A gas of non-interacting identical bosons in volume $V$, with one-particle energy levels $\epsilon_{r}, r=1,2, \ldots, \infty$, is in equilibrium at temperature $T$ and chemical potential $\mu$. Let $n_{r}$ be the number of particles in the $r$ th one-particle state. Write down an expression for the grand partition function $\mathcal{Z}$. Write down an expression for the probability of finding a given set of occupation numbers $n_{r}$ of the one-particle states. Hence determine the mean occupation numbers $\bar{n}_{r}$ (the Bose-Einstein distribution). Write down expressions, in terms of the mean occupation numbers, for the total energy $E$ and total number of particles $N$.

Write down an expression for the grand potential $\Omega$ in terms of $\mathcal{Z}$. Given that

$$S=-\left(\frac{\partial \Omega}{\partial T}\right)_{V, \mu}$$

show that $S$ can be written in the form

$$S=k \sum_{r} f\left(\bar{n}_{r}\right)$$

for some function $f$, which you should determine. Hence show that $d S=0$ for any change of the gas that leaves the mean occupation numbers unchanged. Consider a (quasi-static) change of $V$ with this property. Using the formula

$$P=-\left(\frac{\partial E}{\partial V}\right)_{N, S}$$

and given that $\epsilon_{r} \propto V^{-\sigma}(\sigma>0)$ for each $r$, show that

$$P=\sigma(E / V)$$

What is the value of $\sigma$ for photons?

Let $\mu=0$, so that $E$ is a function only of $T$ and $V$. Why does the energy density $\varepsilon=E / V$ depend only on $T ?$ Using the Maxwell relation

$$\left(\frac{\partial S}{\partial V}\right)_{T}=\left(\frac{\partial P}{\partial T}\right)_{V}$$

and the first law of thermodynamics for reversible changes, show that

$$\left(\frac{\partial E}{\partial V}\right)_{T}=T\left(\frac{\partial P}{\partial T}\right)_{V}-P$$

and hence that

$$\varepsilon(T) \propto T^{\gamma}$$

for some power $\gamma$ that you should determine. Show further that

$$S \propto\left(T V^{\sigma}\right)^{\frac{1}{\sigma}} .$$

Hence verify, given $\mu=0$, that $\bar{n}_{r}$ is left unchanged by a change of $V$ at constant $S$.