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{{{cite book |last1=Gorban |first1= Alexander N.|last2= Karlin |first2= Ilya V. |date=2005 |title= Invariant Manifolds for Physical and Chemical Kinetics|url= https://www.academia.edu/17378865|url-access=| location= Berlin, Heidelberg |publisher= Springer|series= Lecture Notes in Physics (LNP, vol. 660)| isbn= 978-3-540-22684-0|doi= 10.1007/b98103|via= |quote=}} [https://archive.org/details/gorban-karlin-lnp-2005 Alt URL]</ref>)|链接=Special:FilePath/StairsOfReduction.svg]]

{{{cite book |last1=Gorban |first1= Alexander N.|last2= Karlin |first2= Ilya V. |date=2005 |title= Invariant Manifolds for Physical and Chemical Kinetics|url= https://www.academia.edu/17378865|url-access=| location= Berlin, Heidelberg |publisher= Springer|series= Lecture Notes in Physics (LNP, vol. 660)| isbn= 978-3-540-22684-0|doi= 10.1007/b98103|via= |quote=}} [https://archive.org/details/gorban-karlin-lnp-2005 Alt URL]</ref>)|链接=Special:FilePath/StairsOfReduction.svg]]

'''玻尔兹曼方程'''或'''玻尔兹曼输运方程(Boltzmann transport equation, BTE)'''是一个描述非热力学平衡状态的热力学系统统计行为的偏微分方程，由'''[[路德维希·玻尔兹曼 Ludwig Edward Boltzmann|路德维希·玻尔兹曼 Ludwig Boltzmann]]'''于1872年提出。<ref name="Encyclopaediaof" /> 这类系统的经典实例是：在空间中具有温度梯度的流体，组成该流体的粒子通过随机但具有偏向性的传输使得热量从较热的区域流向较冷的区域。在现代文献中，玻尔兹曼方程一词通常用于更一般的意义上，指的是描述热力学系统中宏观量变化的任何动力学方程，如能量、电荷或粒子数。

'''玻尔兹曼方程'''或'''玻尔兹曼输运方程(Boltzmann transport equation, BTE)'''是一个描述非热力学平衡状态的热力学系统统计行为的偏微分方程，由'''[[路德维希·玻尔兹曼 Ludwig Edward Boltzmann|路德维希·玻尔兹曼 Ludwig Boltzmann]]'''于1872年提出。<ref name="Encyclopaediaof" /> 这类系统的经典实例是：在空间中具有温度梯度的流体，组成该流体的粒子通过随机但具有偏向性的传输使得热量从较热的区域流向较冷的区域。在现代文献中，玻尔兹曼方程一词通常用于更一般的意义上，指的是描述热力学系统中宏观量变化的任何动力学方程，如能量、电荷或粒子数。

The equation arises not by analyzing the individual positions and momenta of each particle in the fluid but rather by considering a probability distribution for the position and momentum of a typical particle—that is, the probability that the particle occupies a given very small region of space (mathematically the volume element <math>\mathrm{d}^3 \bf{r}</math>) centered at the position <math>\bf{r}</math>, and has momentum nearly equal to a given momentum vector <math> \bf{p}</math> (thus occupying a very small region of momentum space <math>\mathrm{d}^3 \bf{p}</math>), at an instant of time.

The equation arises not by analyzing the individual [[position vector|position]]s and [[momentum|momenta]] of each particle in the fluid but rather by considering a probability distribution for the position and momentum of a typical particle—that is, the [[probability]] that the particle occupies a given [[infinitesimal|very small]] region of space (mathematically the [[volume element]] <math>\mathrm{d}^3 \bf{r}</math>) centered at the position <math>\bf{r}</math>, and has momentum nearly equal to a given momentum vector <math> \bf{p}</math> (thus occupying a very small region of [[momentum space]] <math>\mathrm{d}^3 \bf{p}</math>), at an instant of time.

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，而是着重考虑一个典型粒子的位置和动量的概率分布，即粒子某一时刻在几何空间占据给定位置<math>\bf{r}</math>处小邻域（数学上的体积元<math>\mathrm{d}^3 \bf{r}</math>），以及在动量空间占据给定动量矢量<math> \bf{p}</math>处小邻域（<math>\mathrm{d}^3 \bf{p}</math>）的概率。

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，而是着重考虑一个典型粒子的位置和动量的概率分布，即粒子某一时刻在几何空间占据给定位置<math>\bf{r}</math>处小邻域（数学上的体积元<math>\mathrm{d}^3 \bf{r}</math>），以及在动量空间占据给定动量矢量<math> \bf{p}</math>处小邻域（<math>\mathrm{d}^3 \bf{p}</math>）的概率。

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).

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).

  <math>

  <math>

      《数学》

        《数学》

      \begin{align}

        \begin{align}

      开始{ 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:

        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]