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→The force and diffusion terms “force”项与“diff”项
Suppose at time ''t'' some number of particles all have position '''r''' within element <math> d^3\bf{r}</math> and momentum '''p''' within <math> d^3\bf{p}</math>. If a force '''F''' instantly acts on each particle, then at time ''t'' + Δ''t'' their position will be '''r''' + Δ'''r''' = '''r''' + '''p'''Δ''t''/''m'' and momentum '''p''' + Δ'''p''' = '''p''' + '''F'''Δ''t''. Then, in the absence of collisions, ''f'' must satisfy<syntaxhighlight lang="latex">
Suppose at time ''t'' some number of particles all have position '''r''' within element <math> d^3\bf{r}</math> and momentum '''p''' within <math> d^3\bf{p}</math>. If a force '''F''' instantly acts on each particle, then at time ''t'' + Δ''t'' their position will be '''r''' + Δ'''r''' = '''r''' + '''p'''Δ''t''/''m'' and momentum '''p''' + Δ'''p''' = '''p''' + '''F'''Δ''t''. Then, in the absence of collisions, ''f'' must satisfy<syntaxhighlight lang="latex">
<math>f(\textbf{r}+\frac{\textbf{p}}{m}\Delta t,\textbf{p}+\textbf{F}\Delta t,t+\Delta t)\, d^{3}\textbf{r}\, d^{3}\textbf{p}= f(\textbf{r},\textbf{p},t)\, d^{3}\textbf{r}\, d^{3}\textbf{p}<math>
<math>f(\textbf{r}+\frac{\textbf{p}}{m}\Delta t,\textbf{p}+\textbf{F}\Delta t,t+\Delta t)\, d^{3}\textbf{r}\, d^{3}\textbf{p}= f(\textbf{r},\textbf{p},t)\, d^{3}\textbf{r}\, d^{3}\textbf{p}<math>
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f(\textbf{r}+\frac{\textbf{p}}{m}\Delta t,\textbf{p}+\textbf{F}\Delta t,t+\Delta t)\, d^{3}\textbf{r}\, d^{3}\textbf{p}= f(\textbf{r},\textbf{p},t)\, d^{3}\textbf{r}\, d^{3}\textbf{p}
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<math>\displaystyle f(\textbf{r}+\frac{\textbf{p}}{m}\Delta t,\textbf{p}+\textbf{F}\Delta t,t+\Delta t)\, d^{3}\textbf{r}\, d^{3}\textbf{p}= f(\textbf{r},\textbf{p},t)\, d^{3}\textbf{r}\, d^{3}\textbf{p}<math>
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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