Granular convergence as an iterated local map

When subjected to quasi-statical periodic shearing, granular packs converge to a repetitive behavior. In this repetitive behavior, the packing configuration goes through a finite sequence of states defined by the position of each individual particle. People have tried to capture the dynamics of the system with the use of hysterons with limited success. We propose a different approach. The energy-configuration landscape of this system is complex and the configuration space is high-dimensional. We treat the motion of the particles during shearing as a sequence of transitions between discrete, locally stable configurations, as done previously by [M. Mungan, S. Satry, Phys, K. Dahmen, and I. Regev, Rev. Lett. 123, 178002]. There will be a map from each state to another state corresponding to how a full cycle of shear would affect the state (if the system returns to exactly the same state as it started at the beginning of the shearing cycle, the state would map to itself). If we represent states as points in space and the map as directed edges that connect points with their images we can construct graphs. These graphs represent the evolution of the system through different states as cyclic shear is applied. We propose that the convergence phenomenon results from the qualitative features of these maps. Here we examine the effect of allowing the mapping to include only "local" transitions to nearby configurations.