Higher order statistics in the annulus square billiard: transport and polyspectra

TL;DR

This paper investigates anomalous diffusion in the annulus square billiard, revealing how higher order spectral measures relate to transport properties and providing both analytical estimates and numerical validation.

Contribution

It introduces a novel analysis connecting higher order polyspectra to transport in polygonal billiards, with new analytic estimates for moments growth.

Findings

Anomalous diffusion with algebraic growth of moments

Spectral properties determine transport behavior

Higher order spectra relate to moments via generalized Wiener-Khinchin

Abstract

Classical transport in a doubly connected polygonal billiard, i.e. the annulus square billiard, is considered. Dynamical properties of the billiard flow with a fixed initial direction are analyzed by means of the moments of arbitrary order of the number of revolutions around the inner square, accumulated by the particles during the evolution. An "anomalous" diffusion is found: the moment of order q exhibits an algebraic growth in time with an exponent different from q/2, like in the normal case. Transport features are related to spectral properties of the system, which are reconstructed by Fourier transforming time correlation functions. An analytic estimate for the growth exponent of integer order moments is derived as a function of the scaling index at zero frequency of the spectral measure, associated to the angle spanned by the particles. The n-th order moment is expressed in terms of a multiple-time correlation function, depending on n-1 time intervals, which is shown to be linked to higher order density spectra (polyspectra), by a generalization of the Wiener-Khincin Theorem. Analytic results are confirmed by numerical simulations.