Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 344 / 1 / e

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- Solution e

Solution

Translational invariance makes the pointwise variance independent of

r

. Applying Parseval identity to the stated Fourier convention gives

σ^{2} = ⟨ ∣ ϕ (r) ∣^{2} ⟩_{0} = \frac{1}{V} \sum_{q} ⟨ ∣ ϕ_{q} ∣^{2} ⟩_{0} = \frac{1}{V} \sum_{q} \frac{1}{J ( q )} .

In the thermodynamic limit, the reciprocal-lattice sum becomes

σ^{2} = \int \frac{d ^{d} q}{( 2 π ) ^{d}} \frac{1}{J ( q )} .

At each point

ϕ (r)

is a centered Gaussian random variable, so the supplied absolute third moment gives

g \int d^{d} r ⟨ ∣ ϕ (r) ∣^{3} ⟩_{0} = \frac{4 g V}{2 π} σ^{3} = \frac{4 g V}{2 π} [\int \frac{d ^{d} q}{( 2 π ) ^{d}} \frac{1}{J ( q )}]^{3/2} .

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