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where we have matched the time dependence on both sides of the equation. On
comparing the exponents in (
3.72
) it is evident that
α>
0
.
5 implies a positive long-
time correlation, whereas
5 implies a negative long-time correlation. So let us
summarize the result of this simple theoretical analysis with one eye on Figure
3.3
:
α<
0
.
>α>
.
>β>
;
solid curve:
1
0
5 and 1
0
<α<
.
>β>
dashed curve: 0
0
5 and 2
1.
Thus, we see that the autocorrelation function depicted by the solid line in Figure
3.3
leads to superdiffusive behavior ranging from standard diffusion (
α
=0
.
5) to ballistic
behavior (
= 1). The autocorrelation function corresponding to the dashed curve leads
to subdiffusive behavior ranging from standard diffusion to no motion at all.
It should be stressed that the autocorrelation function for a superdiffusive process
diverges asymptotically, which is to say according to (
3.70
),
D
α
; whereas,
from the autocorrelation function in the subdiffusive case, the diffusion coefficient
remains finite,
D
(
∞
)
=∞
. At early times in the diffusion process the mean-square
value of the dynamic variable
Q
(
∞
)<
∞
increases; then, when the negative part of the
autocorrelation function becomes important, the rate of diffusion decreases. When the
negative tail completely compensates for the positive part of the relaxation process,
the rate of diffusion virtually vanishes. At this late stage further diffusion is rigorously
prevented and the diffusing entity becomes localized. Processes of this kind have been
discovered [
13
] and the theory presented here affords a remarkably straightforward
explanation of them. It is interesting that such complex processes should admit such
a straightforward interpretation.
(
t
)
3.2.2
Linear response theory (LRT)
Large aggregates of people, bugs or particles can arrange themselves into remark-
ably well-organized assemblies. The chemical interactions among particles can produce
stripes, pinwheels or even oscillating colors in time. People fall into step with one
another as they casually walk together and audiences can spontaneously clap their hands
in rhythm. Fireflies can blink in unison even though there is no coordinating external
leader. These kinds of synchronizations are the consequence of the internal dynamics of
complex webs, which we have not taken into account in our discussion so far but take up
in subsequent sections. For the moment we are interested in the simplest response of the
web to a complex environment, both in terms of a stochastic force and a deterministic
perturbation.
Let us return to the Langevin equation for a Brownian particle
dV
(
t
)
=−
λ
V
(
t
)
+
ξ(
t
)
(3.73)
dt
which we wrote down earlier. The formal solution to this equation is
t
e
−
λ(
t
−
t
)
ξ(
t
)
dt
V
(
t
)
=
(3.74)
0
