Legendre structure of κ-thermostatistics revisited in the framework of information geometry

Abstract

Information geometry is a powerful framework in which to study families of probability distributions or statistical models by applying differential geometric tools. It provides a useful framework for deriving many important structures in probability theory by identifying the space of probability distributions with a differentiable manifold endowed with a Riemannian metric. In this paper, we revisit some aspects concerning the κ-thermostatistics based on the entropy Sκ in the framework of information geometry. After introducing the dually flat structure associated with the κ-distribution, we show that the dual potentials derived in the formalism of information geometry correspond to the generalized Massieu function Φκ and the generalized entropy Sκ characterizing the Legendre structure of the κ-deformed statistical mechanics. In addition, we obtain several quantities, such as escort distributions and canonical divergence, relevant for the further development of the theory.