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The Boltzmann equation can be used to determine how physical quantities change, such as [[heat]] energy and [[momentum]], when a fluid is in transport. One may also derive other properties characteristic to fluids such as [[viscosity]], [[thermal conductivity]], and [[electrical conductivity]] (by treating the charge carriers in a material as a gas).<ref name="Encyclopaediaof" /> See also [[convection–diffusion equation]].
The Boltzmann equation can be used to determine how physical quantities change, such as [[heat]] energy and [[momentum]], when a fluid is in transport. One may also derive other properties characteristic to fluids such as [[viscosity]], [[thermal conductivity]], and [[electrical conductivity]] (by treating the charge carriers in a material as a gas).<ref name="Encyclopaediaof" /> See also [[convection–diffusion equation]].
在流体运输过程中，玻尔兹曼方程可以用来确定物理量如何变化，比如热能和动量。人们还可以推导出流体的其他特性，如粘度、热导率和电导率(通过将材料中的载流子当作气体来处理)。<ref name="Encyclopaediaof" /> 参见[[对流扩散方程]]。
The equation is a [[Nonlinear system|nonlinear]] [[integro-differential equation]], and the unknown function in the equation is a probability density function in six-dimensional space of a particle position and momentum. The problem of existence and uniqueness of solutions is still not fully resolved, but some recent results are quite promising.<ref name=":0">DiPerna, R. J.; Lions, P.-L. (1989). "On the Cauchy problem for Boltzmann equations: global existence and weak stability". ''Ann. of Math''. 2. '''130''' (2): 321–366. doi:10.2307/1971423. JSTOR 1971423.
The equation is a [[Nonlinear system|nonlinear]] [[integro-differential equation]], and the unknown function in the equation is a probability density function in six-dimensional space of a particle position and momentum. The problem of existence and uniqueness of solutions is still not fully resolved, but some recent results are quite promising.<ref name=":0">DiPerna, R. J.; Lions, P.-L. (1989). "On the Cauchy problem for Boltzmann equations: global existence and weak stability". ''Ann. of Math''. 2. '''130''' (2): 321–366. doi:10.2307/1971423. JSTOR 1971423.
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>
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 Quantum_field_theory]]的第一原理推导得出。
===General relativity and astronomy 广义相对论和天文学===
===General relativity and astronomy 广义相对论和天文学===