Fractional Fokker-Planck subdiffusion in alternating fields

TL;DR

This paper derives a fractional Fokker-Planck equation for subdiffusion in time-dependent fields, revealing that periodic forces do not alter the long-term scaling of biased subdiffusive currents and diffusion.

Contribution

It introduces a fractional Fokker-Planck framework for subdiffusion under time-periodic forces and demonstrates the invariance of long-term biased subdiffusive behavior to such driving.

Findings

Periodic force does not affect the long-time scaling of biased subdiffusion.

Unbiased subdiffusion can be strongly enhanced by periodic driving.

The derived equation links subdiffusive dynamics to time-dependent force fields.

Abstract

The fractional Fokker-Planck equation for subdiffusion in time-dependent force fields is derived from the underlying continuous time random walk. Its limitations are discussed and it is then applied to the study of subdiffusion under the influence of a time-periodic rectangular force. As a main result, we show that such a force does not affect the universal scaling relation between the anomalous current and diffusion when applied to the biased dynamics: in the long time limit subdiffusion current and anomalous diffusion are immune to the driving. This is in sharp contrast with the unbiased case when the subdiffusion coefficient can be strongly enhanced, i.e. a zero-frequency response to a periodic driving is present.