Sub-diffusion in External Potential: Anomalous hiding behind Normal

TL;DR

This paper introduces a model of sub-diffusion influenced by a persistent external force, revealing how such forces alter particle behavior, transition to apparent ergodicity, and affect stationary states in anomalous diffusion.

Contribution

It presents a novel sub-diffusion model with continuous external force influence, showing how force impacts ergodicity, diffusion coefficient, and stationary distribution in anomalous diffusion.

Findings

Constant force causes transition from non-ergodic to apparent ergodic behavior.

Diffusion coefficient depends on force and anomalous exponent.

Stationary distribution differs from Boltzmann equilibrium.

Abstract

We propose a model of sub-diffusion in which an external force is acting on a particle at all times not only at the moment of jump. The implication of this assumption is the dependence of the random trapping time on the force with the dramatic change of particles behavior compared to the standard continuous time random walk model. Constant force leads to the transition from non-ergodic sub-diffusion to seemingly ergodic diffusive behavior. However, we show it remains anomalous in a sense that the diffusion coefficient depends on the force and the anomalous exponent. For the quadratic potential we find that the anomalous exponent defines not only the speed of convergence but also the stationary distribution which is different from standard Boltzmann equilibrium.