Fractional Calculus Fractional Time Irreversibility Theory of Time

Fractional Evolution Equations and Irreversibility

R. Hilfer

in: Traffic and Granular Flow’99
edited by: D. Helbing and H. Herrmann and M. Schreckenberg and D. Wolf
Springer, Berlin, 215 (2000)
ISBN: 978-3-642-64109-1

submitted on
Monday, September 27, 1999

The paper reviews a general theory predicting the general importance of fractional evolution equations. Fractional time evolutions are shown to arise from a microscopic time evolution in a certain long time scaling limit. Fractional time evolutions are generally irreversible. The infinitesimal generators of fractional time evolutions are fractional time derivatives. Evolution equations containing fractional time derivatives are proposed for physical, economical and traffic applications. Regular non-fractional time evolutions emerge as special cases from the results. Also for these regular time evolutions it is found that macroscopic irreversibility arises in the scaling limit.

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