Statistical characterization of the standard map

TL;DR

This paper provides a detailed numerical analysis of the standard map, revealing how its statistical properties transition from Boltzmann-Gibbs to nonextensive $q$-statistics depending on the parameter $K$, with specific results on Lyapunov exponents and entropy growth.

Contribution

It offers a comprehensive numerical characterization of the standard map's statistical behaviors across different parameter regimes, highlighting the transition between BG and $q$-statistics.

Findings

Large $K$ yields positive Lyapunov exponents and BG statistics with $q$-indices equal to 1.

Small $K$ results in near-zero Lyapunov exponents and $q$-indices indicating nonextensive statistics.

Intermediate $K$ shows coexistence of stable orbits and chaos, with mixed statistical behaviors.

Abstract

The standard map, paradigmatic conservative system in the $(x,p)$ phase space, has been recently shown to exhibit interesting statistical behaviors directly related to the value of the standard map parameter $K$. A detailed numerical description is achieved in the present paper. More precisely, for large values of $K$, the Lyapunov exponents are neatly positive over virtually the entire phase space, and, consistently with Boltzmann-Gibbs (BG) statistics, we verify $q_{\text{ent}}=q_{\text{sen}}=q_{\text{stat}}=q_{\text{rel}}=1$, where $q_{\text{ent}}$ is the $q$-index for which the nonadditive entropy $S_q \equiv k \frac{1-\sum_{i=1}^W p_i^q}{q-1}$ (with $S_1=S_{BG} \equiv -k\sum_{i=1}^W p_i \ln p_i$) grows linearly with time before achieving its $W$-dependent saturation value; $q_{\text{sen}}$ characterizes the time increase of the sensitivity $\xi$ to the initial conditions, i.e., $\xi \sim e_{q_{\text{sen}}}^{\lambda_{q_{\text{sen}}} \,t}\;(\lambda_{q_{\text{sen}}}>0)$, where $e_q^z \equiv[1+(1-q)z]^{1/(1-q)}$; $q_{\text{stat}}$ is the index associated with the $q_{\text{stat}}$-Gaussian distribution of the time average of successive iterations of the $x$-coordinate; finally, $q_{\text{rel}}$ characterizes the $q_{\text{rel}}$-exponential relaxation with time of the entropy $S_{q_{\text{ent}}}$ towards its saturation value. In remarkable contrast, for small values of $K$, the Lyapunov exponents are virtually zero over the entire phase space, and, consistently with $q$-statistics, we verify $q_{\text{ent}}=q_{\text{sen}}=0$, $q_{\text{stat}} \simeq 1.935$, and $q_{\text{rel}} \simeq1.4$. The situation corresponding to intermediate values of $K$, where both stable orbits and a chaotic sea are present, is discussed as well. The present results transparently illustrate when BG or $q$-statistical behavior are observed.