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.

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

Continuum limit and stochastic homogenization of discrete ferromagnetic thin films

Braides, Andrea;
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
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - ANALISI MATEMATICA
English
Con Impact Factor ISI
Dimension reduction; Spin systems; Stochastic homogenization; Γ-convergence; Analysis; Numerical Analysis; Applied Mathematics
https://msp.org/apde/2018/11-2/apde-v11-n2-p06-s.pdf
Braides, A., Cicalese, M., Ruf, M. (2018). Continuum limit and stochastic homogenization of discrete ferromagnetic thin films. ANALYSIS & PDE, 11(2), 499-553 [10.2140/apde.2018.11.499].
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/2108/211698
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