更改

此词条由栗子CUGB翻译整理。[[File:StairsOfReduction.svg|thumb|The place of the Boltzmann kinetic equation on the stairs of model reduction from microscopic dynamics to macroscopic continuum dynamics (illustration to the content of the book<ref name=":2">

此词条由栗子CUGB翻译整理。[[File:StairsOfReduction.svg|thumb|The place of the Boltzmann kinetic equation on the stairs of model reduction from microscopic dynamics to macroscopic continuum dynamics (illustration to the content of the book<ref name=":2">

{{cite book |last1=Gorban |first1= Alexander N.|last2= Karlin |first2= Ilya V. |date=2005 |title= Invariant Manifolds for Physical and Chemical Kinetics|url= https://www.academia.edu/17378865|url-access=| location= Berlin, Heidelberg |publisher= Springer|series= Lecture Notes in Physics (LNP, vol. 660)| isbn= 978-3-540-22684-0|doi= 10.1007/b98103|via= |quote=}} [https://archive.org/details/gorban-karlin-lnp-2005 Alt URL]</ref>)

Gorban, Alexander N.; Karlin, Ilya V. (2005). ''Invariant Manifolds for Physical and Chemical Kinetics''. Lecture Notes in Physics (LNP, vol. 660). Berlin, Heidelberg: Springer. doi:10.1007/b98103. ISBN <bdi>978-3-540-22684-0</bdi>. [https://archive.org/details/gorban-karlin-lnp-2005 Alt URL]</ref>)

玻耳兹曼动力学方程在从微观动力学到宏观连续动力学的模型简化阶梯上的位置(本书内容的说明)<ref name=":2" />|链接=Special:FilePath/StairsOfReduction.svg]]

玻耳兹曼动力学方程在从微观动力学到宏观连续动力学的模型简化阶梯上的位置(本书内容的说明)<ref name=":2" />|链接=Special:FilePath/StairsOfReduction.svg]]

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]]dΩ的元。

===Simplifications to the collision term 对碰撞项的简化===

===Simplifications to the collision term 对碰撞项的简化===

Since much of the challenge in solving the Boltzmann equation originates with the complex collision term, attempts have been made to "model" and simplify the collision term.  The best known model equation is due to Bhatnagar, Gross and Krook.<ref>{{Cite journal|title=A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems|journal=Physical Review|date=1954-05-01|pages=511–525|volume=94|issue=3|doi=10.1103/PhysRev.94.511|first1=P. L.|last1=Bhatnagar|first2=E. P.|last2=Gross|first3=M.|last3=Krook|bibcode=1954PhRv...94..511B}}</ref>  The assumption in the BGK approximation is that the effect of molecular collisions is to force a non-equilibrium distribution function at a point in physical space back to a Maxwellian equilibrium distribution function and that the rate at which this occurs is proportional to the molecular collision frequency.  The Boltzmann equation is therefore modified to the BGK form:

Since much of the challenge in solving the Boltzmann equation originates with the complex collision term, attempts have been made to "model" and simplify the collision term.  The best known model equation is due to Bhatnagar, Gross and Krook.<ref name=":4">{{Cite journal|title=A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems|journal=Physical Review|date=1954-05-01|pages=511–525|volume=94|issue=3|doi=10.1103/PhysRev.94.511|first1=P. L.|last1=Bhatnagar|first2=E. P.|last2=Gross|first3=M.|last3=Krook|bibcode=1954PhRv...94..511B}}</ref>  The assumption in the BGK approximation is that the effect of molecular collisions is to force a non-equilibrium distribution function at a point in physical space back to a Maxwellian equilibrium distribution function and that the rate at which this occurs is proportional to the molecular collision frequency.  The Boltzmann equation is therefore modified to the BGK form:

求解波尔兹曼方程时，许多挑战都来自于其复杂的碰撞项；因此我们会做一些对碰撞项“建模”和简化的尝试。现知最好的模型是由Bhatnagar，Gross和Krook作出的（BGK近似）<ref><cite class="citation journal">Bhatnagar, P. L.; Gross, E. P.; Krook, M. (1954-05-01). </cite></ref>。BGK近似中假设分子的碰撞会迫使一个物理空间中的某一点的非平衡分布函数回到麦克斯韦平衡分布函数，且其发生率正比于分子碰撞频率。于是，波尔兹曼方程可被写作以下的BGK形式：

求解波尔兹曼方程时，许多挑战都来自于其复杂的碰撞项；因此我们会做一些对碰撞项“建模”和简化的尝试。现知最好的模型是由Bhatnagar，Gross和Krook作出的（BGK近似）<ref name=":4" />。BGK近似中假设分子的碰撞会迫使一个物理空间中的某一点的非平衡分布函数回到麦克斯韦平衡分布函数，且其发生率正比于分子碰撞频率。于是，波尔兹曼方程可被写作以下的BGK形式：

:<math>\frac{\partial f}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla f + \mathbf{F} \cdot \frac{\partial f}{\partial \mathbf{p}} = \nu (f_0 - f),</math>

:<math>\frac{\partial f}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla f + \mathbf{F} \cdot \frac{\partial f}{\partial \mathbf{p}} = \nu (f_0 - f),</math>

==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" />  

对于具有多种化学组分的[[混合物]]，我们以 i =1，2，3，……，n 标记各种成分。则对于组分i的方程是：<ref name="Encyclopaediaof" />

对于具有多种化学组分的混合物，我们以 i =1，2，3，……，n 标记各种成分。则对于组分i的方程是：<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>

===Conservation equations 守恒方程===

===Conservation equations 守恒方程===

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

玻尔兹曼方程可用于推导流体动力学中的质量守恒，电量守恒，动量守恒，以及能量守恒定律<ref name="dG1984">{{cite book |last1=de Groot |first1=S.R. |last2=Mazur |first2=P. |title=Non-Equilibrium Thermodynamics |url=http://www.amazon.com/Non-Equilibrium-Thermodynamics-Dover-Books-Physics/dp/0486647412 |accessdate=2013-01-31 |year=1984 |publisher=Dover Publications Inc. |location=New York |isbn=0-486-64741-2 |archive-date=2013-03-28 |archive-url=https://web.archive.org/web/20130328114445/http://www.amazon.com/Non-Equilibrium-Thermodynamics-Dover-Books-Physics/dp/0486647412 |dead-url=no }}</ref>{{rp|p 163}}。对于只含有一种粒子的流体，粒子数密度 <math>n</math> 为：

 

玻尔兹曼方程可用于推导流体动力学中的质量守恒，电量守恒，动量守恒，以及能量守恒定律 <ref name="dG1984" />{{rp|p 163}}。对于只含有一种粒子的流体，粒子数密度 <math>n</math> 为：

:<math>n = \int f \,d^3p.</math>

:<math>n = \int f \,d^3p.</math>

where the last term is zero, since ''A'' is conserved in a collision.

where the last term is zero, since ''A'' is conserved in a collision.

因为 <math>A</math> 在碰撞中守恒，所以最后一项为零。

因为 <math>A</math> 在碰撞中守恒，所以最后一项为零。

Letting <math>A = m</math>, the mass of the particle, the integrated Boltzmann equation becomes the conservation of mass equation:<ref name="dG1984" />{{rp|pp 12,168}}

Letting <math>A = m</math>, the mass of the particle, the integrated Boltzmann equation becomes the conservation of mass equation:<ref name="dG1984" />{{rp|pp 12,168}}

令 <math>A=m</math>，即粒子质量，积分后的玻尔兹曼方程化为质量守恒方程<ref name="dG1984" />{{rp|pp 12,168}}：

令 <math>A=m</math>，即粒子质量，积分后的玻尔兹曼方程化为质量守恒方程<ref name="dG1984" />{{rp|pp 12,168}}：

where <math>\rho = mn</math> is the mass density, and <math>V_i = \langle w_i\rangle</math> is the average fluid velocity.

where <math>\rho = mn</math> is the mass density, and <math>V_i = \langle w_i\rangle</math> is the average fluid velocity.

<math>\rho=mn</math> 为质量密度，<math>V_i=\langle w_i\rangle</math> 为平均流体速度。

<math>\rho=mn</math> 为质量密度，<math>V_i=\langle w_i\rangle</math> 为平均流体速度。

Letting <math>A = p_i</math>, the momentum of the particle, the integrated Boltzmann equation becomes the conservation of momentum equation:<ref name="dG1984" />{{rp|pp 15,169}}

Letting <math>A = p_i</math>, the momentum of the particle, the integrated Boltzmann equation becomes the conservation of momentum equation:<ref name="dG1984" />{{rp|pp 15,169}}

令 <math>A = p_i</math>，即粒子动量，积分后的玻尔兹曼方程化为动量守恒方程<ref name="dG1984" />{{rp|pp 15,169}}：

令 <math>A = p_i</math>，即粒子动量，积分后的玻尔兹曼方程化为动量守恒方程<ref name="dG1984" />{{rp|pp 15,169}}：

===Hamiltonian mechanics 哈密顿力学===

===Hamiltonian mechanics 哈密顿力学===

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

在[[哈密顿力学]]中, 玻尔兹曼方程通常写作

在[[哈密顿力学]]中, 玻尔兹曼方程通常写作

===Quantum theory and violation of particle number conservation 量子理论与粒子数守恒的违背===

===Quantum theory and violation of particle number conservation 量子理论与粒子数守恒的违背===

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">M. Drewes; C. Weniger; S. Mendizabal (8 January 2013). "The Boltzmann equation from quantum field theory". ''Phys. Lett. B''. 718 (3): 1119–1124. arXiv:1202.1301. Bibcode:2013PhLB..718.1119D. doi:10.1016/j.physletb.2012.11.046. S2CID 119253828.</ref>

对于碰撞中粒子数不守恒的相对论量子系统，给出相对论量子玻尔兹曼方程是可能的。这在物理宇宙学中有一些应用<ref name="KolbTurner" />，包括大爆炸核合成中轻元素的形成，暗物质的产生和重子发生。我们并不预先知道量子系统的状态可以用经典相空间密度''f''来表征。然而，对于许多应用来说，定义良好的''f''作为有效玻尔兹曼方程的解是存在的，可以由[[wikipedia:Quantum_field_theory|量子场论]]的第一原理推导得出。

对于碰撞中粒子数不守恒的相对论量子系统，给出相对论量子玻尔兹曼方程是可能的。这在物理宇宙学中有一些应用<ref name="KolbTurner" />，包括大爆炸核合成中轻元素的形成，暗物质的产生和重子发生。我们并不预先知道量子系统的状态可以用经典相空间密度''f'' 来表征。然而，对于许多应用来说，定义良好的''f'' 作为有效玻尔兹曼方程的解是存在的，可以由[[wikipedia:Quantum_field_theory|量子场论]]的第一原理推导得出。

===General relativity and astronomy===

===General relativity and astronomy 广义相对论和天文学===

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]].

Its generalization in [[wikipedia:General_relativity|general relativity]].<ref>Ehlers J (1971) General Relativity and Cosmology (Varenna), R K Sachs (Academic Press NY);Thorne K S (1980) Rev. Mod. Phys., 52, 299; Ellis G F R, Treciokas R, Matravers D R, (1983) Ann. Phys., 150, 487}</ref> is

 

Its generalization in [[wikipedia:General_relativity|general relativity]].<ref name=":5">Ehlers J (1971) General Relativity and Cosmology (Varenna), R K Sachs (Academic Press NY);Thorne K S (1980) Rev. Mod. Phys., 52, 299; Ellis G F R, Treciokas R, Matravers D R, (1983) Ann. Phys., 150, 487}</ref> is

 

:<math>\hat{\mathbf{L}}_\mathrm{GR}=p^\alpha\frac{\partial}{\partial x^\alpha} - \Gamma^\alpha{}_{\beta\gamma}p^\beta p^\gamma\frac{\partial}{\partial p^\alpha},</math>

:<math>\hat{\mathbf{L}}_\mathrm{GR}=p^\alpha\frac{\partial}{\partial x^\alpha} - \Gamma^\alpha{}_{\beta\gamma}p^\beta p^\gamma\frac{\partial}{\partial p^\alpha},</math>

where Γ^α_βγ is the [[wikipedia:Christoffel_symbol|Christoffel symbol]] of the second kind (this assumes there are no external forces, so that particles move along geodesics in the absence of collisions), with the important subtlety that the density is a function in mixed contravariant-covariant (''xⁱ, p_i'') phase space as opposed to fully contravariant (''xⁱ, pⁱ'') phase space.<ref>{{cite journal|last=Debbasch|first=Fabrice|author2=Willem van Leeuwen|title=General relativistic Boltzmann equation I: Covariant treatment|journal=Physica A|volume=388|issue=7|pages=1079–1104|year=2009|bibcode=2009PhyA..388.1079D|doi=10.1016/j.physa.2008.12.023}}</ref><ref>{{cite journal|last=Debbasch|first=Fabrice|author2=Willem van Leeuwen|title=General relativistic Boltzmann equation II: Manifestly covariant treatment|journal=Physica A|volume=388|issue=9|pages=1818–34|year=2009|bibcode=2009PhyA..388.1818D|doi=10.1016/j.physa.2009.01.009}}</ref>

where Γ^α_βγ is the [[wikipedia:Christoffel_symbol|Christoffel symbol]] of the second kind (this assumes there are no external forces, so that particles move along geodesics in the absence of collisions), with the important subtlety that the density is a function in mixed contravariant-covariant (''xⁱ, p_i'') phase space as opposed to fully contravariant (''xⁱ, pⁱ'') phase space.<ref name=":6">{{cite journal|last=Debbasch|first=Fabrice|author2=Willem van Leeuwen|title=General relativistic Boltzmann equation I: Covariant treatment|journal=Physica A|volume=388|issue=7|pages=1079–1104|year=2009|bibcode=2009PhyA..388.1079D|doi=10.1016/j.physa.2008.12.023}}</ref><ref name=":7">{{cite journal|last=Debbasch|first=Fabrice|author2=Willem van Leeuwen|title=General relativistic Boltzmann equation II: Manifestly covariant treatment|journal=Physica A|volume=388|issue=9|pages=1818–34|year=2009|bibcode=2009PhyA..388.1818D|doi=10.1016/j.physa.2009.01.009}}</ref>

 

 

In [[wikipedia:Physical_cosmology|physical cosmology]] the fully covariant approach has been used to study the cosmic microwave background radiation.<ref>Maartens R, Gebbie T, Ellis GFR (1999). "Cosmic microwave background anisotropies: Nonlinear dynamics". Phys. Rev. D. 59 (8): 083506</ref> More generically the study of processes in the [[wikipedia:Early_universe|early universe]] often attempt to take into account the effects of [[wikipedia:Quantum_mechanics|quantum mechanics]] and [[wikipedia:General_relativity|general relativity]].<ref name="KolbTurner" /> In the very dense medium formed by the primordial plasma after the [[wikipedia:Big_Bang|Big Bang]], particles are continuously created and annihilated. In such an environment [[wikipedia:Quantum_coherence|quantum coherence]] and the spatial extension of the [[wikipedia:Wavefunction|wavefunction]] can affect the dynamics, making it questionable whether the classical phase space distribution ''f'' that appears in the Boltzmann equation is suitable to describe the system. In many cases it is, however, possible to derive an effective Boltzmann equation for a generalized distribution function from first principles of [[wikipedia:Quantum_field_theory|quantum field theory]].<ref name="BEfromQFT" /> This includes 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]].

在物理宇宙学中，全协变方法已经应用于宇宙微波背景辐射研究<ref name=":8" />。 更一般地说，对早期宇宙过程的研究常常试图考虑量子力学和广义相对论的影响。<ref name="KolbTurner" /> 大爆炸后，在非常稠密的原始等离子体介质中，粒子不断地产生和湮灭。在这样的环境中，量子相干性和波函数的空间扩展会影响动力学，使人们怀疑玻尔兹曼方程中出现的经典相空间分布''f'' 是否适合描述这一系统。然而，在许多情况下，可以从量子场论的第一原理推导出广义分布函数的有效玻尔兹曼方程。

==方程求解==

==方程求解==

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.

Exact solutions to the Boltzmann equations have been proven to exist in some cases;<ref name=":9">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.

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>

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 name=":10">{{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 name=":11">{{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 name=":12">{{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 name=":13">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(包括有限元和格子玻尔兹曼方法)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" />中的高超音速空气动力学到等离子流<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>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>{{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. 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克努森数幂的渐近展开式来表示(Chapman-Enskog 查普曼-恩斯科格展开式<ref name=":14" />)。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，是希尔伯特第六问题的重要组成部分<ref name=":15" />。

==另见==

==另见==