Nonextensive statistical mechanics, anomalous diffusion and central limit theorems

TL;DR

This paper reviews nonextensive statistical mechanics, explores its connections with anomalous diffusion and central limit theorems, and proposes a new conjectural theorem generalizing independence assumptions.

Contribution

It introduces a conjectural central limit theorem with a generalized independence hypothesis, advancing the theoretical foundation of nonextensive statistical mechanics.

Findings

Connections between nonextensive mechanics and Fokker-Planck equations

Proposed a new central limit theorem conjecture

Generalized concepts of Lyapunov exponents and entropy production

Abstract

We briefly review Boltzmann-Gibbs and nonextensive statistical mechanics as well as their connections with Fokker-Planck equations and with existing central limit theorems. We then provide some hints that might pave the road to the proof of a new central limit theorem, which would play a fundamental role in the foundations and ubiquity of nonextensive statistical mechanics. The basic novelty introduced within this conjectural theorem is the {\it generalization of the hypothesis of independence} of the $N$ random variables being summed. In addition to this, we also advance some nonlinear dynamical (possibly exact) relations which generalize the concepts of Lyapunov exponents, entropy production per unit time, and their interconnection as first proved by Pesin for chaotic systems.