Environmental Engineering Reference
In-Depth Information
Box 2.2: Master equation of a stochastic process driven by
dichotomous Markov noise
In this box, we show the key steps to determine the nonsteady-state master equation of
the stochastic process described by (
2.16
); further details can be found in
Horsthemke
and Lefever
(
1984
).
We introduce the two quantities
P
(
φ,
t
)
=
P
(
φ,
1
,
t
)
+
P
(
φ,
2
,
t
)
,
(B2.2-1)
q
(
φ,
t
)
=
k
2
P
(
φ,
2
,
t
)
−
k
1
P
(
φ,
1
,
t
)
,
(B2.2-2)
where
q
(
φ,
t
) is an auxiliary function, and
P
(
φ,
t
) expresses the time-dependent
probability distribution of
independently of the state of noise.
Adding Eq. (
2.23
)toEq.(
2.25
) and using zero-average condition (
2.8
), we obtain
φ
∂P
∂
t
=−
∂
−
2
−
1
k
1
+
∂
∂φ
P
φ
φ
,
[
f
(
)]
[
qg
(
)]
(B2.2-3)
∂φ
k
2
and, if Eqs. (
2.23
) and (
2.25
) are multiplied by
k
1
and
k
2
, respectively, and then (
2.23
)is
subtracted from (
2.25
), we obtain
∂
f
(
k
2
)
q
q
∂
∂
∂φ
k
2
2
−
k
1
1
t
=
φ
)
+
+
(
k
1
+
k
1
+
k
2
(
2
−
1
)
k
1
k
2
k
1
+
∂
∂
−
x
[
g
(
φ
)
P
]
.
(B2.2-4)
k
2
Using the independent variable
d
φ
η
=
)
,
(B2.2-5)
k
2
2
−
k
1
1
k
1
+
k
2
f
(
φ
)
+
g
(
φ
we can reduce differential equation (
B2.2-4
) to a form that can be analytically integrated
(
Polyanin et al.
,
2002
), leading to
t
e
−
∂
x
f
(
φ
)
+
)
+
k
1
+
k
2
(
t
−
t
)
k
2
2
−
k
1
1
k
1
+
k
2
g
(
φ
q
(
φ,
t
)
=
−∞
(
2
−
1
)
k
1
k
2
k
1
+
∂
∂φ
t
)]d
t
,
×
[
g
(
φ
)
P
(
φ,
(B2.2-6)
k
2
where the statistical independence between the noise
ξ
dn
and the process
φ
(
t
)at
t
→−∞
has been assumed as the initial condition.
Equation (
B2.2-6
) can be substituted into (
B2.2-3
) to obtain the master equation:
∂P
f
(
)
(
φ,
t
)
=−
∂
∂φ
k
2
2
−
k
1
1
φ
)
+
g
(
φ
P
(
φ,
t
)
∂
t
k
1
+
k
2
2
−
1
)
2
(
k
1
+
k
1
k
2
(
∂
g
(
φ
)
+
k
2
)
2
∂φ
t
e
−
∂φ
f
(
x
)
+
g
(
φ
)
+
k
1
+
k
2
(
t
−
t
)
k
2
2
−
k
1
1
k
1
+
×
k
2
−∞
(
2
−
1
)
k
1
k
2
k
1
+
∂
∂φ
t
)]d
t
.
×
[
g
(
φ
)
P
(
φ,
(B2.2-7)
k
2
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