Generalized Fractional Master Equation for Self-Similar Stochastic Processes Modelling Anomalous Diffusion

Gianni Pagnini,Francesco Mainardi,Antonio Mura

doi:10.1155/2012/427383

Abstract

The Master Equation approach to model anomalous diffusion is considered. Anomalous diffusion in complex media can be described as the result of a superposition mechanism reflecting inhomogeneity and nonstationarity properties of the medium. For instance, when this superposition is applied to the time-fractional diffusion process, the resulting Master Equation emerges to be the governing equation of the Erdélyi-Kober fractional diffusion, that describes the evolution of the marginal distribution of the so-called generalized grey Brownian motion. This motion is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion: it is made up of self-similar processes with stationary increments and depends on two real parameters. The class includes the fractional Brownian motion, the time-fractional diffusion stochastic processes, and the standard Brownian motion. In this framework, the M-Wright function (known also as Mainardi function) emerges as a natural generalization of the Gaussian distribution, recovering the same key role of the Gaussian density for the standard and the fractional Brownian motion.

Highlights

Statistical description of diffusive processes can be performed both at the microscopic and at the macroscopic levels
When this superposition is applied to the time-fractional diffusion process, the resulting Master Equation emerges to be the governing equation of the Erdelyi-Kober fractional diffusion, that describes the evolution of the marginal distribution of the so-called generalized grey Brownian motion
In a macroscopic framework, this larger class of self-similar stochastic processes is characterized by a Master Equation that is a fractional differential equation in the Erdelyi-Kober sense

Summary

Introduction

Statistical description of diffusive processes can be performed both at the microscopic and at the macroscopic levels. Schneider 20, 21 , making use of the grey noise theory, introduced a class of selfsimilar stochastic processes termed grey Brownian motion This class provides stochastic models for the slow anomalous diffusion and the corresponding Master Equation turns out to be the time-fractional diffusion equation. This class of self-similar processes has been extended to include stochastic models for both slow and fast anomalous diffusion and it is named generalized grey Brownian motion 22–24. In a macroscopic framework, this larger class of self-similar stochastic processes is characterized by a Master Equation that is a fractional differential equation in the Erdelyi-Kober sense.

The Master Equation and Its Generalization

A Physical Mechanism for Time-Stretching Generalization

The Erdelyi-Kober Fractional Diffusion

Conclusions