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

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.

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