Environmental Engineering Reference
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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 )
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
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)
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