Tsallis Ensemble as an Exact Orthode

TL;DR

This paper demonstrates that the Tsallis ensemble functions as an exact orthode, linking nonextensive thermodynamics with a mechanical model, and identifies the associated entropy with Renyi entropy, providing a consistent theoretical framework.

Contribution

It establishes that Tsallis ensemble is an exact orthode and connects it with Renyi entropy, offering a mechanical foundation for nonextensive thermodynamics.

Findings

Tsallis ensemble is an exact orthode for small Hamiltonian systems.

The entropy associated with this ensemble is Renyi entropy, not Tsallis entropy.

The approach aligns with information-theoretic methods based on Renyi entropy.

Abstract

We show that Tsallis ensemble of power-law distributions provides a mechanical model of nonextensive equilibrium thermodynamics for small interacting Hamiltonian systems, i.e., using Boltzmann's original nomenclature, we prove that it is an exact orthode. This means that the heat differential admits the inverse average kinetic energy as an integrating factor. One immediate consequence is that the logarithm of the normalization function can be identified with the entropy, instead of the q-deformed logarithm. It has been noted that such entropy coincides with Renyi entropy rather than Tsallis entropy, it is non-additive, tends to the standard canonical entropy as the power index tends to infinity and is consistent with the free energy formula proposed in [S. Abe et. al. Phys. Lett. A 281, 126 (2001)]. It is also shown that the heat differential admits the Lagrange multiplier used in non-extensive thermodynamics as an integrating factor too, and that the associated entropy is given by ordinary nonextensive entropy. The mechanical approach proposed in this work is fully consistent with an information-theoretic approach based on the maximization of Renyi entropy.