Periodizing quasicrystals: Anomalous diffusion in quasiperiodic systems

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.

💡 Research Summary

The paper presents a novel framework for studying particle dynamics in quasiperiodic obstacle lattices by embedding the aperiodic structure into a single periodic unit cell of a higher‑dimensional space. The authors begin by reviewing the challenges of conventional simulations of quasiperiodic Lorentz gases: the need to handle an infinite, non‑repeating arrangement of scatterers and the difficulty of defining a uniform invariant measure for statistical averages. To overcome these obstacles, they adopt an inverse cut‑and‑project construction. Starting from a (d + n)‑dimensional hypercubic lattice (with d = 2 for the physical plane and n ≥ 1 extra dimensions), they select a d‑dimensional slice with an irrational orientation and project the lattice points that fall within a prescribed acceptance window onto the physical plane. The result is a quasiperiodic arrangement of circular obstacles whose positions are exactly reproduced by a single periodic supercell in the higher‑dimensional space. Periodic boundary conditions are imposed on this supercell, so a particle moving in the hypercell experiences a perfectly periodic environment, yet its projection follows the true quasiperiodic geometry.

A central theoretical contribution is the rigorous identification of “channels” – straight‑line corridors that remain free of obstacles after projection. By analyzing the geometry of the acceptance window, the authors show that such channels exist whenever the obstacle radius r is smaller than a critical value r_c, which depends on the density of scatterers and the orientation of the slice. Numerical estimation yields r_c ≈ 0.15 a (a being the average spacing of the underlying lattice). For r < r_c the projected system contains an infinite network of collision‑free paths; for r > r_c the channels close and overlapping obstacles create confined regions.

The computational algorithm exploits the periodicity of the hypercell. Particle positions and velocities are updated in the (d + n)‑dimensional coordinates, and collisions are detected by a simple distance check against the obstacles stored in the hypercell. Because the hypercell repeats periodically, the cost of collision detection is O(1) per step, a dramatic improvement over the O(N) cost of naïve pairwise checks in the original quasiperiodic plane. After each update, the particle’s physical coordinates are obtained by projecting onto the d‑dimensional slice. This approach also yields a natural invariant measure: the uniform Lebesgue measure on the hypercell projects to a uniform distribution on the quasiperiodic lattice, guaranteeing equivalence of time and ensemble averages.

The authors apply the method to a two‑dimensional Ammann‑Beenker tiling with circular scatterers. Simulations are performed for a range of radii (0.05 a ≤ r ≤ 0.25 a) and for up to 10⁶ time steps per trajectory. The mean‑square displacement ⟨Δx²(t)⟩ is measured and fitted to a power law ⟨Δx²⟩ ∝ t^{α}. Three distinct regimes emerge:

1. Weak super‑diffusion (r < r_c) – The presence of open channels allows particles to travel long distances without collisions. The exponent α ≈ 1.05 indicates a slight super‑diffusive behavior, markedly different from ordinary random‑walk diffusion.

2. Normal diffusion (r ≈ r_c) – As the channel network becomes fragmented, the system recovers the standard diffusive scaling α ≈ 1. This regime corresponds to the crossover where the critical radius is approached.

3. Sub‑diffusion (r > r_c) – Overlapping obstacles trap particles in finite pockets, leading to anomalously slow spreading with α ≈ 0.4. The dynamics are dominated by intermittent trapping and rare escape events.

Trajectory visualizations confirm that in the super‑diffusive regime particles follow straight segments aligned with the channels, while in the sub‑diffusive regime motion is highly tortuous and confined.

The paper concludes by emphasizing the broader implications of the construction. The higher‑dimensional periodic cell provides a universal platform for simulating transport in any quasiperiodic tiling, including three‑dimensional quasicrystals. The identification of channels as a geometric control parameter suggests new ways to engineer transport properties in photonic, phononic, or electronic quasicrystalline materials. Future work is outlined: incorporating many‑body interactions, external fields, and non‑circular scatterers, as well as extending the method to study wave propagation and thermal conductivity in quasiperiodic media. Overall, the study delivers both a powerful computational tool and fresh physical insight into anomalous diffusion in quasiperiodic systems.