# Aging and Rejuvenation with Fractional Derivatives Page: 2

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of a rearrangement process that may take infinitely long

to complete, leading to the generalized master equation

(GME) of Kenkre et al. [17]. Allegrini et al. [19] were

concerned with how to make the GME stationary and

therefore compatible with the Onsager Principle. Herein,

we extend that discussion to include the connection with

both the fractional calculus and the nonstationary con-

dition.

For simplicity we restrict our discussion to the case

of a two-site system coupled to a heat bath, a problem

closely related to the quantum problem of decoherence

of a qubit, due to coupling to the environment. The con-

nection of the CTRW, a two-state non-Markov master

equation and the fractional calculus has been explored

by Sokolov and Metzler [12]. Herein we go beyond their

discussion and investigate the relationships among the

fractional calculus, a particular generalization of the On-

sager Principle, and one concept of aging. We also show

that the result of Sokolov and Metzler, which refers to

the young state, is an attractor for all the systems that

are partially aged, and not infinitely aged. We call this

process rejuvenation.

A. Beyond the Onsager Principle

We approach the subject of fractional derivatives from

a perspective similar to that of Sokolov and Metzler [12].

More specifically, we address the problem of the connec-

tion between the GME [17] and the stationary version of

the CTRW [15]. The GME considered by Allegrini et al.

[19] is the two-site version of the following equation

dp(t) _- (t- t')Kp(t')dt', (1)

where p (t) is the m-dimensional population vector of m

sites, K is a transition matrix between the sites and (t)

is the memory kernel. The CTRW prescription for this

process yields

p (t) = f dt'/n (t') T (t - t') Mnp (0) . (2)

n=O

Note that ~, (t) is the probability that n jumps occurred

and that the last jump took place at time t = t', implying

the renewal theory relation

4n (t) = n-1 (t -t') i/1 (t') dt', (3)

where 41 (t) is the waiting time distribution function

S(t) introduced into CTRW and o0 (t) = (t). While

M is the transition matrix connecting the sites after onejump has occurred, the probability that no jump occurs

in the time interval (0, t) is(4)

The waiting time distribution function and the mem-

ory kernel can be related to one another by taking the

Laplace transform of the GME (1) and the CTRW (2).

This comparison, after some algebra [19], yields(5)

1 (u)

(u)K =~~1 ) (M- I),

1 - 4;()where I is the m x m unit matrix and the Laplace trans-

form of the function f (t) is f (u). Here, as in Allegrini

et al. [19], we limit our discussion to the two-state case

whereM =( l)

and

1

-1thereby reducting (5) to

(6)

(7)

(8)

1 u (U)

This relation between the Laplace transform of the mem-

ory kernel and the Laplace transform of the waiting time

distribution function was first obtained by Kenkre et al.

[17] and is reviewed by Montroll and West [20].

In the case when the lattice has only two sites,

a left and a right site, the random walker corre-

sponds to a dichotomous signal , with the values

((t) = -1, for the left site, and ((t) 1, for the

right site. For the sake of simplicity, we assume the

two states to have the same statistical weight. Also

in the two-state CTRW, if we adopt a discrete time

representation, the motion of the random walker cor-

responds to a symbolic sequence {(}, with the form

{+++++++-+ --++++-------....},

which shows a significant persistence of both states.

The waiting time distribution 4 (t) is the distribution

of the patches filled with either +'s or -'s. We assume

symmetry between the two states and a finite first

moment of 4 (t) making it possible for us to define the

autocorrelation function for the fluctuations ( (t)< (o)((t) >

((t) =

< 2 >(9)

W (t) - (t') dt'.

Jt

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Aquino, Gerardo; Bologna, Mauro; Grigolini, Paolo & West, Bruce J. Aging and Rejuvenation with Fractional Derivatives, paper, February 2, 2008; (digital.library.unt.edu/ark:/67531/metadc174699/m1/2/: accessed October 15, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.