Models for characterizing the transition among anomalous diffusions with different diffusion exponents

TL;DR

This paper develops models based on CTRW theory to characterize transitions among different anomalous diffusion behaviors, providing analytical solutions, stochastic representations, and applications with external potentials.

Contribution

It introduces a novel CTRW-based framework using Mittag-Leffler functions to model diffusion transitions and derives analytical and numerical results.

Findings

Mean squared displacement trends are analytically derived and numerically verified.

Stochastic process representations are constructed and positivity of PDFs is proved.

Analytical solutions for models with external harmonic potentials are obtained.

Abstract

Based on the theory of continuous time random walks (CTRW), we build the models of characterizing the transitions among anomalous diffusions with different diffusion exponents, often observed in natural world. In the CTRW framework, we take the waiting time probability density function (PDF) as an infinite series in three parameter Mittag-Leffler functions. According to the models, the mean squared displacement of the process is analytically obtained and numerically verified, in particular, the trend of its transition is shown; furthermore the stochastic representation of the process is presented and the positiveness of the PDF of the position of the particles is strictly proved. Finally, the fractional moments of the model are calculated, and the analytical solutions of the model with external harmonic potential are obtained and some applications are proposed.