Embedding quasicrystals in a periodic cell: dynamics in quasiperiodic structures

Area of Science:

• Physics
• Materials Science
• Dynamical Systems

Background:

• Quasiperiodic structures lack translational symmetry, complicating simulations of particle dynamics.
• Existing methods for quasiperiodic structures often rely on projection techniques, which can be computationally intensive.
• Understanding particle transport in these complex lattices is crucial for various applications.

Purpose of the Study:

• To introduce a novel construction for "periodizing" quasiperiodic lattices.
• To develop an efficient algorithm for simulating particle dynamics in these structures.
• To investigate particle transport phenomena, including the existence of collision-free channels and diffusion regimes.

Main Methods:

• Embedding quasiperiodic lattices into higher-dimensional unit cells.
• Reversing the projection method commonly used for quasilattice formation.
• Calculating critical obstacle radii for particle channels in a Penrose tiling.
• Analyzing particle trajectories to identify diffusion behaviors.

Main Results:

• Demonstrated a method to "periodize" quasiperiodic lattices, enabling efficient simulations.
• Established a natural definition of uniform distribution within quasiperiodic structures.
• Proved the generic existence of particle channels up to a critical obstacle radius.
• Identified superdiffusion in the presence of channels and subdiffusion when obstacles overlap.

Conclusions:

• The "periodization" construction provides a powerful tool for studying dynamics in quasiperiodic systems.
• The identified particle channels significantly influence transport properties.
• This work offers insights into the fundamental physics of disordered and quasiperiodic materials.