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Area of Science:

  • Information geometry
  • Statistical mechanics
  • Theoretical physics

Background:

  • A link between generalized Legendre transforms and non-dually flat statistical manifolds explains the prevalence of Rényi's divergence and entropy in physics.
  • Existing research offers limited insight into the nature and physical implications of this relationship.

Purpose of the Study:

  • To explore the deformation of Legendre transforms using symplectic geometry and complexification.
  • To develop a deeper, intuitive understanding of the connection between Legendre transforms and physical systems.
  • To establish a unified framework for analyzing physical systems not adequately described by classical information-theoretic measures.

Main Methods:

  • Deformation of Legendre transforms.
  • Application of symplectic geometry.
  • Complexification techniques.

Main Results:

  • Revealed consequences of deforming Legendre transforms.
  • Established a novel common framework for understanding physical systems.
  • Provided a principled and unified approach to systems beyond classic information theory.

Conclusions:

  • The deformation of Legendre transforms offers new insights into their relationship with statistical manifolds.
  • This work unifies the understanding of diverse physical systems using a novel information-theoretic framework.
  • The findings pave the way for analyzing complex physical phenomena not captured by traditional information-theoretic quantities.