Diffusion and Relaxation Controlled by Tempered \alpha-stable Processes

TL;DR

This paper introduces a tempered lpha-stable process model to describe anomalous diffusion and relaxation, overcoming infinite-moment issues and bridging subdiffusion with normal diffusion.

Contribution

It derives a general framework for anomalous diffusion and relaxation using tempered lpha-stable processes, providing explicit equations and encompassing subdiffusion as a special case.

Findings

Derived explicit Fokker-Planck equation for the process.

Obtained mean square displacement and relaxation functions.

Model bridges subdiffusion and normal diffusion regimes.

Abstract

We derive general properties of anomalous diffusion and nonexponential relaxation from the theory of tempered \alpha-stable processes. Its most important application is to overcome the infinite-moment difficulty for the \alpha-stable random operational time \tau. The tempering results in the existence of all moments of \tau. The subordination by the inverse tempered \alpha-stable process provides diffusion(relaxation) that occupies an intermediate place between subdiffusion (Cole-Cole law) and normal diffusion (exponential law). Here we obtain explicitly the Fokker-Planck equation, the mean square displacement and the relaxation function. This model includes subdiffusion as a particular case.