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Advances in Mathematical Physics
Volume 2013 (2013), Article ID 498789, 7 pages
http://dx.doi.org/10.1155/2013/498789
Research Article

Complexity and the Fractional Calculus

1Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, TX 76203-1427, USA
2Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 6-D, Arica, Chile
3Information Science Directorate, Army Research Office, Research Triangle Park, NC 27709, USA

Received 21 March 2013; Accepted 28 March 2013

Academic Editor: Dumitru Baleanu

Copyright © 2013 Pensri Pramukkul et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study complex processes whose evolution in time rests on the occurrence of a large and random number of events. The mean time interval between two consecutive critical events is infinite, thereby violating the ergodic condition and activating at the same time a stochastic central limit theorem that supports the hypothesis that the Mittag-Leffler function is a universal property of nature. The time evolution of these complex systems is properly generated by means of fractional differential equations, thus leading to the interpretation of fractional trajectories as the average over many random trajectories each of which satisfies the stochastic central limit theorem and the condition for the Mittag-Leffler universality.