Geometric study for the Legendre duality of generalized entropies and its application to the porous medium equation

TL;DR

This paper explores the geometric aspects of Legendre duality in generalized entropies using information geometry, and applies these insights to analyze solutions of the porous medium equation.

Contribution

It introduces a geometric framework for understanding Legendre duality in generalized entropies and applies this to study nonlinear diffusion equations.

Findings

New geometric insights into generalized entropies and Legendre duality

Application of information geometry to the porous medium equation

Results on the behavior of solutions to nonlinear diffusion equations

Abstract

We geometrically study the Legendre duality relation that plays an important role in statistical physics with the standard or generalized entropies. For this purpose, we introduce dualistic structure defined by information geometry, and discuss concepts arising in generalized thermostatistics, such as relative entropies, escort distributions and modified expectations. Further, a possible generalization of these concepts in a certain direction is also considered. Finally, as an application of such a geometric viewpoint, we briefly demonstrate several new results on a behavior of the solution to the nonlinear diffusion equation called the {\em porous medium equation}.