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

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• Rate equations, spatial moments, and concentration profiles for mobile-immobile models with power-law and mixed waiting time distributions
• New Integrable Multi-Lévy-Index and Mixed Fractional Nonlinear Soliton Hierarchies
• Random walks on networks with preferential cumulative damage: Generation of bias and aging
• Squirrels can remember little: A random walk with jump reversals induced by a discrete-time renewal process
• Anomalous diffusion of the heavy quarks through the fractional Langevin equation
• Simulation of the continuous-time random walk using subordination schemes
• A compounded random walk for space-fractional diffusion on finite domains
• 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