= 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 yields
$$
\begin{aligned}
S_q(t)&=e^{-2r_1(q)t}S_q(0)+2\Gamma k_BT\int_0^t e^{-2r_1(q)(t-s)}\,ds\\
&=e^{-2r_1(q)t}S_q(0)+\frac{k_BT}{a_1+\kappa q^2}\left[1-e^{-2r_1(q)t}\right].
\end{aligned}
$$
The initial Gaussian <Boltzmann distribution>, with $a=a_0$, gives $S_q(0)=k_BT/(a_0+\kappa q^2)$ in the same unitary normalization. Thus the <Gaussian Model A quench> has
$$
\boxed{S_q(t)=\frac{k_BT}{a_1+\kappa q^2}+\left[\frac{k_BT}{a_0+\kappa q^2}-\frac{k_BT}{a_1+\kappa q^2}\right]e^{-2\Gamma(a_1+\kappa q^2)t}.}
$$
The <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.
Back to article page
