Complexity and the Fractional Calculus Page: 3
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Advances in Mathematical Physics
regime becomes more extended if the probability of gener-
ating a visible cooperative event decreases as discussed in
Section 2.2.
We assume that the time interval between two consecu-
tive critical events generated by the complex system under
study is given by the waiting-time pdf
1
y(r) oc , with 1 < < 2. (7)
The corresponding cumulative distribution (r) has the form
Tlim m () oc 17, with a - 9- 1 < 1. (8)
The origin of this condition, usually interpreted as a mani-
festation of complexity, can either be the anomalous nature
of the dynamics under investigation [30, 31] or the condition
of criticality [32]. In the former case the property described
by (7) can, for example, be the consequence of diffusing
molecules being trapped for long times in wells with a
random distribution of depths. In the latter and less well
known situation the emergence of temporal complexity is
due to the cooperative action of many interacting units. At
the onset of the cooperation-induced phase transition from
disorder to order, the mean field fluctuates and its non-
stationary waiting-time pdf corresponds to an IPL y(t) [33].
We adopt for the Laplace transform of the time function
f(t) the following notation:
f (u) {f (t) ; u} = dt exp (-ut) f (t). (9)
It is important to stress that to satisfy the long-time limit of
(7) the Laplace transform of (r) has the functional form
(u) = 1- uT +E(u), (10)
with the condition on the subsidiary function E(u)a (u)
lim . u = 0.
u-0o uoathereby yielding for the subsidiary function in (10)
() u )2a
(u) = o 1 + (u/A )a,(15)
satisfying the condition of (11). In other words, the properties
of (10) and (11) are fulfilled by all the waiting-time pdf's with
the scale-free condition of (7). We now show that the waiting-
time pdf corresponding to the sum of a large numbers of
times each of which is generated by the generic pdf y(t) of
(7), not necessarily of the MLF type, are MLF waiting-time
pdf's.
2.2. Imperfect Detection of Events. To make this demonstra-
tion as clear as possible and at the same time provide an
intuitive understanding of the SCLT, let us imagine that the
detector used to monitor the events produced by (7) is not
very accurate and that the probability of perceiving these
events isPs < 1.
(16)
As a consequence of this imperfection a time t between two
consecutive visible events is the sum of m elementary times
derived from the conditionP (m) = Ps( - Ps) ,m
(17)
which is the probability that the first m events after the initial
preparation event are not visible while the (m + 1)th event is
visible. For Ps - 0,P (m) = Ps exp (-mPs),
(18)
thereby implying that the standard deviation is on the same
order as the mean((m2>) (m)2)
(m)2(11)
Therefore the subsidiary function must vanish more rapidly
than us as u -* 0.
Note that the Laplace transform of the MLF survival
probability given by (3) is [20, 22]
ua-1
ML (u) - (12)
ua + AOa
and the relation to the Laplace transform of the waiting-time
pdf is
ML () = 1 ML ) (13)
U
so that with a little algebra we obtainPML (U)
(u/Ao)a + 1
(19)
Thus, the condition
1
(m) - o*00
Ps(20)
of the SCLT is quite different from the condition m - oo of
the Gauss and Levy CLTs, since m has very large fluctuations
around (m) in the traditional argument. To better understand
the new theorem, we note that the probability of generating
at time t an event that is the last of a sequence of m
events occurring at earlier times, im(t), does not satisfy the
condition of generating a MLF stable form for m -* oo.
However, the waiting-time pdf of the time intervals between
visible events, does, for Ps -* O0. The pdf of finding a visible
event a time interval t after an earlier visible event is given by
S(t) (1 - P)m +(t). (21)
m=0
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Pramukkul, Pensri; Svenkeson, Adam; Grigolini, Paolo; Bologna, Mauro & West, Bruce J. Complexity and the Fractional Calculus, article, Date Unknown; [Nasr City, Cairo]. (https://digital.library.unt.edu/ark:/67531/metadc268957/m1/3/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.