Geometric Structures Induced by Deformations of the Legendre Transform

Abstract

The recent link discovered between generalized Legendre transforms and non-dually flat statistical manifolds suggests a fundamental reason behind the ubiquity of Rényi's divergence and entropy in a wide range of physical phenomena. However, these early findings still provide little intuition on the nature of this relationship and its implications for physical systems. Here we shed new light on the Legendre transform by revealing the consequences of its deformation via symplectic geometry and complexification. These findings reveal a novel common framework that leads to a principled and unified understanding of physical systems that are not well-described by classic information-theoretic quantities.

14 pages, 1 figure; version accepted for publication in Entropy

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• Berry Curvature, Triangle Anomalies, and the Chiral Magnetic Effect in Fermi Liquids
• The world according to Renyi: Thermodynamics of multifractal systems
• A generalization of the maximum entropy principle for curved statistical manifolds
• Stochastic deformation of a thermodynamic symplectic structure
• Thermodynamics of exponential Kolmogorov-Nagumo averages
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Cited by in corpus (1)

• Thermodynamics of exponential Kolmogorov-Nagumo averages