Mechanics of geodesics in information geometry and black hole thermodynamics

What is it about?

In this article, we study the mechanics of dynamical aspects of information geometry and apply the results to thermodynamic geometry of black holes. Information geometry treats statistical models as Riemannian manifolds, where gradient flows with a potential function introduce dynamical evolution into the picture. In physics, information geometry is encountered in the form of thermodynamic geometry which usually derives from the entropy acting as the potential function. We compared dual gradient flows from information geometry to equations from classical mechanics, derived Riemannian metrics from them and compared them to optical metrics, studied them from a classical mechanics perspective describing Hamiltonian mechanics using a constraint, and compared the duality to a canonical transformation. We further explored how deformation of the gradient flows leads to a linear modification of the Riemannian metric known as the Randers-Finsler metric, and the distorted results that follow. We have tested the formulation for consistency with and without deformation in the example of the Gaussian model which is a fundamental model in statistics. Finally, we have discuss black hole thermodynamics, where depending on the type of black hole, one deals with either Ruppeiner or Weinhold geometry, which are conformally related to each other. We have applied our results for the Riemannian information metric to describe dynamical evolution of thermodynamic geometry for Kerr and Reissner-Nordström black holes.

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Why is it important?

This is an exhaustive description of the mechanics of time evolving statistical systems described as geodesics on Riemannian manifolds using aspects of classical mechanics. It furthermore describes application to the study of black hole thermodynamics as thermodynamics geometry.