Analysis & PDE

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

Continuum limit and stochastic homogenization of discrete ferromagnetic thin films

Andrea Braides, Marco Cicalese, and Matthias Ruf

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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 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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 49J45: Methods involving semicontinuity and convergence; relaxation 74E30: Composite and mixture properties 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 74Q05: Homogenization in equilibrium problems

$\Gamma$-convergence dimension reduction spin systems stochastic homogenization


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.

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