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 L\'evy-Feller statistics. This means that the fundamental solutions can be interpreted as L\'evy stable probability densities (in the Feller parameterization) with index $\alpha$ ($1<\alpha \le 2$) and skewness $\theta$ ($|\theta| \le 2-\alpha$). 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 ($\theta \ne 0$) 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.