Aging and Rejuvenation with Fractional Derivatives Page: 3
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AGING AND REJUVENATION WITH FRACTIONAL...
that the system is initially in the out of equilibrium state
corresponding to pl(0)- p2(0) + 0. Assuming a regression to
equilibrium in such a way as to retain Eq. (10) we obtain
from the GME (1) using the coupling matrix (7)d ( -t) 2 (t- tt'@ (t')dt'.
dt oPHYSICAL REVIEW E 70, 036105 (2004)
The authors of Ref. [13] determined that the problem of
how to make these processes compatible with the Onsager
principle could be solved by expressing the CTRW in sta-
tionary form, resulting in the GME memory kernel(11)
(15)
u[1 - /i (u)]
- - + =
-2[1- (u)] + u[l + (u)]TThus, the Laplace transform of the autocorrelation function
can be related to the Laplace transform of the memory kernel
byu)+ 2c(u)
where 7 is the average waiting time,
T= tf(t)dt.
o(12)
It is interesting to notice that in the Poisson case, namely,
when i(t) is an exponential function of time, the memory
kernel of the GME, given by Eq. (8) turns out to be equiva-
lent to a Dirac S function of time, thereby implying that the
bath responsible for the fluctuations of the variable S has a
time scale infinitely smaller than the system of interest. In
this specific case, with the help of Eq. (12) we see that the
autocorrelation function Pg(t) decays exponentially with
time. This is a condition behind the ordinary Onsager prin-
ciple. Following the authors of Ref. [13] we want to go be-
yond the ordinary Onsager principle.
The authors of Ref. [13] studied the case where the auto-
correlation function departs from the exponential relaxation
and has the the following time asymptotic property:
1
00(t 2(13)
The waiting time distribution /i(t) corresponding to this au-
tocorrelation function has the following time asymptotic
property [13]:
0(t)t -cc (14)
with / > 2 to fit the stationary condition. At first sight, one
might be surprised about our decision to make these complex
processes obey the Onsager principle. Such processes have
exotic thermodynamical properties, and in some cases they
are even shown to be nonergodic [22] and to produce aging
effects [23,26]. Another interesting effect emerging from
these processes was described in Ref. [24]. These authors
used a fractional Fokker-Planck equation, which is closely
related to the GME used in this paper, and they found that
the response of their GME to external perturbation is quite
different from the response of the corresponding CTRW, in
conflict with the fact that their GME is equivalent to a
CTRW in the absence of perturbation. All these surprising
properties, however, refer to the case < 2, where no invari-
ant measure exists. The case t> 2, under study here, is in
principle compatible with the Onsager principle, and as a
consequence our request is not absurd. Nevertheless, we
shall see that the Onsager principle requires that the system
is infinitely aged, an ideal condition, and that an even appar-
ently negligible departure from this condition yields a strik-
ing effect: a rejuvenation process.The form of the memory kernel given by Eq. (15) is consis-
tent with the equation of motion for the autocorrelation func-
tion (11), and consequently Eq. (15) is equivalent to1(u) 2J u
2 (u) /B. Theoretical waiting time distribution
We have to remind the reader that the stationary autocor-
relation function of S is not related directly to fi(t). Zumofen
and Klafter [25] provided a prescription for deriving the cor-
responding equilibrium autocorrelation function of S from
'(t). Their result rests on the observation that i(t) is an
experimental function, evaluated by observing the time du-
ration of the two states. The connection with renewal theory
is established by assuming that the time duration of a state is
determined by two processes; one is the extraction of a ran-
dom number from a theoretical inverse power-law distribu-
tion *(7T), with the same power index ,a, and the other is a
coin tossing procedure that determines the sign of this lami-
nar region. Thus, a given experimental sojourn time in one of
the two states may correspond to an arbitrarily large number
of drawings and coin tossings. Renewal theory is used to
relate the autocorrelation function ca(t) to the waiting time
distribution function /,*(t). In fact, from the renewal theory
[21] we obtain the following important result:(t)= (t'- t) (t')dt'
/ T(18)
where T is the mean waiting time of the ,*(t)-distribution
density. It is interesting to notice that this equation implies
that the second derivative of the autocorrelation function is
proportional to &/(t),(19)
dt2 ()=T
In Sec. II the departure point of our calculations is given
by the autocorrelation function ca(t) of Eq. (18). In this case
is convenient to assign to this equilibrium autocorrelation
function a simple analytical form. This is done as follows.
First of all we assign to *(t) the following analytical
form:036105-3
(17)
(16)
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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]. (https://digital.library.unt.edu/ark:/67531/metadc67638/m1/3/: accessed April 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.