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

The Bose-Einstein distribution gives one occupation for the nondegenerate ground level and $M$ identical contributions from the excited levels:

At fixed temperature, the excited population is largest as $\mu \uparrow 0$, henceIf $N>N_{ϵ}^{\max }$, the excited levels cannot accommodate all particles. The excess $N-N_{ϵ}^{\max }$ occupies the zero-energy level macroscopically: this is Bose-Einstein condensation.

At the critical temperature $T_B$, the excited states are just saturated and the condensate is about to form, soTherefore

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

Above $T_B$, essentially all $N$ particles occupy the excited levels, soBelow $T_B$, the ground-state condensate carries no energy and $\mu \simeq 0$, henceDifferentiating gives the heat capacityThe displayed formula in the examination paper omits the factor $e^{\beta ϵ}$. Its stated expression follows after the additional high-temperature approximation $\beta ϵ\ll 1$, for which $e^{\beta ϵ}\simeq 1$.