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The difference between the two squared actions is $|\dot{p}+\Gamma \mu {|}^2-|-\dot{p}+\Gamma \mu {|}^2=4\Gamma \dot{p}\cdot \mu$. The common factor cancels, givingThe last equality uses the functional chain rule with the same midpoint convention as the Onsager–Machlup functional.

At equilibrium, microscopic reversibility requires $e^{-F_1/(k_{B}T)}{P}_F=e^{-F_2/(k_{B}T)}{P}_B$, equivalently $P_F/P_B=e^{-(F_2-F_1)/(k_{B}T)}$. Comparing the coefficients of an arbitrary free-energy change gives the model A fluctuation-dissipation relation

Conditioned on $f$, the additive-noise trajectory has the Onsager–Machlup functionalUnder time reversal, $\dot{x}$ changes sign while $x$ and the active force $f$ are even. Subtracting the forward and backward conditional actions givesThe corresponding ratio for an Ornstein–Uhlenbeck path conditioned on its initial endpoint isConsequentlywhereIf stationary endpoint densities are included in the path measures, their ratio cancels the Ornstein–Uhlenbeck boundary term; the displayed convention is the endpoint-conditioned path probability used in the calculation.

For additive Gaussian white noise and a time-even order parameter, a midpoint discretization of the Onsager–Machlup functional evaluates drift derivatives at the midpoint configurations. Reversing a trajectory visits the same midpoints in reverse order, so the noise-to-path Jacobian determinant is invariant under reversal. It may depend on the trajectory, and must be distinguished from the constant normalization of the noise measure.

For nonconserved order-parameter dynamics with $\dot{p}=-\Gamma \mu [p]+f$, the factor at one time step, apart from a path-independent power of the step size, isHere the fields have first been restricted to a finite spatial grid, and ${\mathcal{H}}_n$ is the Hessian matrix of the free energy.

For mixed conserved and nonconserved order-parameter dynamics, the required noises for a joint configuration and flux trajectory arewhere $\mathsf{D},\mathsf{G}$ represent the divergence and gradient, and $\Delta =\mathsf{DG}$ represents the Laplacian. The Jacobian matrix isTaking its Schur complement gives the joint factorFor periodic boundary conditions, discretizing the two spatial operators compatibly gives $\mathsf{D}=-{\mathsf{G}}^{\mathsf{T}}$, so $\Gamma I-M\Delta$ is the positive relaxation operator when $\Gamma ,M>0$. Reversal leaves ${\mathcal{H}}_n$ unchanged and reverses the sign of the flux. The product of the joint factors is consequently the same for both histories, justifying its cancellation from the joint path probability of an order parameter and its flux. The functional chain rule used in the action ratio holds in the continuum limit of this midpoint convention.