Environmental Engineering Reference

In-Depth Information

We can determine the integration constant
C
as a normalization constant by impos-

ing the condition that the integral
2
of
p
(

φ

) over its domain (see Subsection
2.2.3.3
)

be equal to one.

Using the definitions of
f
1
(

φ

)and
f
2
(

φ

) given in Eqs. (
2.15
) and zero-mean con-

dition (
2.8
), we also obtain

)
exp

φ

φ
)

2
−
1
)
g
(

φ

)

1

τ
c

f
(

p
(

φ

)

=

C
(

−

φ
)
d

,

(2.32)

(

φ

(

φ

where

(

φ

)

=

[
f
(

φ

)

+

1
g
(

φ

)][
f
(

φ

)

+

2
g
(

φ

)]

.

(2.33)

We refer again to the four examples introduced in Subsection
2.2.3.1
:

Example 2.1:
f
1
,
2
(

φ

) are defined as in Eqs. (
2.17
). The resulting steady-state pdf,

)
k
1
−
1

k
2
−
1

p
(

φ

)

∝

(1

−
φ

φ

,

(2.34)

is a standard beta distribution (
Johnson et al.
,
1994
) with parameters
k
1
and
k
2
.

Example 2.2:
f
1
,
2
(

φ

) are defined as in Eqs. (
2.18
). The resulting steady-state pdf,

p
(

φ

)

∝

exp[

−

(
k
1
−

k
2
)

φ

]

,

(2.35)

is an exponential distribution with parameter
k
1
−

k
2
.

Example 2.3:
f
1
,
2
(

φ

) are defined as in Eqs. (
2.19
). The resulting steady-state pdf is

φ
−

−

k
1

1
−
a

2

φ

−

a

φ
+

a

a

k
2
−

1

p
(

φ

)

∝

φ

.

(2.36)

(

φ
−

a
)(1

−
φ

)

1

−
φ

Example 2.4:
f
(

φ

) and
g
(

φ

) are defined as in Eqs. (
2.20
), and a symmetric noise (i.e.,

1
=−
2
=

and
k
1
=

k
2
=

k
) is assumed. The resulting steady-state pdf is

2
k
β

k

+
β
(

k

β
−

∝−
φ

2
(

+
β
−
φ

)

φ
+
−
β

)

2

−
β

p
(

φ

)

.

(2.37)

φ

[(

φ
−
β

)
2

−

2
]

The plots of these pdfs are provided in the following subsection, after the methods

for determining the domain of the steady-state pdf are described. In the next two

subsections we discuss the domain of the pdf and its behavior at the boundaries; an

analysis of the modes of the pdf is made in Chapter 3 within the context of the theory

of noise-induced transitions.

2.2.3.3 Domain of the steady-state probability distribution

The domain of the steady-state pdf,
p
(

φ

), i.e., the range of values within which

the asymptotic dynamics of

φ

are confined, depends on the stationary points of the

functions
f
1
,
2
(

φ

) and on their stability. We recall that a stationary point

φ
st
of dynamics

2
The pdf should in general be denoted with a capitalized variable as a subscript [in our case
p
(

φ

)]. However, for

the sake of simplicity, we omit this subscript whenever it is not essential.

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