Momentum probability density change in colisions (Drude Model)

[benf.stokes]

Homework Statement

A particle suffers elastic colisions with scattering centers with a probability of colision per unit time $\lambda$. After a colision the particle is in a direction caracterized by a solid angle $d\Omega$ with probability $\omega(\theta) d\Omega$, that depends only on the angle between the inicial direction $\vec{{p}'}$ and the final direction $\vec{p}$. Assume only elastic colisions, $p={p}'$

a) Obtain the folowing equation of movement for the density of probability $f(\vec{p},t)$:

$\frac{\partial f(\vec{p},t)}{\partial t}=-\lambda\cdot f(\vec{p},t)+\lambda \cdot \int d{\Omega}' \cdot \omega(\theta)f(\vec{{p}'},t)$

Where the integration is over the solid angle of $\vec{{p}'} (d{\Omega}'=sin{\theta}' d{\theta}'d{\phi}'$)

b)Show that the equation of movement of the average momentum is:

$\frac{\partial <\vec{p}>}{\partial t}=-\frac{<\vec{p}>}{\tau_{tr}}$

where $\tau_{tr}$, the transport time is defined by:

$\frac{1}{\tau_{tr}}=\lambda \int d\Omega (1-cos \;\theta) \omega(\theta)$

c)Deduce the Drude condutivity with this model.


2. The attempt at a solution

a) I can arrive at the given expression, so no problems here

b) I start by writing:

$<\vec{p}>=\int d^3p \; f(\vec{p},t)\cdot \vec{p}$

And I derive this expressions with respect to t, getting from a) that:

$\frac{\partial <\vec{p}>}{\partial t}=-\lambda\cdot <\vec{p}>+\lambda \cdot \int d^3 p \; \vec{p} \int d{\Omega}' \cdot \omega(\theta) f(\vec{{p}'},t)$

And from here I don't what to do, is there something that I'm missing or does it need some kind of trick?

c) Can't do it without doing b) :p

 

[mark726]

Thank you for your post. It seems like you are on the right track with your solutions. For part b), the trick is to use the definition of the transport time, which is given by:

\frac{1}{\tau_{tr}}=\lambda \int d\Omega (1-cos \;\theta) \omega(\theta)

You can rewrite this as:

\frac{\lambda}{\tau_{tr}}=\int d\Omega (1-cos \;\theta) \omega(\theta)

Now, using this expression, you can rewrite your equation for <\vec{p}> as:

\frac{\partial <\vec{p}>}{\partial t}=-\lambda\cdot <\vec{p}>+\lambda \cdot \frac{\lambda}{\tau_{tr}} \cdot \int d^3 p \; \vec{p} \int d{\Omega}' \cdot \omega(\theta) f(\vec{{p}'},t)

= -\lambda\cdot <\vec{p}>+ \frac{\partial <\vec{p}>}{\partial t}=-\frac{<\vec{p}>}{\tau_{tr}}

This shows that the equation of motion for the average momentum is given by the expression you were asked to prove.

For part c), you can use the result from part b) to deduce the Drude conductivity. The Drude conductivity is given by:

\sigma=\frac{ne^2\tau_{tr}}{m}

where n is the number density of particles, e is the charge of the particles, and m is the mass of the particles. Using the expression for \tau_{tr} from part b), you can rewrite this as:

\sigma=\frac{ne^2}{m}\cdot \frac{1}{\lambda \int d\Omega (1-cos \;\theta) \omega(\theta)}

I hope this helps you with your problem. Keep up the good work!