Correlation Function and Generalized Master Equation of Arbitrary Age Page: 9
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CORRELATION FUNCTION AND GENERALIZED MASTER...
ps (0) = 1/s.
It is obvious that the inverse Laplace transform of this last
expression yields the initial condition for the correlation
function 4((a)(0)= 1, as it must be, based on the definition of
Eq. (1) when t1 = t and = + 1, a dichotomous variable. Us-
ing this initial condition in (50), it follows that
a)(t) = 1- dti ti) =T (t), (52)
which, as we know from Sec. II, coincides with the exact
expression for the to-old correlation function of Ref. [14].
C. Approximation through iterative expansion,
and truncation
Although we established that Eq. (43) is exact, due to its
implicit nature, it cannot be used to obtain an analytic func-
tion without carrying out a reasonable approximation. The
approximation we select rests on replacing <';)(t) in the
right-hand side of Eq. (43) with the function A(0(t), which
denotes a correlation function of uncertain age, which is,
however, younger than the to-old correlation function defined
by Eq. (43). This correlation function reads
A('(t)= (t) - (t+ (53)
1 - (t)
The easiest way to derive (53) is to adopt the language of
conditional probabilities. The numerator of this expression is
the probability of not finding an event between 0 and t, mi-
nus the probability of not finding an event between -t, and t.
If we call A the condition of no event in the interval
(0, t), and B the condition of at least one event in the interval
(-t, 0], then the numerator of the right-hand side of (53) can
be identified with the joint probability P(A, B). On the other
hand, the denominator of (53) is simply P(B), the probability
of the event B occuring, and therefore the function A"t,)(t)
can be identified, as it should, with the conditional probabi-
lily P(A B), namely with the probability of finding no event
between 0 and t (i.e., it is a correlation function), given an
event between -t, and 0 (i.e., it has an age younger than ta).
Finally, by plugging Eq. (53) into Eq. (43) we obtain
~1a(t) = T*(t+ t) + [1 T- *(t] t) - (t+ t)
1 - t(ta)
(54)
This expression is not exact, but it is analytic. Moreover, the
form of Eq. (43) suggests an iterative approach, and we can
therefore refine our result, by replacing (t')(t) with A(t)(t),
after an arbitrary number of iterations. In Fig. 1 we check the
accuracy of the first-order approximation to the exact expres-
sion given by (54), by comparing it to the exact prediction of
Eq. (40). The curves in Fig. 1 represent a numerical treat-
ment of (40) with different values of t3. In this example t
=2.5 and T= 1.5 (as said, X= 1). The highest curve in the
figure represents the stationary correlation function, namelyPHYSICAL REVIEW E 71, 066109 (2005)
1 10 100
FIG. 1. The to-old correlation function, F,(t), for different val-
ues of ta. Curves represent a numerical integration of (40) with z
5/3 and T= 1.5, while dots correspond to the approximate formula
(54).
t= m. The correlation function 4()'(t) has a faster decay,
with decreasing values of t3. Finally, the lowest curve repre-
sents the correlation function with zero age, namely T* (t).
The symbols in Fig. 1 overlaying the continuous curves rep-
resent the calculations using Eq. (54) with various ages. We
see perfect agreement for t < (t) and for ta > (t). The agree-
ment remains good for intermediate values, with discrepan-
cies comparable to the numerical round-off errors.
V. PHYSICAL CONSEQUENCES OF THE RESULTS
OBTAINED
The results of the earlier sections establish that the GME
of arbitrary age, proposed in this paper, Eq. (30), fits the
requirement of complete equivalence with the current litera-
ture on non-Poisson renewal processes. It is important to
stress that a significant model, generating non-Poisson statis-
tics from a renewal process, has been proposed by Bouchaud
and others [25,26]. This model has raised wide interest and
has recently been applied to a variety of pheonomena
[27-31].
The Bouchaud trap model refers to the spin-glass dynam-
ics, which is modeled with a set of branches, with different
energies E, having a distributionp(E) = pgexp(- /gE),
where Tg 1//3g is the glass transition temperature. For each
branch this model assumes the Arrhenius prescriptiont= to exp(P/E),
where T= 1/3 denotes the sample temperature, and t the
time necessary to overcome the barrier of intensity E, by
means of thermal fluctuations.
Thus, using the relationship between the energy distribu-
tion and the waiting-time distribution,066109-9
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Allegrini, Paolo; Aquino, Gerardo; Grigolini, Paolo; Palatella, Luigi; Rosa, Angelo & West, Bruce J. Correlation Function and Generalized Master Equation of Arbitrary Age, article, June 10, 2005; [College Park, Maryland]. (https://digital.library.unt.edu/ark:/67531/metadc40401/m1/9/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.