On Hilfer's objection to the fractional time diffusion equation

E. Barkai; I. M. Sokolov

doi:10.1016/j.physa.2006.05.022

On Hilfer's objection to the fractional time diffusion equation

E. Barkai, I. M. Sokolov

Department of Physics - at Bar-Ilan University

Humboldt University of Berlin

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

Hilfer [Physica A 329 (2003) 35] claims to give an example of a continuous time random walk (CTRW) model with long-tailed waiting time probability density that approaches a Gaussian behavior in the continuum limit. Rigorous limit theorems, derived previously, show however that in the limit of long-time such a CTRW converges to a non-Gaussian behavior. We discuss two types of continuum limits for the CTRW model: the fractional continuum limit and the one introduced by Hilfer. We show that the fractional limit yields the correct long-time behavior of the CTRW, while Hilfer's continuum limit does not. We discuss a general approach to find a continuum limit of the CTRW process.

Original language	English
Pages (from-to)	231-236
Number of pages	6
Journal	Physica A: Statistical Mechanics and its Applications
Volume	373
DOIs	https://doi.org/10.1016/j.physa.2006.05.022
State	Published - 1 Jan 2007

Bibliographical note

Funding Information:
EB thank the Israel Science Foundation and the center of complexity in Jerusalem, for financial support.

Funding

EB thank the Israel Science Foundation and the center of complexity in Jerusalem, for financial support.

Funders	Funder number
Israel Science Foundation

Keywords

Continuous time random walks
Continuum limit
Fractional diffusion equation

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10.1016/j.physa.2006.05.022

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AU - Sokolov, I. M.

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AB - Hilfer [Physica A 329 (2003) 35] claims to give an example of a continuous time random walk (CTRW) model with long-tailed waiting time probability density that approaches a Gaussian behavior in the continuum limit. Rigorous limit theorems, derived previously, show however that in the limit of long-time such a CTRW converges to a non-Gaussian behavior. We discuss two types of continuum limits for the CTRW model: the fractional continuum limit and the one introduced by Hilfer. We show that the fractional limit yields the correct long-time behavior of the CTRW, while Hilfer's continuum limit does not. We discuss a general approach to find a continuum limit of the CTRW process.

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JO - Physica A: Statistical Mechanics and its Applications

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ER -

On Hilfer's objection to the fractional time diffusion equation

Abstract

Bibliographical note

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Keywords

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