Saturation particle number for Bose-Einstein condensation ($N_c(T)={\int }_0^{\infty }g(ϵ)/(e^{ϵ/(k_{B}T)}-1)dϵ$)

With the ground-state energy set to zero, the chemical potential of an ideal Bose gas cannot exceed zero. The limiting population of excited states as it approaches zero is the saturation particle number, provided the integral converges. Additional particles occupy the ground state macroscopically. For a three-dimensional uniform box, $N_c=\zeta (3/2)V/{\lambda }^3$ with thermal wavelength $\lambda =h/\sqrt{2\pi m{k}_{B}T}$.