Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 303 / 1 / b / ii

Table of contents
- Solution ii

Solution

Write

A (k, T) = γ k^{2} + μ^{2} (T)

and use units with

k_{B} = 1

. The Gaussian functional integral gives the fluctuation free-energy density, up to terms linear in

T

that do not affect the heat capacity,

f (T) = \frac{T}{2} \int \frac{d ^{d} k}{( 2 π ) ^{d}} lo g \frac{A ( k , T )}{T} .

Since the heat capacity per volume is

c = - T d^{2} f / d T^{2}

,

c = \frac{1}{2} \int \frac{d ^{d} k}{( 2 π ) ^{d}} [1 - \frac{2 T μ ˙ ^{2} + T ^{2} μ ¨ ^{2}}{γ k ^{2} + μ ^{2}} + \frac{T ^{2} ( μ ˙ ^{2} ) ^{2}}{( γ k ^{2} + μ ^{2} ) ^{2}}],

where a dot denotes

d / d T

. Thus

g (T) = 2 T \frac{d μ ^{2}}{d T} + T^{2} \frac{d ^{2} μ ^{2}}{d T ^{2}}, h (T) = T^{2} (\frac{d μ ^{2}}{d T})^{2} .

For

μ^{2} = a (T - T_{c})

, these reduce to

g (T) = 2 a T

and

h (T) = a^{2} T^{2}

.

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