Time--Evolving Statistics of Chaotic Orbits of Conservative Maps in the Context of the Central Limit Theorem

TL;DR

This paper investigates how the probability distributions of sums of chaotic orbit iterates in conservative maps evolve over time, revealing transitions from q-Gaussians to Gaussians as orbits explore larger chaotic regions.

Contribution

It provides numerical evidence of time-evolving statistical behaviors in chaotic conservative maps, connecting nonextensive statistics with classical ergodic theory.

Findings

QSS with q-Gaussian distributions in thin chaotic layers

Transition from q-Gaussian to Gaussian distributions as N increases

Orbits diffuse to larger chaotic domains, increasing ergodicity

Abstract

We study chaotic orbits of conservative low--dimensional maps and present numerical results showing that the probability density functions (pdfs) of the sum of $N$ iterates in the large $N$ limit exhibit very interesting time-evolving statistics. In some cases where the chaotic layers are thin and the (positive) maximal Lyapunov exponent is small, long--lasting quasi--stationary states (QSS) are found, whose pdfs appear to converge to $q$--Gaussians associated with nonextensive statistical mechanics. More generally, however, as $N$ increases, the pdfs describe a sequence of QSS that pass from a $q$--Gaussian to an exponential shape and ultimately tend to a true Gaussian, as orbits diffuse to larger chaotic domains and the phase space dynamics becomes more uniformly ergodic.