Homework Help: Solid state physics: From nonequilibrium distribution function to Boltzmann equation

1. Jan 24, 2013

Denver Dang

1. The problem statement, all variables and given/known data
Hi.

I have a course where I am supposed to show how to get to the Boltzmann equation from a nonequilibrium distrubution function.
At the moment I'm kinda lost to how this is done, so hopefully a hint or two would be of some help :)

2. Relevant equations
The nonequilibrium distribution function:

$$g\left( \mathbf{k},t \right)={{g}^{0}}\left( \mathbf{k} \right)+\int_{-\infty }^{t}{dt'}{{e}^{-\left( t-t' \right)\,/\tau \left( \mathrm{ }\varepsilon\text{ }\left( \mathbf{k} \right) \right)}}\left( -\frac{\partial f}{\partial \mathrm{ }\varepsilon\text{ }} \right)\times \mathbf{v}\left( \mathbf{k}\left( t' \right) \right)\cdot \left[ -e\mathbf{E}\left( t' \right)-\nabla \mu \left( t' \right)-\frac{\mathrm{ }\varepsilon\text{ }\left( \mathbf{k} \right)-\mu }{T}\nabla T\left( t' \right) \right]$$
where g0 is the local equilibrium disibution function and f being the Fermi function.

The Boltzmann equation is given by:

$$\frac{\partial g}{\partial t}+\mathbf{v}\cdot \frac{\partial g}{\partial \mathbf{r}}+\mathbf{F}\cdot \frac{1}{\hbar }\frac{\partial g}{\partial \mathbf{k}}={{\left( \frac{\partial g}{\partial t} \right)}_{coll}}$$

3. The attempt at a solution
As far as I have been told, I should be able to come to the Boltzmann equation from the first equation with the use of these equations:

$${{\left( \frac{dg\left( \mathbf{k} \right)}{dt} \right)}_{\mathrm{coll}}}=-\int{\frac{d\mathbf{k}'}{{{\left( 2\mathrm{ }\pi\text{ } \right)}^{3}}}{{W}_{\mathrm{k},\mathrm{k}'}}\left[ g\left( \mathbf{k} \right)-g\left( \mathbf{k}' \right) \right]}$$

and:

$$\int{\frac{d\mathbf{k}'}{{{\left( 2\mathrm{ }\pi\text{ } \right)}^{3}}}{{W}_{\mathrm{k},\mathrm{k}'}}\left[ g\left( \mathbf{k} \right)-g\left( \mathbf{k}' \right) \right]}=\frac{1}{\tau \left( \mathbf{k} \right)}\left[ g\left( \mathbf{k} \right)-{{g}^{0}}\left( \mathbf{k} \right) \right]$$

These last two equations seems to be somewhat related to the relaxtion-time-approximation.
But how I get from the first equation on the very top, to the Boltzmann equation, well, there I'm kinda lost tbh :/

So anyone with an idea?

It's from the book: "Solid state physics by Ashcroft and Mermin" if that helps.

Last edited: Jan 24, 2013