Homogeneous ideal Bose gas condensation ($T_c=2\pi {\hslash }^2[n/\zeta (3/2){]}^{2/3}/(m{k}_B)$)

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 $n_{\mathrm{ex}}={\lambda }_T^{-3}{\mathrm{Li}}_{3/2}(z)$, with quantum gas fugacity $z=e^{\mu /(k_{B}T)}\le 1$. 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 $U=(3/2)V{k}_{B}T{\lambda }_T^{-3}{\mathrm{Li}}_{5/2}(z)$, with $z\to 1$ 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.

For quadratic three-dimensional dispersion below condensation, the excited-state thermal energy is proportional to $T^{5/2}$. Differentiating at fixed particle number and volume gives the displayed heat capacity, with limiting value approximately $1.926N{k}_B$ at $T_c^{-}$. The macroscopic ground occupation carries no excitation energy in the shifted reference. The exponent differs from the linear-phonon result of an interacting gas.

A hard-wall cube has positive one-particle ground energy $E_0=3{\pi }^2{\hslash }^2/(2m{L}^2)$. Condensation corresponds to the absolute chemical potential approaching $E_0$, not zero. Shifting the energy reference makes the limiting chemical potential zero, while the exact shifted excited-state density of states also shifts. Keeping its bulk square-root form is a large-system approximation valid when $k_{B}T\gg E_0$. A periodic cube instead has zero ground energy.

At fixed temperature, the continuum excited population of a three-dimensional ideal Bose gas reaches a finite maximum as the chemical potential approaches the ground energy. Adding particles beyond that maximum creates macroscopic ground occupation. Thus $N^{\prime }=\min (N,N_c)$ and $N_0=\max (0,N-N_c)$ to leading thermodynamic-limit order. Finite systems retain nonmacroscopic ground occupation below the nominal threshold.