Vol. 11, No. 2, 2018

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Continuum limit and stochastic homogenization of discrete ferromagnetic thin films

Andrea Braides, Marco Cicalese and Matthias Ruf

Vol. 11 (2018), No. 2, 499–553
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) nonperiodic 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 a magnetic thin system obtained by a random deposition mechanism.

Keywords
$\Gamma$-convergence, dimension reduction, spin systems, stochastic homogenization
Mathematical Subject Classification 2010
Primary: 49J45, 74E30, 60K35, 74Q05
Milestones
Received: 9 April 2017
Revised: 9 July 2017
Accepted: 5 September 2017
Published: 17 October 2017
Authors
Andrea Braides
Dipartimento di Matematica
Università di Roma “Tor Vergata”
Roma
Italy
Marco Cicalese
Zentrum Mathematik
Technische Universität München
Garching
Germany
Matthias Ruf
Zentrum Mathematik
Technische Universität München
Garching
Germany