# Ashcroft & Mermin, Solid State Physics, Chapter 16, Problem 2

1. Mar 14, 2013

### Denver Dang

Solving Boltzmann equation

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

Taken from Ashcroft & Mermin, Solid State Physics, Chapter 16, Problem 2:

"A metal is perturbed by a spatially uniform electric field and temperature gradient. Making the relaxation-time-approximation (16.9) (where g0 is the local equilibrium distribution appropriate to the imposed temperature gradient), solve the Boltzmann equation (16.13) to linear order in the field and temperature gradient, and verify that the solution is identical to (13.43)."

2. Relevant equations

Eq. 16.9:
$${{\left( \frac{\partial g}{\partial t} \right)}_{coll}}\simeq -\frac{\left[ g\left( \mathbf{k} \right)-{{g}_{0}}\left( \mathbf{k} \right) \right]}{\tau \left( \mathbf{k} \right)}$$
Eq. 16.13:
$$\frac{\partial g}{\partial t}+\mathbf{v}\cdot \frac{\partial g}{\partial \mathbf{r}}+\mathbf{F}\cdot \frac{1}{\hbar }\frac{\partial g}{\partial \mathbf{k}}={{\left( \frac{\partial g}{\partial t} \right)}_{coll}}$$
Eq. 13.43:
$$g\left( \mathbf{k} \right)={{g}_{0}}\left( \mathbf{k} \right)+\tau \left( \mathrm{ }\!\!\varepsilon\!\!\text{ }\left( \mathbf{k} \right) \right)\left( -\frac{\partial f}{\partial \mathrm{ }\!\!\varepsilon\!\!\text{ }} \right)\mathbf{v}\left( \mathbf{k} \right)\cdot \left[ -e\mathbf{\varepsilon }+\frac{\mathrm{ }\!\!\varepsilon\!\!\text{ }\left( \mathbf{k} \right)-\mu }{T}\left( -\nabla T \right) \right]$$

3. The attempt at a solution

Well, I'm kinda lost. I put Eq. 16.9 into 16.13, and then what?
Can't really see how I get from the Boltzmann equation, with the relaxation-time-approximation, to 13.43 :/

So anyone who can help?

Last edited: Mar 14, 2013

Can you offer guidance or do you also need help?

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