Embedding Quasicrystals in a Periodic Cell: Dynamics in Quasiperiodic Structures

Abstract

We introduce a construction to "periodize" a quasiperiodic lattice of obstacles, i.e., embed it into a unit cell in a higher-dimensional space, reversing the projection method used to form quasilattices. This gives an algorithm for simulating dynamics, as well as a natural notion of uniform distribution, in quasiperiodic structures. It also shows the generic existence of channels, where particles travel without colliding, up to a critical obstacle radius, which we calculate for a Penrose tiling. As an application, we find superdiffusion in the presence of channels, and a subdiffusive regime when obstacles overlap.

Horizons and free-path distributions in quasiperiodic Lorentz gases

“…Therefore, we expect weak super-diffusion if the obstacles are small enough, which agrees with numerical results founded in [26]. However, the numerical results are not completely convincing, especially when the obstacles are very small.…”

“…There are several methods to produce quasiperiodic arrays, not all of which producing the same tiling [56]; one of the most popular is the projection method [29,57,58]. Reversing this method, it is possible to simulate and analyze the dynamics of a quasiperiodic LG as a periodic billiard, as two of the current authors previously showed [26]. Throughout this paper, we take the interaction potential to be that of the hard sphere.…”

“…We summarize the method for constructing finite-range quasiperiodic potentials introduced in [26]. The main idea is to produce an n-dimensional periodic potential, with n > m, based on the projection method [29,57,58], with noninteracting classical particles moving inside.…”

“…The construction described in the previous section was originally designed to allow efficient numerical simulation of quasiperiodic Lorentz gases. Nonetheless, it also provides a powerful tool to analyze the geometric structure of these systems [26]: here we use it to prove the generic existence of channels in quasiperiodic LGs; specific cases were studied in [26].…”

Exposing local symmetries in distorted driven lattices via time-averaged invariants

“…In other words, spatial symmetries hold strictly only locally rather than globally in real systems. There are in fact many scenarios where a system is naturally made up of different domains in which distinct spatial symmetries hold locally, striking examples being quasicrystals [12][13][14] or large molecules [15,16]. The quest of finding symmetry remnants in stationary states of systems with globally broken but locally retained symmetries recently led to a generic treatment of one-dimensional (1D) local symmetries in terms of invariant two-point currents [17].…”