Describe the physical relevance of the microcanonical, canonical and grand canonical ensembles. Explain briefly the circumstances under which all ensembles are equivalent.

The Gibbs entropy for a probability distribution $p(n)$ over states is

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

By imposing suitable constraints on $p(n)$, show how maximising the entropy gives rise to the probability distributions for the microcanonical and canonical ensembles.

A system consists of $N$ non-interacting particles fixed at points in a lattice. Each particle has three states with energies $E=-\epsilon, 0,+\epsilon$. If the system is at a fixed temperature $T$, determine the average energy $E$ and the heat capacity $C$. Evaluate each in the limits $T \rightarrow \infty$ and $T \rightarrow 0$.

Describe a configuration of the system that would have negative temperature. Does this system obey the third law of thermodynamics?