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Tsallis entropy induced metrics and CAT(k) spaces

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  • Kalogeropoulos, Nikos

Abstract

Generalizing the group structure of the Euclidean space, we construct a Riemannian metric on the deformed set Rqn 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.

Suggested Citation

  • Kalogeropoulos, Nikos, 2012. "Tsallis entropy induced metrics and CAT(k) spaces," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(12), pages 3435-3445.
    • Handle: RePEc:eee:phsmap:v:391:y:2012:i:12:p:3435-3445
    DOI: 10.1016/j.physa.2012.02.013
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    Cited by:

    1. Creaco, Anthony J. & Kalogeropoulos, Nikolaos, 2019. "Irreversibility from staircases in symplectic embeddings," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 513(C), pages 497-509.
    2. Balankin, Alexander S. & Bory-Reyes, Juan & Shapiro, Michael, 2016. "Towards a physics on fractals: Differential vector calculus in three-dimensional continuum with fractal metric," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 444(C), pages 345-359.

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