Particle transport in open polygonal billiards: a scattering map

TL;DR

This paper investigates particle transport in open polygonal billiards, deriving an exact scattering map and analytical expressions for ballistic propagation speeds, revealing complex dynamical behaviors relevant to anomalous transport phenomena.

Contribution

It introduces a novel analytical framework using interval exchange transformations to describe particle scattering and propagation in polygonal billiard channels, advancing understanding of their complex dynamics.

Findings

Derived an exact scattering map for polygonal billiard channels.

Obtained an analytical expression for ballistic front speeds.

Analyzed the symbolic hierarchy of trajectories in transport phenomena.

Abstract

Polygonal billiards exhibit a rich and complex dynamical behavior. In recent years polygonal billiards have attracted great attention due to their application in the understanding of anomalous transport, but also at the fundamental level, due to its connections with diverse fields in mathematics. We explore this complexity and its consequences on the properties of particle transport in infinitely long channels made of the repetitions of an elementary open polygonal cell. Borrowing ideas from the Zemlyakov-Katok construction, we construct an interval exchange transformation classified by the singular directions of the discontinuities of the billiard flow over the translation surface associated to the elementary cell. From this, we derive an exact expression of a scattering map of the cell connecting the outgoing flow of trajectories with the unconstrained incoming flow. The scattering map is defined over a partition of the coordinate space, characterized by different families of trajectories. Furthermore, we obtain an analytical expression for the average speed of propagation of ballistic modes, describing with high accuracy the speed of propagation of ballistic fronts appearing in the tails of the distribution of the particle displacement. The symbolic hierarchy of the trajectories forming these ballistic fronts is also discussed.