Reworking of the Drude model using scattering statistics

[Aaron young]

Homework Statement

The problem I have been set is to rework the Drude model using clearly defined scattering statistics.

Homework Equations

The Drude model as we have been given it is in terms of momentum
$\vec{p}(t+dt)=(1-\frac{dt}{\tau})(\vec{p}(t)-q\vec{E}(t)dt)+(\frac{dt}{\tau})(0)$
Where that last term represents the contribution of the scattered electrons to the total momentum of the electrons in the system.

The Attempt at a Solution

My attempts so far have focused on trying to use Fermi-Dirac statistics to somehow derive the momentum of a scattered electron (ie. one at thermal velocity) as a function of the average energy of an electron at thermal velocity. I have the horrible feeling I have been barking up completely the wrong tree however, so I am now making an attempt to somehow integrate classical elastic scattering off of nuclei into the equation. I don't know how well this will work though.

If anyone has been given a similar assignment in the past or has an idea what direction sounds most right to be going in some suggestions would be greatly appreciated.

 

[Aaron young]

Based on a question in a different thread that seems to be similar I have done the following

Probability of scattering per unit time = $\lambda$
direction after scattering characterised by the solid angle $d\Omega '$
The probability of a given angle after scattering is given by $\omega (\theta)d\Omega '$
were $\theta$ is the angle between the incident and scattered momenta, $\vec{p}$ and $\vec{p}'$.

From this I have said that
$\frac{\partial}{\partial t}f(\vec{p},t)=(1-\lambda)unscatteredthing+\lambda scatteredthing$
which I have written as
$\frac{\partial}{\partial t}f(\vec{p},t)=f(\vec{p},t)-\lambda f(\vec{p},t)+\lambda \int \omega (\theta) f(\vec{p}',t) d\Omega'$

What I have gotten agrees with what is in the question containing the prompt I went off except for the first term, which does not feature in the version from the hint in that question.