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states vanish as the noise intensity (i.e., the variance of the noise) drops below a
critical value depending on the specific spatiotemporal stochastic model considered.
At the same time, in some cases these noise-induced transitions have been found to
be
reentrant
, in that the ordered phase is destroyed when the noise intensity exceeds
another threshold value. In these cases, the noise has a constructive effect only when
the variance is within a certain interval of values. Smaller or larger values of the
variance correspond to conditions in which noise is either too weak or too strong to
induce ordered states.
We concentrate on the case of noise with an isotropic covariance structure given by
sC
|
−
r
|
,
|
−
t
|
r
t
(
r
,
t
)
ξ
(
r
,
t
)
ξ
=
,
(5.2)
d
τ
c
r
|
,
|
t
|
where
s
is the noise intensity,
C
(
) is the correlation function, and
d
and
τ
c
are the noise-correlation length and time, respectively. When
d
|
r
−
t
−
0,
the case of spatially and temporally uncorrelated (i.e., white) noise is recovered. Most
of the research on noise-induced pattern formation considered cases in which
→
0and
τ
→
ξ
m
and
ξ
a
are modeled as white-Gaussian-noise terms. Only a few papers investigated
the impact of colored Gaussian noises, and even fewer studies explored the case of
spatiotemporal dynamics driven by dichotomous noises and white shot noises. For
this reason, unlike in the previous part of the topic, in this chapter we place more
emphasis on dynamics driven by Gaussian noise.
5.1.2.2 Models of spatial coupling
Several mathematical models are frequently used to express spatial coupling among
different points in a field. Because the main focus of this chapter is on pattern for-
mation, we propose to classify the mathematical operators that represent the spatial
coupling as
pattern-forming
and
non-pattern-forming
operators. We define
pattern
forming
as those operators that are able to generate periodic patterns in a determinis-
tic univariate systemwhen suitable parameter values (e.g., the strength
D
of the spatial
coupling) and shape of the function
f
(
) are used (see Appendix B). In contrast, non-
pattern-forming operators may induce spatial coherence, but in a deterministic setting
they do not produce patterns characterized by a clear dominant length scale. Multi-
scale patterns (see Subsection
5.1.1
) may emerge under suitable conditions, but these
are typically transient or oscillatory in noise-free conditions.
A typical example of a non-pattern-forming operator is the Laplacian operator,
φ
2
2
φ
=
∂
φ
+
∂
φ
2
L
[
φ
]
=∇
y
2
,
(5.3)
∂
x
2
∂
which is often used to represent the effect of a diffusive process on a random field. A
possible schematic representation of the effect of
2
on one-dimensional (1D) spatial
dynamics is reported in Fig.
5.6
(a); the initial condition is a field where the values of
∇
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