 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-1/36b/c/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Write

A = 3 + 3/ (2 n)

. Since

lo g Z_{N}

contains

- N A lo g β

E = - \partial_{β} lo g Z_{N} = N A k_{B} T,

(Δ E)^{2} = \partial_{β}^{2} lo g Z_{N} = N A (k_{B} T)^{2}, C_{V} = N A k_{B} .

Consequently

\frac{Δ E}{E} = \frac{1}{N A} ⟶ 0

in the thermodynamic limit: relative energy fluctuations vanish.

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