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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 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.

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，即粒子某一时刻位于给定位置<math>\bf{r}</math>处占据小邻域（数学上的体积元<math>\mathrm{d}^3 \bf{r}</math>），在动量空间占据给定动量矢量<math> \bf{p}</math>的小邻域（<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。

方程的导出不是通过分析流体中每个粒子的单独位置和动量，而是通过考虑一个典型粒子的位置和动量的概率分布——即粒子某一时刻位于给定位置的小邻域（数学上的体积元<math>\mathrm{d}^3 \bf{r}</math>）、在动量空间占据给定动量矢量<math> \bf{p}</math>的小邻域（<math>\mathrm{d}^3 \bf{p}</math>）的概率。一个给定的非常小的空间区域的概率(数学上是体积元素 < math > mathrm { d } ^ 3 bf { r } </math >) ，动量几乎等于给定的动量矢量 < math > (因此在瞬间占据了一个非常小的动量空间 mathrm { d }3 bf/math >)。

方程的导出不是通过分析流体中每个粒子的单独位置和动量，而是通过考虑一个典型粒子的位置和动量的概率分布——即粒子某一时刻位于给定位置的小邻域（数学上的体积元<math>\mathrm{d}^3 \bf{r}</math>）、在动量空间占据给定动量矢量<math> \bf{p}</math>的小邻域（<math>\mathrm{d}^3 \bf{p}</math>）的概率。一个给定的非常小的空间区域的概率(数学上是体积元素 < math > mathrm { d } ^ 3 bf { r } </math >) ，动量几乎等于给定的动量矢量 < math > (因此在瞬间占据了一个非常小的动量空间 mathrm { d }3 bf/math >)。

  <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]