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For nonconserved order-parameter dynamics, the positive kinetic coefficient $\Gamma$ converts the functional derivative of the free energy into a relaxation rate through $\dot{p}=-\Gamma \delta F/\delta p$. Microscopic reversibility fixes the accompanying Gaussian white noise variance by the Model A fluctuation-dissipation relation.

With constant positive kinetic coefficient $\Gamma$ and mobility $M$, two independent relaxation channels giveThe deterministic mobility operator is $\Gamma -M{\nabla }^2$. Each channel separately satisfies microscopic reversibility when its noise obeys the corresponding Model A fluctuation-dissipation relation or fluctuation-dissipation relation for a conserved flux.

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.

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

Set the noise strengths to the model A fluctuation-dissipation relation and the fluctuation-dissipation relation for a conserved flux:The preceding logarithmic ratio becomesThe last two terms combine as ${\partial }_i({\mu }_j{W}_{ij})$. They integrate to zero under the periodic boundary conditions, by integration by parts. The remaining term is $F_2-F_1$ by the functional chain rule, henceAfter including the equilibrium initial weights, the forward and reversed histories have equal probability. Thus microscopic reversibility is satisfied by the two channels together, with each channel's own thermal noise strength.

Thermal molecular motion causes the coarse droplet radius to fluctuate as well as drift down $F(R)$, so its effective equation must be an overdamped Langevin dynamics. With $⟨\Lambda (t)\Lambda (t^{\prime })⟩=\delta (t-t^{\prime })$, detailed balance with equilibrium density proportional to $e^{-F/(k_{B}T)}$ imposes the Fluctuation-dissipation theorem. The Model A fluctuation-dissipation relation givesThe factor of mobility ensures that the diffusion in $R$ and the dissipative drift have the same equilibrium Gibbs distribution.

Subtracting the two quadratic actions givesThe functional chain rule identifies the last integral as $F_2-F_1$, so

Microscopic time-reversal invariance implies detailed balance. The equilibrium probability density of a configuration with free energy $F$ obeysThereforeComparison for arbitrary endpoint free energies yields the Model A fluctuation-dissipation relation