For a specified trajectory, the <nonconserved order-parameter dynamics> equation determines the noise realization
$$
\mathbf f_F
=\dot{\mathbf p}
+\Gamma\frac{\delta F}{\delta\mathbf p}.
$$
The forward <Onsager--Machlup path probability for Model A dynamics> is therefore
$$
P_F[\mathbf p]
=N_F\exp\left[
-\frac1{2\sigma^2}
\int_{t_1}^{t_2}dt\int d\mathbf r\,
\left|
\dot{\mathbf p}
+\Gamma\frac{\delta F}{\delta\mathbf p}
\right|^2
\right].
$$

Assume the order-parameter field is even under <time-reversal symmetry> and its free-energy functional is time-reversal invariant. The reversed path is
$$
\mathbf p^R(t)=\mathbf p(t_1+t_2-t).
$$
Its time derivative changes sign, so
$$
P_B[\mathbf p^R]
=N_B\exp\left[
-\frac1{2\sigma^2}
\int_{t_1}^{t_2}dt\int d\mathbf r\,
\left|
-\dot{\mathbf p}
+\Gamma\frac{\delta F}{\delta\mathbf p}
\right|^2
\right].
$$
For additive <Gaussian white noise>, the trajectory-to-noise Jacobian is the same in the two directions. We assume it and all path-independent normalization factors are absorbed into equal constants $N_F=N_B$. A time-reversal-odd order parameter would require the corresponding parity transformation as well.

Solution

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

Initial equilibrium at $a_I$ gives
$$
\langle p_{\mathbf q,i}(t_1)p_{-\mathbf q,j}(t_1)\rangle
=\delta_{ij}\frac{k_BT}{a_I+\kappa q^2}.
$$
The initial field and later noise are independent, so their cross terms vanish. Define
$$
\Delta t=t_2-t_1.
$$
With the supplied <Gaussian white noise> covariance, the noise contribution is
$$
2k_BT\Gamma\delta_{ij}
\int_{t_1}^{t_2}ds\,e^{-2r(q)(t_2-s)}
=\delta_{ij}\frac{k_BT}{a_F+\kappa q^2}
\left(1-e^{-2r(q)\Delta t}\right),
$$
Using the decay rate found in part (c), it follows that
$$
\langle p_{\mathbf q,i}(t_2)p_{-\mathbf q,j}(t_2)\rangle
=\delta_{ij}\left[
\frac{k_BT}{a_I+\kappa q^2}e^{-2r(q)\Delta t}
+\frac{k_BT}{a_F+\kappa q^2}
\left(1-e^{-2r(q)\Delta t}\right)
\right].
$$
This is the covariance interpolation in a <Gaussian Model A quench>: each mode forgets its initial equilibrium with relaxation time $1/r(q)$ and approaches the final equilibrium variance. Larger-$q$ modes relax faster.

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

Gaussian white noise is a generalized Gaussian process with zero mean and covariance proportional to a <Dirac delta function> in time, and possibly in space. It is the formal time derivative of <Brownian motion> and has independent fluctuations on disjoint time intervals.

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 Incoming links: Gaussian white noise