Fluctuation of gauge field for general nonlinear Fokker-Planck equation and covariant version of Fisher information matrix

TL;DR

This paper explores the relationship between gauge fields in nonlinear Fokker-Planck equations and Fisher information, proposing a covariant approach that reveals internal geometric structures and flux correlations.

Contribution

It introduces a covariant Fisher information matrix for gauge fields in nonlinear Fokker-Planck equations, linking gauge theory with nonequilibrium dynamics.

Findings

Decomposition of gauge field fluctuations into three parts.

Covariant Fisher information matrix relates flux component correlations.

The covariant Fisher information is not gauge invariant.

Abstract

We clarify a strong link between general nonlinear Fokker-Planck equations with gauge fields associated with nonequilibrium dynamics and the Fisher information of the system. The notion of Abelian gauge theory for the non-equilibrium Fokker-Planck equation has proposed in the literature, in which the associated curvature represents internal geometry. We present the fluctuation of the gauge field can be decomposed into three parts. We further show that if we define the Fisher information matrix by using a covariant derivative then it gives correlation of the flux components but it is not gauge invariant.