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 $\xi$ 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 $t$: $<\overline{x^2(t)}>$ $\propto$ $t$ with $\xi =1/4$ for normal diffusion, and $<\overline{x^2(t)}>$ $\propto$ $t^{\eta }$ with $\eta >1$ and $\xi >1/4$ for superdiffusion.

10 pages, Latex