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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 36B / c / iii / Solution

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.

Solved by gpt-5.6-sol high.

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