- #1
Ai52487963
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Homework Statement
Show that the multiplicity of an Einstein solid with large N and q is
[tex]\frac{\left(\frac{q+N}{q}\right)^q\left(\frac{q+N}{N}\right)^N}{\sqrt{2\pi q\left(q+N\right)/N}}[/tex]
Homework Equations
[tex]N! \approx N^N e^{-N} \sqrt{2 \pi N}[/tex]
The Attempt at a Solution
Well, I've done thus so far:
[tex]
\Omega(N,q) = \frac{(q+N-1)!}{q!(N-1)!} \approx \frac{(q+N)!}{q!N!}
ln(\Omega) = ln(q+N)! - lnq! - lnN
\par
\approx (q+N)ln(q+N) - (q+N) - qlnq+q - NlnN + N = (q+N)ln(q+N) - qlnq - NlnN
[/tex]
I feel like I'm close, but I've no idea where to go from here.