Tsallis entropy induced metrics and CAT(k) spaces

Abstract

Generalizing the group structure of the Euclidean space, we construct a Riemannian metric on the deformed set \ ${\mathbb{R}}_q^n$ \ induced by the Tsallis entropy composition property. We show that the Tsallis entropy is a "hyperbolic analogue" of the "Euclidean" Boltzmann/Gibbs/Shannon entropy and find a geometric interpretation for the nonextensive parameter $q$. We provide a geometric explanation of the uniqueness of the Tsallis entropy as reflected through its composition property, which is provided by the Abe and the Santos axioms. For two, or more, interacting systems described by the Tsallis entropy, having different values of $q$, we argue why a suitable extension of this construction is provided by the Cartan/Alexandrov/Toponogov metric spaces with a uniform negative curvature upper bound.

23 pages, Standard LaTeX2e, Accepted for publication in Physica A

References in corpus (7)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• A possible deformed algebra and calculus inspired in nonextensive thermostatistics
• Phase transitions and configuration space topology
• Thermodynamic curvature measures interactions
• Ensemble equivalence for general many-body systems
• Distributivity and deformation of the reals from Tsallis entropy
• Recent developments in superstatistics

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• Weak Chaos from Tsallis Entropy
• Irreversibility from staircases in symplectic embeddings
• Moduli of curve families and (quasi-)conformality of power-law entropies
• Nilpotence in Physics: the case of Tsallis entropy
• Tsallis entropy composition and the Heisenberg group
• The ${\tau }_q$-Fourier transform: covariance and uniqueness
• Non-linear Fokker-Planck equations from conformal metrics and scalar curvature