Particle saturation in a homogeneous Bose gas ($N_c(T)=V\zeta (3/2)/{\lambda }_T^3$)

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.