2016/12/08 by Andrea Braides, Marco Cicalese, Matthias Ruf
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Ferromagnetism #Homogenization (climate) #Lattice (music) #Limit (mathematics) #Scaling #Scaling limit #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Thin film #math.AP #msc:49J45 #msc:49J55 #msc:49Q20 #msc:49S05 #msc:82B20 #msc:82B24
paper · pdf · doi:10.2140/apde.2018.11.499
published as Analysis & PDE 11 (2018) 499-553 · 43 pages, 2 figures
arxiv created 2016/12/08 · openalex created_date 2017/01/06 · openalex publication_date 2017/11/17 · arxiv updated 2018/03/16 · openalex updated_date 2026/08/05
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 [math] , we perform a [math] -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.