Calculus of Variations and Geometric Measure Theory

A. Braides - V. Chiadò Piat - M. Solci

Discrete double-porosity models for spin systems

created by braidesa on 01 Dec 2015
modified on 02 Jul 2016


Published Paper

Inserted: 1 dec 2015
Last Updated: 2 jul 2016

Journal: Math. Mech. Complex Syst.
Volume: 4
Pages: 79-102
Year: 2016
Doi: 10.2140/memocs.2016.4.79


We consider spin systems between a finite number $N$ of ``species'' or ``phases'' partitioning a cubic lattice $\mathbb{Z}^d$. We suppose that interactions between points of the same phase are coercive, while between point of different phases (or, possibly, between points of an additional ``weak phase'') are of lower order. Following a discrete-to-continuum approach we characterize the limit as a continuum energy defined on $N$-tuples of sets (corresponding to the $N$ strong phases) composed of a surface part, taking into account homogenization at the interface of each strong phase, and a bulk part which describes the combined effect of lower-order terms, weak interactions between phases, and possible oscillations in the weak phase.