Environmental Engineering Reference
In-Depth Information
Box 5.3: Generalized mean-field theory
The mean-field theory is typically used to provide an approximated solution of
stochastic partial differential equations for spatially extended systems. The method is
valuable mainly for a qualitative analysis of (stochastic) spatiotemporal dynamics
(
van den Broeck et al.
,
1994
;
van den Broeck
,
1997
;
Buceta and Lindenberg
,
2003
;
Porporato and D'Odorico
,
2004
).
The mean-field technique adopts a finite-difference representation of stochastic
spatiotemporal dynamics (
5.1
):
d
φ
i
d
t
=
f
(
φ
i
)
+
g
(
φ
i
)
ξ
m
,
i
+
Dl
(
φ
i
,φ
j
)
+
ξ
a
,
i
,
(B5.3-1)
where for sake of simplicity we have set
F
(
t
)
=
0. In Eq. (
B5.3-1
),
φ
i
,
ξ
m
,
i
, and
ξ
a
,
i
are
the values of
φ
,
ξ
m
, and
ξ
a
at site
i
, respectively;
i
runs across all the discretization cells,
φ
i
,φ
j
) expresses the spatial
coupling between cell
i
and its neighbors (see Subsection
5.1.2.4
). A general expression
for
l
(
j
∈
nn
(
i
) refers to the neighbors of the
i
th cell, and
l
(
φ
i
,φ
j
)is
j
∈
nn
(
i
)
w
j
φ
j
,
l
(
φ
i
,φ
j
)
=
w
i
φ
i
+
(B5.3-2)
where
w
j
are weighting factors. Expressions for these weighting factors are
specified in Subsection
5.1.2.4
for some commonly used spatial couplings.
The analytical solution of Eq. (
B5.3-1
) is hampered by the fact that the dynamics of
φ
i
are coupled to those of the neighboring points. In fact, the spatial interaction term in
(
B5.3-1
) depends on the values
w
i
and
in the neighborhood of
i
. To avoid this obstacle,
the mean-field approach assumes that (i) the variables
φ
j
of
φ
φ
j
can be approximated by the
local ensemble mean
φ
j
, and (ii) there is a link between
φ
j
and the ensemble average
φ
i
in the
i
th cell.
Typically we are interested in the stability and instability conditions of the
homogeneous basic state with respect to periodic perturbations. For this reason, the
i
th
site is placed on a local maximum of the field, and the pattern is approximated by a
harmonic function,
φ
j
=
φ
i
cos[
k
·
(
r
i
−
r
j
)]
,
(B5.3-3)
where
k
=
k
y
) and
k
x
and
k
y
are the two wave numbers along the
x
and
y
axes,
respectively. It follows that the function
l
(
(
k
x
,
φ
i
,φ
j
)inEq.(
B5.3-1
) is approximated as
l
(
φ
i
,φ
j
)
≈
l
h
(
φ
i
,
φ
i
,
k
x
,
k
y
)
,
(B5.3-4)
where
l
h
(
) is a function whose structure depends on the spatial coupling considered.
Under assumption (
B5.3-4
), dynamics (
B5.3-1
)of
·
φ
i
do not depend anymore on
those of the neighboring points, and it is possible to determine exact expressions for
the steady-state probability distributions,
p
st
(
φ
;
φ
i
,
k
x
,
k
y
), of
φ
by the methods
described in Chapter 2.
p
st
(
φ
;
φ
i
,
k
x
,
k
y
) will necessarily depend on a number of
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