Environmental Engineering Reference
In-Depth Information
Box 3.2: Correlation between noise and
φ
Consider the Langevin equation for a process driven by Gaussian white noise,
Eq. (
2.80
). By taking the expectation of both sides of Eq. (
2.80
) we obtain
+
g
(
ξ
gn
,
d
φ
d
t
=
f
(
φ
)
φ
)
(B3.2-1)
which is sometimes called the
macroscopic equation
of the dynamical system (e.g.,
van
Kampen
,
1992
). We obtain the stationary states of the deterministic system by setting
d
φ
/
d
t
=
0 and
ξ
gn
=
0inEq.(
B3.2-1
), i.e., as solutions to
f
(
φ
)
=
0. When the
φ
function
f
(
is obtained; in
all other cases, higher-order moments will also influence the solution of the steady-state
macroscopic equation (see
van Kampen
,
1992
, pp. 122-127). A Taylor expansion of
f
(
φ
) is a linear one, a closed-form equation for the average
φ
), truncated to the first order, will provide an approximated, closed-form equation
for the steady-state average of the process
.
We are interested here in considering the role of noise. When the noise is additive, i.e.,
φ
const, the term
g
(
ξ
gn
in Eq. (
B3.2-1
) factorizes and becomes uniformly equal
g
(
φ
)
=
φ
)
to zero because
ξ
gn
=
0. In formulas,
g
(
ξ
gn
=
ξ
gn
=
0. As a consequence,
additive Gaussian white noise is not able to modify the deterministic stationary states of
the system, i.e., to induce transitions in the steady-state average of the process. The
same considerations apply when the noise is multiplicative, but the corresponding
Langevin equation is interpreted following Ito's convention. We showed in Subsection
(
2.3.4
) that Ito's convention assigns the rule that
g
(
φ
)
g
(
φ
)
φ
) should be calculated with the
value of
just before the jump (
van Kampen
,
1981
). As a consequence, the noise term
turns out to be independent of
φ
ξ
gn
vanishes (see
van Kampen
,
1992
, p. 231).
The situation is different when Statonovich's convention is adopted: In this case the
noise turns out to be correlated to
, and
g
(
φ
φ
)
φ
, with (
van Kampen
,
1992
, p. 231)
s
gn
d
g
(
)
g
(
ξ
gn
=
s
gn
g
(
)
,
φ
)
φ
)
g
(
φ
=
φ
)
g
(
φ
(B3.2-2)
φ
d
which is sometimes called
Novikov's theorem
(e.g.,
Garcia-Ojalvo and Sancho
,
1999
). It
is clear that in this case the noise may have profound effects on the stationary states of
the system, which are now found as solutions of
s
gn
g
(
)
=
f
(
φ
+
φ
φ
0.
Noise-induced transitions in the steady-state average of the process are therefore
possible in this case. Note that this equation has a form similar to that of Eq. (
3.48
),
except for the sign of the second term. Examples of application of this equation are
given in Chapter 5.
This noise-induced effect is not peculiar to Gaussian white noise. In fact, with other
types of noise we would find qualitatively similar results (i.e., that multiplicative noise
may induce phase transitions in the dynamics). However, with other forms of noise the
problem of computing the correlation between noise and
)
)
g
(
,asinEq.(
B3.2-2
), becomes
rather intractable. If the noise is white and its cumulants are delta correlated (see
van Kampen
,
1992
, p. 237), the following expression can be obtained after some
φ
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