Global existence and stability for the modified Mullins–Sekerka and surface diffusion flow

<abstract><p>In this survey we present the state of the art about the asymptotic behavior and stability of the <italic>modified Mullins</italic>–<italic>Sekerka flow</italic> and the <italic>surface diffusion flow</italic> of smooth sets, mainly due to E. Acerbi, N. Fusco, V. Julin and M. Morini. First we discuss in detail the properties of the nonlocal Area functional under a volume constraint, of which the two flows are the gradient flow with respect to suitable norms, in particular, we define the <italic>strict stability</italic> property for a critical set of such functional and we show that it is a necessary and sufficient condition for minimality under $ W^{2, p} $–perturbations, holding in any dimension. Then, we show that, in dimensions two and three, for initial sets sufficiently "close" to a smooth <italic>strictly stable critical</italic> set $ E $, both flows exist for all positive times and asymptotically "converge" to a translate of $ E $.</p></abstract>

Justification for the use of Markovian Langevin statistics in the modelling of activated surface diffusion.

Low temperature surface diffusion is driven by the thermally activated hopping of adatoms between adsorption sites. Helium spin-echo techniques, capable of measuring the sub-picosecond motion of individual adatoms, have enabled the bench-marking of many important adsorbate-substrate properties. The well-known Markovian Langevin equation has emerged as the standard tool for the interpretation of such experimental data, replacing adatom-substrate interactions with stochastic white noise and linear friction. However, the consequences of ignoring the colored noise spectrum and non-linearities inherent to surface systems are not known. Through the computational study of three alternative models of adatom motion, we show that the hopping rate and jump distributions of an adatom are fixed to within a few percent by the potential energy surface and a new generalized energy exchange rate parameter alone, independent of the model used. This result justifies the use of the Markovian Langevin equation, regardless of the true statistical nature of adatom forces, provided results are quoted in terms of the new energy exchange rate parameter. Moreover, numerous mechanisms for the effect of noise correlations and non-linear friction on the energy exchange rate are proposed which likely contribute to activated surface diffusion and activated processes more generally.