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
( T (t T)
T (t + 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
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:
2 J a (t + 1
(t + 1)+1
t' 1 dt' +2 t+
FIG. 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.
so that inverting this equation we have the formal ex-
pression for the slow part of the memory kernel
2r (1 - p) (ti+ 1)+1
/1 D 1- [/t3 , + 1 (t)] .
2r (1 - /)
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 to
which can be solved by means of the fractional calculus
. We use the Riemann-Liouville (RL) definition of the
1 f (t') dt'
F (9) Jo (t -t')'
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. ,
oa (t) - ) (t+ 1)2/ < 1
2F(1 - p)r(l + p) (t + 1)2 '
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) +
III. THE EMERGENCE OF FRACTIONAL
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
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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 October 16, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.