Aging and Rejuvenation with Fractional Derivatives Page: 3
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S- - 2. (22)
Thus, in the case ,u < 3, the auto-correlation function of
the fluctuations is not integrable.
Zumofen and Klafter , in addition to explaining
with clear physical arguments the connection between
0(t) and 4*(t), established that the Laplace transforms
of the two functions are related one to the other by
1 + (u)
first exit times. On the other hand, we extend the ap-
proach to systems of any age and reveal the phenomenon
of a continuous time random walk with rejuvenation. To
accomplish this dual role we rely heavily on the results
recently obtained by Barkai  and, to a lesser extent,
the results of Allegrini et al. . However, this allows
us to reveal some aspects of aging and rejuvenation de-
pendent on the order of fractional derivatives, which were
not previously identified.
II. THE INVERSE LAPLACE TRANSFORM OF
(23) THE MEMORY KERNEL
This important relation allows us to establish a connec-
tion between T and T*, which turns out to be
S= 27T. (24)
The equivalence between Eq. (14) and Eq. (16) rests
on the key property of Eq. (23). We note that if we make
the choice of Eq. (13) then the waiting time distribu-
tion b* loses the simple analytical form of Eq. (19), and
viceversa. On the same token, the choice of the analyti-
cal form of Eq. (13) for 0(t) makes the auto-correlation
function #e(t) lose the analytical form of Eq. (21). How-
ever, using the property of Eq. (23), it is straigthfor-
ward to prove that 0(t) with the form of (13) yields, for
the auto-ocorrelation function ( (t), the following time
tD (t) ' (25)
Thus, whatever choice is made, either the analytical form
of Eq. (13) or the analytical form of Eq. (19), in both
cases the two waiting time distributions maintain the
same time asymptotic behavior, with the same ,p. So
do the two different expressions for the equilibrium auto-
autocorrelation functions, the time asymptotic equiva-
lence being the property that matters to study the emer-
gence of fractional derivative.
Herein, using the inverse Laplace transform of (16) we
determine the unknown memory kernel # (t), making it
possible to discuss how to express the GME in terms
of fractional derivatives. The case where 2 < ,u < 3
is compared to the recent work of Sokolov and Metzler
. We find that the index of the fractional deriva-
tive is 3 - ,u, rather than ,u - 2, as predicted by Sokolov
and Metzler. We prove that this difference in index is
due to the fact that we adopt a stationary condition,
while Sokolov and Metzler do not. We also prove that
in the case of a finite, rather than infinite age, our GME
makes a transition from the (3 - ,u)-th to the (,u - 2)-th
order. The stationary case becomes stable only in the
limiting case of infinite age. Thus, on the one hand we
shed light on the meaning of the work of Allegrini et al.
, which is proven to be a subordination to a Markov
master equation through the stationary distribution of
To establish the form of the unknown memory kernel
(t), we make a few preliminary observations. First of
all, we note that through Eq. (16) we establish a direct
connection with the correlation function ( (t) and that
this correlation function is, in turn, directly related to
the waiting time distribution 4*(t), through Eq. (17).
Thus, with no loss of generality for the reasons illus-
trated in Section 1, it is convenient to refer ourselves to
*b*(t) rather than to 4(t). For simplicity, we set T* = 1
throughout the section. Thus, the Laplace transform of
the autocorrelation function is :
u (u)= (1 -/) (e E
where 0 < 3 < 1, given the fact that we are consider-
ing 2 < ,u < 3, and E_ 1 is the generalized exponential
function . Thus, #4 (u) diverges as u - 0 and Eq.
(12) yields (0) 0= . We explore the opposite limit,
u - oc using Eq. (14), which yields ( (u) = = .
In the time representation, the latter limit is equivalent
to l (t) 2 (t for t O0. Therefore, we segment the
Laplace transform of the GME memory kernel into two
parts as follows
S(u) = 2+ (u) .
The first term models the short-time limit, while the sec-
ond term is responsible for the long-time behavior. In
the time representation we have
Note that this division of the memory kernel into a white-
noise contribution and a slow term corresponds to a sim-
ilar partition made by Fuliiski .
Thus, for the time evolution equation of the correlation
function of ((t), we derive the following equation
dt (t) - 2 (t - t') (t') dt'. (29)
D (t = (* + (t) .
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Aquino, Gerardo; Bologna, Mauro; Grigolini, Paolo & West, Bruce J. Aging and Rejuvenation with Fractional Derivatives, paper, February 2, 2008; (https://digital.library.unt.edu/ark:/67531/metadc174699/m1/3/: accessed March 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.