Environmental Engineering Reference
In-Depth Information
Small-correlation expansion.
An often-adopted approach consists of dealing with colored
noises close to the white-noise limit, i.e., for small values of
τ
ou
. Thus, starting from exact
t
)
master equation (
2.90
), the term
φ
(
t
)
/ξ
cn
(
t
) is expanded into a power series in (
t
−
truncated to the first order. We obtain (
Hanggi
,
1989
;
Hanggi and Jung
,
1995
)
)
1
)
f
g
p
(
∂
p
(
φ,
t
)
=−
∂
∂φ
s
ou
∂
∂φ
)
∂
∂φ
[
f
(
φ
)
p
(
φ,
t
)]
+
g
(
φ
g
(
φ
+
τ
ou
g
(
φ
φ,
,
(2.100)
t
)
∂
t
which is the standard small-
τ
ou
approximation of the master equation; therefore the non-
Markovian process
(
t
) has been approximated by a Markovian process characterized by
Fokker-Planck equation (
2.100
), whose solution is
φ
C
p
(
φ
)
=
|
g
(
φ
)
{
1
+
τ
ou
g
(
φ
)[
f
(
φ
)
/
g
(
φ
)]
}|
exp
φ
)d
φ
f
(
×
,
(2.101)
φ
)
φ
)[
f
(
φ
)
φ
)]
}
s
ou
g
2
(
{
1
+
τ
ou
g
(
/
g
(
φ
where
C
is a normalization constant. This expression of
p
(
φ
)[Eq.(
2.101
)] can also be
τ
ou
expansion. The main weakness of this
obtained with other methods based on a truncated
τ
ou
expansion is a singular perturbation expansion, so that the relative
importance of the neglected terms is not easy to estimate. Moreover, the approximation is not
homogeneous for all values of
approach is that the
φ
, and this introduces artificial boundaries at the point where
)]
=
τ
ou
g
(
1. Thus, if the pdf [Eq. (
2.101
)] tends to zero at these boundaries,
the approximation can be considered reasonable, whereas if it diverges other approximation
schemes need to be adopted. To overcome these shortcomings some authors refined the
approach based on the small-
φ
)[
f
(
φ
)
/
g
(
φ
τ
ou
approximation. In particular, we recall the steady-state pdf
determined by
Fox
(
1986
),
)]
|
C
|
1
−
τ
ou
g
(
φ
)[
f
(
φ
)
/
g
(
φ
p
(
φ
)
=
|
g
(
φ
)
|
exp
φ
φ
)
φ
)[
f
(
φ
)
φ
)]
}
f
(
{
1
−
τ
ou
g
(
/
g
(
×
d
,
(2.102)
s
ou
g
2
(
φ
)
φ
and the pdf obtained by
Sancho and San Miguel
(
1989
),
)
f
(
f
(
2
φ
)
1
2
s
ou
φ
)
)
e
−
τ
ou
G
(
φ
)
p
(
φ
)
=
Cp
0
(
φ
,
G
(
φ
)
=
g
(
φ
+
,
(2.103)
g
(
φ
)
g
(
φ
)
where
C
is the normalization factor and
p
0
(
φ
) is the pdf in the case of white noise (i.e.,
τ
ou
=
0).
Equations (
2.102
) and (
2.103
) give in general good results, though their validity remains
limited to (i) small-correlation times and (ii) small values of
s
ou
. Moreover we can
observe that in all the approximated expressions the white-noise case (according to the
Stratonovich interpretation of the Langevin equation) is recovered when
τ
ou
/
0.
Decoupling approximation.
This approach was proposed by
Hanggi et al.
(
1985
) to provide
an expression of
p
(
τ
ou
→
) whose validity is not limited to the case of small-autocorrelation scales
for the noise term. The starting point is to decouple the correlation present in master equation
φ
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