Stochastic transport and fractional derivatives

A systematic derivation of macroscopic equations describing superdiffusion (8) and subdiffusion (20) over a wide range of physical processes is given on the basis of general microscopic characteristics of the motion of individual particles. It is shown that fractional derivatives are a necessary component of these equations, the time derivatives being of a different type from the derivatives with respect to the spatial variables. The simplest properties of the equations are investigated-specifically, how quickly a universal self-similar profile emerges from an arbitrary initial particle distribution. O 1995 American Institute c?f Physics.