Environmental Engineering Reference
In-Depth Information
This rather intricate integrodifferential equation shows that, in general,
φ
(
t
) is not a
Markovian process. In fact, the probability distribution of
φ
at time
t
depends on the
integral between
. Equation (
B2.2-7
) can be analytically
solved in only very simple cases (
Bena
,
2006
). An important case is the so-called
persistent diffusion on a line (
f
(
−∞
and
t
of a function of
φ
1) that has interesting applications
in chemistry and physics (
van den Broeck
,
1990
;
Bena
,
2006
).
φ
)
=
0 and
g
(
φ
)
=
In steady-state conditions the temporal derivatives in Eqs. (
2.23
)and(
2.25
)are
equal to zero and the forward Kolmogorov equations become
∂
∂φ
[
P
(
φ,
1
)
f
1
(
φ
)]
+
P
(
φ,
1
)
k
1
−
P
(
φ,
2
)
k
2
=
0
,
∂
∂φ
[
P
(
φ,
2
)
f
2
(
φ
)]
+
P
(
φ,
2
)
k
2
−
P
(
φ,
1
)
k
1
=
0
.
(2.26)
By summing up Eqs. (
2.26
) and integrating with respect to
φ
, we obtain
f
1
(
φ
)
P
(
φ,
2
)
=−
P
(
φ,
1
)
)
,
(2.27)
f
2
(
φ
where the integration constant is set to zero. Equation (
2.27
) inserted into the first of
Eqs. (
2.26
) leads to
∂
∂φ
f
1
(
φ
)
[
P
(
φ,
1
)
f
1
(
φ
)]
+
P
(
φ,
1
)
k
1
+
P
(
φ,
1
)
)
k
2
=
0
.
(2.28)
φ
f
2
(
The integration of (
2.28
) provides the probability distribution
)
exp
k
1
f
1
(
d
φ
C
f
1
(
k
2
f
2
(
P
(
φ,
1
)
=
−
φ
)
+
,
(2.29)
φ
)
φ
φ
where
C
is an integration constant. Equation (
2.29
) can be set in (
2.27
) to obtain
f
2
(
x
)
exp
k
1
f
1
(
d
φ
C
k
2
f
2
(
P
(
φ,
2
)
=−
−
φ
)
+
.
(2.30)
φ
)
φ
We now use these two joint distributions to determine the marginal steady-state
pdf
p
(
φ
) for the state variable
φ
,as
p
(
φ
)
=
P
(
φ,
1
)
+
P
(
φ,
2
)(
Pawula
,
1977
;
Kitahara et al.
,
1980
;
van den Broeck
,
1983
):
=
C
1
f
1
(
exp
k
1
f
1
(
d
φ
1
f
2
(
k
2
f
2
(
p
(
φ
)
)
−
−
φ
)
+
.
(2.31)
φ
φ
)
φ
)
φ
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