Environmental Engineering Reference
In-Depth Information
Box 3.1: Probabilistic potential
In Chapter 2 we showed that the steady-state pdf for dynamical system governed by a
stochastic differential equation forced by multiplicative white Gaussian noise is, under
the Stratonovich interpretation,
)
exp
1
φ
φ
)
g
2
(
C
g
(
f
(
φ
=
.
p
(
)
φ
)
d
(B3.1-1)
φ
s
gn
φ
This pdf is often written in the form
C
exp
−
V
(
φ
)
p
(
φ
)
=
,
(B3.1-2)
s
gn
where
V
(
φ
) is defined as the
probabilistic potential
and can be expressed as
φ
)
g
2
(
f
(
φ
+
V
(
φ
)
=−
φ
)
d
s
gn
log
g
(
φ
)
.
(B3.1-3)
φ
In particular, if
g
(
φ
)
=
1 (i.e., when the noise is additive), the potential becomes
φ
)d
φ
V
(
φ
)
=−
f
(
(B3.1-4)
φ
and coincides - apart from an irrelevant integration constant - with the deterministic
potential
V
(
)].
The idea of using the probabilistic potential to detect the transitions is based on the
following rationale: Once the pdf
p
(
φ
) of the underlying deterministic process [(i.e.,
f
(
φ
)
=
d
V
(
φ
)
/
d
φ
) is obtained for a generic stochastic equation
driven by one of the noises considered in this chapter (dichotomous noise, shot noise,
etc.), it is possible to obtain the probabilistic potential
φ
) of an equivalent stochastic
process drinen by additive white Gaussian noise that would give the same pdf. In fact,
by using Eq. (
B3.1-2
), we obtain
V
(
φ
V
φ
∝−
φ
.
(
)
log
p
(
)
(B3.1-5)
In this way the crests and hollows (or
basins of attraction
) of the probabilistic potential
V
). Therefore the behavior of the
stochastic dynamical system can be immediately visualized as the trajectory of a
hypothetical ball that moves on a surface that has the same shape as the probabilistic
potential. When the ball is forced by additive white Gaussian noise, it will tend to
remain within its basin of attraction and it will move to another basin of attraction (if
any) as an effect of strong fluctuations in the random driver. Structural changes in the
behavior of the system (i.e., noise-induced transitions) are visualized as changes in the
number and the location of the basin of attractions.
(
φ
) coincide with the modes and antimodes of
p
(
φ
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