Linearization Principle: The Geometric Origin of Nonlinear Fokker-Planck Equations

TL;DR

This paper introduces a geometric derivation of nonlinear Fokker-Planck equations using the Linearization Principle, linking Tsallis statistics to anomalous diffusion and providing a thermodynamic framework with a duality between dynamic and thermodynamic indices.

Contribution

It presents a novel geometric approach to derive nonlinear Fokker-Planck equations, connecting Tsallis statistics with a duality between indices, and proves an H-theorem within this framework.

Findings

Stationary states are q-Gaussians minimizing a generalized free energy.

The framework describes anomalous diffusion without ad-hoc nonlinear drift forces.

The derived equations satisfy an H-theorem and apply to harmonic oscillator and free particle.

Abstract

Anomalous diffusion and power-law distributions are observed in various complex systems. To provide a consistent dynamical foundation for these phenomena, we present a geometric derivation of the nonlinear Fokker-Planck equation by introducing the Linearization Principle directly at the dynamical stage. By identifying the generalized chemical potential as the natural dynamical ansatz, we construct a general thermodynamic framework where the drift term remains linear in the probability density, preserving the standard form of the Einstein relation. Within this framework, we show that the $q$-deformed geometry, corresponding to Tsallis statistics, exhibits a fundamental duality between the dynamic index $q$ and the thermodynamic index $2-q$: the stationary state is a $q$-Gaussian distribution that minimizes a free energy functional defined by a generalized entropy of index $2-q$. We prove the $H$-theorem for the derived equation and demonstrate its application to the harmonic oscillator and the free particle. This framework describes anomalous diffusion without relying on ad-hoc constraints or phenomenological nonlinear drift forces.