On the fractal dimension of orbits compatible with Tsallis statistics

Abstract

In a previous paper (1) it was shown how, for a dynamical system, the probability distribution function of the sojourn-times in phase-space, defined in terms of the dynamical orbits (up to a given observation time), induces unambiguously a statistical ensemble in phase-space. In the present paper it is shown which is the p.d.f. of the sojourn-times corresponding to a Tsallis ensemble (this, by the way, requires the solution of a problem of a general character, disregarded in paper (1)). In particular some qualitative properties, such as the fractal dimension, of the dynamical orbits compatible with the Tsallis ensembles are indicated.

Compatibility between dynamics and Tsallis statistics

“…It turns out that F T s (n) doesn't have a closed expression in terms of known functions, except for some special cases, but expression for F T s (n) can however be given as a series expansion (details can be find in ref. [13], see also [14]).…”

Maximum or minimum entropy generation for open systems?

“…In order to establish a correspondence between dynamics and Tsallis distribution one has to solve an analytical problem by an exact correspondence between dynamics and Tsallis distribution, given only for continuous-time dynamical systems and not for mappings. In the case of a mapping, such a correspondence is obtained by introducing a suitable limiting procedure (Carati 2008). In 1988 Tsallis postulated the physical relevance of one-parameter generalization of the entropy (Tsallis 1988), introducing a generalization of standard thermodynamics and of Boltzmann-Gibbs statistical mechanics.…”

Introduction to Nonextensive Statistical Mechanics

“…However, a clear-cut physical interpretation in terms of the dynamics and occupancy geometry within the full phase-space ⌫ is still lacking. Some important hints can be found in [55][56][57].…”

“…Let us anticipate that it has been recently shown[55][56][57][58] that, if we impose a Poissonian distribution for visitation times in phase-space, in addition to the first and second principles of thermodynamics, we obtain the BG functional form for the entropy. If a conveniently deformed Poissonian distribution is imposed instead, we obtain the S q functional form.…”