Infinite Invariant Density Determines Statistics of Time Averages for Weak Chaos

TL;DR

This paper investigates how infinite invariant densities influence the statistical behavior of time averages in weakly chaotic systems, revealing new distributional properties for non-integrable observables and establishing key identities.

Contribution

It introduces a novel connection between infinite invariant densities and the distribution of time averages for non-integrable observables in weak chaos.

Findings

Distribution of time averages linked to infinite invariant density

Identified identities between amplitude ratios

Extended understanding of weak chaos statistics

Abstract

Weakly chaotic non-linear maps with marginal fixed points have an infinite invariant measure. Time averages of integrable and non-integrable observables remain random even in the long time limit. Temporal averages of integrable observables are described by the Aaronson-Darling-Kac theorem. We find the distribution of time averages of non-integrable observables, for example the time average position of the particle. We show how this distribution is related to the infinite invariant density. We establish four identities between amplitude ratios controlling the statistics of the problem.