A Dynamical Approach to Non-Extensive Thermodynamics

TL;DR

This paper extends thermodynamic formalism to non-extensive systems using a $q$-generalization inspired by Tsallis entropy, establishing new relations and properties for $q$-pressure and $q$-equilibrium states.

Contribution

It introduces a $q$-thermodynamic formalism, linking it to classical concepts, and proves existence, uniqueness, and differentiability results for $q$-equilibrium states and related operators.

Findings

Established the existence and uniqueness of $q$-equilibrium states.

Proved the differentiability of the $q$-pressure.

Linked $q$-pressure with $(2-q)$-transfer operators and classical equilibrium states.

Abstract

We develop a non-extensive thermodynamic formalism for the one-sided shift on a finite alphabet, inspired by Tsallis' generalization of Boltzmann entropy in statistical physics. We introduce notions of $q$-entropy, $q$-pressure, and $q$-transfer operators which extend the classical thermodynamic formalism when $q=1$. We prove a Bowen-type relation linking the $q$-pressure with a $(2-q)$-Ruelle transfer operator and show that $q$-equilibrium states correspond to classical equilibrium states for a related potential. We establish the existence and uniqueness of $q$-equilibrium states for Lipschitz potentials, prove the differentiability of the $q$-pressure, and obtain variational principles for both the $q$-pressure and a related asymptotic pressure. Finally, we study cohomological equations associated with $(2-q)$-transfer operators and prove the differentiable dependence of their solutions on the potential, yielding an alternative construction of eigenfunctions for classical Ruelle operators.