Nonextensive statistics: theoretical, experimental and computational evidences and connections

Abstract

The domain of validity of standard thermodynamics and Boltzmann-Gibbs statistical mechanics is discussed and then formally enlarged in order to hopefully cove r a v ariety o f anomalous systems. The generalization concerns nonextensive systems, where nonextensivity is understood in the thermodynamical sense. This generalization was rst proposed in 1988 inspired by the probabilistic description of multifractal geometries, and has been intensively studied during this decade. In the present e ort, after introducing some historical background, we brie y describe the formalism, and then exhibit the present status in what concerns theoretical, experimental and computational evidences and connections, as well as some perspectives for the future. In addition to these, here and there we point out various possibly relevant questions, whose answer would certainly clarify our current understanding of the foundations of statistical mechanics and its thermodynamical implications.

On a Nonlinear Master Equation and the Haken-Kelso-Bunz Model

“…with [a] + = a for a ≥ 0 and [a] + = 0 for a < 0 [18,46,48]. β is determined by the condition N i=1 p i,st = 1.…”

“…4.11], and Landsberg [33]. With the advent of an one-parametric nonextensive entropy proposed by Tsallis [46], generalized thermodynamical concepts have been developed aiming at descriptions of nonextensive systems characterized by equilibrium statistics different from the extensive Boltzmann-Gibbs-Shannon statistics [1,3,4,6,34,35,37,48]. In this context, transient stochastic processes that converge to generalized equilibrium distributions have been described in terms of Fokker-Planck equations which are nonlinear with respect to their probability densities [7,8,10,13,15,16,18,20,36,38,49].…”

A unified regularization theory: the maximum non-extensive entropy principle

“…A non-extensive formulation for the entropy has been proposed by Tsallis [8,9]. Recently, the non-extensive entropic form (S q ) was used as a new regularization operator [10], using only q = 0.5.…”

“…A UNIFIED REGULARIZATION THEORY where, in the limit N p → ∞, S q diverges if q ≤ 1, and saturates at 1/(q − 1) if q > 1 [9]. The equiprobability condition leads the regularization function defined by the operator S q (p), given by Eq.…”

“…Figure 1 shows the functional form for the Tsallis' entropy for several values of q. For q < 5/3, the standard central limit theorem applies, implying that if p i is written as a sum of M random independent variables, in the limit case M → ∞, the probability density function for p i in the distribution space is the normal (Gaussian) distribution [9]. However, for 5/3 < q < 3 the Levy-Gnedenko's central limit theorem applies, resulting for M → ∞ the Levy distribution as the probability density function for the random variable p i .…”