(a) State the Bose-Einstein distribution formula for the mean occupation numbers $n_{i}$ of discrete single-particle states $i$ with energies $E_{i}$ in a gas of bosons. Write down expressions for the total particle number $N$ and the total energy $U$ when the singleparticle states can be treated as continuous, with energies $E \geqslant 0$ and density of states $g(E)$.

(b) Blackbody radiation at temperature $T$ is equivalent to a gas of photons with

$$g(E)=A V E^{2}$$

where $V$ is the volume and $A$ is a constant. What value of the chemical potential is required when applying the Bose-Einstein distribution to photons? Show that the heat capacity at constant volume satisfies $C_{V} \propto T^{\alpha}$ for some constant $\alpha$, to be determined.

(c) Consider a system of bosonic particles of fixed total number $N \gg 1$. The particles are trapped in a potential which has ground state energy zero and which gives rise to a density of states $g(E)=B E^{2}$, where $B$ is a constant. Explain, for this system, what is meant by Bose-Einstein condensation and show that the critical temperature satisfies $T_{c} \propto N^{1 / 3}$. If $N_{0}$ is the number of particles in the ground state, show that for $T$ just below $T_{c}$

$$N_{0} / N \approx 1-\left(T / T_{c}\right)^{\gamma}$$

for some constant $\gamma$, to be determined.

(d) Would you expect photons to exhibit Bose-Einstein condensation? Explain your answer very briefly.