(a) The entropy of a thermodynamic ensemble is defined by the formula

$$S=-k \sum_{n} p(n) \log p(n)$$

where $k$ is the Boltzmann constant. Explain what is meant by $p(n)$ in this formula. Write down an expression for $p(n)$ in the grand canonical ensemble, defining any variables you need. Hence show that the entropy $S$ is related to the grand canonical partition function $\mathcal{Z}(T, \mu, V)$ by

$$S=k\left[\frac{\partial}{\partial T}(T \log \mathcal{Z})\right]_{\mu, V}$$

(b) Consider a gas of non-interacting fermions with single-particle energy levels $\epsilon_{i}$.

(i) Show that the grand canonical partition function $\mathcal{Z}$ is given by

$$\log \mathcal{Z}=\sum_{i} \log \left(1+e^{-\left(\epsilon_{i}-\mu\right) /(k T)}\right)$$

(ii) Assume that the energy levels are continuous with density of states $g(\epsilon)=A V \epsilon^{a}$, where $A$ and $a$ are positive constants. Prove that

$$\log \mathcal{Z}=V T^{b} f(\mu / T)$$

and give expressions for the constant $b$ and the function $f$.

(iii) The gas is isolated and undergoes a reversible adiabatic change. By considering the ratio $S / N$, prove that $\mu / T$ remains constant. Deduce that $V T^{c}$ and $p V^{d}$ remain constant in this process, where $c$ and $d$ are constants whose values you should determine.