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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. [[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 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. [[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 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'''[[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" />。

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==