Fick's law is extensively adopted as a model for standard diffusion process
es. However, requiring separation of scales, it is not suitable for describ
ing non-local transport processes. We discuss a generalized non-local Fick'
s law derived from the space-fractional diffusion equation generating the L
ivy-Feller statistics. This means that the fundamental solutions can be int
erpreted as Levy stable probability densities (in the Feller parameterizati
on) with index alpha (1 < alpha greater than or equal to 2) and skewness th
eta (\theta\ less than or equal to 2 - alpha). We explore the possibility o
f defining an equivalent local diffusivity by displaying a few numerical ca
se 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) 2
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