A system consisting of non-interacting bosons has single-particle levels uniquely labelled by $r$ with energies $\epsilon_{r}, \epsilon_{r} \geq 0$. Show that the free energy in the grand canonical ensemble is

$$F=k T \sum_{r} \log \left(1-e^{-\beta\left(\epsilon_{r}-\mu\right)}\right) .$$

What is the maximum value for $\mu$ ?

A system of $N$ bosons in a large volume $V$ has one energy level of energy zero and a large number $M \gg 1$ of energy levels of the same energy $\epsilon$, where $M$ takes the form $M=A V$ with $A$ a positive constant. What are the dimensions of $A ?$

Show that the free energy is

$$F=k T\left(\log \left(1-e^{\beta \mu}\right)+A V \log \left(1-e^{-\beta(\epsilon-\mu)}\right)\right)$$

The numbers of particles with energies $0, \epsilon$ are respectively $N_{0}, N_{\epsilon}$. Write down expressions for $N_{0}, N_{\epsilon}$ in terms of $\mu$.

At temperature $T$ what is the maximum number of bosons $N_{\epsilon}^{\max }$ in the normal phase (the state with energy $\epsilon$ )? Explain what happens when $N>N_{\epsilon}^{\max }$.

Given $N$ and $T$ calculate the transition temperature $T_{B}$ at which Bose condensation occurs.

For $T>T_{B}$ show that $\mu=\epsilon\left(T_{B}-T\right) / T_{B}$. What is the value of $\mu$ for $T<T_{B}$ ?

Calculate the mean energy $E$ for (a) $T>T_{B}$ (b) $T<T_{B}$, and show that the heat capacity of the system at constant volume is

$$C_{V}=\left\{\begin{array}{cl} \frac{1}{k T^{2}} \frac{A V \epsilon^{2}}{\left(e^{\beta \epsilon}-1\right)^{2}} & T<T_{B} \\ 0 & T>T_{B} \end{array}\right.$$