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Figure 7.9.
One line is twice the response to a perturbation whose intensity is half that of the perturbation
generating the other [
5
]. It is clear that the three data sets coincide. Redrawn with permission.
repeat the experiment, generating several perturbations of different intensity, and then
rescale the results with the perturbation amplitude. The scaled results are illustrated by
Figure
7.9
. This experiment superimposes one curve on top of the other, thereby estab-
lishing that
for small perturbations
the response of the web is linear in the perturbation
intensity.
7.3.4
Theoretical interpretation
The experimental realization of a non-stationary LRT given in the last subsection com-
pels us to address this problem from a theoretical point of view. The first theoretically
interesting question is how to derive (
7.140
) from LRT. On the basis of (
7.73
)wemay
make the conjecture that the new LRT is realized by
d
dt
t
)
=
t
),
χ(
t
,
C
(
t
,
(7.144)
in terms of the two-time autocorrelation function. However, it is also possible to make
the choice
d
dt
C
t
)
=−
t
).
χ(
t
,
(
t
,
(7.145)
In the stationary case the two choices are equivalent, but in the non-stationary case they
are not. A discussion of which is the most convenient choice from the two alterna-
tives was presented by the authors [
4
,
7
], and in the earlier work [
8
]aswell.Herewe
show how to relate these choices to the rate of event generation
R
given by (
7.140
),
which is an essential step in our experiment-based assessment of the non-stationary
LRT. The LRT of a process whose dynamics are dominated by events rather than being
driven by the Hamiltonian of the web
S
, which, even when it exists, is unknown to us,
must be based on the information on the events produced by the ensemble. The central
prescription is given by the age-specific rate of event production [
23
]:
(
t
)
)
=
ψ(
)
t
(
)
.
g
t
(7.146)
(
t
In Poisson processes
g
.
The fully turbulent regime produces a high rate of events of this kind, but the tran-
sition to the defect-mediated turbulent regime makes this high
g
decrease in time.
(
t
)
=
g
is time-independent, implying that
(
t
)
=
exp
(
−
gt
)
