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Consider particles described by ''f'', each experiencing an ''external'' force '''F''' not due to other particles (see the collision term for the latter treatment).

Consider particles described by ''f'', each experiencing an ''external'' force '''F''' not due to other particles (see the collision term for the latter treatment).

 

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<math> \mathbf{r}+\Delta \mathbf{r}= \textbf{r}+\frac{\textbf{p}}{m}\Delta t</math> and momentum '''p''' + Δ'''p''' = '''p''' + '''F'''Δ''t''. Then, in the absence of collisions, ''f'' must satisfy

 

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

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