Some remarks on Rényi relative entropy in a thermostatistical framework

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 Rényi 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 Rényi 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 Rényi 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 Rényi relative entropy as far as its minimization (which is equivalent to the maximization of Rényi entropy) is considered in order to obtain a stationary equilibrium distribution since Rényi relative entropy does not conform to Shore-Johnson axioms.