Integro-differential diffusion equation for continuous time random walk

TL;DR

This paper derives an integro-differential diffusion equation for continuous time random walks applicable to various waiting time distributions, analyzing diffusion behaviors including normal and subdiffusive regimes with different probability density functions.

Contribution

It introduces a generalized integro-differential equation for CTRW and explores its implications for different waiting time distributions, including exponential and power-law types.

Findings

Normal diffusion with Gaussian distribution for exponential waiting times.

Subdiffusive behavior with non-Gaussian distribution for combined power-law and Mittag-Leffler waiting times.

The equation captures diverse diffusion regimes based on waiting time PDFs.

Abstract

In this paper we present an integro-differential diffusion equation for continuous time random walk that is valid for a generic waiting time probability density function. Using this equation we also study diffusion behaviors for a couple of specific waiting time probability density functions such as exponential, and a combination of power law and generalized Mittag-Leffler function. We show that for the case of the exponential waiting time probability density function a normal diffusion is generated and the probability density function is Gaussian distribution. In the case of the combination of a power-law and generalized Mittag-Leffler waiting probability density function we obtain the subdiffusive behavior for all the time regions from small to large times, and probability density function is non-Gaussian distribution.