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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
    [itex]\overline{n}_{s}=\frac{1}{e^{\alpha+\betaε_{r}}-1}[/itex]


    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

    [itex]ln(Z)=-\alpha-ln(1-e^{-\alpha-\betaε_{r}})[/itex]

    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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