## Analysis & PDE

• Anal. PDE
• Volume 11, Number 2 (2018), 499-553.

### Continuum limit and stochastic homogenization of discrete ferromagnetic thin films

#### 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 $Γ$-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.

#### Article information

Source
Anal. PDE, Volume 11, Number 2 (2018), 499-553.

Dates
Revised: 9 July 2017
Accepted: 5 September 2017
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513774512

Digital Object Identifier
doi:10.2140/apde.2018.11.499

Mathematical Reviews number (MathSciNet)
MR3724495

Zentralblatt MATH identifier
1379.49045

#### Citation

Braides, Andrea; Cicalese, Marco; Ruf, Matthias. Continuum limit and stochastic homogenization of discrete ferromagnetic thin films. Anal. PDE 11 (2018), no. 2, 499--553. doi:10.2140/apde.2018.11.499. https://projecteuclid.org/euclid.apde/1513774512

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