math>m</math>为粒子质量。 右边的一项用于描述粒子间相互碰撞产生的影响；如果此项为零，则说明粒子之间没有碰撞。无碰撞情况下,个体碰撞被长程聚合相互作用(例如库仑相互作用)所取代，此时的玻尔兹曼方程常被称为[[wikipedia:Vlasov equation|'''弗拉索夫方程式 Vlasov Equation''']]。

这里，<math>F (\mathbf{r}, t)</math> 为流体中作用在粒子上的力场，<math>m</math>为粒子质量。 右边的一项用于描述粒子间相互碰撞产生的影响；如果此项为零，则说明粒子之间没有碰撞。无碰撞情况下,个体碰撞被长程聚合相互作用(例如库仑相互作用)所取代，此时的玻尔兹曼方程常被称为[[wikipedia:Vlasov equation|'''弗拉索夫方程 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.

相反，数值方法(包括'''[[wikipedia:Finite_elements|有限元 Finite Elements]]'''和'''[[格子玻尔兹曼方法|格子玻尔兹曼方法 Lattice Boltzmann Methods]]''')经常用来帮助人们寻找各种形式的玻尔兹曼方程的近似解。应用范围覆盖稀薄气流<ref name=":10" /><ref name=":11" />中的'''[[wikipedia:Hypersonic_speed|高超音速空气动力学 Hypersonic Aerodynamics]''']到等离子流<ref name=":12" />。电动力学中，玻尔兹曼方程可以应用于电导率的计算，其结果与半经典结果一致<ref name=":13" />。

相反，数值方法(包括'''[[wikipedia:Finite_elements|有限元 Finite Elements]]'''和'''[[格子玻尔兹曼方法|格子玻尔兹曼方法 Lattice Boltzmann Methods]]''')经常用来帮助人们寻找各种形式的玻尔兹曼方程的近似解。应用范围覆盖稀薄气流<ref name=":10" /><ref name=":11" />中的'''[[wikipedia:Hypersonic_speed|高超音速空气动力学 Hypersonic Aerodynamics]''']到等离子流<ref name=":12" />。电动力学中，玻尔兹曼方程可以应用于电导率的计算，其结果与半经典结果一致<ref name=":13" />。

Close to [[wikipedia:Non-equilibrium_thermodynamics#Local_thermodynamic_equilibrium|local equilibrium]], solution of the Boltzmann equation can be represented by an [[wikipedia:Asymptotic_expansion|asymptotic expansion]] in powers of [[wikipedia:Knudsen_number|Knudsen number]] (the [[wikipedia:Chapman–Enskog_theory|Chapman-Enskog]] expansion<ref name=":14">Sydney Chapman; Thomas George Cowling The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases, Cambridge University Press, 1970. ISBN 0-521-40844-X </ref>). The first two terms of this expansion give the [[wikipedia:Euler_equations_(fluid_dynamics)|Euler equations]] and the [[wikipedia:Navier-Stokes_equations|Navier-Stokes equations]]. The higher terms have singularities. The problem of developing mathematically the limiting processes, which lead from the atomistic view (represented by Boltzmann's equation) to the laws of motion of continua, is an important part of [[wikipedia:Hilbert's_sixth_problem|Hilbert's sixth problem]].<ref name=":15">{{cite journal|doi=10.1098/rsta/376/2118|volume=376|year=2018|journal=Philosophical Transactions of the Royal Society A|title=Theme issue 'Hilbert's sixth problem'|issue=2118|doi-access=free}}</ref>

Close to [[wikipedia:Non-equilibrium_thermodynamics#Local_thermodynamic_equilibrium|local equilibrium]], solution of the Boltzmann equation can be represented by an [[wikipedia:Asymptotic_expansion|asymptotic expansion]] in powers of [[wikipedia:Knudsen_number|Knudsen number]] (the [[wikipedia:Chapman–Enskog_theory|Chapman-Enskog]] expansion<ref name=":14">Sydney Chapman; Thomas George Cowling The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases, Cambridge University Press, 1970. [[index.php?title=Special:BookSources/052140844X|ISBN 0-521-40844-X]] </ref>). The first two terms of this expansion give the [[wikipedia:Euler_equations_(fluid_dynamics)|Euler equations]] and the [[wikipedia:Navier-Stokes_equations|Navier-Stokes equations]]. The higher terms have singularities. The problem of developing mathematically the limiting processes, which lead from the atomistic view (represented by Boltzmann's equation) to the laws of motion of continua, is an important part of [[wikipedia:Hilbert's_sixth_problem|Hilbert's sixth problem]].<ref name=":15">{{cite journal|doi=10.1098/rsta/376/2118|volume=376|year=2018|journal=Philosophical Transactions of the Royal Society A|title=Theme issue 'Hilbert's sixth problem'|issue=2118|doi-access=free}}</ref>

在接近局部均衡的情况下，玻尔兹曼方程的解可以用克努森数幂的'''[[wikipedia:Asymptotic_expansion|渐近展开 Asymptotic Expansion]]'''来表示('''[[wikipedia:Chapman–Enskog_theory|查普曼-恩斯科格 Chapman-Enskog]]'''展开式<ref name=":14" />)。展开式的前两项给出了'''[[wikipedia:Euler_equations_(fluid_dynamics)|欧拉方程 Euler Equations]]'''和'''[[wikipedia:Navier-Stokes_equations|纳维-斯托克斯方程 Navier-Stokes Equations]]'''。较高阶项有奇点。从原子观点(以玻尔兹曼方程为代表)到连续体运动定律的极限过程的数学推导问题，是'''[[wikipedia:Hilbert's_sixth_problem|希尔伯特第六问题Hilbert's Sixth Problem]]'''的重要组成部分<ref name=":15" />。

在接近局部均衡的情况下，玻尔兹曼方程的解可以用克努森数幂的'''[[wikipedia:Asymptotic_expansion|渐近展开 Asymptotic Expansion]]'''来表示('''[[wikipedia:Chapman–Enskog_theory|查普曼-恩斯科格 Chapman-Enskog]]'''展开式<ref name=":14" />)。展开式的前两项给出了'''[[wikipedia:Euler_equations_(fluid_dynamics)|欧拉方程 Euler Equations]]'''和'''[[wikipedia:Navier-Stokes_equations|纳维-斯托克斯方程 Navier-Stokes Equations]]'''。较高阶项有奇点。从原子观点(以玻尔兹曼方程为代表)到连续体运动定律的极限过程的数学推导问题，是'''[[wikipedia:Hilbert's_sixth_problem|希尔伯特第六问题Hilbert's Sixth Problem]]'''的重要组成部分<ref name=":15" />。

==另见==

==另见==

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<!--*[[BGK equation]]

<!--*[[BGK equation]]

*[[linear Boltzmann equation]]

*[[linear Boltzmann equation]]

*[[neutron transport equation]]

*[[neutron transport equation]]

*[[photon transport equation]]-->

*[[photon transport equation]]-->

* [[H-theorem]]

* [[wikipedia:H-theorem|H定理 H-theorem]]

* [[Fokker&ndash;Planck equation]]

* [[wikipedia:Fokker&ndash;Planck equation|福克-普朗克方程Fokker&ndash;Planck equation]]

* [[Williams spray equation|Williams-Boltzmann equation]]

* [[wikipedia:Williams spray equation|威廉姆斯-玻尔兹曼方程 Williams-Boltzmann equation]]

* [[Navier&ndash;Stokes equations]]

* [[wikipedia:Vlasov equation#The Vlasov&ndash;Poisson equation|弗拉索夫-泊松方程 Vlasov&ndash;Poisson equation]]

* [[Vlasov equation#The Vlasov&ndash;Poisson equation|Vlasov&ndash;Poisson equation]]

 

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