A boundary integral formalism for stochastic ray tracing in billiards

TL;DR

This paper introduces a boundary integral method for modeling uncertain ray trajectories in billiards, bridging deterministic and random flow descriptions, with applications in acoustics, seismology, and quantum mechanics.

Contribution

It presents a novel boundary integral framework for propagating uncertain flows, providing a systematic interpolation between deterministic and stochastic dynamics.

Findings

Efficient discretisation of uncertain billiard dynamics in rectangular domains.

Framework effectively models uncertainty in ray tracing.

Applicable to various physical wave and particle transport problems.

Abstract

Determining the flow of rays or particles driven by a force or velocity field is fundamental to modelling many physical processes, including weather forecasting and the simulation of molecular dynamics. High frequency wave energy distributions can also be approximated using flow or transport equations. Applications arise in underwater and room acoustics, vibro-acoustics, seismology, electromagnetics, quantum mechanics and in producing computer generated imagery. In many practical applications, the driving field is not known exactly and the dynamics are determined only up to a degree of uncertainty. This paper presents a boundary integral framework for propagating flows including uncertainties, which is shown to systematically interpolate between a deterministic and a completely random description of the trajectory propagation. A simple but efficient discretisation approach is applied to model uncertain billiard dynamics in an integrable rectangular domain.