Periodizing quasicrystals: Anomalous diffusion in quasiperiodic systems

TL;DR

This paper presents a new method for modeling quasiperiodic systems with obstacles, revealing channels that enable unique diffusion behaviors, and offers an efficient simulation algorithm for these complex structures.

Contribution

It introduces a novel construction embedding quasiperiodic lattices into higher-dimensional periodic cells, enabling analysis of channels and diffusion in quasiperiodic systems.

Findings

Identification of channels allowing collision-free particle travel

Discovery of weak super-diffusion in the presence of channels

Observation of sub-diffusion when obstacles overlap

Abstract

We introduce a construction to embed a quasiperiodic lattice of obstacles into a single unit cell of a higher-dimensional space, with periodic boundary conditions. This construction transparently shows the existence of channels in these systems,in which particles may travel without colliding, up to a critical obstacle radius. It provides a simple and efficient algorithm for numerical simulation of dynamics in quasiperiodic structures, as well as giving a natural notion of uniform distribution (measure) and averages. As an application, we simulate diffusion in a two-dimensional quasicrystal, finding three different regimes, in particular atypical weak super-diffusion in the presence of channels, and sub-diffusion when obstacles overlap.