(a) Define the canonical partition function $Z$ for a system with energy levels $E_{n}$, where $n$ labels states, given that the system is in contact with a heat reservoir at temperature $T$. What is the probability $p(n)$ that the system occupies state $n$ ? Starting from an expression for the entropy $S=k_{B} \partial(T \ln Z) / \partial T$, deduce that

$$S=-k_{B} \sum_{n} p(n) \ln p(n)$$

(b) Consider an ensemble consisting of $W$ copies of the system in part (a) with $W$ very large, so that there are $W p(n)$ members of the ensemble in state $n$. Starting from an expression for the number of ways in which this can occur, find the entropy $S_{W}$ of the ensemble and hence re-derive the expression $(*)$. [You may assume Stirling's formula $\ln X ! \approx X \ln X-X$ for $X$ large. $]$

(c) Consider a system of $N$ non-interacting particles at temperature $T$. Each particle has $q$ internal states with energies

$$0, \mathcal{E}, 2 \mathcal{E}, \ldots,(q-1) \mathcal{E}$$

Assuming that the internal states are the only relevant degrees of freedom, calculate the total entropy of the system. Find the limiting values of the entropy as $T \rightarrow 0$ and $T \rightarrow \infty$ and comment briefly on your answers.