Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 337 / 2 / f

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- Solution f

Solution

The Fluctuation-dissipation theorem in this convention reads

S (ω, k) = \frac{ϱ ( ω , k )}{1 - e ^{- ω / T}} .

In the classical limit

∣ ω ∣ ≪ T

, this becomes

S (ω, k) ≃ \frac{T}{ω} ϱ (ω, k) = \frac{π T}{s ω _{k}} [δ (ω - ω_{k}) + δ (ω + ω_{k})] .

The inverse temporal Fourier transform is therefore

C (t, k) = \int \frac{d ω}{2 π} e^{- iω t} S (ω, k) = \frac{T}{s ω _{k}} cos (ω_{k} t) = \frac{T}{ρ _{s} k ^{2}} cos (ω_{k} t) .

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