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Particle transport in open polygonal billiards: A scattering map.

Jordan Orchard¹, Federico Frascoli¹, Lamberto Rondoni^2,3

¹Department of Mathematics, School of Science, Computing and Engineering Technologies, Swinburne University of Technology, H38, PO Box 218, Hawthorn, Victoria 3122, Australia.

Chaos (Woodbury, N.Y.)

|December 17, 2024

Summary

Polygonal billiards reveal complex particle transport in channels. Researchers derived a scattering map and average speed for ballistic modes, enhancing understanding of anomalous transport.

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Area of Science:

Dynamical systems
Mathematical physics
Transport phenomena

Background:

Polygonal billiards display complex dynamics with applications in anomalous transport and mathematics.
Understanding particle transport in channels formed by repeating polygonal cells is crucial.

Purpose of the Study:

To analyze particle transport in infinitely long channels composed of repeating polygonal cells.
To derive a scattering map and analytical expressions for transport properties.

Main Methods:

Construction of an interval exchange transformation based on billiard flow discontinuities.
Derivation of a scattering map connecting incoming and outgoing trajectories.
Analytical calculation of the average speed of ballistic modes.

Main Results:

An exact scattering map for the polygonal cell was derived.
An analytical expression for the average speed of ballistic modes was obtained.
High-accuracy description of ballistic front propagation and trajectory hierarchy.

Conclusions:

The study provides a detailed analytical framework for particle transport in polygonal billiards.
The findings offer insights into anomalous transport and the mathematical underpinnings of complex dynamical systems.