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

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Solution

The other diagonal entry of the same inverse gives

⟨ p (q) p (- q)⟩ = \frac{a + κ q ^{2} + γ q ^{4}}{ν ( a + κ q ^{2} + γ q ^{4} )}

(1)

or, more revealingly,

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

(2)

The first term is the uncoupled local orientational fluctuation. The second shows that every nonuniform composition fluctuation induces a correlated surfactant-polarization fluctuation. It vanishes at

q = 0

, as the coupling contains a gradient, and is enhanced at wavevectors where the composition structure factor is large.

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

Solution

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