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For quadratic free energy with positive $a,\kappa$ and kinetic coefficient $\Gamma$, a deterministic initial Dirac delta distribution evolves under nonconserved order-parameter dynamics to the meanTaking the expectation removes the centered Gaussian white noise; the resulting diffusion-reaction equation is solved by the heat kernel. The integrated mean is $A{e}^{-\Gamma at}$, so spreading does not conserve the initial excess. The connected Fourier mode covariance grows from zero to $k_{B}T/(a+\kappa q^2)$ as $1-e^{-2\Gamma (a+\kappa q^2)t}$ by the Itô isometry. A realization retains equilibrium fluctuations after its mean has decayed. Pointwise continuum white-noise quantities require a coarse-graining cutoff.

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 polar molecular field is the functional derivative of the free energy with respect to the polar order parameter. In the positive-derivative convention, dissipative dynamics is $D_t\mathbf{p}=-\Gamma \mathbf{h}$ with positive kinetic coefficient. For a quartic local term and an isotropic gradient penalty,Some authors instead define the molecular field with a minus sign, which changes the associated stress and evolution conventions.