Environmental Engineering Reference
In-Depth Information
Box 2.4: The master equation for processes driven by
Gaussian white noise
The master equation for a process driven by Gaussian white noise is known as the
Fokker-Planck equation; the derivation of the Fokker-Planck equation is a standard
problem treated in several textbooks on stochastic processes, including those of
Parzen
(
1967
),
Gardiner
(
1983
), and
van Kampen
(
1992
), among others. The interested reader
is referred to these topics for details. The Fokker-Planck equation corresponding to
Langevin equation (
2.80
) under the Stratonovich interpretation is
∂
g
(
t
)]
p
(
φ,
t
)
=−
∂
[
f
(
φ
)
p
(
φ,
t
)]
s
gn
∂
∂φ
∂
∂φ
+
φ
)
[
g
(
φ
)
p
(
φ,
.
(B2.4-1)
∂
t
∂φ
This equation has the standard form of a convection-diffusion equation, with diffusion
coefficient
s
gn
. The steady-state solution of Eq. (
B2.4-1
)is(
2.83
). Transient solutions in
the form of time-dependent probability distributions exist in particular cases, including
the Ornstein-Uhlenbeck (O-U) process [represented by Eq. (
2.80
) with
f
(
φ
)
=−
γφ
and
g
(
1]: In this case the transient solution is a Gaussian distribution with average
φ
0
e
−
γ
t
and variance 2
s
gn
/γ
1
φ
)
=
e
−
2
γ
t
, where
φ
0
is the initial state of the system.
Under Ito's interpretation, the Fokker-Planck equation corresponding to Langevin
equation (
2.80
)is
−
2
∂
p
(
φ,
t
)
=−
∂
[
f
(
φ
)
p
(
φ,
t
)]
s
gn
∂
)
2
p
(
+
2
[
g
(
φ
φ,
t
)]
,
(B2.4-2)
∂
t
∂φ
∂φ
with the steady-state solution
)
2
exp
φ
φ
)
s
gn
g
(
C
g
(
f
(
p
(
φ
)
=
φ
)
2
d
,
(B2.4-3)
φ
φ
where
C
is an integration constant that ensures that the pdf has unit area. Equation
(
B2.4-3
) differs from (
2.83
) only for the presence of the exponent 2 in the function
g
(
φ
)
in the denominator.
To exemplify the application of Eq. (
2.83
), we report the solutions for the two
examples previously defined.
Example 2.7:
The steady-state pdf is
C
exp
8
4
φ
−
φ
p
(
φ
)
=
.
(2.85)
4
s
gn
Example 2.8:
Under the Stratonovich interpretation, the steady-state pdf is
exp
+
φ
3
C
φ
4
φ
=
−
.
p
(
)
(2.86)
2
s
gn
φ
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