= Solution
<Bose-Einstein condensation> is macroscopic occupation of a single lowest-energy one-particle state. For a homogeneous three-dimensional <ideal gas> of conserved massive <bosons>, set that <energy> to zero. The excited-state population is
$$
N_{\rm ex}=\frac{V}{\lambda_T^3}g_{3/2}(z),\qquad
z=e^{\mu/(k_BT)}\le1,\qquad
\lambda_T=\sqrt{\frac{2\pi\hbar^2}{mk_BT}}.
$$
Here $g_{3/2}(z)=\sum_{r\ge1}z^r/r^{3/2}$. As $\mu\to0^-$, the <excited states> can hold only $V\zeta(3/2)/\lambda_T^3$ particles. Additional particles occupy the <ground state>. Thus
$$
\boxed{T_c=\frac{2\pi\hbar^2}{mk_B}\left[\frac{n}{\zeta(3/2)}\right]^{2/3},\qquad
\frac{N_0}{N}=1-\left(\frac T{T_c}\right)^{3/2}\quad(T<T_c).}
$$
The isolated <ground state> must be counted separately from the continuum <density of states>; otherwise one misses the condensate. A dilute atomic gas is a physical realization of the quantum-degeneracy regime, but trapping geometry and interactions modify the homogeneous ideal-gas formulas.<Statistical Physics by David Tong>[§3.5.3: saturation of excited states, separate ground-state occupation and ideal-gas condensation temperature]{ref}
\Image[/past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2012/paper-1-tsp2012-condensation.png]
{title=Ideal Bose condensate fraction}
{description=Original homogeneous three-dimensional ideal-gas condensate fraction and maximum thermal population, normalized to the critical temperature. No trapped-gas or experimental data are implied.}
In the homogeneous <ideal gas>, one and two dimensions do not have finite-temperature condensation in the <thermodynamic limit> because the low-energy population integral diverges. Finite size or a trap changes that conclusion's assumptions. Condensation and <superfluidity> are distinct: the ideal quadratic gas has zero <Landau critical velocity>, while interactions can support a sound mode and superfluid flow. Neither zero <viscosity> nor phase-rigid flow follows from the occupation formula alone.
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