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→The force and diffusion terms “force”项与“diff”项
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{{NumBlk|2=<math>\begin{align}
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{{NumBlk|2=<math>\begin{align}
dN_{coll}= \left ( \frac{\partial f}{\partial t} \right )_{coll}\Delta td^{3}\textbf{r}\, d^{3}\textbf{p}
dN_{coll}&= \left ( \frac{\partial f}{\partial t} \right )_{coll}\Delta td^{3}\textbf{r}\, d^{3}\textbf{p}
&= 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}
&= 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}
&=\Delta f d^{3}\textbf{r}\, d^{3}\textbf{p}</math>|3={{EquationRef|1}}|：}}
&=\Delta f d^{3}\textbf{r}\, d^{3}\textbf{p}
\end{align}</math>|3={{EquationRef|1}}|：}}
where Δ''f'' is the ''total'' change in ''f''. Dividing ({{EquationNote|1}}) by <math> d^3\bf{r}</math> <math> d^3\bf{p}</math> Δ''t'' and taking the limits Δ''t'' → 0 and Δ''f'' → 0, we have{{NumBlk|2=<math>\frac{d f}{d t} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}</math>|3={{EquationRef|2}}|：}}
where Δ''f'' is the ''total'' change in ''f''. Dividing ({{EquationNote|1}}) by <math> d^3\bf{r}</math> <math> d^3\bf{p}</math> Δ''t'' and taking the limits Δ''t'' → 0 and Δ''f'' → 0, we have{{NumBlk|2=<math>\frac{d f}{d t} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}</math>|3={{EquationRef|2}}|：}}