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Universality classes for the Fisher metric derived from relative group entropy

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  • Gomez, Ignacio S.
  • Portesi, Mariela
  • Borges, Ernesto P.

Abstract

We consider the Fisher metric which results from the Hessian of the relative group entropy, that we call group Fisher metric. In particular, the metrics corresponding to the Boltzmann–Gibbs, Tsallis, Kaniadakis and Abe universality classes are obtained. We prove that the scalar curvature derived from the group Fisher metric results in a multiple of the Boltzmann–Gibbs one, with the factor of proportionality given by the local properties of the group entropy. We analyze, for the Tsallis universality class, the 2D correlated model that presents a softening and strengthening of the scalar curvature, and we illustrate with the canonical ensemble of a pair of interacting harmonic oscillators as well as a quartic harmonic oscillator.

Suggested Citation

  • Gomez, Ignacio S. & Portesi, Mariela & Borges, Ernesto P., 2020. "Universality classes for the Fisher metric derived from relative group entropy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 547(C).
    • Handle: RePEc:eee:phsmap:v:547:y:2020:i:c:s0378437119321284
    DOI: 10.1016/j.physa.2019.123827
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    References listed on IDEAS

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    1. Portesi, M. & Plastino, A. & Pennini, F., 2006. "Information measures based on Tsallis’ entropy and geometric considerations for thermodynamic systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 365(1), pages 173-176.
    2. Ali, S.A. & Cafaro, C. & Kim, D.-H. & Mancini, S., 2010. "The effect of microscopic correlations on the information geometric complexity of Gaussian statistical models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(16), pages 3117-3127.
    3. Kim, D.-H. & Ali, S.A. & Cafaro, C. & Mancini, S., 2012. "Information geometry of quantum entangled Gaussian wave-packets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(19), pages 4517-4556.
    4. Gomez, Ignacio S., 2017. "Notions of the ergodic hierarchy for curved statistical manifolds," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 484(C), pages 117-131.
    5. Borges, Ernesto P., 2004. "A possible deformed algebra and calculus inspired in nonextensive thermostatistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 340(1), pages 95-101.
    6. Portesi, M. & Pennini, F. & Plastino, A., 2007. "Geometrical aspects of a generalized statistical mechanics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 373(C), pages 273-282.
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    Cited by:

    1. Cristina-Liliana Pripoae & Iulia-Elena Hirica & Gabriel-Teodor Pripoae & Vasile Preda, 2022. "Fisher-like Metrics Associated with ϕ -Deformed (Naudts) Entropies," Mathematics, MDPI, vol. 10(22), pages 1-26, November.

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