 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-4/35c/c/solution

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

Minimization gives

m = 0

for

T > T_{c}

and

m = \pm a (T_{c} - T) / b

below. Thus

F_{e q} = F_{0} (T \geq T_{c}), F_{e q} = F_{0} - \frac{a ^{2} ( T _{c} - T ) ^{2}}{4 b} (T < T_{c}) .

Using

S = - F_{T}

and

C = - T F_{TT}

, entropy is continuous but the low-temperature heat capacity has the additional

T a^{2} / (2 b)

. The jump is

T_{c} a^{2} / (2 b)

, so the transition is second order.

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