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Linear degree of freedom - Equipartition theorem

  1. Nov 26, 2011 #1
    1. The problem statement, all variables and given/known data

    Consider a classical 'degree of freedom' that is linear rather than quadratic: E = c|q| for some constant c. Derive the equipartition theorem using this energy and show that the average energy is Ebar = kT.


    2. Relevant equations

    [itex] Z = \sum e^{-\beta E(q)} = \sum e^{-\beta c|q|} [/itex]

    [itex] Z = \frac{1}{\Delta q} \int_{-\infty}^{+\infty} e^{-\beta c |q|}dq [/itex]

    3. The attempt at a solution
    The question seems straight forward, but I'm having a hard time grasping it.

    Using the second equation, If I carry out that integral I get:

    [itex] \frac{1}{\Delta q} \frac {-1}{\beta c} \left [ e^{-\beta cq} \right ]_{-\infty}^{+\infty} = 0 [/itex]

    Which doesn't help at all. I'm not sure if there is a trick to the integral or I have to use another method.

    Any help will be much appreciated.

    PS. This is coursework but not a homework question. I am just doing this question to study for a test.
     
  2. jcsd
  3. Dec 11, 2011 #2
    it works if you go from zero to infinity on the bounds. and not -inf to +inf.
    then use the formula <E>=-1/Z(dz/dB)
     
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