Codex @codex  0

Put $u_j={\dot{p}}_j+{\partial }_i{W}_{ij}$. The additive independent noises give the forward actionFor 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 giveAt the two thermal noise strengths, periodic boundary conditions cancel the two spatial terms by integration by parts, leaving $-(F_2-F_1)/(k_{B}T)$. Thus the joint dynamics satisfies microscopic reversibility.

Write $E(t)=\delta A/\delta x(t)$ for the functional derivative of the action. Under time reversal in classical mechanics, $\dot{x}$ changes sign whereas $E$ is even by assumption. Thus the forward and reversed Gaussian white noise histories associated with the same geometric path areThe Onsager--Machlup path probability therefore givesHere the path probabilities are densities conditional on their respective initial states; they are not separately normalized bridges conditioned on both endpoints. The noise-to-path Jacobian determinant must be treated with a consistent discretization: it cancels when it is invariant under time reversal in classical mechanics. This is the usual additive-noise Langevin dynamics convention, including a constant-mass Underdamped Langevin dynamics or the time-reversal invariance of a path Jacobian in midpoint Overdamped Langevin dynamics. A completely arbitrary velocity-dependent Lagrangian would require specifying that measure rather than deducing its cancellation solely from the parity of $E$.

Write ${\mu }_j=\delta F/\delta p_j$ and $\int ={\int }_{t_1}^{t_2}dt\int d\mathbf{r}$. Eliminating the noise gives $f=\dot{p}+\Gamma \mu$. The Onsager--Machlup path probability is thereforeThese are conditional path densities given the corresponding starting configuration. For a time-even vector order parameter, reversal reads the configurations backward, changes $\dot{p}$ to $-\dot{p}$, and keeps $\mu$ unchanged on each corresponding configuration.

The two Gaussian white noise measures have the same normalization. With a common midpoint discretization, the noise-to-path Jacobian also agrees under reversal by time-reversal invariance of a path Jacobian. The common factor $\mathcal{K}[p]$ may include that path-dependent Jacobian; it need not be a universal constant. This makes the shared-factor assertion precise while leaving the requested ratio unaffected.

Put $u_j={\dot{p}}_j+{\partial }_i{W}_{ij}$. The noises required by a joint trajectory are $f_j=u_j+\Gamma {\mu }_j$ and $N_{ij}=W_{ij}+M{\partial }_i{\mu }_j$. Their independent Gaussian white noise weights give the joint path probability of an order parameter and its flux:For the physically reversed path, $p$ is time-even and the transport flux $W$ is time-odd. Therefore $u\mapsto -u$, $W\mapsto -W$, andThe common factor includes the two noise normalizations and the midpoint Jacobian determinant. Its calculation must include both relaxation channels. On a finite spatial grid, let $\mathsf{D}$ and $\mathsf{G}$ represent the divergence and gradient, with $\Delta =\mathsf{DG}$ the discrete Laplacian, and let ${\mathcal{H}}_n=\partial \mu /\partial p$ be the Hessian matrix of $F$ at the midpoint of time step $n$. Differentiating the two required noises with respect to the next configuration and the interval flux gives the Jacobian matrixUsing the Schur complement, its Jacobian determinant, apart from the path-independent power of $\Delta t$, isReversal visits the same midpoint configurations in reverse order, so the product of these factors is unchanged. The joint time-reversal invariance of a path Jacobian therefore involves the complete relaxation operator $\Gamma I-M\Delta$.

Taking the action difference givesRepeated component indices are summed. The second squared norm sums over both flux indices. This is mixed conserved and nonconserved order-parameter dynamics; reversing the order parameter history while leaving the flux unreversed would give the wrong probability ratio.