Environmental Engineering Reference

In-Depth Information

2

with the constraint that

<β

/

4 [otherwise the potential corresponding to
f
2
(

φ

)

=

φ

(

β
−
φ

)

−

would no longer have a minimum at finite values of

φ

, and the process

would diverge].

We can obtain the modes and antimodes of the pdf by applying Eq. (
3.3
) to model

(
3.11
). In this case only the terms associated with the deterministic dynamics and

the noise autocorrelation - i.e., the first and the last addenda on the left-hand side of

Eq. (
3.3
) - are different from zero. Thus it is found that the modes and antimodes of

φ

are solutions of

m
)
1

(

β
−

2

φ
m
)

φ

β
−
φ

+

=

.

m
(

0

(3.14)

k

The deterministic steady state

φ
=
β

is always either a minimum or a maximum of

p
(

φ

), depending on the sign of the second-order derivative,

φ
=
β
∝
β

d
2
p
(

φ

)

(

β
−

k
)

,

(3.15)

φ

2

d

k

φ
=

φ

whereas the other deterministic steady state

0 is outside the domain of
p
(

)

[see domain (
3.13
)]. The value

2 is another extremum, provided that it

is included within the boundaries of domain (
3.13
); this happens when
q

φ
=

(
k
+
β

)

/

<

k

<

p
;

the second-order derivative at this point is

φ
=

d
2
p
(

k
2

2

φ

)

−
β

+
β
2
∝

.

(3.16)

d

φ

2

2
k

k

To complete the qualitative study of the shape of the pdf we investigate its behavior

near the boundaries as indicated in Chapter 2. We obtain

lim

2
p
(

φ

)

→∞

(

→

0)

if

k

<

p
(
k

>

p
)

,

φ
→

(

β
+

p
)

/

lim

2
p
(

φ

)

→∞

(

→

0)

if

k

<

q
(
k

>

p
)

.

φ
→

(

β
+

q
)

/

We are now able to understand the possible qualitative behaviors of the steady-state

pdf of

in the dynamics modeled by Eq. (
3.11
). As shown in Fig.
3.2
, the pdf can

assume four different shapes. In particular, we note that (i) the deterministic steady

state is a mode of the stochastic dynamics only if
k

φ

>β

, otherwise it becomes an

antimode; and (ii)
p
(

) is unimodal in only one case, whereas in all other cases it

is bimodal and exhibits two preferential states, with one of them being at one of the

boundaries of the domain. When the dichotomous noise switches frequently (i.e., for

high values of
k
) and the noise amplitude (i.e.,

φ

) is small, the most visited (i.e.,

preferential) state coincides with the deterministic steady state (i.e.,

), and

the stochastic forcing induces random fluctuations around this state. However, when

φ
m
,
det

=
β

increases, a noise-induced transition occurs, and a second mode appears close to the

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