Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 172

In this chapter we concentrate mainly on the case of dynamics in which the state of
the system is determined by only one state variable, . These spatiotemporal dynamics

can in general be modeled by a stochastic partial differential equation expressing the
temporal variability of

φ

φ at any point, r = ( x , y ), as the sumof five terms: (i) a function

f ( φ ) of local conditions [i.e., of the value of φ at ( x , y )]; (ii) a multiplicative-noise

φ ξ , L φ

term, g ( ], accounting for the spatial interactions with

the other points of the domain, (iv) a term expressing the effect of a time-dependent
forcing, F ( t ), which can in general be modulated by a function h (

) m ( r t ); (iii) a term, D [

φ ) of the local state

of the system, and (v) an additive-noise component ξ a ( r , t ). Therefore the dynamics

are

∂φ
∂ t =

f ( φ ) + g ( φ ) ξ m ( r , t ) + D L [ φ ] + h ( φ ) F ( t ) + ξ a ( r , t ) , (5.1)

where is a (differential or integral) operator expressing the spatial coupling of

the dynamics and D is a parameter expressing the strength of the spatial coupling.
Depending on the specific stochastic model, some components in Eq. ( 5.1 ) can be
absent. In the following discussion, we use the subscripts a and m only when essential
to distinguish the dynamic role of the noise (additive or multiplicative).
The pattern-formation mechanism is sometimes called breaking of ergodicity .The
meaning of this expression can be understood from general equation ( 5.1 ). The spatial
coupling

L

t ), at a certain point r ∗ ,
from spanning across the same phase space that would be explored by the correspond-
ing zero-dimensional system (i.e., in the case of dynamics with no spatial coupling).
Therefore the spatial coupling imposes a constraint on the dynamics and can mod-
ify the portion of the phase space that is explored by the spatiotemporal system. In
this sense, spatially extended systems can exhibit breaking of ergodicity because the
probability distribution of

( r ∗ , L prevents the dynamics of the state variable φ

is different from that of the dynamics with no spatial
coupling. This means that, when each point in the domain fluctuates independently of
its neighbors (and of the rest of the field), the spatial average at a given time is the same
as the temporal average at a certain point (i.e., the system is ergodic). Conversely, in
the presence of spatial coupling, the spatial average of the field can be different from
the temporal average at a given point. Thus the spatial coupling breaks the ergodicity
of the system.
The role of noise can be also interpreted as an external input of energy that is
dissipated by the dynamical system. This explains why noise-induced patterns are
sometimes called dissipative structures ( Manneville , 1990 ).

φ

5.1.2.1 Models of the noise term
The key aspect in the dynamical systems investigated in this chapter is that patterned
states are noise induced, i.e., they are induced by random fluctuations and do not occur
in the deterministic counterpart of the dynamics. In fact, these symmetry-breaking