The Canonical and Grand Canonical Ensembles

Derive the canonical (Boltzmann) distribution for a system in contact with a heat reservoir.

Define the canonical partition function Z and relate it to the Helmholtz free energy F = -k_B T ln Z.

Derive the grand canonical distribution for a system in contact with a heat and particle reservoir.

Define the grand partition function Ξ and relate it to the grand potential Φ = -k_B T ln Ξ.

Derive expressions for thermodynamic observables (energy, entropy, pressure, particle number) from the appropriate partition function.

Derive the formulae for energy and particle number fluctuations in the canonical and grand canonical ensembles, respectively.

State and derive the classical equipartition theorem from the canonical partition function.