## Aging and Rejuvenation with Fractional Derivatives

Description: Article on aging and rejuvenation with fractional derivatives. February 2, 2008 UNT College of Arts and Sciences UNT Scholarly Works Total Uses: 39 Past 30 days: 0 Yesterday: 0
Creator (Author): Aquino, Gerardo University of North Texas Bologna, Mauro University of North Texas Grigolini, Paolo University of North Texas; Università di Pisa; Istituto dei Processi Chimico Fisici del Consiglio Nazionale delle Ricerche West, Bruce J. United States. Army Research Office Creation: February 2, 2008 Article on aging and rejuvenation with fractional derivatives. Department: Physics Department: Center for Nonlinear Science This is the author manuscript version of an article published in Physical Review E. Abstract: We discuss a dynamic procedure that makes the fractional derivative emerge in the time asymptotic limit of non-Poisson processes. We find that two-state fluctuations, with an inverse power-law distribution of waiting times, finite first moment and divergent second moment, namely with the power index μ in the interval 2 < μ < 3, yields a generalized master equation equivalent to the sum of an ordinary Markov contribution and of a fractional derivative term. We show that the order of the fractional derivative depends on the age of the process under study. If the system is infinitely old, the order of the fractional derivative, ord = μ - 2. If the system is prepared at time -tₐ < 0 and the observation begins at time t = 0, we derive the following scenario. For times 0 < t << tₐ the system is satisfactorily described by the fractional derivative with ord = 3 - μ. Upon time increase the system undergoes a rejuvenation process that in the time limit t >> tₐ yields ord = μ - 2. The intermediate time regime is probably incompatible with a picture based on fractional derivatives, or, at least, with a mono-order fractional derivative. 11 p.: ill. aging order | fractional derivatives | Onsager principle | fractional operators arXiv: cond-mat/0311314 UNT College of Arts and Sciences UNT Scholarly Works ARK: ark:/67531/metadc174699 Paper Text Access: Public