True Ground State Population of Ideal Bose-Einstein Condensate at Critical Temperature

Question

I'm supposed to demonstrate that although we make the assumption in an ideal BEC that the ground state population follows

$N_0 = N\left[1-\left(\frac{T}{T_c}\right)^{3/2}\right]$

in reality the true ground state population at $T = T_c$ is not zero, and is

$N_0 = \sqrt{\frac{2\pi^{1/2}N}{\zeta(3/2)}} + \mathcal{o}(\sqrt{N})$.

The hint included is to use $N = \frac{e^{\beta \mu}}{1 - e^{\beta\mu}} + \frac{V}{\lambda_D^3}g_{3/2}\left(e^{\beta\mu}\right)$.

I've been racking my brain trying to figure this out. I assumed I was supposed to consider the limit of small temperature above the critical temperature. I set $g_{3/2}\left(e^{\beta\mu}\right) \to g_{3/2}(1) = \zeta(3/2)$ and I tried expanding the exponents to first order in $\beta \mu$ for small values of $\mu$ but I can't seem to figure out the correct expression. I'm using Landau and found an expression for $\mu$ which was determined by expanding about $T_c$ but it didn't seem to be terribly useful to me here either... Any ideas?

Answer 1

Around $\mu = 0$, you can say $$\beta \mu \approx -\frac{1}{N_0} $$ and use the first two terms in the series expansion of the polylogarithm: $$ g_{s}(\exp[X]) = \Gamma(1-s)(-X)^{s-1} + \sum_{k=0}^\infty \frac{\zeta(s-k)}{k!} X^k $$ Solving for: $$ N_0 = \frac{e^{\beta \mu}}{1-e^{\beta \mu}} $$ Should give you nearly the desired result: $$ N_0 \approx \left(\frac{2 N \sqrt{\pi}}{\zeta(3/2)}\right)^{2/3} $$

I don't promise that's right, but its at least on the right track.