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Note that we have used the fact that the phase space volume element <math> d^3\bf{r}</math> <math> d^3\bf{p}</math> is constant, which can be shown using [[Hamilton's equations]] (see the discussion under [[Liouville's theorem (Hamiltonian)|Liouville's theorem]]). However, since collisions do occur, the particle density in the phase-space volume <math> d^3\bf{r}</math> '<math> d^3\bf{p}</math> changes, so
Note that we have used the fact that the phase space volume element <math> d^3\bf{r}</math> <math> d^3\bf{p}</math> is constant, which can be shown using [[Hamilton's equations]] (see the discussion under [[Liouville's theorem (Hamiltonian)|Liouville's theorem]]). However, since collisions do occur, the particle density in the phase-space volume <math> d^3\bf{r}</math> '<math> d^3\bf{p}</math> changes, so
这里，注意到相空间元 <math> d^3\mathbf{r}\,d^3\mathbf{p}</math> 是恒定的这个事实可以从[[wikipedia:Hamilton's equations|哈密顿方程 Hamilton's Equations]]（见[[wikipedia:Liouville's theorem (Hamiltonian)|刘维尔定理 Liouville's Theorem（哈密顿力学）]]）得知。然而，由于存在碰撞，相空间元 <math> d^3\mathbf{r}\,d^3\mathbf{p}</math> 中的粒子密度是可变的，所以有：{{NumBlk|2=<math>\begin{align}
这里，注意到相空间元 <math> d^3\mathbf{r}\,d^3\mathbf{p}</math> 是恒定的这个事实可以从[[wikipedia:Hamilton's equations|哈密顿方程 Hamilton's Equations]]（见[[wikipedia:Liouville's theorem (Hamiltonian)|刘维尔定理 Liouville's Theorem]]）得知。然而，由于存在碰撞，相空间元 <math> d^3\mathbf{r}\,d^3\mathbf{p}</math> 中的粒子密度是可变的，所以有：{{NumBlk|2=<math>\begin{align}
dN_{coll} &= \left ( \frac{\partial f}{\partial t} \right )_{coll}\Delta td^{3}\textbf{r}\, d^{3}\textbf{p}\\[5pt]
dN_{coll} &= \left ( \frac{\partial f}{\partial t} \right )_{coll}\Delta td^{3}\textbf{r}\, d^{3}\textbf{p}\\[5pt]
& = f\left ( \textbf{r}+\frac{\textbf{p}}{m}\Delta t,\textbf{p}+\textbf{F}\Delta t,t+\Delta t \right )\, d^{3}\textbf{r}\, d^{3}\textbf{p}- f(\textbf{r},\textbf{p},t)\, d^{3}\textbf{r}\, d^{3}\textbf{p}\\[5pt]
& = f\left ( \textbf{r}+\frac{\textbf{p}}{m}\Delta t,\textbf{p}+\textbf{F}\Delta t,t+\Delta t \right )\, d^{3}\textbf{r}\, d^{3}\textbf{p}- f(\textbf{r},\textbf{p},t)\, d^{3}\textbf{r}\, d^{3}\textbf{p}\\[5pt]