The standard map: From Boltzmann-Gibbs statistics to Tsallis statistics

TL;DR

This paper demonstrates how the standard map transitions from Boltzmann-Gibbs to Tsallis statistics, clarifying the conditions under which each statistical framework applies to conservative dynamical systems.

Contribution

It provides a clear illustration of the transition between Boltzmann-Gibbs and Tsallis statistics in the standard map, highlighting their respective domains of validity.

Findings

Standard map exhibits a transition from Boltzmann-Gibbs to Tsallis statistics.

Domains of validity for both statistical approaches are clearly identified.

Results demonstrate the applicability of Tsallis statistics in non-ergodic, strongly correlated regimes.

Abstract

As well known, Boltzmann-Gibbs statistics is the correct way of thermostatistically approaching ergodic systems. On the other hand, nontrivial ergodicity breakdown and strong correlations typically drag the system into out-of-equilibrium states where Boltzmann-Gibbs statistics fails. For a wide class of such systems, it has been shown in recent years that the correct approach is to use Tsallis statistics instead. Here we show how the dynamics of the paradigmatic conservative (area-preserving) standard map exhibits, in an exceptionally clear manner, the crossing from one statistics to the other. Our results unambiguously illustrate the domains of validity of both Boltzmann-Gibbs and Tsallis statistics.