Abstract

Fractional kinetic equations employ non-integer calculus to model anomalous relaxation and diffusion in many systems. While this approach is well explored, it so far failedto describe an important class of transport in disordered systems. Motivated by workon contaminant spreading in geological formations we propose and investigate a fractional advection-diffusion equation describing the biased spreading packet. While usualtransport is described by diffusion and drift, we find a third term describing symmetrybreaking which is omnipresent for transport in disordered systems. Our work is based oncontinuous time random walks with a finite mean waiting time and a diverging variance,a case that on the one hand is very common and on the other was missing in the kaleidoscope literature of fractional equations. The fractional space derivatives stem from longtrapping times while previously they were interpreted as a consequence of spatial L´evyflights.