更改

===一般形式===

===一般形式===

The general equation can then be written as<ref name="McGrawHill">McGraw Hill Encyclopaedia of Physics (2nd Edition), C. B. Parker, 1994, {{ISBN|0-07-051400-3}}.</ref>

The general equation can then be written as<ref name=":3">McGraw Hill Encyclopaedia of Physics (2nd Edition), C. B. Parker, 1994, <nowiki>ISBN 0-07-051400-3</nowiki>.</ref>

玻尔兹曼方程的一般形式可以写作：

玻尔兹曼方程的一般形式可以写作：

</math>

</math>

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<ref name=":3" />。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 [[wikipedia:Diffusion|diffusion]] of particles, and "coll" is the [[wikipedia:Collision|collision]] term – accounting for the forces acting between particles in collisions. Expressions for each term on the right side are provided below.<ref name=":3" />

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=":3">McGraw Hill Encyclopaedia of Physics (2nd Edition), C. B. Parker, 1994, <nowiki>ISBN 0-07-051400-3</nowiki>.</ref>.

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'''.

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<ref name=":3" />。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。

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

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

==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="Encyclopaediaof2">Encyclopaedia of Physics (2nd Edition), [[Rita G. Lerner|R. G. Lerner]], G. L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3.</ref>  

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

: <math>\frac{\partial f_i}{\partial t} + \frac{\mathbf{p}_i}{m_i} \cdot \nabla f_i + \mathbf{F} \cdot \frac{\partial f_i}{\partial \mathbf{p}_i} = \left(\frac{\partial f_i}{\partial t} \right)_\text{coll},</math>

: <math>\frac{\partial f_i}{\partial t} + \frac{\mathbf{p}_i}{m_i} \cdot \nabla f_i + \mathbf{F} \cdot \frac{\partial f_i}{\partial \mathbf{p}_i} = \left(\frac{\partial f_i}{\partial t} \right)_\text{coll},</math>

==方程求解==

==方程求解==

Exact solutions to the Boltzmann equations have been proven to exist in some cases;<ref>{{cite journal|doi=10.1090/S0894-0347-2011-00697-8|arxiv=1011.5441|title=Global Classical Solutions of the Boltzmann Equation without Angular Cut-off|authors=Philip T. Gressman, Robert M. Strain|journal=Journal of the American Mathematical Society|volume=24|issue=3|year=2011|page=771|s2cid=115167686}}</ref> this analytical approach provides insight, but is not generally usable in practical problems.

Exact solutions to the Boltzmann equations have been proven to exist in some cases;<ref>Philip T. Gressman, Robert M. Strain (2011). "Global Classical Solutions of the Boltzmann Equation without Angular Cut-off". ''Journal of the American Mathematical Society''. '''24''' (3): 771. arXiv:1011.5441. doi:10.1090/S0894-0347-2011-00697-8. S2CID&nbsp;115167686.</ref> this analytical approach provides insight, but is not generally usable in practical problems.

这种分析方法提供了洞察力，但在实际问题中通常不能使用。

这种分析方法提供了洞察力，但在实际问题中通常不能使用。

Instead, [[wikipedia:Numerical_methods_in_fluid_mechanics|numerical methods]] (including [[wikipedia:Finite_elements|finite elements]] and [[wikipedia:Lattice_Boltzmann_methods|lattice Boltzmann methods]]) are generally used to find approximate solutions to the various forms of the Boltzmann equation. Example applications range from [[wikipedia:Hypersonic_speed|hypersonic aerodynamics]] in rarefied gas flows<ref>{{Cite journal|title=A discontinuous finite element solution of the Boltzmann kinetic equation in collisionless and BGK forms for macroscopic gas flows|url=https://cronfa.swan.ac.uk/Record/cronfa6256|journal=Applied Mathematical Modelling|date=2011-03-01|pages=996–1015|volume=35|issue=3|doi=10.1016/j.apm.2010.07.027|first1=Ben|last1=Evans|first2=Ken|last2=Morgan|first3=Oubay|last3=Hassan}}</ref><ref>{{Cite journal|last1=Evans|first1=B.|last2=Walton|first2=S.P.|date=December 2017|title=Aerodynamic optimisation of a hypersonic reentry vehicle based on solution of the Boltzmann–BGK equation and evolutionary optimisation|journal=Applied Mathematical Modelling|volume=52|pages=215–240|doi=10.1016/j.apm.2017.07.024|issn=0307-904X|url=https://cronfa.swan.ac.uk/Record/cronfa34688}}</ref> to plasma flows.<ref>{{Cite journal|title=Numerical Solution of the Boltzmann Equation I: Spectrally Accurate Approximation of the Collision Operator|journal=SIAM Journal on Numerical Analysis|date=2000-01-01|issn=0036-1429|pages=1217–1245|volume=37|issue=4|doi=10.1137/S0036142998343300|first1=L.|last1=Pareschi|first2=G.|last2=Russo|citeseerx=10.1.1.46.2853}}</ref> An application of the Boltzmann equation in electrodynamics is the calculation of the electrical conductivity - the result is in leading order identical with the semiclassical result.<ref>H.J.W. Müller-Kirsten, Basics of Statistical Mechanics, Chapter 13, 2nd ed., World Scientific (2013), {{ISBN|978-981-4449-53-3}}.</ref>

Instead, [[wikipedia:Numerical_methods_in_fluid_mechanics|numerical methods]] (including [[wikipedia:Finite_elements|finite elements]] and [[wikipedia:Lattice_Boltzmann_methods|lattice Boltzmann methods]]) are generally used to find approximate solutions to the various forms of the Boltzmann equation. Example applications range from [[wikipedia:Hypersonic_speed|hypersonic aerodynamics]] in rarefied gas flows<ref>{{Cite journal|title=A discontinuous finite element solution of the Boltzmann kinetic equation in collisionless and BGK forms for macroscopic gas flows|url=https://cronfa.swan.ac.uk/Record/cronfa6256|journal=Applied Mathematical Modelling|date=2011-03-01|pages=996–1015|volume=35|issue=3|doi=10.1016/j.apm.2010.07.027|first1=Ben|last1=Evans|first2=Ken|last2=Morgan|first3=Oubay|last3=Hassan}}</ref><ref>{{Cite journal|last1=Evans|first1=B.|last2=Walton|first2=S.P.|date=December 2017|title=Aerodynamic optimisation of a hypersonic reentry vehicle based on solution of the Boltzmann–BGK equation and evolutionary optimisation|journal=Applied Mathematical Modelling|volume=52|pages=215–240|doi=10.1016/j.apm.2017.07.024|issn=0307-904X|url=https://cronfa.swan.ac.uk/Record/cronfa34688}}</ref> to plasma flows.<ref>{{Cite journal|title=Numerical Solution of the Boltzmann Equation I: Spectrally Accurate Approximation of the Collision Operator|journal=SIAM Journal on Numerical Analysis|date=2000-01-01|issn=0036-1429|pages=1217–1245|volume=37|issue=4|doi=10.1137/S0036142998343300|first1=L.|last1=Pareschi|first2=G.|last2=Russo|citeseerx=10.1.1.46.2853}}</ref> An application of the Boltzmann equation in electrodynamics is the calculation of the electrical conductivity - the result is in leading order identical with the semiclassical result.<ref>H.J.W. Müller-Kirsten, Basics of Statistical Mechanics, Chapter 13, 2nd ed., World Scientific (2013), <nowiki>ISBN 978-981-4449-53-3</nowiki>. </ref>

相反，数值方法(包括有限元)通常用于寻找各种形式的玻尔兹曼方程的近似解。示例应用范围从稀薄气流中的高超音速空气动力学到等离子流。

相反，数值方法(包括有限元)通常用于寻找各种形式的玻尔兹曼方程的近似解。示例应用范围从稀薄气流中的高超音速空气动力学到等离子流。

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>Sydney Chapman; Thomas George Cowling [https://books.google.com/books?id=JcjHpiJPKeIC&hl=en&source=gbs_navlinks_s 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>{{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>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>{{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。