= Solution
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 notation
$$
\mathcal J(\mathbf r;[\phi])=-\Gamma\frac{\delta F}{\delta\phi(\mathbf r)}\mathcal P[\phi]-\frac{\sigma^2}{2}\frac{\delta\mathcal P}{\delta\phi(\mathbf r)}.
$$
The equilibrium <Boltzmann distribution> $\mathcal P_{\rm eq}\propto e^{-\beta F}$ has $\delta\mathcal P_{\rm eq}/\delta\phi=-\beta(\delta F/\delta\phi)\mathcal P_{\rm eq}$. <Detailed balance> for this time-even field requires zero configuration-space current. Therefore
$$
\mathcal J_{\rm eq}=\left(-\Gamma+\frac{\beta\sigma^2}{2}\right)\frac{\delta F}{\delta\phi}\mathcal P_{\rm eq}=0,\qquad \boxed{\sigma^2=2\Gamma k_BT.}
$$
This 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 $\dot\phi+\Gamma\delta F/\delta\phi$ and $-\dot\phi+\Gamma\delta F/\delta\phi$. 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.
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