Fluctuation Statistics and Diffusive Properties of the 2D Triangular Lorentz Gas in the Finite-Horizon Regime

TL;DR

This paper investigates the chaotic dynamics and diffusion properties of the 2D triangular Lorentz gas in the finite-horizon regime, revealing universal spectral behavior and confirming normal diffusion with a linear variance scaling.

Contribution

It introduces a spectral analysis of trajectory lengths and demonstrates universal spectral properties, along with confirming normal diffusion in the finite-horizon Lorentz gas.

Findings

Universal behavior in level spacing distribution and spectral rigidity.

Variance of trajectory length scales linearly with number of collisions.

Normal diffusion persists with a central limit theorem in the finite-horizon regime.

Abstract

We study chaotic behavior and diffusion in the 2D periodic Lorentz gas in the finite-horizon regime. The dynamical observable which we consider is the length of single particle's trajectories, which moves in a triangular array of rigid disks. To analyze the chaoticity of this system, we construct a matrix of the length of trajectories and perform a level spacing analysis of the spectrum of this matrix. We find that a universal behavior occurs both in level spacing distribution and spectral rigidity. In order to examine diffusion in this classical system, we investigate the variance of the length of trajectories versus number of collisions to disks. In the case where there is a finite-horizon, such a variance scales linearly with number of bounces. This shows that a normal diffusion exists and a central limit theorem is maintained in this regime.