Environmental Engineering Reference
In-Depth Information
D
m
5
1
b
m
0
a
2.5
0.5
m
0
s
gn
s
gn
2.5
5
7.5
10
2.5
5
7.5
10
12.5
Figure 5.32. Results of the mean-field analysis of the VPT model with diffusive
spatial coupling: (a) Curve delimiting the phase transition (the vertical dashed line
marks the threshold
s
c
=
1), (b) behavior of the order parameter
m
as a function of
the noise intensity for
D
=
5.
It is worth recalling that the existence of phase transitions does not necessarily
imply the occurrence of patterns. Pattern formation needs to be investigated through
numerical simulations of Eq. (
5.61
). Using periodic boundary conditions and random
initial conditions, these simulations show (Fig.
5.33
) the emergence of patterns with
features that are qualitatively similar to those already discussed for prototype model
(
5.58
). Regions with strong spatial coherence occur, but do not exhibit any obvious
periodicity. Such regions tend to fade out after some time. Thus, after a transient
(whose duration increases with the domain size), the field appears to be practically
homogeneous with average
m
63. This value is close to the one calculated with
the mean-field analysis for the same noise intensity (see Fig.
5.32
). The final value
of the order parameter
m
is then different from
=
0
.
0 and can be either positive
or negative, depending on the initial conditions. The numerical simulations therefore
confirm that the diffusive VPT model exhibits noise-induced phase transitions and
irregular transient patterns and that no stable periodic structures emerge.
Figure
5.33
shows the probability density distributions corresponding to the sim-
ulated fields. Such pdf's are substantially unimodal (only a weak bimodality occurs
in some realizations). Therefore coherence regions are not due to a temporal noise-
induced transition in the local stochastic dynamics (like those described in Chap-
ter 3), which change the shape of the local pdf from unimodal to bimodal.
φ
=
0
5.6 The role of the temporal autocorrelation of noise
This section investigates the role of colored noise in spatially extended systems. In
particular, we focus on the same prototype models as those described in the previous
sections, but we model the random fluctuations by using DMN instead of Gaussian
white noise. As remarked in Chapter 2, DMN is a simple form of colored noise,
characterized by the autocovariance function
2
−
1
)
2
(
k
1
k
1
k
2
(
ξ
dn
(
t
)
e
−|
t
−
t
|
(
k
1
+
k
2
)
ξ
dn
(
t
)
=
.
(5.68)
+
k
2
)
2
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