Consider an ideal quantum gas with one-particle states $|i\rangle$ of energy $\epsilon_{i}$. Let $p_{i}^{\left(n_{i}\right)}$ denote the probability that state $|i\rangle$ is occupied by $n_{i}$ particles. Here, $n_{i}$ can take the values 0 or 1 for fermions and any non-negative integer for bosons. The entropy of the gas is given by

$$S=-k_{B} \sum_{i} \sum_{n_{i}} p_{i}^{\left(n_{i}\right)} \ln p_{i}^{\left(n_{i}\right)}$$

(a) Write down the constraints that must be satisfied by the probabilities if the average energy $\langle E\rangle$ and average particle number $\langle N\rangle$ are kept at fixed values.

Show that if $S$ is maximised then

$$p_{i}^{\left(n_{i}\right)}=\frac{1}{\mathcal{Z}_{i}} e^{-\left(\beta \epsilon_{i}+\gamma\right) n_{i}}$$

where $\beta$ and $\gamma$ are Lagrange multipliers. What is $\mathcal{Z}_{i}$ ?

(b) Insert these probabilities $p_{i}^{\left(n_{i}\right)}$ into the expression for $S$, and combine the result with the first law of thermodynamics to find the meaning of $\beta$ and $\gamma$.

(c) Calculate the average occupation number $\left\langle n_{i}\right\rangle=\sum_{n_{i}} n_{i} p_{i}^{\left(n_{i}\right)}$ for a gas of fermions.