Environmental Engineering Reference
In-Depth Information
(
2.90
). In this way the main mathematical difficulty is removed. Thus, a Fokker-Planck-like
equation is formally obtained, and at steady state the pdf of
φ
is obtained as
exp
1
φ
−
τ
ou
(
f
−
fg
/
φ
)
g
2
(
C
g
)
f
(
φ
=
,
p
(
)
φ
)
d
(2.104)
|
g
(
φ
)
|
s
ou
φ
where the ensemble averages are taken over the probability distribution
p
(
φ
) [i.e.,
· =
p
(
).
To avoid this difficulty, we usually approximate the ensemble averages by taking them over
the probability distribution
p
gn
(
φ
)d
φ
]. It follows that it is generally impossible to obtain an explicit expression of
p
(
φ
φ
) corresponding to the white-noise approximation (i.e.,
τ
ou
→
0).
Approximated form (
2.104
) corresponds to the usual steady-state pdf for Gaussian white
noise with an effective noise strength
s
ou
s
eff
(
τ
)
=
)
.
(2.105)
1
−
τ
ou
(
f
−
fg
/
g
f
<
Notice that, for globally stable physical systems (i.e.,
0) and additive noise (i.e.,
g
(
s
ou
; therefore, in these cases, noise
correlation entails a reduction of the effective noise strength with the consequent sharpening
of the pdf with respect to the white-noise case.
We recall that Eq. (
2.104
) is valid not only for small-noise-correlation scales but also
for moderate-to-strong values of
φ
)
=
const), we obtain 0
<
s
eff
(
τ
ou
)
<
s
eff
(
τ
ou
=
0)
=
τ
ou
; however, it requires small values of the noise strength
s
ou
. The decoupling approximation is homogeneous for all values of
φ
and no artificial
boundaries are introduced.
Unified colored-noise approximation (UCNA).
A weak point of the decoupling approx-
imation approach is that it works for small values of
s
ou
. Therefore it is not well suited
to investigate noise-induced transitions (see Chapter 3) that usually emerge for moderate-
to-strong noise intensities. To overcome this restriction,
Jung and Hanggi
(
1987
) proposed
another approach known as the unified colored-noise approximation (UCNA), which pro-
vides the pdf of
φ
in the form (see also
Hanggi and Jung
,
1995
)
C
)]
1
−
τ
ou
g
(
φ
)[
f
(
φ
)
/
g
(
φ
p
(
φ
)
=
g
(
φ
)
exp
φ
φ
)[1
φ
)[
f
(
φ
)
φ
)]
]
f
(
−
τ
ou
g
(
/
g
(
×
d
.
(2.106)
s
ou
g
2
(
φ
)
φ
This expression is valid for both small- and moderate-to-large-correlation times
τ
ou
with
the constraint that
1
τ
ou
+
√
τ
ou
)
g
(
φ
)
f
(
−
φ
)
+
f
(
φ
>
0
.
(2.107)
φ
g
(
)
Notice that, even though the theory developed by these authors (as well as the resulting
time-dependent Fokker-Planck equations) is substantially different from those based on
the small-autocorrelation-scale approximations, the steady-state distribution of
given by
(
2.106
) is precisely the same as the one obtained by Fox within the small-correlation theory
φ
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