Abstract
In this paper, we present a review of recent developments on the -deformed statistical mechanics in the framework of the information geometry. Three different geometric structures are introduced in the -formalism which are obtained starting from three, not equivalent, divergence functions, corresponding to the -deformed version of Kullback–Leibler, “Kerridge” and Brègman divergences. The first statistical manifold derived from the -Kullback–Leibler divergence form an invariant geometry with a positive curvature that vanishes in the limit. The other two statistical manifolds are related to each other by means of a scaling transform and are both dually-flat. They have a dualistic Hessian structure endowed by a deformed Fisher metric and an affine connection that are consistent with a statistical scalar product based on the -escort expectation. These flat geometries admit dual potentials corresponding to the thermodynamic Massieu and entropy functions that induce a Legendre structure of -thermodynamics in the picture of the information geometry.
Highlights
In the study of complex systems, long tailed probability distributions are often discussed.Anomalous statistical behavior is largely observed in physical systems plagued by long-range interactions or long-time memory effects as well as in non-physical systems, such as biological, social and economic ones
In this paper, inspired by a recent work of Zhang and Naudts [66,67], we study the κ-distribution and its associated statistical manifold in the framework of information geometry, revisiting some results already known [68,69,70] and deriving new geometrical structures obtained from three different, not equivalent, divergence functions corresponding to the κ-deformed version of Brègman, “Kerridge”
The three statistical manifolds derived from different version of divergence function are studied in Section 4 and in particular, in Section 4.1 we introduce the statistical manifold derived from the κ-Kullback–Leibler divergence that is an invariant geometry with a positive curvature, while in Sections 4.2 and 4.3 we introduced two dually-flat geometries with their Hessian structures obtained, respectively, from the κ-Kerridge and κ-Brègman divergences
Summary
In the study of complex systems, long tailed probability distributions are often discussed. In this paper, inspired by a recent work of Zhang and Naudts [66,67], we study the κ-distribution and its associated statistical manifold in the framework of information geometry, revisiting some results already known [68,69,70] and deriving new geometrical structures obtained from three different, not equivalent, divergence functions corresponding to the κ-deformed version of Brègman, “Kerridge”. In the framework of κ-deformed formalism escort distributions has been previously introduced in [68] to investigate a dually flat geometry in the κ-distribution space and than a double escort distribution has been defined in [70] to study a second dually flat geometry, which is based on the escort expectation instead of the standard expectation In both these geometric structures a kind of fluctuation-response relation, that could be relevant in the framework of the non-equilibrium κ-deformed statistical mechanics, has been deduced by using, respectively, escort and double escort expectations. The κ-Brègman divergence (18) has been introduced and studied in [37]
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