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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>

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>

其中 Γ^α_βγ 代表第二类'''[[wikipedia:Christoffel_symbol|克里斯托费尔符号 Christoffel Symbol]''']（这里假定没有外力，所以粒子在没有碰撞时沿着短程线运动），巧妙地传递出重要的讯息：密度是逆变-协变(xⁱ, p_i)混合相空间内的函数，而不是完全的逆变 (xⁱ, pⁱ)相空间<ref name=":6" /><ref name=":7" />。

其中 Γ^α_βγ 代表第二类'''[[wikipedia:Christoffel_symbol|克里斯托费尔符号 Christoffel Symbol]]'''（这里假定没有外力，所以粒子在没有碰撞时沿着短程线运动），巧妙地传递出重要的讯息：密度是逆变-协变(xⁱ, p_i)混合相空间内的函数，而不是完全的逆变 (xⁱ, pⁱ)相空间<ref name=":6" /><ref name=":7" />。

In [[wikipedia:Physical_cosmology|physical cosmology]] the fully covariant approach has been used to study the cosmic microwave background radiation.<ref name=":8">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]].

In [[wikipedia:Physical_cosmology|physical cosmology]] the fully covariant approach has been used to study the cosmic microwave background radiation.<ref name=":8">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]].

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>

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" />。

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. 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" />。

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==