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

TL;DR

This work develops a general framework for interacting hysterons by linking transition graphs (t-graphs) to microscopic, state-dependent switching fields $U_i^\pm(S)$. It introduces scaffolds to organize the combinatorial space and derives design inequalities that impose a partial order on the switching fields, enabling systematic realizability tests of t-graphs. By constructing all t-graphs for small numbers of hysterons ($n=2,3$) through scaffolds and finite binary trees, the authors quantify the design-space volume and show how avalanches and Garden-of-Eden states shape memory pathways. The framework enables rational design of memory effects in frustrated materials and provides a path toward extended models and finite-state-machine representations of dynamic responses, with practical implications for metamaterials and driven disordered systems.

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.

Figures (9)

• Figure 1: Interacting hysterons and the physical systems they inhabit. a) The response to driving with a field $U$ (left) and graph representation (right) of a single hysteron with switching fields $u_i^\pm$. The phase of the hysteron is expressed in a binary variable $s_i$. b) Abstract representation of hysterons interacting through their embedding medium. c) Interacting hysterons (local rearrangements) in a sheared amorphous solid regev2019. d) Interacting hysterons (mountain/valley folds) in a compressed crumpled sheet shohat2022memory.
• Figure 2: Scaffolds and switching fields. (a) Graphical representation of the set of $n=2$ switching fields: hysterons in state $s_i=0$ ($s_i=1$) have an up (down) switching field indicated by purple (gold) arrows. (b) Example of a given set of switching fields (boxes; number indicates hysteron index) and corresponding stability ranges (black lines) -- in the dashed regions, the state is unstable. (c) Corresponding scaffold, with the $2^{n+1}-2$ critical hysterons $k^\pm(S)$ indicated. (d) Alternative yet equivalent graphical representation of the scaffold showing 'passages', i.e., tentative transitions to a landing state $S^{(1)}$ that would occur upon flipping each critical hysterons $k^\pm(S)$ (see text). (e) Example of a subset of $n=3$ switching fields for the Preisach model where $\Delta_i^\pm(S)=0$. Since $U_2^+(000)=U_2^+(001)=u_2^+$, and $U_3^+(000)=U_3^+(001)=u_3^+$, hysterons 2 and 3 flip in the same order for both states, and since $k^+(000) = 3$, $k^+(100)=3$. (f) Graphical representation of the corresponding scaffold. (g) Example of a subset of $n=3$ switching fields, for the states $(000)$ and $(100)$, for the general model where $\Delta_i^\pm(S) \ne 0$. Since the switching fields $\{U_i^\pm(S)\}$ are independent, the order in which hysterons 2 and 3 flip may be different ('scrambled') between states $(000)$ and $(100)$. (h) Graphical representation of the corresponding scrambled scaffold.
• Figure 3: Construction of transitions and full transition graph from the set of switching fields. (a) Schematic of the three scenarios for the landing state $S^{(1)}$ at driving field $U^\pm(S^{(0)})$: (i) $S^{(1)}$ is stable; (ii) a single hysteron in state $S^{(1)}$ is unstable; (iii) $S^{(1)}$ multiple hysterons in $S^{(1)}$ are unstable (see text). (b) Two transitions ($l=1$ transition $(10)\downarrow (00)$ and $l=2$ transition $(00)\uparrow(01)\uparrow(11)$) for the set of switching fields shown in Fig. \ref{['fig:graph_construction_transitions']}b. (c) The $l=1$ transition $(10)\downarrow(00)$ follows the scaffold (faded) (d) The avalanche transition $(00)\uparrow(01)\uparrow(11)$ follows the scaffold. (e) The tentative transition $(01)\downarrow(00)\uparrow(10)$ is incompatible with the scaffold, as the transition requires that $k^+(00) = 1$, while the scaffold specifies that $k^+(00) = 2$. (f) Full t-graph for the set of switching fields shown in panel b.
• Figure 4: The two types of ill-defined transitions. (a) Set of switching fields that leads to a self-loop $(00)\uparrow (01)\uparrow (11)\downarrow (10) \downarrow (00)\uparrow \dots$. (b) Graphical representation for this self-loop. (c) Example of a (subset of) switching fields that leads to a race condition, due to the instability of multiple hysterons in state $(111)$ at $U^+(011)$. (d) Graphical representation for this race condition: we cannot draw a tentative down transition because the order in which hysterons 2 and 3 flip is not well defined.
• Figure 5: Design inequalities. (a) Example of an $n=2$ target t-graph. (b) The design inequalities are organized in three groups pertaining to the initial state $S^{(0)}$, intermediate states $S^{(1)}, \dots, S^{(l-1)}$, and final state $S^{(l)}$. (c) Length-1 transition $(10)\downarrow (00)$. (d) Length-2 transition $(00)\uparrow (01)\uparrow(11)$. (e) The design inequalities for the target graph shown in panel (a). We label transitions by their starting state $S^{(0)}$ and their up or down direction: for example, the up transition starting from state $(00)$ is labeled 00U, and the down transition starting from state $(10)$ is denoted 10D (see also panels (c) and (d)). Note that all inequalities are organized by transition, and can be further separated by whether they are given by the initial state (scaffold), intermediate state (if present), or final state inequalities.