Horizons and free path distributions in quasiperiodic Lorentz gases

TL;DR

This paper investigates the structure and dynamics of quasiperiodic Lorentz gases, revealing regimes with infinite and finite horizons, and characterizing free path distributions with novel power-law exponents.

Contribution

It introduces an embedding method for simulating quasiperiodic Lorentz gases and uncovers new regimes with distinct free path distribution exponents.

Findings

Infinite horizon regime with power-law distribution exponent -3.

Finite horizon regime with distribution exponent -5.

Efficient simulation algorithm for quasiperiodic structures.

Abstract

We study the structure of quasiperiodic Lorentz gases, i.e., particles bouncing elastically off fixed obstacles arranged in quasiperiodic lattices. By employing a construction to embed such structures into a higher dimensional periodic hyperlattice, we give a simple and efficient algorithm for numerical simulation of the dynamics of these systems. This same construction shows that quasiperiodic Lorentz gases generically exhibit a regime with infinite horizon, that is, empty channels through which the particles move without colliding, when the obstacles are small enough; in this case, the distribution of free paths is asymptotically a power law with exponent -3, as expected from infinite-horizon periodic Lorentz gases. For the critical radius at which these channels disappear, however, a new regime with locally-finite horizon arises, where this distribution has an unexpected exponent of -5, previously observed only in a Lorentz gas formed by superposing three incommensurable periodic lattices in the Boltzmann-Grad limit where the radius of the obstacles tends to zero.