Pre-asymptotic corrections to fractional diffusion equations

TL;DR

This paper investigates the limitations of fractional diffusion models in describing early-time particle transport in complex media, proposing a modified equation to account for pre-asymptotic effects validated by simulations.

Contribution

It introduces a new modified transport equation that captures pre-asymptotic corrections to fractional diffusion, enhancing modeling accuracy for initial particle dynamics.

Findings

Pre-asymptotic corrections are significant depending on microscopic dynamics.

A modified transport equation effectively models early-time anomalous diffusion.

Monte Carlo simulations validate the proposed corrections.

Abstract

The motion of contaminant particles through complex environments such as fractured rocks or porous sediments is often characterized by anomalous diffusion: the spread of the transported quantity is found to grow sublinearly in time due to the presence of obstacles which hinder particle migration. The asymptotic behavior of these systems is usually well described by fractional diffusion, which provides an elegant and unified framework for modeling anomalous transport. We show that pre-asymptotic corrections to fractional diffusion might become relevant, depending on the microscopic dynamics of the particles. To incorporate these effects, we derive a modified transport equation and validate its effectiveness by a Monte Carlo simulation.