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第15行： − − 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). − − 玻尔兹曼方程可以用来确定流体在运输过程中物理量如何变化，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)。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>{{cite journal | last1=DiPerna | first1= R. J. |last2 = Lions | first2 = P.-L. | title= On the Cauchy problem for Boltzmann equations: global existence and weak stability | journal= Ann. of Math. |series= 2 | volume=130 | pages= 321–366 | year=1989 | doi=10.2307/1971423 | issue=2| jstor= 1971423 }}+ − − </ref><ref>{{cite journal |author1=Philip T. Gressman |authorlink1=Philip Gressman |author2=Robert M. Strain |name-list-style=amp|year=2010 |title= Global classical solutions of the Boltzmann equation with long-range interactions |journal= Proceedings of the National Academy of Sciences |volume=107 |pages= 5744–5749 | doi = 10.1073/pnas.1001185107 |bibcode = 2010PNAS..107.5744G |arxiv = 1002.3639 |issue= 13 |pmid=20231489 |pmc=2851887}}</ref> − − 玻尔兹曼方程是一个非线性积分微分方程，方程中的未知函数是位置和动量六维空间中的一个概率密度函数。方程解的存在唯一性仍然是未完全解决的问题，但是近期的一些结果是很有希望的。 第72行：
第67行： − + − 《数学》+ − + − + − + − \begin{align}+ − + − 开始{ align }+ − + − is the number of molecules which ''all'' have positions lying within a volume element <math> d^3\bf{r}</math> about '''r''' and momenta lying within a [[momentum space]] element <math> \mathrm{d}^3\bf{p}</math> about '''p''', at time ''t''.<ref>{{Cite book |last=Huang |first=Kerson |year=1987 |title=Statistical Mechanics |url=https://archive.org/details/statisticalmecha00huan_475 |url-access=limited |location=New York |publisher=Wiley |isbn=978-0-471-81518-1 |page=[https://archive.org/details/statisticalmecha00huan_475/page/n65 53] |edition=Second }}</ref> [[Integration (calculus)|Integrating]] over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region:+
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玻尔兹曼方程并不分析流体中每个粒子的单个位置和动量，而是只考虑一类粒子的位置和动量的概率分布，此类粒子某一时刻在几何空间占据以给定位置<math>\bf{r}</math>为中心的小邻域（数学上的体积元<math>\mathrm{d}^3 \bf{r}</math>），且其动量几乎与给定动量矢量<math> \bf{p}</math>相等，在动量空间占据非常小的区域<math>\mathrm{d}^3 \bf{p}</math>。
玻尔兹曼方程并不分析流体中每个粒子的单个位置和动量，而是只考虑一类粒子的位置和动量的概率分布，此类粒子某一时刻在几何空间占据以给定位置<math>\bf{r}</math>为中心的小邻域（数学上的体积元<math>\mathrm{d}^3 \bf{r}</math>），且其动量几乎与给定动量矢量<math> \bf{p}</math>相等，在动量空间占据非常小的区域<math>\mathrm{d}^3 \bf{p}</math>。
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]].
玻尔兹曼方程可以用来确定流体在运输过程中物理量如何变化，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.
</ref><ref name=":1">Philip T. Gressman & Robert M. Strain (2010). "Global classical solutions of the Boltzmann equation with long-range interactions". ''Proceedings of the National Academy of Sciences''. '''107''' (13): 5744–5749. arXiv:1002.3639. Bibcode:2010PNAS..107.5744G. doi:10.1073/pnas.1001185107. PMC 2851887. <nowiki>PMID 20231489</nowiki>.</ref>
玻尔兹曼方程是非线性积分微分方程，方程中的未知函数是位置和动量在六维空间中的概率密度函数。方程解的存在唯一性仍然是未完全解决的问题，但是一些研究显示解决这一问题是很有希望的。<ref name=":0" /><ref name=":1" />
==Overview 概述==
==Overview 概述==
<math>
<math>
《数学》
\begin{align}
开始{ align }
is the number of molecules which ''all'' have positions lying within a volume element <math> d^3\bf{r}</math> about '''r''' and momenta lying within a [[momentum space]] element <math> \mathrm{d}^3\bf{p}</math> about '''p''', at time ''t''.<ref>{{Cite book |last=Huang |first=Kerson |year=1987 |title=Statistical Mechanics |url=https://archive.org/details/statisticalmecha00huan_475 |url-access=limited |location=New York |publisher=Wiley |isbn=978-0-471-81518-1 |page=[https://archive.org/details/statisticalmecha00huan_475/page/n65 53] |edition=Second }}</ref> [[Integration (calculus)|Integrating]] over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region:
N & = \int\limits_\mathrm{momenta} \text{d}^3\mathbf{p} \int\limits_\mathrm{positions} \text{d}^3\mathbf{r}\,f (\mathbf{r},\mathbf{p},t) \\[5pt]
N & = \int\limits_\mathrm{momenta} \text{d}^3\mathbf{p} \int\limits_\mathrm{positions} \text{d}^3\mathbf{r}\,f (\mathbf{r},\mathbf{p},t) \\[5pt]