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Prove the 3 definitions of entropy are equivalent (stat. mechanics)

  1. Jul 9, 2014 #1
    1. The problem statement, all variables and given/known data

    [tex]
    S(E,V) = kln(\Gamma(E) )\\
    S(E,V) = kln(\omega(E) )\\
    S(E,V) = kln(\Sigma(E) )\\
    [/tex]

    S entropy, k Boltzmann's constant. Prove these 3 are equivalent up to an additive constant.

    2. Relevant equations

    [tex]
    \Gamma(E) = \int_{E<H<E+\Delta}^{'}dpdq\\
    \Gamma(E)=\omega\Delta \\
    \Delta << E\\

    \Sigma(E) = \int_{H<E}^{'}dpdq\\
    \omega = \frac{\partial \Sigma}{\partial E}\\
    [/tex]

    H is the system's Hamiltonian and E is an arbitrary energy. These are integrations over all the p and q's, I wrote them like that to abbreviate.

    3. The attempt at a solution

    Using the 1st definition I can get to the 2nd one, but I can't reach at sigma's definition.
    [tex]

    kln(\Gamma(E)) = kln(\omega\Delta) = kln(\omega) + kln(\Delta)\\
    ln(\Delta) << ln(\omega) => S = kln(\omega)\\
    [/tex]
     
    Last edited: Jul 9, 2014
  2. jcsd
  3. Jul 12, 2014 #2
    I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
     
  4. Jul 13, 2014 #3
    I would say ##\Gamma (E)=\Sigma (E+\Delta) -\Sigma (E)=\Delta \frac{\partial \Sigma}{\partial E}+...##
     
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