leoneri
- 19
- 0
Hi
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}<br /> - f(\mathbf{x},\mathbf{p},t)d\mathbf{x}\,d\mathbf{p} =<br /> \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}<br /> + \frac{\partial f}{\partial \mathbf{x}} \cdot \frac{\mathbf{p}}{m}<br /> + \frac{\partial f}{\partial \mathbf{p}} \cdot \mathbf{F}<br /> = \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.
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}<br /> - f(\mathbf{x},\mathbf{p},t)d\mathbf{x}\,d\mathbf{p} =<br /> \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}<br /> + \frac{\partial f}{\partial \mathbf{x}} \cdot \frac{\mathbf{p}}{m}<br /> + \frac{\partial f}{\partial \mathbf{p}} \cdot \mathbf{F}<br /> = \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.
