Nonextensive statistics: Theoretical, experimental and computational evidences and connections

TL;DR

The article argues that Boltzmann-Gibbs statistics is not universal and introduces nonextensive thermostatistics based on the entropy S_q. It surveys the formalism (generalized entropy and canonical ensemble) and a wide range of theoretical, experimental, and computational evidences showing that a single parameter q captures diverse anomalous behaviors such as Levy diffusion, Zipf-Mandelbrot scaling, and non-Maxwellian distributions. The review covers applications across physics, astronomy, biology, economics, and beyond, highlighting both successful fittings and open questions. It also discusses long-range interactions, self-organized criticality, and optimization, suggesting that nonextensive statistics provides a unifying framework for complex systems and identifying key directions for future research.

Abstract

The domain of validity of standard thermodynamics and Boltzmann-Gibbs statistical mechanics is discussed and then formally enlarged in order to hopefully cover a variety of anomalous systems. The generalization concerns {\it nonextensive} systems, where nonextensivity is understood in the thermodynamical sense. This generalization was first proposed in 1988 inspired by the probabilistic description of multifractal geometries, and has been intensively studied during this decade. In the present effort, after introducing some historical background, we briefly 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