Discrete double-porosity models for spin systems

TL;DR

This paper develops a discrete-to-continuum framework for spin systems with multiple phases, capturing complex inter-phase interactions and homogenization effects in the continuum limit.

Contribution

It introduces a novel continuum energy model for multi-phase spin systems, incorporating interface homogenization and lower-order interaction effects.

Findings

Characterizes the continuum limit as an energy on N-tuples of sets.

Includes surface and bulk energy contributions.

Accounts for homogenization and oscillations at phase interfaces.

Abstract

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.