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I am trying to make the following equation to get Boltzmann equation which I write below.

[tex]

f(\mathbf{x}+\frac{\mathbf{p}}{m}dt,\mathbf{p} + \mathbf{F}dt,t+dt) \,d\mathbf{x}\,d\mathbf{p}

- f(\mathbf{x},\mathbf{p},t)d\mathbf{x}\,d\mathbf{p} =

\left. \frac{\partial f(\mathbf{x},\mathbf{p},t)}{\partial t} \right|_{\mathrm{coll}} \, d\mathbf{x} \, d\mathbf{p} \, dt

[/tex]

Boltzmann equation:

[tex]

\frac{\partial f}{\partial t}

+ \frac{\partial f}{\partial \mathbf{x}} \cdot \frac{\mathbf{p}}{m}

+ \frac{\partial f}{\partial \mathbf{p}} \cdot \mathbf{F}

= \left. \frac{\partial f}{\partial t} \right|_{\mathrm{coll}}

[/tex]

So, should I do partial differentiation? But how to do that since I do not know the function exactly.. Can someone give me some guidances? Many thanks before.

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# Boltzmann equation

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