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APS/123-QED

Aging and Rejuvenation with Fractional Derivatives

Gerardo Aqulino', lauro Bologna 1. Paolo Grigolini'2:'3, Bruce J. W1est 4

1 Center for Nonlinear Science. University of North Texas, P.O. Box 311497. Denton, Texas 76203-1427

2Dipartimento di Fisica dell'Universita di Pisa and INFM. via Bonarroti 2, 56127 Pisa, Italy

3Istituto d:i Processi Chimico Fisiec del CNR Area della Ricera di Pisa,. Via G. Mor"uzzi 1, 56124 Pisa. Italy

Mathematics Division. Army Research Office, Research Triangle Park, NC 27709, USA

(Dated: February 2, 2008)

We discuss a. dynamic procedure that makes the fractional derivatives emerge in the time asymnp-

totic 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 tlhe

power index p in the interval 2 < p < 3, yields a generalized master equation equivalent to the

sum of an ordinary 1MIarkov contribution and of a fractional derivative term. I Ge 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- p. A brand new

system is characterized by the degree ord = 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 << t,, the

system is satisfactorily described by the fractional derivative with ord = 3 - p. Upon time increase

the system undergoes a rejuvenation process that in the time limit t >> t,, yields ord = i - 2. The

intermediate time regime is probably incompatible with a pictures based on fractional derivatives,

or, at least, with a mono-order fractional derivative.

PACS numbers: 05.40.Fb, 05.60.Cd, 02.50.FvI. INTRODUCTION

The fractional calculus has recently received a great

deal of attention in the physics literature, through the

publications of books [1, 2], review articles [3. 4]. as well

as an ever increasing munber of research papers. some of

which are quoted here [5. 6. 7, 8. 9, 10, 11, 12]. The blos-

soming interest in the fractional calculus is due, in part,

to the fact it provides one of the dynamical foundations

for fractal stochastic processes [2. 4]. The adoption of the

fractional calculus by the physics community was inhib-

ited historically because there was no clear experimental

evidence for its need. TIhe disciplines of statistical physics

and thermodynamics were thought to be sufficient for de-

scribing complex physical phenomena solely with the use

and modifications of analytic functions. This view was

supported by the successes of sl(uch physicists as Lars On-

sager, who through the use of simple physical arguments

was able to relate apparently independent transport pro-

cesses to one another. even though these processes are

associated with quite different physical phenomena [13].

His general arguments rested on three assumptions: 1)

microscopic dynamics have time-reversal symmetry : 2)

fluctuations of the heat bath decay at the same rate as

do macroscopic deviations from equlilibrilum and 3) phys-

ical systems are aged. NWe refer to assumption 2 as the

Onsager Principle and show that it is tied up with as-

sinnption 3.

Onsager's arguments focused on a systern that is in

contact 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 per-turbations of the bath to equilibrium is instantaneous.

This means that the energy absorbed from the system

of interest by the bath, through macroscopic dissipa-

tion, is distributed over the bath degrees of freedom on

a very much shorter time scale than the relaxation time

of the system. This property is summarized in the well

known fluct nation -dissipatioii theorem., which has even

been generalized to the case where the fluctuations in

the bath do not regress instantaneously [14].

The dynamics of the physical variables to which the

Onsager Principle apply are described by two different

kinds of equations: 1) the Langevin equation, a stochas-

tic 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 randloim walk (CTRW) approach of Montroll and

Weiss [15]. The CTR W formnalisin describes a random

walk in which the walker pauses after each jump for a.

sojourn specified by a waiting time distribution function.

It was shown by Bedeaux et al. [16] that the Markov

niaster equation is equ'ivaleCnt to CTRWX if the waiting

time distribution is Poissonian. However, when the wait-

ing time distribution is not exponential, the case we con-

sider he re, the equdivalence between the two approaches is

only maiitntained by generalizing to the non-MNarkov minas-

ter equation, the so-called GME [17]. Recently. Metzler

[18] argued that the GME unifies the fractional calculus

and CTRW.

Allegrini ct al. [19] have shown that to create a master

equation compatible with the Onsager Principle requires

that the system be entangled with the heat bath in such

a way that the bath does not regress to equilibrium in-

finitely fast. The system-bath entanglement is the result

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Aquino, Gerardo; Bologna, Mauro; Grigolini, Paolo & West, Bruce J. Aging and Rejuvenation with Fractional Derivatives, paper, February 2, 2008; (digital.library.unt.edu/ark:/67531/metadc174699/m1/1/: accessed March 23, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.