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PHYSICAL REVIEW E 70, 036105 (2004)

Aging and rejuvenation with fractional derivatives

Gerardo Aquino,1 Mauro Bologna,1 Paolo Grigolini,1,2'3 and Bruce J. West4

1 Center for Nonlinear Science, University of North Texas, P O. Box 311427, Denton, Texas 76203-1427, USA

2Dipartimento di Fisica dell'University di Pisa and INFM, via Buonarroti 2, 56127 Pisa, Italy

3Istituto dei Processi Chimico Fisici del CNR, Area della Ricerca di Pisa, Via G. Moruzzi 1, 56124 Pisa, Italy

4Mathematics Division, Army Research Office, Research Triangle Park, North Carolina 27709, USA

(Received 18 March 2004; published 10 September 2004)

We discuss a dynamic procedure that makes 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 ,c in the interval

2 < ,u < 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, , is given by =3

- ,. A brand new system is characterized by the degree t=/p-2. If the system is prepared at time -t,< 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 = 3- ,. Upon time increase the system undergoes a

rejuvenation process that in the time limit t> t, yields =, u-2. The intermediate time regime is probably

incompatible with a picture based on fractional derivatives, or, at least, with a mono-order fractional derivative.DOI: 10.1103/PhysRevE.70.036105

I. INTRODUCTION

The fractional calculus has recently received a great deal

of attention in the physics literature, through the publication

of books [1,2] and review articles [3,4], as well as an ever

increasing number of research papers, some of which are

quoted here [5-12]. The blossoming interest in the fractional

calculus is due, in part, to the fact that it provides one of the

dynamical foundations for fractal stochastic processes [2,4].

The adoption of the fractional calculus by the physics com-

munity was inhibited historically because there was no clear

experimental evidence for its need. The disciplines of statis-

tical physics and thermodynamics were thought to be suffi-

cient for describing complex physical phenomena solely with

the use and modifications of analytic functions. This view

was supported by the successes of such physicists as On-

sager, who through the use of simple physical arguments was

able to relate apparently independent transport processes to

one another, even though these processes are associated with

quite different physical phenomena [14]. His general argu-

ments rested on three assumptions: (1) microscopic dynam-

ics have time-reversal symmetry; (2) fluctuations of the heat

bath decay at the same rate as do macroscopic deviations

from equilibrium; and (3) physical systems are aged. We

refer to assumption 2 as the Onsager principle and show that

it is tied up with assumption 3.

Onsager's arguments focused on a system that is in con-

tact with a heat bath sufficiently long that the bath has come

to thermal equilibrium and consequently the system is aged.

In statistical physics we know that the bath is responsible for

both fluctuations and dissipation, and if the fluctuations are

white the regression of perturbations of the bath to equilib-

rium is instantaneous. This means that the energy absorbed

from the system of interest by the bath, through macroscopic

dissipation, is distributed over the bath degrees of freedom

on a very much shorter time scale than the relaxation time ofPACS number(s): 02.50.Ey, 05.40.Fb, 05.60.Cd

the system. This property is summarized in the well known

fluctuation-dissipation theorem, which has even been gener-

alized to the case where the fluctuations in the bath do not

regress instantaneously [15].

The dynamics of the physical variables to which the On-

sager principle apply are described by two different kinds of

equations: (1) the Langevin equation, a stochastic differential

equation for the dynamical variable and (2) the phase space

equation for the probability density. Two distinct methods

have been developed to describe the phase space evolution of

the probability density: the master equation introduced by

Pauli and the continuous time random walk (CTRW) ap-

proach of Montroll and Weiss [16]. The CTRW formalism

describes a random walk in which the walker pauses after

each jump for a sojourn specified by a waiting time distribu-

tion function. It was shown by Bedeaux et al. [17] that the

Markov master equation is equivalent to a CTRW if the wait-

ing time distribution is Poissonian. However, when the wait-

ing time distribution is not exponential, the case we consider

here, the equivalence between the two approaches is main-

tained only by generalizing to the non-Markov master equa-

tion, the so-called generalized master equation (GME) [18].

Recently, Metzler [19] argued that the GME unifies the frac-

tional calculus and the CTRW.

Allegrini et al. [13] have shown that creating a master

equation compatible with the Onsager principle requires that

the system be entangled with the bath in such a way as to

realize a condition of stable thermodynamic equilibrium.

This system-bath entanglement is the result of a rearrange-

ment process that may take an infinitely long time to com-

plete, leading to the replacement of the GME of Kenkre et al.

[18], which corresponds to the nonstationary condition, with

a new GME compatible with the stationary condition, and

consequently with the Onsager principle. Herein, we extend

that discussion to include the connection with the fractional

calculus in both infinitely and not infinitely aged condition.1539-3755/2004/70(3)/036105(11)/$22.50

70 036105-1

02004 The American Physical Society

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Aquino, Gerardo; Bologna, Mauro; Grigolini, Paolo & West, Bruce J. Aging and Rejuvenation with Fractional Derivatives, article, September 10, 2004; [College Park, Maryland]. (digital.library.unt.edu/ark:/67531/metadc67638/m1/1/: accessed July 25, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.