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Momentum probability density change in colisions (Drude Model)

  1. Apr 3, 2013 #1
    1. The problem statement, all variables and given/known data

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

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

    [itex]\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)[/itex]

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

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

    [itex]\frac{\partial <\vec{p}>}{\partial t}=-\frac{<\vec{p}>}{\tau_{tr}}[/itex]

    where [itex]\tau_{tr}[/itex], the transport time is defined by:

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

    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:

    [itex]<\vec{p}>=\int d^3p \; f(\vec{p},t)\cdot \vec{p}[/itex]

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

    [itex]\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)[/itex]

    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
     
    Last edited: Apr 3, 2013
  2. jcsd
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