Equivalence of two thermostatistical formalisms based on the Havrda&Charvat-Daroczy-Tsallis entropies

TL;DR

This paper proves the equivalence between two thermostatistical formalisms based on Havrda-Charvat-Daroczy-Tsallis entropies, showing they predict identical equilibrium states when using reciprocal q-values, but also highlights the failure of transitivity of equilibrium in these models.

Contribution

It demonstrates the formal equivalence of two Tsallis-based entropy formalisms and discusses their thermodynamic properties.

Findings

Equivalence of the two formalisms when using reciprocal q-values.

Both formalisms predict the same expectation values for all observables.

The transitivity of equilibrium property fails in these formalisms.

Abstract

We show that the latest thermostatistical formalism based on the Havrda & Charvat-Dar\'oczy-Tsallis entropy $S_q [ \rho ] = (q-1)^{-1}(1- tr (\rho^q))$ proposed by Tsallis, Mendes and Plastino is {\em equivalent} to the first one proposed by Tsallis in 1988. Here, equivalent means: {\em the ``equilibrium'' state predicted by either formalism using $q$ leads to the same expectation values for all observables as that predicted by the other formalism using $1/q$''}. We also point out once again that the basic property of {\em transitivity of equilibrium} (e.g., the $0^{th}$ Law of Thermodynamics) fails in these formalisms.