Macroscopic current in fractional anomalous diffusion

Fick’s law – which relates the macroscopic current with the density in ordinary diffusion – is generalized to fractional diffusion, where either the waiting time distribution or the displacement probability density exhibit power-law decays. These long-tailed distributions give rise to (respectively, temporal or spatial) nonlocality in connection between current and density. In the case of power-law displacement distributions, the current is shown to also display such decays in the spatial coordinate. An illustrative example is explicitly worked out in a one-dimensional system.