Edwards Wilkinson: an interface with thermal fluctuations:

Consider domain wall $h(r,t)$ fluctuating at equilibrium at the temparature $T$ . Here $t$ is time, $r$ defines the d-dimensional coordinate of the interface and $h$ is the scalar height field. Hence, the domain wall separating two phases in a film has $d=1,r\in {\cal {R}}$ , in a solid instead $d=2,r\in {\cal {{R}^{2}}}$ .

Two assumptions are done:

Overhangs, pinch-off are neglected, so that $h(r,t)$ is a scalar univalued function.
The dynamics is overdamped, so that we can neglect the inertial term.

Derivation

The Langevin equation of motion is

\partial _{t}h(r,t)=-\mu {\frac {\delta E_{pot}}{\delta h(r,t)}}+\eta (r,t)

The first term $-\delta E_{pot}/\delta h(r,t)$ is the elastic force trying to smooth the interface, the mobility $\mu$ is inversily proportional to the viscosity. The second term is the Langevin Gaussian noise defined by the correlations

\langle \eta (r,t)\rangle =0,\;\langle \eta (r',t')\eta (r,t)\rangle =2dD\delta ^{d}(r-r')\delta (t-t')

The symbol $\langle \ldots \rangle$ indicates the average over the thermal noise. The diffusion constant is fixed by the Eistein relation (fluctuation-dissipation theorem):