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第88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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The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.
'''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 Boltzmann]]'''于1872年提出。
'''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。
The '''Boltzmann equation''' or '''Boltzmann transport equation''' ('''BTE''') describes the statistical behaviour of a [[thermodynamic system]] not in a state of [[Thermodynamic equilibrium|equilibrium]], devised by [[Ludwig Boltzmann]] in 1872.<ref name="Encyclopaediaof">Encyclopaedia of Physics (2nd Edition), R. G. Lerner, G. L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3.</ref>
The '''Boltzmann equation''' or '''Boltzmann transport equation''' ('''BTE''') describes the statistical behaviour of a [[thermodynamic system]] not in a state of [[Thermodynamic equilibrium|equilibrium]], devised by [[Ludwig Boltzmann]] in 1872.<ref name="Encyclopaediaof">Encyclopaedia of Physics (2nd Edition), R. G. Lerner, G. L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3.</ref>
<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]