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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