Complexity Characterization in a Probabilistic Approach to Dynamical Systems Through Information Geometry and Inductive Inference

TL;DR

This paper explores how information geometry and inductive inference can characterize the complexity of dynamical systems by analyzing their probabilistic structures on curved statistical manifolds, linking microscopic details to macroscopic behavior.

Contribution

It reviews the Maximum Relative Entropy formalism and develops an information geometrodynamical approach to chaos, introducing new complexity measures based on geometric properties.

Findings

Sectional curvature and Jacobi field intensity serve as complexity indicators.

Information geometrodynamical entropy quantifies chaos and regularity.

Application to theoretical models demonstrates the approach's effectiveness.

Abstract

Information geometric techniques and inductive inference methods hold great promise for solving computational problems of interest in classical and quantum physics, especially with regard to complexity characterization of dynamical systems in terms of their probabilistic description on curved statistical manifolds. In this article, we investigate the possibility of describing the macroscopic behavior of complex systems in terms of the underlying statistical structure of their microscopic degrees of freedom by use of statistical inductive inference and information geometry. We review the Maximum Relative Entropy (MrE) formalism and the theoretical structure of the information geometrodynamical approach to chaos (IGAC) on statistical manifolds. Special focus is devoted to the description of the roles played by the sectional curvature, the Jacobi field intensity and the information geometrodynamical entropy (IGE). These quantities serve as powerful information geometric complexity measures of information-constrained dynamics associated with arbitrary chaotic and regular systems defined on the statistical manifold. Finally, the application of such information geometric techniques to several theoretical models are presented.