Measuring subdiffusion parameters

TL;DR

This paper introduces a method to determine subdiffusion parameters from experimental data using a membrane system, and applies it to analyze sugar transport in gel solvent, confirming subdiffusive behavior.

Contribution

It provides a fully analytic solution to the fractional subdiffusion equation and demonstrates how to extract subdiffusion parameters from experimental measurements.

Findings

Confirmed subdiffusive transport of sugars in gel solvent

Precisely determined the subdiffusion parameter α<1

Measured subdiffusion coefficient D_α

Abstract

We propose a method to extract from experimental data the subdiffusion parameter $\alpha$ and subdiffusion coefficient $D_\alpha$ which are defined by means of the relation $<x^2> =2D_\alpha/\Gamma(1+\alpha) t^\alpha$ where $<x^2>$ denotes a mean square displacement of a random walker starting from $x=0$ at the initial time $t=0$. The method exploits a membrane system where a substance of interest is transported in a solvent from one vessel to another across a thin membrane which plays here only an auxiliary role. Using such a system, we experimentally study a diffusion of glucose and sucrose in a gel solvent. We find a fully analytic solution of the fractional subdiffusion equation with the initial and boundary conditions representing the system under study. Confronting the experimental data with the derived formulas, we show a subdiffusive character of the sugar transport in gel solvent. We precisely determine the parameter $\alpha$, which is smaller than 1, and the subdiffusion coefficient $D_\alpha$.