Environmental Engineering Reference
In-Depth Information
zero-mean WSN). These limits lead to
)
exp
φ
φ
)
1
f
(
p
(
φ
)
=
C
−
φ
)
d
.
(2.69)
f
(
φ
)
−
λα
g
(
φ
[
f
(
φ
)
−
λα
g
(
φ
)]
α
g
(
φ
We can obtain the same solution by determining and solving the master equation for
WSN (see Box 2.3).
The derivation of the master equation and the steady-state probability distribution
in Box 2.3 holds under the Stratonovich interpretation of Langevin equation (
2.57
)
because the ordinary rules of calculus are used in the transformation to an additive
equation. The master equation corresponding to Langevin equation (
2.57
), interpreted
with Ito's convention is (
Denisov et al.
,
2009
)
p
H
φ
−
φ
g
(
d
φ
φ
,
∂
p
(
φ,
t
)
=−
∂
∂φ
p
(
t
)
φ
.
φ,
φ
−
λ
φ,
+
λ
[
p
(
t
)
f
(
)]
p
(
t
)
∂
t
|
g
(
φ
)
|
φ
)
0
(2.70)
1] this master equation becomes
equivalent to Eq. (
B2.3-7
) because there is no difference between Ito's and Stra-
tonivich's interpretation when the noise is additive (
van Kampen
,
1981
). Unfortu-
nately, no analytical solution is known of Eq. (
2.70
)when
g
(
As expected, when the noise is additive [
g
(
φ
)
=
φ
)
=
1, not even under
steady-state conditions.
The domain of the steady-state pdf (under the Stratonovich interpretation) can be
inferred from the general rule presented for the case of DMN, i.e., that the boundaries
of the domain are defined by the stable points of the dynamics d
φ/
=
φ
d
t
f
1
(
)and
φ/
=
φ
d
d
t
f
2
(
). In the case of WSN the stable points correspond to the zeros of
f
(
φ
)
−
λα
g
(
φ
)and
g
(
φ
). The same conditions already described for DMN apply
also to the case of WSN.
In the case of “externally” imposed bounds [i.e., bounds not emerging from dy-
namics (
2.57
) of the state variable
],
Rodriguez-Iturbe et al.
(
1999b
) showed that the
Markovianity of the process is preserved. In this case the pdf of
φ
may have a spike
(or “atom” of probability) at the bound that can be calculated as explained in the case
of dichotomous noise.
We can now refer to the examples presented in Subsection
2.3.4.1
to show some
applications of Eq. (
2.69
).
Example 2.5:
In this case the pdf is
φ
2
√
λ
e
−
α
−
φ
φ
2
√
αλ
p
(
φ
)
=
B
1
√
α
,
(2.71)
2
where
B
1
[
·
] is the first-order modified Bessel function. A plot of
p
(
φ
) is shown in Fig.
2.12
.
Example 2.6:
in this case the pdf of
φ
is
1
λ
1
+
1
e
−
φ
φ
−
2
−
1
λ
1
α
1
α
φ
=
+
,
p
(
)
(2.72)
where
[
·
] is the gamma function. An example of
p
(
φ
) is shown in Fig.
2.12
.
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