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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-344/3/a/solution