Transition Graphs of Interacting Hysterons: Structure, Design, Organization and Statistics

TL;DR

This paper develops a comprehensive framework linking transition graphs of interacting hysterons to their microscopic parameters, enabling analysis of memory effects and design of complex media with tailored responses.

Contribution

It introduces a systematic scaffold-based approach to structure transition graphs and connect their topology to microscopic hysteron parameters, advancing understanding of memory in multistable media.

Findings

Framework for linking transition graph topology to microscopic parameters

Tools for analyzing the realizability of transition graphs

Insights into designing materials with specific memory effects

Abstract

Transition graphs capture the memory and sequential response of multistable media, by specifying their evolution under external driving. Microscopically, collections of bistable elements, or hysterons, provide a powerful model for these materials, with recent work highlighting the crucial role of hysteron interactions. Here, we introduce a general framework that links transition graphs and the microscopic parameters of interacting hysterons. We first introduce a systematic framework, based on so-called scaffolds, which structures the space of transition graphs and provides tools to deal with their combinatorial explosion. We then connect the topology of transition graphs to partial orders of the microscopic parameters. This allows us to understand the statistical properties of transition graphs, as well as determine whether a given graph is realizable, i.e. compatible with the hysteron framework. Our approach paves the way for a deeper theoretical understanding of memory effects in complex media and opens a route to rationally design pathways and memory effects in materials.