Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 215

mean-field assumption ( B5.3-4 )is

2

l h ( φ i , φ i , k x , k y ) =

2 {

[cos( k x ) + cos( k y )] φ i − 2 φ i } . (5.63)

It follows that discretized equation ( B5.3-1 )is
d

2 cos(

k y ) φ

φ i
d t

2 D 4 D

.
(5.64)
Because this analysis assumes the i th cell to be at a maximum of the (possible) wavy
pattern - i.e.,

= φ + φ ξ + + − φ + ξ

f ( i ) g ( i ) k x ) cos( m , i i i a , i

2

0 - the term proportional to the ensemble average in Eq. ( 5.64 )
(i.e. the third addendum) has the maximum destabilizing effect on the dynamics of

φ i ≥

φ i

when [cos( 0. Thus, in the

case of Laplacian spatial coupling, the most unstable modes correspond to a phase
transition with no stable patterns. This phase transition can be explored with the
standard mean-field assumption ( B5.4-1 ), which for a generic diffusive model leads
to

k x ) + cos( k y )] is maximum, namely when k x = k y =

d φ i

d t =

4 D

f ( φ i ) + g ( φ i ) ξ m , i +

2 ( m

− φ i ) + ξ a , i . (5.65)

Transitions can therefore be investigated by use of the discretized version of model
( 5.61 ), namely

d φ i

d t =− φ i (1

4 D

i ) 2

2 2

+ φ + (1 + φ

i )

ξ gn +

2 ( m

− φ i ) . (5.66)

The corresponding steady-state pdf is (see Chapter 2)

exp

s gn m φ i − φ

2
i
s gn

2
i

φ

D mD arctan( φ i )

− + +

2
i

1 + φ s gn

p ( φ i , m ) = , (5.67)

+ φ

2

2(1

i )

which allows us to impose the self-consistency condition, m = −∞

p ( φ i , m )d φ i ,and

to explore the existence of solutions with m values different from 0( van den

Broeck et al. , 1994 ). Figure 5.32 (a) shows the line marking the transition between the
region of the parameter space

φ 0 =

0 is the only solution of Eq. ( 5.67 )
and the region where new solutions appear. Figure 5.32 (b) shows the emergence of
the phase transition as a function of s gn for a given D . It can be observed that the
diffusive VPT model exhibits multiple solutions of the order parameter for suitable
values of the noise intensity s gn and strength of the spatial coupling D . Moreover, this
phase transition is clearly a second-order (nonequilibrium) phase transition because
it occurs with no discontinuity in the order parameter m . Finally, the figure shows
that the transition is reentrant, in that the stable state of the system switches back to
the initial (disordered) state m

{ s gn , D } where m =

0 for large values of s gn . Thus the emergence of a
new phase requires both spatial coupling (i.e., D needs to exceed a noise-dependent
critical value) and intermediate noise intensities.

=