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