Aging and Rejuvenation with Fractional Derivatives Page: 9
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AGING AND REJUVENATION WITH FRACTIONAL...
done while the bath is drifting toward a condition that will
never be reached. This aging effect affects the form of the
first exit time distribution, whose index is /- 1 rather than /1.
However, after the first exit the trajectories are injected back
with a uniform probability, and thus all the ensuing jumps
are determined by the ordinary waiting time distribution i(t).
It would be desirable to have an equation of motion with
a fractional operator order that changes as a function of time
from 3 - a to a-2. However, there are technical and concep-
tual difficulties that make it difficult, if not impossible, to
realize this goal. In fact, according to the perspective adopted
in this paper, the order of the fractional operator is estab-
lished using time asymptotic arguments. Thus, if ta< o we
can associate a time t< to with the order 6=- -2 of the frac-
tional derivative. This is so because for t< to the calculations
would be virtually equivalent to those done in Appendix A. It
is not clear how to proceed when t is of the order of ta, this
being the first reason why assigning a fractional derivative
order to any time might be difficult. There also exists a
physical reason that might make it impossible to move from
the order 3-1a to 1-2. Physically, this extremely extended
transition process might involve a mixture of fractional de-
rivatives of different orders. It has been shown [31] that the
Levy walk does not have well-defined scaling, due to aging
effects. Similarly, the adoption of a fractional time deriva-
tive, with time-dependent order, might be inadequate to ex-
plore the regime of transition from the order 3-1a to the
order /-2. In conclusion, the fractional operator and its or-
der reflect a stable condition, of a brand new or infinitely old
system. The regime of transition from the dynamic to any of
these two thermodynamic regimes, and the regime of transi-
tion from the earlier to the latter thermodynamic regime, is
not yet a fully understood physical condition, an issue calling
for further investigation.
It is interesting to notice that, even if we select 1 > 2 and
consequently adopt a condition compatible with the station-
ary condition, the effective index of the first exit distribution
is located in the nonstationary region if /3< 1. This is prob-
ably the reason why the memory kernel seems to share the
same properties as those adopted in Refs. [6,10,32] to pro-
duce subdiffusion. Notice that the baths used by Lutz [6] and
Pottier [32] have properties quite different from the subordi-
nation perspective of Ref. [10], even though the relaxation
process stemming from subordination [10] is quite similar to
that produced by the non-Ohmic baths of Lutz and Pottier.
We hope that the present work might help in understanding
the connection between the two perspectives. This is another
subject for future research.
ACKNOWLEDGMENTS
G. A. and P. G. gratefully acknowledge financial support
from ARO through Grant No. DAAD190210037.
APPENDIX APHYSICAL REVIEW E 70, 036105 (2004)
using fi(t) rather than V,*(t). Thus, according to the procedure
adopted in this paper, we use the analytical form of Eq. (26).
For the reader's convenience, we rewrite this expression
here,T , -1
f(t) = ( - 1)(T+ _) "(Al)
It is important to stress that in the text we made the choice
of Eq. (20) for V,*(t), in this way determining, through Eq.
(24), a form for V(t) that departs from Eq. (Al). However,
both choices lead to the same time asymptotic behavior of
Eq. (14) for both the waiting time distributions.
We note that for u--o we get a finite value (oo)= (c
-1) /T corresponding to /,(0) in t space. As in Sec. II, we
separate the kernel into two contributions: c(t)=[(
-1)/ T]S(t)+?a(t), and insert the Laplace transform of sepa-
ration into Eq. (8), to write
1- (u)r 't,)
1 - i(u)
We introduce /D(u) as the Laplace transform of the distribu-
tion's derivative i,'(t)=-/p(/i- 1) /-1(T+t) +1; using the
Laplace transform of an inverse power law and substituting it
into Eq. (A2) we have() - ( -1)2 (1 - u )(eU - E )
_ U - 1)F(- ,) u(e - E
u1 -- 1)F(1 - pu)(e"- E"z)'(A3)
where, as usual, for simplicity we have set T- 1. Anticipating
the convolution form of the solution we cross-multiply to
obtain
[U - ( - 1)r(1 - p)(e"- E" )] a(u) = ( - 1)[g
- 1)F(1 -/)(e- E~ _)] - (- 1i)u(e"- E). (A4)
Using the relation [2]" (raa
J +exp[- ut]dt= sin ra(E
and setting a= - 1, we construct
1 t
f( - 1) 0
It , -I
-i j) t' a(t- t')dt',
o0 '-eua), a> -1,
(A5)
(sin - 1)F(1
7T"(A6)
In this appendix we show how to obtain the order of the
fractional operator in the GME for 1 <u <3, using the
Laplace transform form for P(t) given by Eq. (8). We arewhere R(t) is the inverse Laplace transform of the right side
of Eq. (A4). Using the well known recurrence relation of the
F function, we can combine terms in Eq. (A6) to obtain036105-9
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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/m1/9/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.