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Information Geometry of κ-Exponential Families: Dually-Flat, Hessian and Legendre Structures
Antonio M Scarfone1, Hiroshi Matsuzoe2, Tatsuaki Wada3
1Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche (ISC-CNR), c/o Politecnico di Torino, 10129 Torino, Italy.
Entropy (Basel, Switzerland)
|December 3, 2020
Summary
This study reviews kappa-deformed statistical mechanics using information geometry. It introduces three geometric structures based on deformed divergences, revealing insights into kappa-thermodynamics and information geometry.
Area of Science:
- Statistical Mechanics
- Information Geometry
- Thermodynamics
Background:
- The study of statistical mechanics has been extended by incorporating concepts from information geometry.
- Deformed statistical mechanics, particularly kappa-deformed systems, offers alternative frameworks for analyzing complex systems.
- Information geometry provides a powerful lens for understanding the geometric properties of statistical models.
Purpose of the Study:
- To review recent advancements in kappa-deformed statistical mechanics within the information geometry framework.
- To introduce and analyze novel geometric structures derived from kappa-deformed divergences.
- To explore the implications of these geometric structures for kappa-thermodynamics.
Main Methods:
- Development of three distinct geometric structures using kappa-deformed Kullback-Leibler, Kerridge, and Bregman divergences.
- Analysis of the curvature properties of the statistical manifold derived from kappa-Kullback-Leibler divergence.
- Investigation of the dualistic Hessian structure, deformed Fisher metric, and affine connection for the other two manifolds.
Main Results:
- The kappa-Kullback-Leibler divergence yields an invariant geometry with positive curvature, vanishing as kappa approaches 0.
- Two other statistical manifolds, related by scaling, are dually-flat with a deformed Fisher metric and affine connection.
- These geometries support a Legendre structure for kappa-thermodynamics, linked to Massieu and entropy functions.
Conclusions:
- The integration of kappa-deformed statistical mechanics and information geometry reveals rich geometric structures.
- These structures provide a deeper understanding of generalized statistical properties and thermodynamic relations.
- The findings offer a new perspective on the interplay between deformation parameters and geometric properties.
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