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

For a specified trajectory, the nonconserved order-parameter dynamics equation determines the noise realizationThe forward Onsager--Machlup path probability for Model A dynamics is therefore

Assume the order-parameter field is even under time-reversal symmetry and its free-energy functional is time-reversal invariant. The reversed path isIts time derivative changes sign, soFor 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.