Environmental Engineering Reference

In-Depth Information

Example 2.4:

φ

(
t
) follows logistic-type deterministic dynamics
f
(

φ

), perturbed by a di-

chotomous noise modulated by a linear
g
(

φ

):

f
(

φ

)

=
φ

(

β
−
φ

)

,

g
(

φ

)

=
φ,

(2.20)

where

β>

0 is a parameter. An example, calculated with

β
=

1, is shown in Fig.
2.3
(d).

2.2.3.2 Derivation of the steady-state probability density function

This subsection is devoted to obtaining the steady-state probability density function

(pdf ) for the process described by Langevin equation (
2.16
). The standard procedure

typically followed to address this task involves (i) deriving the master equations for

the process, i.e., the forward differential equations that relate the state probabilities

at different points in time; (ii) taking the limit as
t

in the master equation

to attain statistically steady-state conditions; and (iii), solving the resulting forward

differential equation to find the steady-state pdf. The detailed derivation of the steady-

state probability distribution of

→∞

φ

following this approach is described in Box 2.2. In

this subsection, we describe a simpler approach in a way that the nonexpert reader

can more easily follow how the solution of Langevin equation (
2.16
) is determined.

Consider the probability that, at time
t
+
t
, the state variable takes a value con-

tained within the interval [

φ,φ
+

d

φ

] and the noise is in state

ξ
dn
=
1
. These

conditions may be attained either when

ξ
dn
=
1
at time
t
, no jumps of

ξ
dn
occur in

[
t

,

t

+

t
], and the value of

φ

at time
t
is

φ
−

f
1
(

φ

)

t
, or when at time
t
we have

ξ
dn
=
2
, the random variable

ξ
dn
shifts from

2
to

1
in the interval [
t

,

t

+

t
], and

f
∗
(

t
,where
f
∗
(

the state variable at time
t
is

φ
−

φ

)

φ

) is a suitable combination of

f
1
(

φ

)and
f
2
(

φ

) to account for the fact that in the interval [
t

,

t

+

t
] both functions

contribute to determine the trajectory of

φ

. Thus the joint probability that

ξ
dn
=
1

and

φ

is comprised within [

φ,φ
+
φ

] can be expressed as

P

[

φ,

1
;
t

+

t
]d

φ
=

(1

−

k
1

t
)

P

[

φ
−

f
1
(

φ

)

t

,

1
;
t
]d[

φ
−

f
1
(

φ

)

t
]

f
∗
(

f
∗
(

+

k
2

t

P

[

φ
−

φ

)

t

,

2
;
t
]d[

φ
−

φ

)

t
]

.

(2.21)

The first term on the right-hand-side of Eq. (
2.21
) is the product of three factors:

(i) the probability that the noise

ξ
dn
remains in the state

ξ
dn
=
1
in the interval

(
t

,

t

+

t
). This probability is 1 minus the probability
k
1

t
that a jump occurs from

1
to

2
in the same interval; (ii) the joint probability

P

that at time
t
noise is equal

to

1
and the state variable is at

φ
−
φ

,where

φ
=

f
1
(

φ

)

t
from Eqs. (
2.15
); and

(iii) the infinitesimal amplitude of the interval d[

φ
−

f
1
(

φ

)

t
]. Similarly, the second

term represents the probability that

ξ
dn
=
2
, the state variable is equal to

φ
−
φ

at

time
t
, and a jump from

2
to

1
occurs during the interval (
t

,

t

+

t
). This jump

in

ξ

dn
occurs with probability
k
2

t
. Note that, because the jump may occur at any

f
∗
(

t
,where
f
∗
(

time during (
t

,

t

+

t
), in this case

φ

is expressed as

φ
=

φ

)

φ

)

is a combination of
f
1
(

φ

)and
f
2
(

φ

) (see
Horsthemke and Lefever
,
1984
, Eq. 9.22).

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