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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