Environmental Engineering Reference

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domain. The possible boundaries of the domain, [

φ
−
,φ
+
], are then all pairs of stable

stationary points that are not separated by an unstable point.

As noted, the stable points are the boundaries of the domain only if neither one of

the two dynamics has an unstable point between them. For example, to the right of

the domain
D
2
(see Fig.
2.4
) there are two minima (one for each potential function),

but the maximum of potential
V
2
(

φ

) existing between these minima prevents the

long-term persistence of the particle in this interval. Note that the same criteria for

the determination of the extremes of the steady-state domain apply when the minima

of the potential are at

. Finally, it should be noted that if the stable points

were coincident the pdf would reduce to a Dirac delta function centered in the two

overlapping stable points.

The previous discussion focused on the emergence of boundaries intrinsic to the

dynamics. However, boundaries can also be externally imposed. For example, a fre-

quent case in the biogeosciences is when the variable

±∞

φ

is positive valued or it has

a boundary at a certain threshold value

φ
th
. This corresponds to changing the poten-

tial of the deterministic dynamics by setting
V
1
(

)and
V
2
(

φ

)

=

V
1
(

φ

φ

)

=

V
2
(

φ

)for

φ
≤
φ
th
,and
V
1
(

and
V
2
(

φ
th
is assumed to be an

upper bound). The general rule presented in this section to determine the boundaries

of the domain can now be applied to the modified potentials
V
1
(

φ

)

→∞

φ

)

→∞

for

φ>φ
th
(if

)and
V
2
(

). The

presence of the external bound may create a new minimum in the potential and affect

the original boundaries of the domain if the sign of the derivative of
V
1
(

φ

φ

φ

)and
V
2
(

φ

)

is negative at

th
.

An important property of the functions
f
1
(

φ

) emerges from these analyses

of the boundaries of the domain: In all the cases discussed before (Fig.
2.4
), one of the

two potentials monotonically decreases whereas the other monotonically increases

inside each of the possible domains. As a consequence, inside the domain we have

f
1
(

φ

)and
f
2
(

φ

φ

)

≥

0and
f
2
(

φ

)

≤

0[or
f
1
(

φ

)

≤

0and
f
2
(

φ

)

≥

0], with the equal sign occurring

at the boundaries. Thus

always increases when the noise is in one state, whereas it

always decreases when it is in the other.

Finally, a very particular situation is represented in Fig.
2.5
. In this case two stable

points are always separated by an unstable point. The particle therefore cannot remain

in the long term inside any of these domains, and it will necessarily continue to move

from the left to the right of the

φ

axis. A noise-induced drift is therefore generated,

a situation that is known in the literature as the
ratchet effect
; see
Bena et al.
(
2003
)

and Chapter 3.

To clarify the conditions determining the boundaries of the domain, we use exam-

ples similar to those presented in the previous subsections:

φ

Example 2.1:
Both
f
1
(

have a single stable fixed point. The

boundaries of the domain correspond to the minima of the two potentials,
V
1
(

φ

)

=

1

−
φ

and
f
2
(

φ

)

=−
φ

φ

)

=−
φ
+

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