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Homework Help: 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

    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]


    [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
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