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

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- Solution c

Solution

After the sudden parameter change, the functional derivative is

\frac{δ F}{δ p} = a_{F} p - κ \nabla^{2} p .

Each Cartesian component of each Fourier mode consequently obeys

\overset{p}{˙}_{q, i} = - r (q) p_{q, i} + f_{q, i}, r (q) = Γ (a_{F} + κ q^{2}) .

This is an Ornstein-Uhlenbeck process. The integrating-factor method gives

p_{q, i} (t) = p_{q, i} (t_{1}) e^{- r (q) (t - t_{1})} + \int_{t_{1}}^{t} d t^{'} f_{q, i} (t^{'}) e^{- r (q) (t - t^{'})} .

Because both coefficients are positive, every mode has positive decay rate.

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