# Aging and Rejuvenation with Fractional Derivatives

11 p.

### Creation Information

Aquino, Gerardo; Bologna, Mauro; Grigolini, Paolo & West, Bruce J. September 10, 2004.

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#### UNT College of Arts and Sciences

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## What

11 p.

### Notes

Copyright 2004 American Physical Society. The following article appeared in Physical Review E 70, 70:3; http://pre.aps.org/abstract/PRE/v70/i3/e036105

Abstract: We discuss a dynamic procedure that makes a fractional derivatives 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, yield a generalized master equation equivalent to the sum of an ordinary Markov contribution and 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, o, is given by o=3-μ. A brand new system is characterized by the degree o=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 o=3-μ. Upon time increase the system undergoes a rejuvenation process that in the time limit t⪢tₐ yields o=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.

### Source

• Physical Review E, 2004, College Park: American Physical Society 70:3

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### Publication Information

• Publication Title: Physical Review E
• Volume: 70
• Issue: 3
• Peer Reviewed: Yes

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## When

### Creation Date

• September 10, 2004

### Added to The UNT Digital Library

• Nov. 24, 2011, 12:20 a.m.

### Description Last Updated

• March 27, 2014, 3:07 p.m.

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