# Aging and Rejuvenation with Fractional Derivatives Page: 3

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where

S- - 2. (22)

Thus, in the case ,u < 3, the auto-correlation function of

the fluctuations is not integrable.

Zumofen and Klafter [22], 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 by4 *(u)

2 (u)

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 [23] and, to a lesser extent,

the results of Allegrini et al. [19]. 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 KERNELThis 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

asymptotic behavior:

1

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

[12]. 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.

[19], which is proven to be a subordination to a Markov

master equation through the stationary distribution ofTo 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 [24]:u (u)= (1 -/) (e E

ul- ('E(26)

where 0 < 3 < 1, given the fact that we are consider-

ing 2 < ,u < 3, and E_ 1 is the generalized exponential

function [2]. 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 followsS(u) = 2+ (u) .

27(27)

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(28)

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 [25].

Thus, for the time evolution equation of the correlation

function of ((t), we derive the following equation

t

dt (t) - 2 (t - t') (t') dt'. (29)

0- (t)

D (t = (* + (t) .

2T*

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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.