Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 344 / 3 / e

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- Solution e

Solution

Let

t_{m i n} = min (t_{2}, t_{3}), t_{m a x} = max (t_{2}, t_{3}) .

Evolution from the earlier time to the later time multiplies the earlier field by the deterministic decay factor, plus fresh noise independent of that field. The unequal-time correlation function is therefore

⟨ p_{q, i} (t_{2}) p_{- q, j} (t_{3})⟩ = δ_{ij} e^{- r (q) ∣ t_{3} - t_{2} ∣} [\frac{k _{B} T}{a _{I} + κ q ^{2}} e^{- 2 r (q) (t_{m i n} - t_{1})} + \frac{k _{B} T}{a _{F} + κ q ^{2}} (1 - e^{- 2 r (q) (t_{m i n} - t_{1})})] .

Equivalently,

⟨ p_{q, i} (t_{2}) p_{- q, j} (t_{3})⟩ = δ_{ij} [\frac{k _{B} T}{a _{F} + κ q ^{2}} e^{- r (q) ∣ t_{3} - t_{2} ∣} + (\frac{k _{B} T}{a _{I} + κ q ^{2}} - \frac{k _{B} T}{a _{F} + κ q ^{2}}) e^{- r (q) (t_{2} + t_{3} - 2 t_{1})}] .

When both observation times are many relaxation times after the quench, the second term vanishes and the stationary Ornstein-Uhlenbeck process covariance remains:

⟨ p_{q, i} (t_{2}) p_{- q, j} (t_{3})⟩ ⟶ δ_{ij} \frac{k _{B} T}{a _{F} + κ q ^{2}} e^{- r (q) ∣ t_{3} - t_{2} ∣} .

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