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**I'm trying to show that conductivity of a metal in uniform Electric field is:**

$$

\sigma=\int \frac{d\textbf{k}}{4\pi^3}\left (- \frac{\partial f}{\partial \epsilon} \right )\textbf{v(k)u(k)}

$$

where u(k) is a solution to the integral equation

$$

\textbf{v(k)}=\int \frac{d\textbf{k'}}{(2\pi)^3}W_{kk'}\textbf{[u(k)-u(k')]}

$$

$$

g(\textbf{k})=f(\textbf{k})+\delta g(\textbf{k})

$$

\delta g(k) is of order of E.

I also want to derive to linear order in E an integral equation obeyed by \delta g.

I wrote:

$$

-\frac{e}{\hbar}\frac{\partial f}{\partial E} \textbf{v.E}=\int \frac{d\textbf{k'}}{(2\pi)^3}W_{kk'}[\delta\textbf{g(k)}-\delta\textbf{g(k')}]

$$

so what is the next step?

this is problem 4 of chapter 16 Ashcroft and Mermin. I don't have any idea for part b of the problem too. Any help is appreciated.

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