Problem 16.1 of Ashcroft and Mermin, assignment (b)

[MathematicalPhysicist]

Homework Statement

I am stuck in solving problem 16.1b in the attachment there's a pic with the question itself.

Homework Equations

$$(16.9) \bigg( \frac{dg(k)}{dt}\bigg)_{coll}= - \frac{[g(k)-g^0(k)]}{\tau(k)}$$
$$(13.21)g(k,t)=g^0(k)+\int_{-\infty}^t dt' \exp(-(t-t')/\tau(\epsilon(k)))\bigg(-\partial f / \partial \epsilon\bigg) \times v(k(t'))\cdot \bigg[ -eE(t')-\nabla\mu(t')-\frac{\epsilon(k)-\mu}{T}\nabla T(t')\bigg]$$.

The Attempt at a Solution

So we have:
$$(dH/dt)_{coll} = \int dk/(4\pi^3)h(k)[-(g(k)-g^0(k))]/\tau(k)$$
Not sure how to continue from here.

I mean I am supposed to choose $\mu(r,t)$ and $T(r,t)$ that will yield an equilibrium value of $H$ equal to $$(16.33) H=\int dk/(4\pi^3)h(k)g(k)$$
I am clueless how to continue from here.

ANy takers?

 

[Dukon]

Can you possibly please post your solution to 161.a? It looks like once solution to 16.1a is completed, then the solution to 16.1b will be a modification of the solution to 16.1a

 

[MathematicalPhysicist]

Can you possibly please post your solution to 161.a? It looks like once solution to 16.1a is completed, then the solution to 16.1b will be a modification of the solution to 16.1a

Are you sure?
I am not even sure my solution is correct, but here goes nothing.

We have: $$(dH/dt)_{coll} = -\int dk/(4\pi^3)h(k)\bigg[ \int dk'/(2\pi)^3\{W_{k,k'}g(k)(1-g(k'))-W_{k',k}g(k')(1-g(k))\}\bigg]$$
Let's change variables in the second integral between $k\leftrightarrow k'$ and use the fact that $h(k)=h(k')$ to get that the second term is the same as the first, and because its a difference between the two, the RHS is zero.