# Aging and Rejuvenation with Fractional Derivatives Page: 4

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Using Eq. (9) and substituting into it the explicit ex-

pression of T* as a function of ,u, after some algebra, we

obtain( T (t T)

t+T) T(t+T)/3t

T (t + T)(30)

t

The two terms on the left-hand side Eq. (30) are positive.

Due to the negative sign on the right-hand term of this

equation we conclude that it might well be that (a (t) is

always negative.

Let us concentrate on the case 3 < 1: using the auto-

correlation function Il (t) of Eq. (21) (with T* = 1) and

using the change of time variable t' + 1 - t', we rewrite

Eq. (30) in the form:t+l

2 J a (t + 1

0

/3t

(t + 1)+11

t' 1 dt' +2 t+

o0FIG. 1: The slow component of the memory kernel ((t),

(a(t), as a function of time. The black dots denote the re-

sult of the numerical inversion of the expression in Laplace

transform resulting from Eqs. (14) and (27) for 3 = 0.5, the

continuous line is the analytical approximation given by Eq.

(35).

so that inverting this equation we have the formal ex-

pression for the slow part of the memory kernel1

t') dt'

t'Wa (t)

(31)

1 1t

2r (1 - p) (ti+ 1)+1

/1 D 1- [/t3 , + 1 (t)] .

2r (1 - /)(34)

In the limiting case t - oc we neglect the second term

on the left-hand side of this equation. This is a natural

consequence of the assumption that the memory kernel

must tend to zero with a negative tail, as an inverse power

law. With this assumption it is straightforward to prove

that the modulus of the first term becomes much larger

than that of the second-term on left-hand side of this

equation. The consequences of this crucial assumption

are supported by the numerical results depicted in Fig.

(1). With this assumption Eq. (31) simplifies totdt

0/3t

(t 1)+1'(32)

which can be solved by means of the fractional calculus

[2]. We use the Riemann-Liouville (RL) definition of the

fractional integral1 f (t') dt'

F (9) Jo (t -t')'(33)

which is the anti-derivative of the fractional derivative

with order q, with q < 1. Consequently for 3 < 1 we can

express (32) in terms of the RL fractional integral

D [()] = - 1 /3t

tD [a 2(t)] 21(1 -/) (t + 1)0+1'We denote the correlation function with the inverse

power (p + 1) by the symbol (,I+1 (t). Carrying out

the required calculations on the right-hand side of (34),

we obtain, see for example pg.90 of West et al. [2],t0

oa (t) - ) (t+ 1)2/ < 1

2F(1 - p)r(l + p) (t + 1)2 '(35)

A comparison with a numerical inversion of the kernel is

shown in Fig. (1).

For the sake of completeness, it is worth noticing that

we can proceed in a similar way also in the case 3 > 1. In

this case we find, for the contribution (a(t) of the GME,

the following time asymptotic behavior/3(/- 1) t

oa () t=+1) , > 1.

2 (t + 1) +(36)

III. THE EMERGENCE OF FRACTIONAL

OPERATORS

In this section we show that in the two-site case we are

discussing, the GME has the form of a transport equa-

tion, with two terms on the right-hand side. The first

has the form afforded by the ordinary master equation

and consequently satisfies the Onsager Principle, giv-

ing a relaxation dependent on the average waiting time

of CTRW. The second term corresponds to a fractional

derivative in time, and extends the Onsager Principle to( (t)

-0.002

-0.004-0.006

0100 200

t300

R

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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/m1/4/: accessed December 18, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.