Some remarks on R\'{e}nyi relative entropy in a thermostatistical framework

TL;DR

This paper investigates the physical meaning of Rényi relative entropy within thermostatistics, concluding it aligns with traditional free energy differences only as the parameter approaches 1, indicating its limited applicability for non-Boltzmann-Gibbs systems.

Contribution

It demonstrates that Rényi relative entropy has a meaningful physical interpretation only in the limit as q approaches 1, linking it to classical thermostatistical concepts.

Findings

Rényi relative entropy aligns with free energy differences only as q approaches 1.

Rényi entropy is an equilibrium entropy only in the Boltzmann-Gibbs limit.

Rényi relative entropy does not satisfy Shore-Johnson axioms for stationary distributions.

Abstract

In ordinary Boltzmann-Gibbs thermostatistics, the relative entropy expression plays the role of generalized free energy, providing the difference between the off-equilibrium and equilibrium free energy terms associated with Boltzmann-Gibbs entropy. In this context, we studied whether this physical meaning can be given to R\'{e}nyi relative entropy definition found in the literature from a generalized thermostatistical point of view. We find that this is possible only in the limit as $q$ approaches to 1. This shows that R\'{e}nyi relative entropy has a physical (thermostatistical) meaning only when the system can already be explained by ordinary Boltzmann-Gibbs thermostatistics. Moreover, this can be taken as an indication of R\'{e}nyi entropy being an equilibrium entropy since any relative entropy definition is a two-probability generalization of the associated entropy definition. We also note that this result is independent of the internal energy constraint employed. Finally, we comment on the lack of foundation of R\'{e}nyi relative entropy as far as its minimization (which is equivalent to the maximization of R\'{e}nyi entropy) is considered in order to obtain a stationary equilibrium distribution since R\'{e}nyi relative entropy does not conform to Shore-Johnson axioms.