(a) A system of non-interacting bosons has single particle states $|i\rangle$ with energies $\epsilon_{i} \geqslant 0$. Show that the grand canonical partition function is

$$\log \mathcal{Z}=-\sum_{i} \log \left(1-e^{-\beta\left(\epsilon_{i}-\mu\right)}\right)$$

where $\beta=1 /(k T), k$ is Boltzmann's constant, and $\mu$ is the chemical potential. What is the maximum possible value for $\mu$ ?

(b) A system of $N \gg 1$ bosons has one energy level with zero energy and $M \gg 1$ energy levels with energy $\epsilon>0$. The number of particles with energies $0, \epsilon$ is $N_{0}, N_{\epsilon}$ respectively.

(i) Write down expressions for $\left\langle N_{0}\right\rangle$ and $\left\langle N_{\epsilon}\right\rangle$ in terms of $\mu$ and $\beta$.

(ii) At temperature $T$ what is the maximum possible number $N_{\epsilon}^{\max }$ of bosons in the state with energy $\epsilon ?$ What happens for $N>N_{\epsilon}^{\max } ?$

(iii) Calculate the temperature $T_{B}$ at which Bose condensation occurs.

(iv) For $T>T_{B}$, show that $\mu=\epsilon\left(T_{B}-T\right) / T_{B}$. For $T<T_{B}$ show that

$$\mu \approx-\frac{k T}{N} \frac{e^{\epsilon /(k T)}-1}{e^{\epsilon /(k T)}-e^{\epsilon /\left(k T_{B}\right)}} .$$

(v) Calculate the mean energy $\langle E\rangle$ for $T>T_{B}$ and for $T<T_{B}$. Hence show that the heat capacity of the system is

$$C \approx \begin{cases}\frac{1}{k T^{2}} \frac{M \epsilon^{2}}{\left(e^{\beta \epsilon}-1\right)^{2}} & T<T_{B} \\ 0 & T>T_{B}\end{cases}$$