Environmental Engineering Reference
In-Depth Information
2
φ
t
1.5
1
0.5
1000 1500 20 0 2500 3000
t
500
0.5
1
1.5
2
Figure 3.19. Example of temporal path of deterministic dynamics (
3.59
)driven
by white Gaussian noise. The stochastic equation was simulated according to the
numerical approach described by
Sancho et al.
(
1982
).
(
t
). In this example we use white
Gaussian noise with intensity
s
gn
. This noise component is additive, and the dynamics
of
We can now introduce the stochastic forcing
ξ
φ
are expressed by
d
d
t
=
φ
2
)
(1
−
φ
+
ξ
gn
(
t
)
.
(3.59)
The effect of the stochastic forcing is twofold: It introduces randomness in the
temporal dynamics of
(
t
), and, more important, this forcing drives the transitions
from a well to another across the potential barrier (which is located at
φ
φ
=
φ
b
). As a
result, the dynamics of the variable
exhibit the pattern shown in Fig.
3.19
,which
shows clear transitions across the barrier. Because these transitions are induced by
the random forcing, their occurrence is random, with a mean passage time equal to
(see
Gardiner
,
1983
;
Wellens et al.
,
2004
)
φ
s
gn
φ
−∞
d
φ
m
,
2
1
s
gn
φ
)
φ
)
s
gn
V
(
V
(
e
−
φ
φ
.
t
c
=
e
d
(3.60)
φ
m
,
1
In particular in the case shown in Fig.
3.19
,wehave
t
c
=
100
.
8 time units, with
s
gn
=
085.
If we assume that the transitions occur rarely with respect to the oscillations within
each well - i.e., the mean passage time is much longer that the intrawell relaxation
time,
0
.
t
c
t
iw
- it is possible to use Kramers's formula (
Gardiner
,
1983
),
2
φ
b
2
φ
m
d
2
V
d
d
2
V
d
−
φ
φ
e
−
V
W
=
,
(3.61)
s
gn
2
π
to determine an approximation of the mean transition rate, i.e.,
W
−
1
t
c
.For
example, in the case shown in Fig.
3.19
(where
t
c
=
100
.
8
t
iw
=
1
/
2), application
of Eq. (
3.61
)gives
t
c
=
84
.
1 time units.
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