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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