= Gaussian Model A impulse response
{c}
{title2=$\langle\phi\rangle=Ae^{-\Gamma at}G_{\Gamma\kappa t}$}
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 mean
$$
\langle\phi(\mathbf r,t)\rangle=\frac{Ae^{-\Gamma at}}{(4\pi\Gamma\kappa t)^{d/2}}e^{-|\mathbf r-\mathbf r'|^2/(4\Gamma\kappa t)}.
$$
Taking the expectation removes the centered <Gaussian white noise>; the resulting diffusion-reaction equation is solved by the <heat kernel>. The integrated mean is $Ae^{-\Gamma at}$, so spreading does not conserve the initial excess. The connected <Fourier mode> covariance grows from zero to $k_BT/(a+\kappa q^2)$ 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.
Back to article page
