Aging and Rejuvenation with Fractional Derivatives Page: 1
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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.
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 . 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 of
PACS 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 .
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 . 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.  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) .
Recently, Metzler  argued that the GME unifies the frac-
tional calculus and the CTRW.
Allegrini et al.  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.
, 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.
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 January 23, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.