Aging and Rejuvenation with Fractional Derivatives Page: 1
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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.Fv
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 .
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-
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 .
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 . 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.  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 . Recently. Metzler
 argued that the GME unifies the fractional calculus
Allegrini ct al.  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 July 22, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.