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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]].

玻尔兹曼方程在星系动力学中也大有可为。在一定的假设下，星系可以近似为连续流体；其质量分布可以表示为 ''f''  。在星系中，不同星体间的物理碰撞鲜有发生。在远长于宇宙年龄的时间内，引力碰撞的影响可以被忽略。

玻尔兹曼方程在星系动力学中也大有可为。在一定的假设下，星系可以近似为连续流体；其质量分布可以表示为 ''f''  。在星系中，不同星体间的物理碰撞鲜有发生。在远长于'''[[wikipedia:Age_of_the_universe|宇宙年龄 Age of the Universe]]'''的时间内，引力碰撞的影响可以被忽略。

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

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

方程在广义相对论中的推广<ref name=":5" />为：

方程在'''[[wikipedia:General_relativity|广义相对论 General Relativity]]'''中的推广<ref name=":5" />为：

:<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 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 代表第二类克里斯托费尔符号（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]].