(i) Let $\rho_{i}$ be the probability that a system is in a state labelled by $i$ with $N_{i}$ particles and energy $E_{i}$. Define

$$s\left(\rho_{i}\right)=-k \sum_{i} \rho_{i} \log \rho_{i} .$$

$s\left(\rho_{i}\right)$ has a maximum, consistent with a fixed mean total number of particles $N$, mean total energy $E$ and $\sum_{i} \rho_{i}=1$, when $\rho_{i}=\bar{\rho}_{i}$. Let $S(E, N)=s\left(\bar{\rho}_{i}\right)$ and show that

$$\frac{\partial S}{\partial E}=\frac{1}{T}, \quad \frac{\partial S}{\partial N}=-\frac{\mu}{T},$$

where $T$ may be identified with the temperature and $\mu$ with the chemical potential.

(ii) For two weakly coupled systems 1,2 then $\rho_{i, j}=\rho_{1, i} \rho_{2, j}$ and $E_{i, j}=E_{1, i}+E_{2, j}$, $N_{i, j}=N_{1, i}+N_{2, j}$. Show that $S(E, N)=S_{1}\left(E_{1}, N_{1}\right)+S_{2}\left(E_{2}, N_{2}\right)$ where, if $S(E, N)$ is stationary under variations in $E_{1}, E_{2}$ and $N_{1}, N_{2}$ for $E=E_{1}+E_{2}, N=N_{1}+N_{2}$ fixed, we must have $T_{1}=T_{2}, \mu_{1}=\mu_{2}$.

(iii) Define the grand partition function $\mathcal{Z}(T, \mu)$ for the system in (i) and show that

$$k \log \mathcal{Z}=S-\frac{1}{T} E+\frac{\mu}{T} N, \quad S=\frac{\partial}{\partial T}(k T \log \mathcal{Z})$$

(iv) For a system with single particle energy levels $\epsilon_{r}$ the possible states are labelled by $i=\left\{n_{r}: n_{r}=0,1\right\}$, where $N_{i}=\sum_{r} n_{r}, E_{i}=\sum_{r} n_{r} \epsilon_{r}$ and $\sum_{i}=\prod_{r} \sum_{n_{r}=0,1}$. Show that

$$\bar{\rho}_{i}=\prod_{r} \frac{e^{-n_{r}\left(\epsilon_{r}-\mu\right) / k T}}{1+e^{-\left(\epsilon_{r}-\mu\right) / k T}} .$$

Calculate $\bar{n}_{r}$. How is this related to a free fermion gas?