Environmental Engineering Reference
In-Depth Information
φ
t
φ
t
2
2.5
2
1
1.5
0
1
0.5
1
10
t
10
t
2
4
6
8
2
4
6
8
(a)
(b)
Figure 2.14. Example of a realizations of processes driven by Gaussian white noise:
(a) Eq. (
2.81
), (b) Eq. (
2.82
). The noise intensity is
s
gn
=
1, and the realization of the
noise is the same in the two panels.
(
t
) for the two processes. From the comparison of
the two panels it is clear that in this case the presence of the multiplicative term has an
even stronger influence than in the case of the shot-noise process (see also Fig.
2.11
).
In fact, the presence of the multiplicative term changes the domain of the process and
the shape of the corresponding probability distribution, as explained in the following
subsections.
Figure
2.14
shows the path of
φ
2.4.4.2 Steady-state pdf and its properties
We obtain the steady-state pdf for a process driven by Gaussian white noise either
from Eq. (
2.32
) by taking limiting values (
2.78
) or from (
2.69
)byusing(
2.79
). We
obtain (e.g.,
Horsthemke and Lefever
,
1984
)
)
exp
φ
φ
)
s
gn
g
(
C
g
(
f
(
p
(
φ
)
=
φ
)
2
d
,
(2.83)
φ
φ
where
C
is an integration constant that ensures that the pdf has unit area. We obtain the
same solution by considering the master equation for the evolution of the probability
density in time, which is reported in Box 2.4.
The domain of steady-state pdf (
2.83
) is bounded by the zeros of
g
(
φ
), i.e., by the
minima of the potentials
V
1
(
φ
)and
V
2
(
φ
) (defined in Subsection
2.2.3.3
), that in this
case are defined by
d
V
1
(
φ
)
d
V
2
(
φ
)
g
(
φ
)
=−
,
g
(
φ
)
=
,
(2.84)
d
φ
d
φ
which can be obtained from Eqs. (
2.38
) by use of limit values (
2.78
). For example,
if the noise is additive, i.e.,
g
(
const, the domain covers the whole real axis.
This is the case for Example 2.7. For Example 2.8,
g
(
φ
)
=
φ
)
=
φ
, which implies that
the dynamics are constrained within the interval [0
,
∞
]or[
−∞
,
0], depending on
whether the initial condition
φ
0
is positive or negative valued.
Search WWH ::
Custom Search