= Homogeneous ideal Bose gas condensation
{title2=$T_c=2\pi\hbar^2[n/\zeta(3/2)]^{2/3}/(mk_B)$}
For a uniform three-dimensional ideal gas of spinless <bosons> with energy $p^2/(2m)$ and ground energy set to zero, the excited number density is $n_{\rm ex}=\lambda_T^{-3}\operatorname{Li}_{3/2}(z)$, with <quantum gas fugacity> $z=e^{\mu/(k_BT)}\leq1$. Its finite maximum $\zeta(3/2)/\lambda_T^3$ gives the displayed <critical temperature>. Below it, particle conservation gives <condensate fraction> $N_0/N=1-(T/T_c)^{3/2}$. The excited-state <internal energy> is $U=(3/2)Vk_BT\lambda_T^{-3}\operatorname{Li}_{5/2}(z)$, with $z\to1$ below the transition. The exponent differs from that of a harmonic trap. Condensation is not by itself a demonstration of <superfluidity>; the ideal quadratic dispersion has vanishing Landau critical speed.
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