Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 36A / b / iv / Solution

For $T>T_B$, the ground-state occupation is negligible to leading order, so $N\simeq N_{ϵ}$. Comparing the number equation with its value at $T_B$ gives

For $T<T_B$, $\mu$ is close to zero andUsing $M=N(e^{ϵ/(k{T}_B)}-1)$ in $N_0=N-N_{ϵ}$ and solving for $\mu$ gives