Numerical Analysis of Conservative Maps: A Possible Foundation of Nonextensive Phenomena

We discuss the sensitivity to initial conditions and the entropy production of low-dimensional conservative maps, focusing on situations where the phase space presents complex (fractal-like) structures. We analyze numerically the standard map as a specific example and we observe a scenario that presents appealing analogies with anomalies detected in long-range Hamiltonian systems. We see how the Tsallis nonextensive formalism handles this situation both from a dynamical and from a statistical mechanics point of view…. In recent years, the Tsallis extension of the Boltzmann-Gibbs (BG) statistical mechanics [9, 26], usually referred to as nonextensive (NE) statistical mechanics, has become an intense and exciting research area (see, e.g., Tsallis [25]). The q-exponential distribution functions that emerge as a consequence of the NE formalism have been applied to an impressive variety of problems, ranging from turbulence, to high-energy physics, epilepsy, protein folding, and financial analysis. Yet, the foundation of this formalism, as well as the definition of its area of applicability, is still not completely understood, and it stands as a present challenge in the affirmation of the whole proposal. An intensive effort is currently being made to investigate this point, precisely in trying to understand: (1) which mechanisms lead to a crisis of the BG formalism; and (2) in these cases, does the NE formalism provide a "way out" to some of the problems? A possible approach to these questions comes from the study of the underlying dynamics that gives the basis for a statistical mechanic treatment of the system. This idea is not new. Einstein, in his critical remark about the validity of the Boltzmann principle [10], was one of the first to call attention to the relevance of a dynamical foundation of statistical mechanics. Another fundamental contribution is Krylov's seminal work [14] on the mixing properties of dynamical systems. In one-dimensional (dissipative) systems, intensive effort has been made to analyze the properties of the systems at the edge of chaos, i.e., at the critical poin that marks the transition between chaoticity and regularity [6, 8, 16, 19, 18, 23, 27].