更改

相反，数值方法（包括[[有限元 Finite Elements]]和[[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">{{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>中的[[高超音速空气动力学 Hypersonic Aerodynamics]]到等离子流<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>。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">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 Finite Elements]]和[[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">{{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>中的[[高超音速空气动力学 Hypersonic Aerodynamics]]到等离子流<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>。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">H.J.W. Müller-Kirsten, Basics of Statistical Mechanics, Chapter 13, 2nd ed., World Scientific (2013). </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来表示（'''[[查普曼-恩斯科格 Chapman-Enskog]]'''展开式<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>）。展开式的前两项给出了'''[[wikipedia:Euler_equations_(fluid_dynamics)|欧拉方程 Euler Equations]]'''和'''[[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'''[[希尔伯特第六问题 Hilbert's Sixth Problem]]'''的重要组成部分<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来表示（[[查普曼-恩斯科格 Chapman-Enskog]]展开式<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>）。展开式的前两项给出了[[欧拉方程 Euler Equations]]和[[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[[希尔伯特第六问题 Hilbert's Sixth Problem]]的重要组成部分<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>。

==另见==

==另见==