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Consequently we see that the influence of the environment on the evolution of the
Brownian particle, that is, the influence of the chaotic booster, is to asymptotically
replace the exponential decay of the velocity correlation that appears in traditional sta-
tistical physics with an inverse power-law decorrelation. This replacement is due to the
divergence of the microscopic time scale resulting in the loss of time-scale separation
between microscopic and macroscopic dynamics. Therefore the chaos on the micro-
scopic scale is transmitted as randomness to the macroscopic level and the derivatives
and integrals of the ordinary calculus are replaced with fractional derivatives and inte-
grals on the macroscopic level. We now briefly discuss some of the formal and useful
properties of the fractional calculus.
5.1.1
Fractional calculus
It is useful to have in mind the formalism of the fractional calculus before interpreting
models using this formalism to explain the dynamics of complex webs. The fractional
calculus dates back to a question l'Hôpital asked Leibniz, the co-inventor with Newton
of the ordinary calculus, in a 1695 letter. L'Hôpital wanted to know how to interpret a
differential operator when the index of the operator is not an integer. Specifically, he
wanted to know how to take the 1/2-power derivative of a function. Rather than quoting
the response of Leibniz, let us consider the ordinary derivative of a monomial, say the
n
th derivative of
t
m
for
m
>
n
:
D
t
[
t
m
t
m
−
n
]=
m
(
m
−
1
)...(
m
−
n
+
1
)
m
!
t
m
−
n
=
,
(5.22)
(
m
−
n
)
!
where the operator
D
t
is the ordinary derivative with respect to
t
and we interpret
D
t
to mean taking this derivative
n
times. Formally the derivative can be generalized by
replacing the ratio of factorials with gamma functions,
(
m
+
1
)
D
t
[
t
m
t
m
−
n
]=
,
(5.23)
(
m
+
1
−
n
)
and, proceeding by analogy, replacing the integers
n
and
m
with the non-integers
α
and
β
:
(β
+
1
)
D
t
[
t
β
]=
t
β
−
α
.
(5.24)
(β
+
1
−
α)
In a rigorous mathematical demonstration this blind replacement of gamma functions
would not be tolerated since the definition of the gamma function needs to be gener-
alized to non-integer arguments. However, we never pretended to be giving a rigorous
mathematical demonstration, so we take solace in the fact that this generalization can
be done and the details of how to do that need not concern us here. But if you are inter-
ested there are texts that address the mathematical subtleties [
24
], excellent topics on the
engineering applications [
14
] and monographs that focus on the physical interpretation
of such non-integer operators [
31
].
