Aging and Rejuvenation with Fractional Derivatives Page: 6
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PHYSICAL REVIEW E 70, 036105 (2004)
/3 t
Da(t)- 2F(1 - )F(1 ) (t+ 1)2' 3<1. (36)
A comparison with 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 in the case /> 1 also. In this
case we find, for the contribution ,a(t) of the GME, the
following time asymptotic behavior:(2 (t+ 1) t
a(t) 2 (t +1)'1/3>1.
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 equation,
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, giving a relaxation dependent
on the average waiting time of the CTRW. The second term
corresponds to a fractional derivative in time, and extends
the Onsager principle to the case of a relaxation with a fat
tail. To obtain these results, we use what we have learned in
the preceding section.
First of all, since we are dealing with the two-site case,
using the form of M and K matrices with z= x= -1 and y
1, we rewrite Eq. (1) in the following form:
d (38)
dtpl(t) = f#(t- 7)[p2(7) - pl(r)]dr, (38)
dto(t) = + a(t)
27(42)
with T= T*/(AU-2)=1/3, thanks to the fact that we set T*
-1. Substituting the decomposition of the memory kernel
into Eq. (38), we obtain
dpl (t) p2(t) 1- ( t)
dt 2 + a(t- )[p2(T) -Pl(T)]dT.
dt 270(43)
(37) Writing Ia(t) as the derivative of an as yet unspecified func-
tion f(t), and using the propertyd t t
d f(t -)g(r)dr= f(0)g(t)+ f'(t- )g(r)dT
dt o o
(44)
with f' (t)= (d/dt) f(t), we obtaindpl(t) p2(t) Pl(t) d t
dt 27* -f()[p2(t) p(t) J 1(t
x [p2() -Pl(T)]dT.T)
(45)
We also found that in the case 0< P3 < 1 the asymptotic be-
havior of the memory kernel is expressed by
2F(1 - P)r(1 + ) (t+ 1)22 1 /p/-2
2F(1 - P)F(1 + P)t-oo T*= 1
(46)p2 (t) D(t- T)(pl(T) -p2(T))dT.
0t(39)
The memory kernel c is related to the autocorrelation func-
tion of the dichotomous variable AC through Eq. (17). Insert-
ing Eq. (17) into the Laplace transform of the set of the
two-site dynamical equations, solving the resulting set of
equations, and taking the corresponding inverse Laplace
transforms yields the solutions1
pi(t) = -{1 - A( (t)[p2(0) - pi(0)]},
1
p2(t) 2[1 (0) - p (0))].Note that these solutions can be combined to yield the gen-
eralized Onsager principle given by Eq. (10) in terms of the
difference in the probabilities.
We now want to find a formal equation of evolution for
the probabilities involving fractional operators. We know
from the preceding section that(40)
(41)Rewriting Eq. (47) as
2F(1 - /3)F(f)( - 1) dt(47)
we identify f(t) with [1/2(1-3)rF(1 -3p)r(3)] t-1 for t-oom.
Choosing (0)= 0 and using the properties of the F function,
we assign to the time asymptotic equation of motion the
form
dpl(t) p (t) - p2(t) 1 d t
dt 27 2F(2 - )F(/) dtJo(48)
- 7) -[p2( 7) -Pl ( ) ]d
or, in terms of the RL fractional integral (34),dp( t)
dtpl(t) - p2(t)
271 D [pl (t)
2F(2 - P)p2(t)].
(49)
The same procedure applied to the equation of motion for
P2(t) yields
dp2(t) p2(t) - pl( t) 1 t
dt 27 2F(2 - ) [(t) 2(t)
(50)036105-6
AQUINO et al.
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Aquino, Gerardo; Bologna, Mauro; Grigolini, Paolo & West, Bruce J. Aging and Rejuvenation with Fractional Derivatives, article, September 10, 2004; [College Park, Maryland]. (https://digital.library.unt.edu/ark:/67531/metadc67638/m1/6/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.