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Derivation of ln Z for Bose-Einstein case

  1. Feb 28, 2012 #1
    1. Problem From Fitzpatrick we need to derive[itex]ln(Z)=αN-\sum ln(1-e^{-\alpha-\betaε_{r}})[/itex] (Equation 8.45)

    2. Relevant equations
    This is claimed to be derived from Equations 8.20, 8.30, and 8.43

    Eq 8.20
    [itex]\overline{n}_{s}=-\frac{1}{\beta}\frac{\partial ln(Z)}{\partial\epsilon_{s}}[/itex]

    Eq 8.30
    [itex]\alpha\cong\frac{\partial ln(Z)}{\partial N}[/itex]

    Eq 8.43

    3. The attempt at a solution
    My initial attempt involved setting the RHS of 8.20 to the RHS of 8.43 and then integrating to solve for ln(Z)
    but this ultimately gave me


    I'm not so concerned with the lack of N and the missing summation since that should come when I apply it to more particles, but as hard as I try everything I do ends up with that -[itex]\alpha[/itex]

    Nevermind, solved it. Needed to consider the contribution from all the different portions of the partial and make sure it met all requirements. Thanks
    Last edited: Feb 28, 2012
  2. jcsd
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