Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 36C / a / Solution

For discrete states $s$ of energy $E_s$, the canonical partition function isFor a classical continuous system the sum is replaced by the appropriately normalized phase-space integral, including the Gibbs factor $1/N!$ for identical particles.

It packages the equilibrium probabilities $P_s=e^{-\beta E_s}/Z$ and hence all canonical thermodynamics. In particular,and $S=k_B(\log Z+\beta ⟨E⟩)$.