Pattern Formation in Mathematical Billiards with Spatial Memory

Many classes of active matter develop spatial memory by encoding information in space, leading to complex

pattern formation. It has been proposed that spatial memory can lead to more efficient navigation and

collective behaviour in biological systems and influence the fate of synthetic systems. This raises important

questions about the fundamental properties of dynamical systems with spatial memory. We present a

framework based on mathematical billiards in which particles remember their past trajectories and react to

them. Despite the simplicity of its fundamental deterministic rules, such a system is strongly non-ergodic

and exhibits highly intermittent statistics, manifesting in complex pattern formation. We show how these

self-memory-induced complexities emerge from the temporal change of topology and the consequent chaos in

the system. We study the billiards' fundamental properties, particularly the long-time behaviour

when the particles are self-trapped in an arrested state. We exploit numerical simulations of several millions

of particles to explore pattern formation and the corresponding statistics in polygonal billiards of different

geometries. Our work illustrates how the dynamics of a single-body system can dramatically change when

particles feature spatial memory and provide a scheme to explore systems with complex memory

kernels further (arXiv:2307.01734).