Nonextensive thermodynamic formalism for chaotic dynamical systems

TL;DR

This paper develops a nonextensive thermodynamic framework for chaotic systems using Tsallis distribution, deriving key structures and applying it to logistic map analysis.

Contribution

It introduces a nonextensive thermodynamic formalism for chaotic systems, explicitly deriving the Legendre transform structure and thermodynamic limit.

Findings

Derived relation between box size and nonextensivity parameter

Verified the formalism with logistic map data

Calculated bit variance within the new framework

Abstract

A nonextensive thermostatic approach to chaotic dynamical systems is developed by expressing generalized Tsallis distribution as escort distribution. We explicitly show the thermodynamic limit and also derive the Legendre Transform structure. As an application, bit variance is calculated for ergodic logistic map. Consistency of the formalism demands a relation between box size ($\epsilon$) and degree of nonextensivity, given as $(1-q)\sim -1/{\rm ln} \epsilon$. This relation is numerically verified for the case of bit variance as well as using basic definition of Tsallis entropy.