3 research outputs found
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 fundamental properties of these billiards and 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 further explore systems with complex memory kernels.Comment: 11 pages, 6 figure