Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 184

Box 5.4: Classic mean-field analysis

The classic mean-field theory can be presented as a simplified version of the
generalized mean field described in Box 5.3. In this case it is assumed that all cells have
the same mean, which coincides with the spatiotemporal mean of the field. This
corresponds to taking k

= 0inEq.( B5.3-3 ), i.e., φ j = φ i = m , with m defined in

Eq. ( 5.15 ). In this case, Eq. ( B5.3-5 ) in Box 5.3 becomes

+∞

F m . m = φ p st ( φ ; m )d φ = (B5.4-1)

−∞

Multiple solutions of Eq. ( B5.4-1 ) indicate the existence of multiple phase transitions
of the system. In this case the mean-field analysis is not used to investigate the
emergence of periodic patterns but the occurrence of phase transitions. This zero-
wave-number version of the mean-field approximation is the standard mean-field
method, and the mean-field analysis that involves k x = 0(or k y = 0) is called

generalized mean-field theory ( Sagues et al. , 2007 ).
The effectiveness of the standard mean-field approximation can be improved by
expressing the values of φ j in the neighborhood of point i as the average between the

spatiotemporal mean and the local value of

φ at point i , namely

1
2 ( m

φ j ≈ + φ i ) . (B5.4-2)

φ j on
the local conditions (see Sagues et al. , 2007 ). On the basis of this analysis we conclude
that the standard mean-field assumption can overestimate the strength of the spatial
coupling, and the diffusivity D in ( B5.4-1 ) could be replaced with a new diffusivity
D n =

This correction of the mean-field approximation accounts for the dependence of

2( Sagues et al. , 2007 ). For sake of simplicity in the examples presented in this
chapter we apply the classical form of the mean-field approximation without using this
correction.
The mean-field framework is also used to investigate different mechanisms of
noise-induced pattern formation. For example, Zhonghuai et al. ( 1998 ) used this method
to study noise-induced phase transitions in generic two-variable systems exhibiting
Turing instability. When a control parameter of the kinetics is perturbed by noise, new
kinds of patterns arise (transition from single-spiral to double-spiral waves). A similar
study was developed by Carrillo et al. ( 2004 ) for the analysis of pattern formation in
chemical reactions and fluid convection. Further, this method was used in Chapter 4 to
investigate the emergence of phase transitions in multivariate (temporal) population
dynamics.

D /