Motion of discrete interfaces in low-contrast random environments

Abstract

We study the asymptotic behavior of a discrete-in-time minimizing movement scheme for square lattice interfaces when both the lattice spacing and the time step vanish. The motion is assumed to be driven by minimization of a weighted random perimeter functional with an additional deterministic dissipation term. We consider rectangular initial sets and lower order random perturbations of the perimeter functional. In case of stationary, α-mixing perturbations we prove a stochastic homogenization result for the interface velocity. We also provide an example which indicates that stationary, ergodic perturbations do not yield a spatially homogenized limit velocity for this minimizing movement scheme.2010 Mathematics Subject Classification. 53C44, 49J55, 49J45.

Nucleation and Growth of Lattice Crystals

“…Note that our discretization approach can be regarded as a "backward" version of Braides et al (2010) if the index k is considered as parameterizing negative time (see Braides 2013, Section 10.2). Other analyses of minimizing movements on lattices related to the perimeter can be found in Braides and Scilla (2013a), Scilla (2014Scilla ( , 2020 and Ruf (2018). We note that checkerboard, stripes and other structures arise in antiferromagnetic systems related to maximization of the perimeter (see Braides and Cicalese 2017 for a variational analysis in terms of -convergence, and the wide literature in Statistical Mechanics, e.g., Giuliani et al 2011;Daneri and Runa 2019).…”

Nucleation and growth of lattice crystals

“…Note that our discretization approach can be regarded as a 'backward' version of [10] if the index k is considered as parameterizing negative time (see [7,Section 10.2]). Other analyses of minimizing movements on lattices related to the perimeter can be found in [11,27,26,28]. We note that checkerboard, stripes and other structures arise in antiferromagnetic systems related to maximization of the perimeter (see [8] for a variational analysis in terms of Γ-convergence, and the wide literature in Statistical Mechanics, e.g.…”

Motion of discrete interfaces on the triangular lattice

“…Indeed, they are very sensitive and may depend on microscopic properties not detected in the limit description, as showed, e.g., by Braides and Scilla [18] in case of periodic media and by Scilla [31] in case of 'low-contrast' periodic media; therein the dependence of the limit velocity on the curvature is described by a homogenized formula quite different with respect to [16]. A random counterpart of the low-contrast setting has been provided by Ruf [30]. Recently, Braides, Cicalese and Yip [13] investigated the case of antiferromagnetic energies, in particular anti-phase boundaries between striped patterns, showing the appearance of some non-local curvature dependence velocity law reflecting the creation of some defect structure on the interface at the microscopic level.…”