Environmental Engineering Reference
In-Depth Information
p
φ
0
b
0.1
a
0.4
0.3
0.05
0.2
0.1
t
φ
0.2
0.4
0.6
0.8
1
4
2
2
4
φ
−
φ
0
obtained as ensemble average of 10
6
realiza-
tions of model (
5.52
). The initial condition is
Figure 5.24. (a) Behavior of
φ
=
0
.
1, and
s
gn
=
4. (b) Example of
the steady-state pdf of model (
5.52
)for
s
gn
=
4.
L
φ
=
This model - though with spatial coupling expressed by diffusion (i.e.,
[
]
2
) - was proposed by
van den Broeck et al.
(
1994
) as an example of a sys-
tem able to exhibit noise-induced phase transitions (see Section
5.5
) and is indicated
here as the VPT model. Subsequently the same authors investigated the properties
of (
5.49
) to show how multiplicative noise can induce patterns in the presence of a
suitable spatial coupling (
Parrondo et al.
,
1996
;
Garcia-Ojalvo and Sancho
,
1999
).
We note that Eq. (
5.49
) can be also interpreted as dynamics driven by both additive
and multiplicative noise; in fact,
g
(
D
∇
φ
2
)
φ
)
ξ
gn
=
(1
+
φ
ξ
gn
can be split into an additive
2
term,
ξ
gn
.
The stable homogeneous state of the underlying deterministic dynamics (which is
found with
s
gn
=
ξ
a
≡
ξ
gn
, and a multiplicative term,
ξ
m
≡
φ
0) is
φ
(
r
,
t
)
=
φ
0
=
0, and the short-term behavior close to
φ
0
is
given by
≈−
φ
1
2
φ
1
=
d
φ
dt
2
2
+
φ
+
+
φ
φ
.
2
s
gn
f
eff
(
)
(5.50)
The condition for the marginal stability is then
φ
0
=−
d
f
eff
d
1
+
2
s
gn
=
0
,
(5.51)
φ
=
.
φ
<
.
which entails
s
c
0
5.
0
is then stable for
s
gn
0
5 and becomes unstable for
>
.
s
gn
5. However, the amplification of the initial perturbation is transient in the
absence of spatial coupling. This fact is demonstrated in Fig.
5.24
(a), which shows
the temporal behavior of the ensemble average of the zero-dimensional stochastic
model we obtain by eliminating the spatial component from Eq. (
5.49
):
0
d
d
t
=−
φ
2
)
2
2
)
(1
+
φ
+
(1
+
φ
ξ
gn
(
t
)
.
(5.52)
Figure
5.24
(a) shows that the growth phase appears only at the beginning (i.e., in the
short term), whereas in the long run the effect of the initial perturbation disappears.
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