I would like to understand why Bose-Einstein condensation can occur only for $d>2$ dimensions. Therefore, I want to know what the entropy of a Bose-Einstein condensate is.

The grand canonical potential of a Bose-Einstein condensate in $d$ dimensions is given by

$$\Phi= -\frac{1}{\beta}\left( g_1(z) + \frac{V}{\lambda^d} g_{\frac{d}{2}+1}(z) \right)$$

with $\beta = (k_B T)^{-1}$, $\lambda = \sqrt{\frac{2\pi\hbar^2}{mk_BT}}$, $z = e^{\beta\mu}$ and $g_n(z)$ the polylogarithmic function defined as $$ g_n(z) = \frac{1}{\Gamma(n)}\int_0^\infty\mathop{dx}\frac{x^{n-1}}{e^x z^{-1}-1}.$$

I would like to know how to calculate the expressions for the entropy of a Bose-Einstein condensate above and below the critical temperature directly from the grand canonical potential. I could not find a derivation of the entropy which is the reason why I'm hoping somebody can show me how to do this. What I know is that $$S = - \frac{\partial\Phi}{\partial T},$$ but my problem is that I don't quite know how to distinguish the case $T < T_\text{critical}$ and $T > T_\text{critical}$. It would be nice if somebody could show me how to perform the calculation (at least the necessary steps).



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