Environmental Engineering Reference
In-Depth Information
Notice how this instability arises as an effect of the spurious drift term in the
macroscopic equation, which would not exist if the noise were additive or Ito's
interpretation were used. In the second case
f
eff
(
φ
0
)
=
0 also in the stochastic dynamics
(i.e., when
s
m
>
0). Thus, using in Eq. (
B5.5-5
) the first-order truncated Taylor's
expansion of
f
eff
(
φ
) around
φ
0
, we find that instability emerges when
φ
0
=−
φ
0
≥
d
2
d
f
eff
d
V
(
φ
)
0
.
(B5.5-7)
φ
d
φ
2
φ
0
yields
In fact, the first-order truncated Taylor expansion of the function
f
eff
around
φ
0
φ
.
d
φ
d
t
d
f
eff
d
≈
(B5.5-8)
φ
Thus the sign of d
f
eff
(
determines the stability or instability of the dynamics
with respect to a (small) perturbation around
φ
0
)
/
d
φ
φ
0
. The noise threshold can therefore be
calculated when condition (
B5.5-7
) is equal to zero.
In both cases noise has to be multiplicative [i.e.,
g
(
) should not be a constant] for a
transition to unstable dynamics to emerge. When the noise is additive the effective
kinetics are
f
eff
(
φ
) and the stable states of (
B5.5-5
) are the same as those of
the deterministic counterpart of the process.
It is also worth stressing the impact of the type of noise and noise interpretation.
When
φ
)
=
f
(
φ
ξ
m
is white Gaussian noise and we interpret Langevin equation (
B5.5-1
)byusing
the Stratonovich rule,
g
S
(
φ
)
=
s
gn
g
(
φ
)
g
(
φ
). In contrast, under Ito's interpretation,
g
S
(
0 (see Box 3.2). These differences play an important role in the short-term
instability presented in this section. In fact, such instability is triggered by spurious drift
[see Eq. (
B5.5-5
)]. Therefore noise-induced instability is possible only under the
Stratonovich interpretation of Langevin's equation, and it cannot occur when Ito's
framework is adopted.
φ
)
=
where
a
is a negative-valued number, the random component is modulated by a
function
g
(
ξ
gn
is zero-mean white Gaussian noise with intensity
s
gn
;
Eq. (
5.34
) is interpreted in the Stratonovich sense. The local dynamics,
φ
)
=
φ
,and
f
(
φ
)
=
3
, and the spatial coupling a la Swift-Hohenberg are the same as those already
used to discuss patterns induced by additive noise. Thus the deterministic dynamics
underlying Eq. (
5.34
) exhibit the behavior described in Section
5.2
. In particular, the
homogeneous stable state, obtained as a solution of
f
(
a
φ
−
φ
0.
The short-term stability of the basic state is investigated as described in Box 5.5.
For model (
5.34
), Eq. (
B5.5-5
) yields
φ
0
)
=
0, is
φ
(
r
,
t
)
=
φ
=
0
d
φ
d
t
3
≈
a
φ
−
φ
+
s
gn
φ
=
f
eff
(
φ
)
.
(5.35)
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