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Crystallinity of the homogenized energy density of periodic lattice systems

Chambolle, Antonin; Kreutz, Leonard (2023), Crystallinity of the homogenized energy density of periodic lattice systems, Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 21, 1. 10.1137/21M1442073

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Type
Article accepté pour publication ou publié
Date
2023
Journal name
Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal
Volume
21
Number
1
Publisher
SIAM - Society for Industrial and Applied Mathematics
Publication identifier
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Author(s)
Chambolle, Antonin
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Kreutz, Leonard
Fachbereich Mathematik und Informatik [Münster] = Fachbereich Mathematik und Informatik [Münster] [FB 10]
Abstract (EN)
We study the homogenized energy densities of periodic ferromagnetic Ising systems. We prove that, for finite range interactions, the homogenized energy density, identifying the effective limit, is crystalline, i.e. its Wulff crystal is a polytope, for which we can (exponentially) bound the number of vertices. This is achieved by deriving a dual representation of the energy density through a finite cell formula. This formula also allows easy numerical computations: we show a few experiments where we compute periodic patterns which minimize the anisotropy of the surface tension.
Subjects / Keywords
Γ-convergence; Ising system; Crystallinity; Wulff Shape