Continuum limit and stochastic homogenization of discrete ferromagnetic thin films

Abstract

We study the discrete-to-continuum limit of ferromagnetic spin systems when the lattice spacing tends to zero. We assume that the atoms are part of a (maybe) non-periodic lattice close to a flat set in a lower dimensional space, typically a plate in three dimensions. Scaling the particle positions by a small parameter ε > 0 we perform a Γ-convergence analysis of properly rescaled interfacial-type energies. We show that, up to subsequences, the energies converge to a surface integral defined on partitions of the flat space. In the second part of the paper we address the issue of stochastic homogenization in the case of random stationary lattices. A finer dependence of the homogenized energy on the average thickness of the random lattice is analyzed for an example of magnetic thin system obtained by a random deposition mechanism.

The Choice of Representative Volumes in the Approximation of Effective Properties of Random Materials

“…defining the X 0 y as indicated in this formula, and setting X m y := 0 for m ≥ 1, we immediately observe that the average F avg (a) is the sum of a family of random variables with multilevel local dependence structure with K := 1. The bound (26) follows immediately from the uniform bound on a (with B := ||a|| L ∞ and arbitrary γ > 0).…”

Motion of discrete interfaces in low-contrast random environments

“…While for static problems the above notions (i) and (ii) are often enough to prove stochastic homogenization results for variational models (see for example [2,3,12,15]), in this minimizing movement setting we make use of mixing properties. More precisely, we require that the random field is α-mixing with…”

“…Our aim is to include dynamical effects in order to describe the curvature-driven motion of magnetic domain walls. The natural approach in the spirit of [3,12] would be to replace the periodic lattice in the definition of P ε by a stationary random lattice εL(ω) with suitable short-range interactions. This seems to be a very challenging problem.…”

Homogenization of random convolution energies

“…Furthermore, discrete counterparts of functionals (), that is, energies of the form where is a ‐dimensional lattice, have been widely investigated (see, for example, [2, 11, 16, 34]) as a discrete approximation of quadratic integral functionals. Such type of functionals or the corresponding operators have been analysed in different ways under various inhomogeneity and randomness assumptions (see, for example, [3, 5, 13, 15, 23, 30, 32, 34]).…”

“…In the work [3], discrete‐to‐continuous ‐limits are investigated for energies defined on random stationary lattices (for random thin‐film energies, see, for example, [13]). The energies admit both nearest neighbours and long distance interactions, it is assumed that the nearest‐neighbour terms satisfy ‐growth conditions and for the long distance terms proper moment conditions are fulfilled.…”