The fractional Fick's law for non-local transport processes

Fick's law is extensively adopted as a model for standard diffusion processes. However, requiring separation of scales, it is not suitable for describing non-local transport processes. We discuss a generalized non-local Fick's law derived from the space-fractional diffusion equation generating the Livy-Feller statistics. This means that the fundamental solutions can be interpreted as Levy stable probability densities (in the Feller parameterization) with index alpha (1 < alpha greater than or equal to 2) and skewness theta (\theta\ less than or equal to 2 - alpha). We explore the possibility of defining an equivalent local diffusivity by displaying a few numerical case studies concerning the relevant quantities (flux and gradient). It turns out that the presence of asymmetry (theta not equal 0) plays a fundamental role: it produces shift of the maximum location of the probability density function and gives raise to phenomena of counter-gradient transport. (C) 2001 Elsevier Science B.V. All rights reserved.