Anomalous Behaviors in Fractional Fokker-Planck Equation

TL;DR

This paper analyzes a fractional Fokker-Planck equation with time-dependent drift forces, deriving the asymptotic behavior of tracer moments and classifying diffusion types based on scaling exponents.

Contribution

It introduces a fractional Fokker-Planck model with temporal power-law dependence and analytically characterizes the long-time behavior of tracer moments.

Findings

Long-time asymptotic behavior of the second moment depends on the scaling exponent  imposed by drift fields.

Normal diffusion occurs with =1/4, where the second moment scales linearly with time.

Superdiffusion is characterized by >=1/4, with the second moment scaling faster than linearly.

Abstract

We introduce a fractional Fokker-Planck equation with a temporal power-law dependence on the drift force fields. For this case, the moments of the tracer from the force-force correlation in terms of the time-dependent drift force fields are discussed analytically. The long-time asymptotic behavior of the second moment is determined by the scaling exponent $\xi$ imposed by the drift force fields. In the special case of the space scaling value $\nu=1$ and the time scaling value $\tau=1$, our result can be classified according to the temporal scaling of the mean second moment of the tracer for large $t$: $< \bar{x^2(t)} >$ $\propto$ $t$ with $\xi={1/4}$ for normal diffusion, and $< \bar{x^2(t)} >$ $\propto$ $t^{\eta}$ with $\eta>1$ and $\xi>{1/4}$ for superdiffusion.