Random walk's models for fractionaldiffusion equation

Fractional diffusion equations are used for mass spreading in inhomogeneous media. They are applied to model anomalous diffusion, where a cloud of particles spreads in a different manner than the classical diffusion equation predicts. Thus, they involve fractional derivatives. Here we present acontinuous variant of Grünwald-Letnikov's formula, which is useful to compute the flux of particles performing random walks, allowing for heavy-tailed jump distributions. In fact, we set a definition of fractional derivatives yielding the operators which enable us to retrieve the space fractional variant of Fick's law, for enhanced diffusion in disordered media, without passing through any partial differential equation for the space and time evolution of the concentration.