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provides interrelated fluctuations and dissipation. In the motion of a Brownian particle
the force fluctuations were found to have Gaussian statistics and to be delta-correlated
in time. The response of the Brownian particle at equilibrium to these fluctuations is
determined by the canonical distribution. However, the statistics of the heat bath need
not be either Gaussian or delta-correlated in character, particularly for non-physical
networks. So another way complexity can enter network models is through modification
of the properties of the fluctuating force in the Langevin equation.
We advocate an approach to complexity that is based on a phase-transition perspec-
tive. Complexity emerges from the interaction among many elements located at the
nodes of a web, and the complex structure of the web makes it possible to realize this
phase transition without using the frequently made approximation that every node of the
network interacts with all the other nodes (all-to-all coupling). We return to this point
in Chapter 5. Nevertheless, the web will not leave a given state for all time, but under-
goes abrupt transitions from one state to another. These transitions may be visible or
they may be hidden by the emergence of macroscopic coherence. In order to establish
a satisfactory command of this complexity condition it is necessary to become familiar
with the notion of renewal and non-renewal statistics.
3.4.1
Poisson statistics
Let us consider the simplest statistical process that can be generated by dynamics,
namely an ordinary Poisson process. Following Akin
et al.
[
2
], we use a dynamical
model based on a particle moving in the positive direction along the
Q
-axis and confined
to the unit interval
I
≡[
0
,
1
]
. The equation of motion is chosen to be the rate equation
dQ
(
t
)
=
γ
Q
(
t
),
(3.185)
dt
where
Statistics are inserted into the deterministic equation (
3.185
) through the
boundary condition; whenever the particle reaches the border
Q
γ
1
.
1, it is injected back
into the unit interval
I
to a random position having uniform probability. Throughout
this discussion we refer to this back injection as an event, an event that disconnects
what happens in one sojourn on the interval from what happens during another.
With this assumption of randomness for back injection the initial value for the
solution
Q
=
(
t
)
after an event is a random variable
Q
(
0
)
=
ξ
(3.186)
with
standing for a random number in the unit interval
I
. Once the initial time has been
set, the sojourn time of the particle in the unit interval is determined by the solution to
(
3.185
),
ξ
e
γ
t
e
γ
t
Q
(
t
)
=
Q
(
0
)
=
ξ
.
(3.187)
We denote by
t
the total time it takes for the particle to reach the border
Q
=
1 from a
random initial point
ξ
.Using(
3.187
) and setting
Q
(τ)
=
1, we obtain
e
−
γτ
,
ξ
=
(3.188)
