Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 344 / 1 / d / Solution

The post-quench Ornstein-Uhlenbeck process has $r_1(q)=\Gamma (a_1+\kappa q^2)$. Its future noise is independent of the initial field. Consequently the cross term between the initial Fourier mode and its future noise integral has zero expectation. The Itô isometry, or the given white-noise covariance, then yieldsThe initial Gaussian Boltzmann distribution, with $a=a_0$, gives $S_q(0)=k_{B}T/(a_0+\kappa q^2)$ in the same unitary normalization. Thus the Gaussian Model A quench hasThe static structure factor interpolates monotonically mode by mode between its two positive equilibrium values. The covariance relaxes at twice the amplitude decay rate. These statements assume that $T$, $\Gamma$ and $\kappa$ are unchanged by the quench; both $a_0,a_1>0$ exclude a spinodal instability.