A physical interpretation for the fractional derivative in Levy diffusion

Abstract-To the authors’ knowledge, previous derivations of the fractional diffusion equation are based on stochastic principles [l], with the result that physical interpretation of the resulting fractional derivatives has been elusive [2]. Herein, we develop a fmctional J%Z Iaw relating solute flux at a given point to what might be called the complete (twosided) fractional derivative of the concentration distribution at the same point. The fractional derivative itself is then identified as a typical superposition integral over the spatial domain of the Levy diffusion process. While this interpretation does not obviously generalize to all applications, it does point toward the search for superposition principles when attempting to give physical meaning to fractional derivatives. @ 2002 Elsevier Science Ltd. All rights reserved.