# Aging and Rejuvenation with Fractional Derivatives

### Description

Article on aging and rejuvenation with fractional derivatives.

11 p.: ill.

### Creation Information

Aquino, Gerardo; Bologna, Mauro; Grigolini, Paolo & West, Bruce J. February 2, 2008.

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### Description

Article on aging and rejuvenation with fractional derivatives.

11 p.: ill.

### Notes

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.

### Source

• arXiv: cond-mat/0311314

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#### UNT Scholarly Works

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### Creation Date

• February 2, 2008

### Added to The UNT Digital Library

• July 24, 2013, 1:20 p.m.

### Description Last Updated

• May 12, 2014, 11:38 a.m.

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