Put $u_j=\dot p_j+\partial_iW_{ij}$. The additive independent noises give the forward action
$$
S_F=\int\left[\frac{|u+\Gamma\mu|^2}{2\sigma^2}
+\frac{|W+M\nabla\mu|^2}{2\sigma_N^2}\right]d\mathbf r\,dt.
$$
For a time-even <order parameter> and time-odd flux, the backward action replaces $u,W$ by $-u,-W$ while keeping $\mu$ fixed on the corresponding configurations. Common normalization and <time-reversal invariance of a path Jacobian> then give
$$
\log\frac{P_F}{P_B}
=-\frac{2\Gamma}{\sigma^2}\int\mu_j(\dot p_j+\partial_iW_{ij})
-\frac{2M}{\sigma_N^2}\int W_{ij}\partial_i\mu_j.
$$
At the two thermal noise strengths, periodic <boundary conditions> cancel the two spatial terms by <integration by parts>, leaving $-(F_2-F_1)/(k_BT)$. Thus the joint dynamics satisfies <microscopic reversibility>.


 Joint path probability of an order parameter and its flux

ID: joint-path-probability-of-an-order-parameter-and-its-flux