Fractional Advection Diffusion Asymmetry Equation, derivation, solution and application

TL;DR

This paper derives and analyzes a fractional advection diffusion equation for non-Markovian anomalous diffusion, applying it to tracer spreading, contamination, and first passage time problems, with a focus on finite mean waiting times.

Contribution

It provides a novel derivation and explanation of fractional space derivatives in a non-Markovian CTRW model with fat-tailed waiting times.

Findings

Derived a fractional advection diffusion asymmetry equation for non-Markovian CTRW.

Applied the model to tracer spreading and contamination in hydrology.

Presented a subordination scheme for finite mean and divergent variance waiting times.

Abstract

The non-Markovian continuous-time random walk model, featuring fat-tailed waiting times and narrow distributed displacements with a non-zero mean, is a well studied model for anomalous diffusion. Using an analytical approach, we recently demonstrated how a fractional space advection diffusion asymmetry equation, usually associated with Markovian L\'evy flights, describes the spreading of a packet of particles. Since we use Gaussian statistics for jump lengths though fat-tailed distribution of waiting times, the appearance of fractional space derivatives in the kinetic equation demands explanations provided in this manuscript. As applications we analyse the spreading of tracers in two dimensions, breakthrough curves investigated in the field of contamination spreading in hydrology and first passage time statistics. We present a subordination scheme valid for the case when the mean waiting time is finite and the variance diverges, which is related to L\'evy statistics for the number of renewals in the process.