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

Hiroki Suyari

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, starting from the growth law $dy/dx = y^q$. By identifying the $q$-logarithm as the natural coordinate system of the state space, we construct a thermodynamic framework where the drift term remains linear in the probability density, preserving the standard form of the Einstein relation. We show the 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.

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

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, starting from the growth law . By identifying the -logarithm as the natural coordinate system of the state space, we construct a thermodynamic framework where the drift term remains linear in the probability density, preserving the standard form of the Einstein relation. We show the duality between the dynamic index and the thermodynamic index : the stationary state is a -Gaussian distribution that minimizes a free energy functional defined by a generalized entropy of index . We prove the -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.
Paper Structure (18 sections, 27 equations, 1 figure)

This paper contains 18 sections, 27 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Geometric Derivation of Nonlinear Dynamics
  3. The Fundamental Growth Law
  4. The Linearization Principle and Natural Coordinates
  5. Generalized Flux and Chemical Potential
  6. The Nonlinear Fokker-Planck Equation
  7. Stationary Solutions and Equilibrium Properties
  8. Stationary State Condition
  9. The $q$-Gaussian Distribution
  10. Generalized Einstein Relation
  11. Entropy and Free Energy: The Dual Nature
  12. Free Energy Functional
  13. The $2-q$ Entropy and $H$-theorem
  14. Duality of Indices
  15. Applications to Specific Systems
  16. ...and 3 more sections

Figures (1)

  • Figure 1: Comparison of stationary distributions $p_{st}(x)$ for a particle in a harmonic potential. The curves are normalized to unity at $x=0$. The solid black line represents the standard Gaussian distribution ($q=1$). The dot-dashed red line shows a heavy-tailed $q$-Gaussian ($q=2.0$), typical for super-diffusive systems. The dashed blue line shows a $q$-Gaussian with compact support ($q=0.5$), typical for sub-diffusive systems.