Diffusion rate of windtree models and Lyapunov exponents

TL;DR

This paper links the diffusion rate in windtree models with Lyapunov exponents of quadratic differentials, providing a new method to compute and analyze the asymptotic behavior of diffusion in these systems.

Contribution

It introduces a novel strategy to compute the generic diffusion rate of windtree models using Lyapunov exponents, connecting dynamical systems and geometric structures.

Findings

Diffusion rate equals the largest Lyapunov exponent of certain quadratic differential strata.

Developed a numerical approach to estimate diffusion rates for various obstacle shapes.

Observed asymptotic behavior of diffusion rates based on obstacle geometry.

Abstract

Consider a windtree model with several parallel arbitrary right-angled obstacles placed periodically on the plane. We show that its diffusion rate is the largest Lyapunov exponent of some stratum of quadratic differentials and exhibit a new general strategy to compute the generic diffusion rate of such models. This result enables us to compute numerically the diffusion rates of a large family of models, and to observe its asymptotic behaviour according to the shape of the obstacles.