Continuum limit and stochastic homogenization of discrete ferromagnetic thin films

TL;DR

This paper investigates the transition from discrete ferromagnetic spin systems to continuum models as the lattice spacing approaches zero, including stochastic homogenization for random lattices, with applications to magnetic thin films.

Contribution

It provides a rigorous Gamma-convergence analysis of interfacial energies and explores stochastic homogenization for non-periodic, random lattice structures in magnetic thin films.

Findings

Convergence of discrete energies to a surface integral in the continuum limit

Analysis of stochastic homogenization for random stationary lattices

Dependence of homogenized energy on lattice thickness in magnetic systems

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 $\varepsilon>0$ we perform a $\Gamma$-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.