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

Search WWH ::

Custom Search