Anomalous Diffusion and Relaxation Close to Thermal Equilibrium: A Fractional Fokker-Planck Equation Approach

TLDR

The study introduces a fractional Fokker‑Planck equation that models a particle’s stochastic evolution under a nonlinear external force and a thermal bath. The equation is illustrated by a harmonic‑potential example, showing how the model captures dynamics in a concrete setting. In the force‑free case the model predicts subdiffusion, satisfies generalized Einstein relations, yields a Boltzmann stationary distribution, and shows single‑mode relaxation following a Mittag‑Leffler decay.

Abstract

We introduce a fractional Fokker-Planck equation describing the stochastic evolution of a particle under the combined influence of an external, nonlinear force and a thermal heat bath. For the force-free case, a subdiffusive behavior is recovered. The equation is shown to obey generalized Einstein relations, and its stationary solution is the Boltzmann distribution. The relaxation of single modes is shown to follow a Mittag-Leffler decay. We discuss the example of a particle in a harmonic potential.