Environmental Engineering Reference
In-Depth Information
where
δ
(
·
) is the Dirac delta function,
φ
(
t
) is the solution of stochastic equation (
2.87
)
for a given noise realization
is a value of the
state variable. In Eq. (
2.89
) the ensemble average is taken over the noise realizations
and the initial conditions.
The master equation for the evolution in time of the pdf, corresponding to Langevin
equation (
2.87
)is(
Hanggi
,
1989
;
Hanggi and Jung
,
1995
)
ξ
cn
(
t
) and a given initial condition, and
φ
)
t
0
t
)
d
t
,
∂
φ,
=−
∂
φ,
φ
∂
∂φ
δ
φ
−
φ
p
(
t
)
p
(
t
)
f
(
)
[
(
(
t
)
)]
−
g
(
φ
C
(
t
−
(2.90)
∂
t
∂φ
ξ
cn
(
t
)
where the notation
z
represents the functional derivative of the generic func-
tional
F
(
z
) with respect to the function
z
.
F
(
z
)
/
2.5.1 Solution for linear Langevin equations
Master equation (
2.90
) can be solved (
Hanggi and Jung
,
1995
) only if
φ
(
t
)
/ξ
cn
(
t
)
does not depend on the process
φ
(
t
), so that the ensemble average in Eq. (
2.90
) reduces
to
)
φ
[
δ
(
φ
(
t
)
−
φ
)]
∂
∂φ
δ
(
t
)
=−
∂
p
(
φ,
t
)
φ
(
t
)
=
−
(
φ
(
t
)
−
φ
cn
(
t
)
.
(2.91)
cn
(
t
)
cn
(
t
)
ξ
ξ
∂φ
ξ
In particular, transformation (
2.91
) is valid for all linear processes (or processes
that can be linearized through changes in variables) that can be written in the form
d
t
=
d
a
+
b
φ
+
c
ξ
cn
(
t
)
.
(2.92)
In this case we obtain
φ
(
t
)
s
)
ce
b
(
t
−
t
)
cn
(
t
)
=
θ
(
t
−
.
(2.93)
ξ
The master equation then becomes a Fokker-Planck-like equation,
t
t
)
e
b
(
t
−
t
)
d
t
∂
2
p
(
∂
p
(
φ,
t
)
a
∂
p
(
φ,
t
)
b
∂φ
p
(
φ,
t
)
φ,
t
)
=−
−
+
C
(
t
−
,
(2.94)
∂
∂φ
∂φ
∂φ
2
t
0
whose solution is
exp
[
φ
−
β
(
t
,φ
0
)]
2
−
2
α
(
t
)
p
(
φ,
t
)
=
√
2
,
(2.95)
πα
(
t
)
where
φ
0
is the initial condition (at
t
=
0),
t
2
c
t
0
e
2
b
(
t
−
t
)
D
(
t
)d
t
,
t
)
e
b
(
t
−
t
)
d
t
,
D
(
t
)
C
(
t
−
α
(
t
)
=
=
(2.96)
0
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