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For a deterministic localized initial peak in an unbounded $d$-dimensional sample, the mean obeys ${\partial }_t⟨\varphi ⟩=\Gamma \kappa {\nabla }^2⟨\varphi ⟩-\Gamma a⟨\varphi ⟩$. The Gaussian Model A impulse response is therefore the heat kernel multiplied by the local relaxation factor:The peak spreads over a distance of order $\sqrt{\Gamma \kappa t}$ and its central height falls both by spreading and by exponential relaxation. Its integrated mean is $A{e}^{-\Gamma at}$, rather than $A$: the equation describes nonconserved order-parameter dynamics, despite the use of the word “density” in this part.

An individual realization also develops thermal fluctuations; it does not tend pointwise to zero. For an initially deterministic field, the connected Fourier mode variance grows as $[k_{B}T/(a+\kappa q^2)](1-e^{-2r(q)t})$, while the deterministic mean decays. It approaches the zero-mean Gaussian equilibrium field at the coarse-graining cutoff. On a periodic or bounded sample use the corresponding boundary-adapted heat kernel; at $\kappa =0$ there is no spatial spreading. The Dirac delta distribution is an ideal initial profile, recovered as a distributional limit; a finite-energy physical peak has a microscopic width.