The fractional Fickʹs law for non-local transport processes

Fickʹs 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 Fickʹs law derived from the space-fractional diffusion equation generating the Lévy–Feller statistics. This means that the fundamental solutions can be interpreted as Lévy stable probability densities (in the Feller parameterization) with index α (1<α 2) and skewness θ (θ 2−α). 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 (θ≠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.