How to Measure Subdiffusion Parameters

Abstract

We propose a method to measure the subdiffusion parameter $\alpha$ and subdiffusion coefficient $D_{\alpha }$ which are defined by means of the relation $<x^2>=\frac{2D_{\alpha }}{\Gamma (1+\alpha )}{t}^{\alpha }$ 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. We experimentally study a diffusion of glucose and sucrose in a gel solvent, and we precisely determine the parameters $\alpha$ and $D_{\alpha }$, using a fully analytic solution of the fractional subdiffusion equation.

short version of cond-mat/0309072, to appear in Phys. Rev. Lett