Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 344 / 1 / f

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- Solution f

Solution

The covariance of a centered multivariate Gaussian distribution is the inverse of its quadratic kernel. Writing

A (q) = a + κ q^{2} + γ q^{4}

, one finds

G (q)^{- 1} = \frac{1}{ν A ( q ) - λ ^{2} q ^{2}} (ν iλ q - iλ q A (q)) .

Consequently the composition static structure factor is

⟨ ϕ (q) ϕ (- q)⟩ = \frac{1}{a + κ q ^{2} + γ q ^{4}} .

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