Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 33C / a / Solution

With $\beta =1/(k_{B}T)$, the canonical partition function and equilibrium probabilities areDifferentiating $T\log Z$, with fixed energy levels, gives $S=k_B\log Z+⟨E⟩/T$. Since $\log p(n)=-\beta E_n-\log Z$, normalization yieldsThe sums include degeneracy by labeling states rather than distinct energy values, and a zero probability contributes zero by continuity.