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2018 Continuum limit and stochastic homogenization of discrete ferromagnetic thin films
Andrea Braides, Marco Cicalese, Matthias Ruf
Anal. PDE 11(2): 499-553 (2018). DOI: 10.2140/apde.2018.11.499


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.


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Andrea Braides. Marco Cicalese. Matthias Ruf. "Continuum limit and stochastic homogenization of discrete ferromagnetic thin films." Anal. PDE 11 (2) 499 - 553, 2018.


Received: 9 April 2017; Revised: 9 July 2017; Accepted: 5 September 2017; Published: 2018
First available in Project Euclid: 20 December 2017

zbMATH: 1379.49045
MathSciNet: MR3724495
Digital Object Identifier: 10.2140/apde.2018.11.499

Primary: 49J45 , 60K35 , 74E30 , 74Q05

Keywords: $\Gamma$-convergence , Dimension reduction , spin systems , Stochastic homogenization

Rights: Copyright © 2018 Mathematical Sciences Publishers


Vol.11 • No. 2 • 2018
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