Published Paper

Inserted: 7 dec 2016
Last Updated: 17 nov 2017

Journal: Analysis & PDE
Volume: 11
Number: 2
Pages: 499-553
Year: 2018
Doi: 10.2140/apde.2018.11.499

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) non-periodic 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 $\varepsilon > 0$ we perform a $\Gamma$-convergence analysis of properly rescaled interfacial-type energies. We show that, up to subsequences, the energies converge to a surface integral dened 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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