Nonextensive Entropies and Sensitivity to Initial Conditions of Complex Systems

Abstract

Tsallis generalized statistics has been successfully applied to describe some relevant features of several natural systems exhibiting a nonextensive character. It is based on an extended form for the entropy, namely S<sub>q</sub> = (1 — Σ<sub>q</sub>p<sup>Q</sup><sub>q</sub>)/(Q —1), where q is a parameter that measures the degre of nonextensivity (q→1 for the traditional Boltzmann-Gibbs statistics). A series of recent works have shown that the power-law sensitivity to initial conditions in a complex state provides a natural link between the g-entropic parameter and the scaling properties of dynamical attractors. These results contribute to the growing set of theoretical and experimental evidences that Tsallis statistics can be a natural frame for studying systems with a fractal-like structure in phase space. Here, the main ideas underlying this relevant aspect are reviewed. The starting point is the weak sensitivity to initial conditions exhibited by low-dimensional dynamical systems at the onset of chaos. It is shown how general scaling arguments can provide a direct relation between the entropic index q and the scaling exponents associated with the multifractal critical attractor. These works shed light in the elusive problem concerning the connection between the g-entropic parameter of Tsallis statistics and the underlying microscopic dynamics of nonextensive systems.... Inspired on the scaling properties of multifractals, Tsallis introduced a generalized entropy with the aim of extending the usual statistical mechanics and thermodynamics [37].