Environmental Engineering Reference
In-Depth Information
p
φ
0.0002
0.0001
φ
0.5
1
Figure 5.41. Occurrence of a noise-induced transition in the purely temporal compo-
nent of model (
5.84
) (i.e.,
D
=
0) when the noise strength
s
gn
crosses the threshold
s
c
=
4
a
. The dotted and continuous pdf's refer to
s
gn
=
3 and
s
gn
=
30, respectively
(
a
1 in both cases). The areas of the pdf's appear to be different because of the
truncation of the left peak.
=
To show what happens when
g
(
φ
0
)
=
0, we consider the model
∂φ
∂
D
(
k
0
+∇
2
)
2
t
=−
a
φ
+
φ
(1
−
φ
)
ξ
gn
(
t
)
−
φ,
(5.84)
interpreted according to Ito. The purely temporal version of model (
5.84
) [i.e.,
d
φ/
d
t
=−
a
φ
+
φ
(1
−
φ
)
ξ
gn
] shows a noise-induced transition for
s
c
=
4
a
. In fact,
the steady-state pdf is
exp
a
s
gn
−
2
(1
−
φ
)
a
s
gn
(1
p
(
φ
)
=
−
,
φ
∈
]0
,
1[
,
(5.85)
a
s
gn
+
2
−
φ
)
φ
and always has a mode for
φ
→
0, whereas a new mode occurs at
s
gn
−
3
4
+
4
a
16
s
gn
φ
=
(5.86)
when
s
gn
>
s
c
(Fig.
5.41
shows an example of this noise-induced bimodality). It is
instructive to take a closer look at the causes of this bimodality. In fact, its emergence
is different from the one observed in model (
5.76
). In that case, the function
g
(
φ
)
exhibited a decrease as
0. Thus a sufficiently strong noise
was able to unlock the deterministic stable state, and two modes emerged at values of
φ
φ
moved away from
φ
=
) tended to compensate. Differently, in model
(
5.84
) bimodality is due to cooperation between the noise and the natural boundaries
at
where the actions of noise and
f
(
φ
φ
=
0and
φ
=
1. When the noise is sufficiently strong, the system tends to move
away from
φ
=
0, but the boundary at
φ
=
1 prevents the system from visiting the
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