Aging and Rejuvenation with Fractional Derivatives Page: 7
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AGING AND REJUVENATION WITH FRACTIONAL...
The difference between these two equations yieldsd
d[p(t) -P2(t)]
dtp1(t)- p2( t) 1
91- D : [p1(
7 r(2 - tp)t)
(51)clearly showing the two kinds of contribution to the gener-
alized Onsager principle. The first term gives the relaxation
of the perturbation away from equilibrium at the macro-
scopic rate required by Onsager. The second term gives the
additional slow relaxation in the form of the fractional inte-
gral.
IV. AGING ORDER
We have to remark that the condition of Eq. (12) refers to
the stationary condition explicitly considered by Klafter and
Zumofen [29]. In this section we prove that there is a con-
nection between a system's age and the order of the frac-
tional derivative in the relaxation process. A sign of the de-
pendence of the fractional derivative order on age is given by
the discrepancy between the results of Sec. III and Ref. [12].
Let us compare Eq. (49) to Eq. (16) of Ref. [12]. We obtain
the fractional index 1-/3 rather than /3 as in the work of
Sokolov and Metzler. In Appendix A we prove that our time
asymptotic approach to fractional derivatives, in the nonsta-
tionary case studied by Sokolov and Metzler, yields the same
index as they obtain [12]. Thus, the discrepancy between our
prediction and the prediction of Sokolov and Metzler de-
pends on the fact that we consider a condition consistent
with the Onsager principle, whereas Sokolov and Metzler do
not. Furthermore, if the system is not infinitely aged, a sort
of rejuvenation process is expected to take place that will
lead to the fractional order of Sokolov and Metzler.
To support our remarks concerning the relation between
aging and the order of the fractional operator, here we dis-
cuss how to define a waiting time distribution of any age.
The authors of Ref. [13] have shown that the waiting time
distribution '(t) of Eq. (14) is obtained from the following
dynamic model. A particle moves in an the interval I
[0, 1] driven by the equation of motiondy dt= af ,
PHYSICAL REVIEW E 70, 036105 (2004)
at t=-ta+ 71 and ending at t=- ta+ r + 72, and so on. The
waiting time distribution of age ta, denoted by cta(t), is de-
termined by the first of these time intervals overlapping with
t> 0. The time length of that overlap is the time length
whose distribution determines q (t). We make the assump-
tion that the beginning of the first time interval overlapping
with t> 0 occurs with equal probability at any point between
t=-ta and t= 0. The validity of this assumption is discussed
in Appendix B, which establishes that this assumption is very
good for to+ 0 and t+ oc. In between the asymptotic limits
the resulting prediction is not exact. However, since it yields
simple analytical formulas, we adopt this simplifying as-
sumption for any age. Thus, we have thatt f (t+ y)dy
ta(t)= s\ {
where gt) is the normalization factor defined by
where g(ta) is the normalization factor defined byg(t) o
o(53)
(54)
and P(t) is the probability that no event occurs throughout
the time interval of length t. Using for #(t), according to the
prescription adopted in this paper, the analytical form of Eq.
(26), it is easy to prove that Eq. (53) can be written in the
form(t+ 7)(- ) - (t+ + t')(- L)
, (t) = (4 - 2) (2-/) _(a (2-/)(55)
This formula proves that for t< to the index of the distribu-
tion is a- 1, whereas for t> to the index becomes 1a. This
result for the age-dependent waiting time distribution func-
tion agrees with the predictions by Barkai [26] and by the
authors of Ref. [13]. Notice that the formula Eq. (55) is
equivalent to drawing the initial condition for y from an aged
distribution of this variable.
Here, we are in a position to evaluate the waiting time
index at a generic time, where we write ctq(t) as(52)
with z> 1. When the particle reaches the border y= 1, it is
injected back to an initial condition between y= 0 and y= 1
with uniform probability. The age of the CTRW is deter-
mined by the distribution of first exit times. The ordinary
CTRW is based on identifying this distribution with fi(t).
This means that the CTRW is equivalent to assuming that the
system is prepared in a flat distribution at t= 0, which coin-
cides with the beginning of the observation process.
Let us discuss now the consequence of beginning the ob-
servation a significant time after the preparation. Let us
imagine that the system is prepared in a flat distribution at a
time t=-t < 0, and that the observation begins at t=0. This
means that the flat distribution begins producing a sequence
of time intervals of size 7, according to the distribution of
Eq. (14); more precisely, the time interval T1 beginning at t
=-ta and ending at t =-t+ '1, the time interval T2 beginning(56)
A( T, t )
(t+ T)efrt)"Using Eq. (55) we arrive at the following formula for the
time dependence of the effective power-law index:Aef( t)
(57)
ln[(t+ T) )- (t+ T+ ta)(-)]
In[t+ T]Figure 2 illustrates the regression of the effective power-
law index to At with two different ages, and shows clearly
that the regression is slower for older systems. This formula
does more than explain the discrepancy between Eq. (49)
and Eq. (16) of Ref. [12]. In fact, it shows that it is possible
to build a GME that at short times follows the prescription
of our GME and at long times moves into the basin of at-
traction of Sokolov and Metzler. This is certainly the case if
t> -00036105-7
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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/m1/7/: accessed April 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.