Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2007-05-06
"Anomalous Transport: Foundations and Applications", edited by R. Klages, G. Radons and I.M Sokolov, as Chapter 4, pp. 93-127,
Physics
Condensed Matter
Statistical Mechanics
36 pages, 3 figures (5 files eps). Invited lecture by R. Gorenflo at the 373. WE-Heraeus-Seminar on Anomalous Transport: Exper
Scientific paper
We show the asymptotic long-time equivalence of a generic power law waiting time distribution to the Mittag-Leffler waiting time distribution, characteristic for a time fractional CTRW. This asymptotic equivalence is effected by a combination of "rescaling" time and "respeeding" the relevant renewal process followed by a passage to a limit for which we need a suitable relation between the parameters of rescaling and respeeding. Turning our attention to spatially 1-D CTRWs with a generic power law jump distribution, "rescaling" space can be interpreted as a second kind of "respeeding" which then, again under a proper relation between the relevant parameters leads in the limit to the space-time fractional diffusion equation. Finally, we treat the `time fractional drift" process as a properly scaled limit of the counting number of a Mittag-Leffler renewal process.
Gorenflo Rudolf
Mainardi Francesco
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