Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 36A / b / v

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- Solution v

Solution

Above

T_{B}

, essentially all

N

particles occupy the excited levels, so

⟨ E ⟩ ≃ N ϵ, C ≃ 0 (T > T_{B}) .

Below

T_{B}

, the ground-state condensate carries no energy and

μ ≃ 0

, hence

⟨ E ⟩ ≃ \frac{M ϵ}{e ^{β ϵ} - 1} (T < T_{B}) .

Differentiating gives the heat capacity

C = \frac{\partial ⟨ E ⟩}{\partial T} ≃ \frac{M ϵ ^{2} e ^{β ϵ}}{k T ^{2} ( e ^{β ϵ} - 1 ) ^{2}} (T < T_{B}) .

The displayed formula in the examination paper omits the factor

e^{β ϵ}

. Its stated expression follows after the additional high-temperature approximation

β ϵ ≪ 1

, for which

e^{β ϵ} ≃ 1

.

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