Non-linear Fokker-Planck equations from conformal metrics and scalar curvature

TL;DR

This paper explores a geometric origin for non-linear Fokker-Planck equations, linking their non-linearity to conformal deformations of the underlying metric and the scalar curvature in a mesoscopic system context.

Contribution

It proposes a novel geometric interpretation of non-linear Fokker-Planck equations via conformal metric deformations and scalar curvature considerations.

Findings

Non-linear terms arise from conformal metric deformations.

Scalar curvature influences the form of the equations.

Geometric perspective connects to $q$-entropy systems.

Abstract

We present an argument which intends to explore a potential geometric origin of a class of non-linear Fokker-Planck equations related to the mesoscopic behavior of systems conjecturally described by the $q$-entropy. We argue that the appearance of the non-linear term(s) in such equations can be ascribed to the fact that the effective mesoscopic metric describing the behavior of the underlying system may not be the originally chosen one, but a conformal deformation of it. Motivated by Liouville's theorem, we highlight the role played by the scalar curvature of conformally related metrics in establishing such a non-linear Fokker-Planck equation.