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.      Search WWH ::

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