Aging and Rejuvenation with Fractional Derivatives

TL;DR

This paper explores how fractional derivatives naturally emerge in non-Poisson processes with two-state fluctuations, showing that the derivative order depends on the system's age and describing a rejuvenation process over time.

Contribution

It introduces a dynamic framework linking the age of a system to the order of fractional derivatives in its governing equations, revealing age-dependent rejuvenation effects.

Findings

Fractional derivative order depends on system age and process parameters.

Rejuvenation causes a transition from one fractional order to another over time.

Intermediate regimes may not be well-described by single-order fractional derivatives.

Abstract

We discuss a dynamic procedure that makes the 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 mu in the interval 2<mu <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, is given by ord=3-mu . A brand new system is characterized by the degree ord=mu -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-mu . Upon time increase the system undergoes a rejuvenation process that in the time limit t>>ta yields ord=mu -2. The intermediate time regime is probably incompatible with a picture based on fractional derivatives, or, at least, with a mono-order fractional derivative.