Interacting Hysterons with Asymptotically Small or Large Spans

TL;DR

This paper explores the behavior of interacting hysterons in two key limits, revealing how their parameters influence collective responses and establishing connections to spin models in complex systems.

Contribution

It provides a detailed analysis of hysteron models in two physically relevant limits and shows how they can be represented by interacting spins, deepening understanding of their collective behavior.

Findings

Limited pathways when hysteron span dominates

Hysterons behave as spins when span is small

Hysterons can be mapped to pairs of interacting spins

Abstract

Models of interacting hysteretic elements, called hysterons, capture the sequential response and complex memory effects in a wide range of complex systems and can guide the design of intelligent metamaterials. However, even simple models with few hysterons feature a bewildering number and variety of behaviors. Here we study the hysteron model in two physically relevant limits, where {the} response {of a hysteron system} is easier to understand. First, when the hysteron span - the gap between its two hysteretic transitions - dominates all other scales, the range of pathways encoded in transition graphs (t-graphs) becomes limited because many avalanches {are} absent. Second, when the hysteron span becomes vanishingly small, hysterons behave as interacting binary spins, {which require avalanches in order to} exhibit nontrivial pathways. Finally we show that hysterons can be mimicked by pairs of strongly interacting spins, {such} that collections of $n$ interacting hysterons can be mapped to $2n$ interacting spins, albeit {via} highly specific interactions. {Altogether,} our work provides a deeper understanding of the role of the hysteron parameters on their collective behavior, and points to connections and differences between spin- and hysteron-based models of complex matter.