# Derivation of ln Z for Bose-Einstein case

1. Feb 28, 2012

### VnHorn

1. Problem From Fitzpatrick we need to derive$ln(Z)=αN-\sum ln(1-e^{-\alpha-\betaε_{r}})$ (Equation 8.45)

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

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

Eq 8.30
$\alpha\cong\frac{\partial ln(Z)}{\partial N}$

Eq 8.43
$\overline{n}_{s}=\frac{1}{e^{\alpha+\betaε_{r}}-1}$

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

$ln(Z)=-\alpha-ln(1-e^{-\alpha-\betaε_{r}})$

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 -$\alpha$

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