Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 214

γ

1

s gn 0.8

k

0 5 1 1.5 2

s gn 0.5

1

s gn 0.2

2

Figure 5.31. Dispersion relation of the VPT model with diffusive spatial coupling
for three values of noise standard deviation, s gn ( D = 1).

5.5.1 The VPT model with diffusive spatial coupling
The VPT model with diffusive spatial coupling (see also Subsection 5.4.3 ),

∂φ
∂

2 ) 2 2 ) 2

=− φ + φ + + φ ξ + ∇ φ,

(1 (1 D (5.61) gn

t

is the original form of the VPT model ( van den Broeck et al. , 1994 ), which was
proposed as one of the first examples of a spatiotemporal system exhibiting noise-
induced (nonequilibrium) phase transitions.
In Subsection 5.4.3 , we demonstrated that the temporal component of the VPT
model shows a short-term instability when the noise intensity s gn exceeds the threshold
s c

= .

5. The dispersion relation is very similar to the one determined for prototype
model ( 5.58 ), in that both models have about the same linearized structure. In fact,
the dispersion relation for model ( 5.61 )is

0

Dk 2 γ ( k ) = 2 s gn − 1 − (5.62)

(Fig. 5.31 shows some examples). Apart from an inessential factor of 2 in the noise
intensity, the stability analysis of the basic state 0 of the ensemble leads to the

same conclusions as those discussed for prototype model ( 5.58 ): (i) the threshold
s c =

φ 0 =

5 dete cted by the sh ort-term instability is confirmed, (ii) a range of wave
numbers [0

0 .

, (2 s gn −

1) / D ] becomes unstable when s gn > s c , and (iii) the dispersion

relation has a peak at k max = 0.

It is instructive to explore the behavior of the diffusive VPT model by means of
the generalized mean-field technique. When the spatial coupling is expressed by the
diffusion term, the most unstable modes are k x 0 and correspond to the phase

transition of the order parameter m . In fact, in the case of Laplacian spatial coupling,

= k y =