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 0-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.

Continuum limit and stochastic homogenization of discrete ferromagnetic thin films / Braides, A.; Cicalese, M.; Ruf, M.. - In: ANALYSIS & PDE. - ISSN 2157-5045. - 11:2(2018), pp. 499-553. [10.2140/apde.2018.11.499]

Continuum limit and stochastic homogenization of discrete ferromagnetic thin films

Braides, A.;Cicalese, M.;Ruf, M.
2018-01-01

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 0-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.
2018
11
2
499
553
https://doi.org/10.2140/apde.2018.11.499
https://arxiv.org/abs/1612.02775
Braides, A.; Cicalese, M.; Ruf, M.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/139459
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