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arXiv · 2603.08896

A Dynamical Approach to Non-Extensive Thermodynamics

Artur O. Lopes·Paulo Varandas
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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.

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Artur O. Lopes, Paulo Varandas. 2026-03-09. A Dynamical Approach to Non-Extensive Thermodynamics. https://arxiv.org/abs/2603.08896

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