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 .     Search WWH ::

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