1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Solid state physics: From nonequilibrium distribution function to Boltzmann equation

  1. Jan 24, 2013 #1
    1. The problem statement, all variables and given/known data
    Hi.

    I have a course where I am supposed to show how to get to the Boltzmann equation from a nonequilibrium distrubution function.
    At the moment I'm kinda lost to how this is done, so hopefully a hint or two would be of some help :)


    2. Relevant equations
    The nonequilibrium distribution function:

    [tex]g\left( \mathbf{k},t \right)={{g}^{0}}\left( \mathbf{k} \right)+\int_{-\infty }^{t}{dt'}{{e}^{-\left( t-t' \right)\,/\tau \left( \mathrm{ }\varepsilon\text{ }\left( \mathbf{k} \right) \right)}}\left( -\frac{\partial f}{\partial \mathrm{ }\varepsilon\text{ }} \right)\times \mathbf{v}\left( \mathbf{k}\left( t' \right) \right)\cdot \left[ -e\mathbf{E}\left( t' \right)-\nabla \mu \left( t' \right)-\frac{\mathrm{ }\varepsilon\text{ }\left( \mathbf{k} \right)-\mu }{T}\nabla T\left( t' \right) \right][/tex]
    where g0 is the local equilibrium disibution function and f being the Fermi function.

    The Boltzmann equation is given by:

    [tex]\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}}[/tex]


    3. The attempt at a solution
    As far as I have been told, I should be able to come to the Boltzmann equation from the first equation with the use of these equations:

    [tex]{{\left( \frac{dg\left( \mathbf{k} \right)}{dt} \right)}_{\mathrm{coll}}}=-\int{\frac{d\mathbf{k}'}{{{\left( 2\mathrm{ }\pi\text{ } \right)}^{3}}}{{W}_{\mathrm{k},\mathrm{k}'}}\left[ g\left( \mathbf{k} \right)-g\left( \mathbf{k}' \right) \right]}[/tex]

    and:

    [tex]\int{\frac{d\mathbf{k}'}{{{\left( 2\mathrm{ }\pi\text{ } \right)}^{3}}}{{W}_{\mathrm{k},\mathrm{k}'}}\left[ g\left( \mathbf{k} \right)-g\left( \mathbf{k}' \right) \right]}=\frac{1}{\tau \left( \mathbf{k} \right)}\left[ g\left( \mathbf{k} \right)-{{g}^{0}}\left( \mathbf{k} \right) \right][/tex]

    These last two equations seems to be somewhat related to the relaxtion-time-approximation.
    But how I get from the first equation on the very top, to the Boltzmann equation, well, there I'm kinda lost tbh :/

    So anyone with an idea?

    It's from the book: "Solid state physics by Ashcroft and Mermin" if that helps.
     
    Last edited: Jan 24, 2013
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted