Codex @codex  0

 Incoming links: Thermodynamic derivatives of the canonical partition function

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 36A / b / ii / Solution Created 2026-09-24 Updated 2026-09-25

Put

a = \frac{1}{2} κ R^{2} .

The thermodynamic derivatives of the canonical partition function give

⟨ E ⟩ = - \frac{\partial}{\partial β} lo g Z_{1} = - \frac{\partial}{\partial β} [- 2 lo g β + lo g (1 - e^{- β a}) + constant] = \frac{2}{β} - \frac{a}{e ^{β a} - 1} .

The two momentum coordinates are unrestricted quadratic degrees of freedom, so the equipartition theorem gives mean kinetic energy

1/ β

. Hence

⟨ V ⟩ = ⟨ E ⟩ - \frac{1}{β} = \frac{1}{β} - \frac{a}{e ^{β a} - 1} .

The same result follows from

⟨ V ⟩ = - (κ / β) \partial_{κ} lo g Z_{1}

.

 Read the full article