Analytic solution of the fractional advection diffusion equation for the time-of-flight experiment in a finite geometry

TL;DR

This paper derives an exact analytical solution to the fractional advection diffusion equation in finite geometry, providing insights into time-of-flight experiments and confirming known theoretical results.

Contribution

It presents a novel closed-form Laplace space solution for the fractional advection diffusion equation in finite geometry, applicable to time-of-flight experimental analysis.

Findings

The solution involves an infinite series of Fourier modes with Mittag-Leffler decay.

The derived current expression matches the -2 slope sum in log-log plots, consistent with CTRW models.

Sensitivity analysis shows how experimental parameters influence current and particle density.

Abstract

A general analytic solution to the fractional advection diffusion equation is obtained in plane parallel geometry. The result is an infinite series of spatial Fourier modes which decay according to the Mittag-Leffler function, which is cast into a simple closed form expression in Laplace space using the Poisson summation theorem. An analytic expression for the current measured in a time-of-flight experiment is derived, and the sum of the slopes of the two respective time regimes on logarithmic axes is demonstrated to be -2, in agreement with the well known result for a continuous time random walk model. The sensitivity of current and particle number density to variation of experimentally controlled parameters is investigated in general, and the results applied to analyze selected experimental data.