Environmental Engineering Reference
In-Depth Information
The probability of occurrence of two or more jumps can be neglected in Eq. (
2.21
)
because it is supposed that
t
is small.
Using a Taylor's expansion truncated to the first order we have
1
P
[
φ,
1
;
t
+
t
]d
φ
=
k
2
t
P
[
φ,
2
;
t
]d
φ
t
)
t
1
t
d
−
∂P
[
φ,
1
;
t
]
∂φ
−
∂
f
1
(
φ
)
+
(1
−
k
1
P
[
φ,
1
;
t
]
f
1
(
φ
)
φ.
∂φ
(2.22)
Notice how Eq. (
2.22
) is independent of the form of the function
f
∗
(
φ
) describing
the trajectory of
φ
in correspondence to jump occurrences. Rearranging the terms,
dividing by d
φ
and
t
, and taking the limit for
t
→
0, we finally obtain the forward
Kolmogorov equation:
∂P
(
φ,
1
,
t
)
∂
=−
∂
∂φ
[
P
(
φ,
1
,
t
)
f
1
(
φ
)]
−
P
(
φ,
1
,
t
)
k
1
+
P
(
φ,
2
,
t
)
k
2
.
(2.23)
t
Analogously, we can write for the probability that at time
t
+
t
the state variable
is within (
φ,φ
+
d
φ
)and
ξ
=
2
,
dn
f
∗∗
(
f
∗∗
(
P
[
φ,
2
;
t
+
t
]d
φ
=
k
1
t
P
[
φ
−
φ
)
t
,
1
;
t
]d[
φ
−
φ
)
t
]
+
(1
−
k
2
t
)
P
[
φ
−
f
2
(
φ
)
t
,
2
;
t
]d[
φ
−
f
2
(
φ
)
t
]
,
(2.24)
where
f
∗∗
(
φ
φ
,
+
) describes the trajectory of
in the interval (
t
t
t
) in the case in
ξ
dn
switches from
1
to
2
in that interval. After a Taylor expansion for
which
t
→
0 we obtain the second forward Kolmogorov equation:
∂P
(
φ,
,
t
)
=−
∂
∂φ
2
[
P
(
φ,
2
,
t
)
f
2
(
φ
)]
−
P
(
φ,
2
,
t
)
k
2
+
P
(
φ,
1
,
t
)
k
1
.
(2.25)
∂
t
We refer the interested reader to Box 2.2 for the derivation of the full master
equation in the time-dependent case. Here we concentrate on steady-state solutions
of (
2.16
).
1
P
[
φ
−
f
1
(
x
)
t
,
1
;
t
] can be expanded in a Taylor's series around
t
=
0:
t
=
0
t
∂P
[
φ
−
f
1
(
φ
)
t
,
1
;
t
]
P
[
φ
−
f
1
(
φ
)
t
,
1
;
t
]
=
P
[
φ,
1
;
t
]
+
∂
t
z
=
φ
t
=
0
∂P
[
z
,
1
;
t
]
∂
z
∂
z
∂
t
=
P
[
φ,
1
;
t
]
+
t
−
∂P
[
φ,
1
;
t
]
∂φ
=
P
[
φ,
1
;
t
]
f
1
(
φ
)
t
,
where the series has been truncated to the first order and
z
=
φ
−
f
1
(
φ
)
t
.
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