(i) A system of $N$ distinguishable non-interacting particles has energy levels $E_{i}$ with degeneracy $g_{i}, 1 \leqslant i<\infty$, for each particle. Show that in thermal equilibrium the number of particles $N_{i}$ with energy $E_{i}$ is given by

$$N_{i}=g_{i} e^{-\beta\left(E_{i}-\mu\right)}$$

where $\beta$ and $\mu$ are parameters whose physical significance should be briefly explained.

A gas comprises a set of atoms with non-degenerate energy levels $E_{i}, 1 \leqslant i<\infty$. Assume that the gas is dilute and the motion of the atoms can be neglected. For such a gas the atoms can be treated as distinguishable. Show that when the system is at temperature $T$, the number of atoms $N_{i}$ at level $i$ and the number $N_{j}$ at level $j$ satisfy

$$\frac{N_{i}}{N_{j}}=e^{-\left(E_{i}-E_{j}\right) / k T}$$

where $k$ is Boltzmann's constant.

(ii) A system of bosons has a set of energy levels $W_{a}$ with degeneracy $f_{a}, 1 \leqslant a<\infty$, for each particle. In thermal equilibrium at temperature $T$ the number $n_{a}$ of particles in level $a$ is

$$n_{a}=\frac{f_{a}}{e^{\left(W_{a}-\mu\right) / k T}-1} \text {. }$$

What is the value of $\mu$ when the particles are photons?

Given that the density of states $\rho(\omega)$ for photons of frequency $\omega$ in a cubical box of side $L$ is

$$\rho(\omega)=L^{3} \frac{\omega^{2}}{\pi^{2} c^{3}}$$

where $c$ is the speed of light, show that at temperature $T$ the density of photons in the frequency range $\omega \rightarrow \omega+d \omega$ is $n(\omega) d \omega$ where

$$n(\omega)=\frac{\omega^{2}}{\pi^{2} c^{3}} \frac{1}{e^{\hbar \omega / k T}-1}$$

Deduce the energy density, $\epsilon(\omega)$, for photons of frequency $\omega$.

The cubical box is occupied by the gas of atoms described in Part (i) in the presence of photons at temperature $T$. Consider the two atomic levels $i$ and $j$ where $E_{i}>E_{j}$ and $E_{i}-E_{j}=\hbar \omega$. The rate of spontaneous photon emission for the transition $i \rightarrow j$ is $A_{i j}$. The rate of absorption is $B_{j i} \epsilon(\omega)$ and the rate of stimulated emission is $B_{i j} \epsilon(\omega)$. Show that the requirement that these processes maintain the atoms and photons in thermal equilibrium yields the relations

$$B_{i j}=B_{j i}$$

and

$$A_{i j}=\left(\frac{\hbar \omega^{3}}{\pi^{2} c^{3}}\right) B_{i j}$$