= Solution
For a deterministic localized initial peak in an unbounded $d$-dimensional sample, the mean obeys $\partial_t\langle\phi\rangle=\Gamma\kappa\nabla^2\langle\phi\rangle-\Gamma a\langle\phi\rangle$. The <Gaussian Model A impulse response> is therefore the <heat kernel> multiplied by the local relaxation factor:
$$
\boxed{\langle\phi(\mathbf r,t)\rangle=\frac{A e^{-\Gamma at}}{(4\pi\Gamma\kappa t)^{d/2}}\exp\left[-\frac{|\mathbf r-\mathbf r'|^2}{4\Gamma\kappa t}\right],\qquad t>0.}
$$
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 $[k_BT/(a+\kappa q^2)](1-e^{-2r(q)t})$, 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.
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