Anomalous Behaviors in Fractional Fokker-Planck Equation

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 $ξ$ imposed by the drift force fields. In the special case of the space scaling value $ν=1$ and the time scaling value $τ=1$, 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 $ξ={1/4}$ for normal diffusion, and $< \bar{x^2(t)} >$ $\propto$ $t^η$ with $η>1$ and $ξ>{1/4}$ for superdiffusion.