Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 183

forcing significantly modifies some prominent statistical feature of the system. In the
case of spatial patterns the signature of noise-induced phenomenon is a modification
of the spatial-correlation function (or the structure function), but noise-induced mod-
ifications of other statistical descriptors of the random field (e.g., mean, modes, or
variance) can also be very relevant.
Usually modifications of order parameters (which are global quantities used to
describe general aspects of the spatial distribution of the state variable

φ

) are known
as phase transitions . In zero-dimensional systems (see Chapter 3), we considered
the mode of the pdf as the order parameter. In fact, in pure temporal systems the
mean can be poorly representative of transitions, whereas the modes indicate the most
frequently visited states of the dynamics and are therefore particularly suitable to rep-
resent structural changes of the system's behavior. Moreover, analytical expressions
for the modes can be determined (Chapter 3), and this allows a theoretical analysis
of transitions. Conversely, in spatiotemporal systems exact analytical results are very
rare, and this hampers the analysis of the modes. For this reason, the simplest order
parameter - i.e., the spatiotemporal average of the state variable at steady state - is
more frequently used 2 because analytical approximated techniques have been devel-
oped for the analysis of this order parameter (see Box 5.4). In this case, the order
parameter is

M ( t )
A ,

m = (5.15)

where the overbar indicates the temporal mean, and a noise-induced phase transition
occurs when the random component is able to change the value of m with respect to
the basic homogeneous steady state

0 . 3
Notice that the occurrence of a nonequilibrium (i.e., order-forming) phase transition
is neither a necessary nor a sufficient condition for noise-induced pattern formation.
In fact, a nonequilibrium phase transition implies that noise is able to change the value
of the order parameter, but not that ordered geometrical structures necessarily appear.
Conversely, a number of cases exist in which patterns occur but m remains unchanged
with respect to the disordered case. In the thermodynamics literature, fields with m =
φ 0 are sometimes called ordered phases (hence the use of the term order parameter
to indicate m ) and nonequilibrium phase transitions often seem to be used, implying
pattern occurrence. For the sake of clarity, in the following discussion we keep the
occurrence of patterns distinct from the existence of a phase transition in the mean.

φ

2 Another possible order parameter that is (less frequently) used is

J ( t )
A

2 ( x ,

J st = with J ( t ) =

D φ

t )d x ,

where J st recalls the flux of convective heat ( Garcia-Ojalvo and Sancho , 1999 ).

3

m is often used as an order parameter because of its correspondence to the so-called magnetization used in the
physics literature in the study of equilibrium systems.