Use and Abuse of a Fractional Fokker-Planck Dynamics for Time-Dependent Driving

TL;DR

This paper critically examines the limitations of the fractional Fokker-Planck equation in modeling subdiffusive dynamics under time-dependent forces, proposing a modified equation valid for specific force types and questioning its general applicability.

Contribution

The authors derive a rigorous modified FFPE for dichotomously alternating forces and demonstrate its validity through numerical validation, highlighting the failure of the standard FFPE in general cases.

Findings

Modified FFPE valid for dichotomous forces

Subdiffusion enhanced under certain time-dependent forces

Standard FFPE fails for arbitrary time-dependent fields

Abstract

We investigate a subdiffusive, fractional Fokker-Planck dynamics occurring in time-varying potential landscapes and thereby disclose the failure of the fractional Fokker-Planck equation (FFPE) in its commonly used form when generalized in an {\it ad hoc} manner to time-dependent forces. A modified FFPE (MFFPE) is rigorously derived, being valid for a family of dichotomously alternating force-fields. This MFFPE is numerically validated for a rectangular time-dependent force with zero average bias. For this case subdiffusion is shown to become enhanced as compared to the force free case. We question, however, the existence of any physically valid FFPE for arbitrary varying time-dependent fields that differ from this dichotomous varying family.