Minimal descriptions of cyclic memories

Many materials that are out of equilibrium can 'learn' one or more inputs that are repeatedly applied. Yet, a common framework for understanding such memories is lacking. Here, we construct minimal representations of cyclic memory behaviours as directed graphs, and we construct simple physically motivated models that produce the same graph structures. We show how a model of worn grass between park benches can produce multiple transient memories-a behaviour previously observed in dilute suspensions of particles and charge-density-wave conductors-and the Mullins effect. Isolating these behaviours in our simple model allows us to assess the necessary ingredients for these kinds of memory, and to quantify memory capacity. We contrast these behaviours with a simple Preisach model that produces return-point memory. Our analysis provides a unified method for comparing and diagnosing cyclic memory behaviours across different materials.