Anomalous Behaviors in Fractional Fokker-Planck Equation

Original · EN

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: < x²(t) > ∝ t with ξ=1/4 for normal diffusion, and < x²(t) > ∝ tη with η>1 and ξ>1/4 for superdiffusion.