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Continuum limit and
stochastic homogenization of discrete ferromagnetic thin
films
Andrea Braides, Marco Cicalese and Matthias Ruf |
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Vol. 11 (2018), No. 2, 499–553
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Abstract |
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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 , we perform a -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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Keywords
$\Gamma$-convergence, dimension reduction, spin systems,
stochastic homogenization
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Mathematical Subject Classification 2010
Primary: 49J45, 74E30, 60K35, 74Q05
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Milestones
Received: 9 April 2017
Revised: 9 July 2017
Accepted: 5 September 2017
Published: 17 October 2017
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Authors |
| Andrea Braides | |
| Dipartimento di Matematica Università di Roma “Tor Vergata” Roma Italy |
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| Marco Cicalese | |
| Zentrum Mathematik Technische Universität München Garching Germany |
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| Matthias Ruf | |
| Zentrum Mathematik Technische Universität München Garching Germany |
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