Information Technology Reference
In-Depth Information
remains constant again as we saw in Section
7.2.2
. It is evident that in the long-time
regime (
7.97
) is replaced by
g
min
ω
(
t
)
=
ε
cos
(ω
t
+
).
(7.138)
The transition from
g
0
to
g
min
also has the effect of changing the phase
φ
into the phase
, whose explicit value depends on the details of the transition process, and this change
is not discussed here.
Let us now consider the case in which the probability of event production after an
earlier event,
g
, rather than being constant, is time-dependent. We use the time-
dependent event generator (
3.233
) from Section 3.5.2 and identify the occurrence of
an event with the occurrence of failure. Consider the case when the exponent in (
3.233
)
is given by
(
t
)
η
=−
η<
−
1 would lead to an asymptotic time with no
event at all, and, consequently, to a vanishing response. Thus, it is convenient to con-
sider the singularity condition
1. The condition
1. Let us imagine such a collection of experiments
that are of such high quality as to create a Gibbs ensemble of webs, each of them char-
acterized by the occurrence of an event at
t
η
=−
0. In this case the Gibbs ensemble is
an out-of-equilibrium web that will try to regress to equilibrium. The signature of this
process of regression to equilibrium is given by the function
R
=
(
t
)
, the number of events
produced per unit time by the network
S
prepared at
t
0. The concept of preparation
is fundamental to understanding the non-stationary nature of the condition
=
2. We
assume that our statistical analysis is based on the preparation of infinitely many real-
izations of the sequence
μ<
ξ
s
(
t
)
with the condition that all these realizations have an event
at time
t
0.
The guidelines to presenting the new LRT in an understandable format to as wide an
audience as possible are indicated by the following three steps: preparation, perturbation
and experimentation.
Preparation
. There are webs whose non-equilibrium nature is determined by a cas-
cade of renewal events with a rate
R
=
(
t
)
decreasing in time, as shown in a subsequent
section, according to
1
t
2
−
μ
,
R
(
t
)
∝
(7.139)
after a proper experimental failure rate. Here we use liquid crystals as exemplars of
physical networks belonging to this group [
57
]. The time decay in the number of events
per unit time is the prediction of the hyperbolic form of the waiting-time distribution
function. The mathematical origin of the cascade implied by the inverse power-law
decay in (
7.139
) is a well-known [
28
] property of renewal events with a survival proba-
bility
(namely the probability of waiting longer than a time
t
for an event to occur)
decaying as
t
1
−
μ
, with
(
t
)
μ<
2, for
t
→∞
. In the experiment illustrated subsequently
is generated by the dynamics of interacting defects that are prepared at a time pre-
ceding the application of a weak perturbation to
S
. In the absence of perturbation the
variable
R
(
t
)
ξ
s
is time-independent, in spite of the perennial out-of-equilibrium condition
represented by the ever-drifting quantity
R
ensuing preparation. This non-ordinary
condition is incompatible with the Hamiltonian-based treatments, where events are not
(
t
)
