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Here $\Gamma >0$ is a kinetic coefficient, and the additive Gaussian white noise must be centered. A rigorous starting point is a finite spatial discretization or finite set of modes of the coarse-grained field. For each field coordinate, the Fokker-Planck probability current has the continuum notationThe equilibrium Boltzmann distribution ${\mathcal{P}}_{\mathrm{eq}}\propto e^{-\beta F}$ has $\delta {\mathcal{P}}_{\mathrm{eq}}/\delta \varphi =-\beta (\delta F/\delta \varphi ){\mathcal{P}}_{\mathrm{eq}}$. Detailed balance for this time-even field requires zero configuration-space current. ThereforeThis is the Model A fluctuation-dissipation relation. Conversely, that noise strength makes the equilibrium current vanish and gives reversible relaxational dynamics when the equilibrium measure is normalizable.

Alternatively, the Onsager--Machlup path probability for Model A dynamics compares the squared required noise histories $\dot{\varphi }+\Gamma \delta F/\delta \varphi$ and $-\dot{\varphi }+\Gamma \delta F/\delta \varphi$. Their difference gives $\log ({\mathbb{P}}_F/{\mathbb{P}}_B)=-2\Gamma \Delta F/{\sigma }^2$, using the functional chain rule. The midpoint convention and time-reversal invariance of a path Jacobian justify cancellation of the common Jacobian. Comparing with the conditional path ratio gives the same result. The spatial white-noise continuum is a coarse-grained idealization: retain a microscopic cutoff when pointwise fluctuations or ultraviolet divergences matter.

For a specified field trajectory, the required normalized noise isand time reversal changes only $\dot{\mathbf{p}}$ to $-\dot{\mathbf{p}}$. Assuming equal additive-noise Jacobians, the difference of the two Onsager--Machlup actions givesThe functional chain rule identifies the first term as $-\beta \Delta F$, soThe forcing performs generalized work $W_Y=\int \mathbf{Y}\cdot \dot{\mathbf{p}}$, and $W_Y-\Delta F$ is the heat dissipated into the bath. The formula is therefore the field-theory form of local detailed balance and quantifies nonequilibrium entropy production.

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.