Statistical characterization of discrete conservative systems: The web map

TL;DR

This study investigates the statistical behavior of the two-dimensional web map, revealing how ergodic and non-ergodic regimes correspond to Boltzmann-Gibbs and q-statistics, respectively, with detailed analysis of distributions and fractal structures.

Contribution

It provides a numerical characterization of the web map's statistical properties across different ergodic regimes, highlighting the transition from Gaussian to q-Gaussian distributions.

Findings

For small and large K, distributions are q-Gaussian and Gaussian, respectively.

Intermediate K values show non-Gaussian distributions linked to fractal structures.

Long-term distributions are characterized by kurtosis and fractal dimension analysis.

Abstract

We numerically study the two-dimensional, area preserving, web map. When the map is governed by ergodic behavior, it is, as expected, correctly described by Boltzmann-Gibbs statistics, based on the additive entropic functional $S_{BG}[p(x)] = -k\int dx\,p(x) \ln p(x)$. In contrast, possible ergodicity breakdown and transitory sticky dynamical behavior drag the map into the realm of generalized $q$-statistics, based on the nonadditive entropic functional $S_q[p(x)]=k\frac{1-\int dx\,[p(x)]^q}{q-1}$ ($q \in {\cal R}; S_1=S_{BG}$). We statistically describe the system (probability distribution of the sum of successive iterates, sensitivity to the initial condition, and entropy production per unit time) for typical values of the parameter that controls the ergodicity of the map. For small (large) values of the external parameter $K$, we observe $q$-Gaussian distributions with $q=1.935\dots$ (Gaussian distributions), like for the standard map. In contrast, for intermediate values of $K$, we observe a different scenario, due to the fractal structure of the trajectories embedded in the chaotic sea. Long-standing non-Gaussian distributions are characterized in terms of the kurtosis and the box-counting dimension of chaotic sea.