Fractional advection-diffusion-asymmetry equation

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 failed to describe an important class of transport in disordered systems. Motivated by work on contaminant spreading in geological formations we propose and investigate a fractional advection-diffusion equation describing the biased spreading packet. While usual transport is described by diffusion and drift, we find a third term describing symmetry breaking which is omnipresent for transport in disordered systems. Our work is based on continuous 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 long trapping times while previously they were interpreted as a consequence of spatial L{é}vy flights.

17 pages

References in corpus (4)

• Fractional Fokker-Planck Equations for Subdiffusion with Space-and-Time-Dependent Forces
• Anomalous heat transport in one dimensional systems: a description using non-local fractional-type diffusion equation
• Scaling properties of field-induced superdiffusion in Continous Time Random Walks
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• Dissipative systems fractionally coupled to a bath
• Subdiffusion of heavy quark in hot QCD matter by the fractional Langevin equation
• Non-normalizable quasi-equilibrium states under fractional dynamics
• Asymmetric random walks with bias generated by discrete-time counting processes