Noise-induced phenomena in zero-dimensional systems - Noise-induced Phenomena in the Environmental Sciences - page 52

changes of the steady-state pdf demonstrate how noise can create order and then have
a “constructive” role in the dynamics.
The first studies on noise-induced transitions emerged within the context of radio
circuits ( Kuznetsov et al. , 1965 ), population dynamics ( May , 1973 ), and enzyme
systems ( Hahn et al. , 1974 ). Theoretical and experimental evidence of this effect of
noise was then provided in these and other fields ( Horsthemke and Lefever , 1984 ).
The expression transitions is used to recall the phase transitions typical of the
equilibrium systems. In systems out of equilibrium, such as those considered in this
chapter, the maxima of the steady-state pdf of the state variable correspond to the most
probable and preferentially observed states of the system (i.e., the modes of the state
variable). These states can be interpreted as the “phases,” i.e., as macroscopic stable
steady states of the system. The aim of this section is to show how some stochastic
dynamics may undergo (noise-induced) nonequilibrium transitions as the variance or
autocorrelation scales of noise vary across suitable threshold values. These transitions
are evidenced by qualitative changes in the macroscopic behavior of the system and
by the consequent emergence of new phases created by noise.
The most important indicators of these transitions are changes in the maxima and
minima - i.e., the modes and antimodes - of the probability density function of the
state variable. 1 In fact, modes and antimodes provide important information about the
shape of the pdf and the preferential states of the system. Another useful tool to detect
noise-induced transitions is the probabilistic potential , defined in Box 3.1.
The basic idea is that the minima of the probabilistic potential, as defined in
Eq. ( B3.1-5 ), represent the noise-induced preferential states of the dynamics of .By

comparing these minima with those of the deterministic potential, we are able to detect
the occurrence of noise-induced transitions. In the following subsections we discuss
the dependence of modes and antimodes (i.e., minima of probabilistic potential)
on the properties of random forcing, considering the four types of noise introduced in
the previous chapter. In the fourth chapter we capitalize on these properties to explain
the fundamental role of noise in some natural systems.

φ

3.2.1 Noise-induced transitions driven by dichotomous Markov noise
To recognize and investigate noise-induced transitions, we first need to analyze the
deterministic counterpart of the dynamics and assess how the states of the system
change in the presence of the stochastic forcing. This task can be tricky in the case of
DMN because of the two different possible applications of DMN. In Subsection 2.2.2
we stressed that stochastic models forced by dichotomous noise can be formulated

1 Moments could be a valid alternative, which in fact are considered in Chapter 5 for spatiotemporal systems.
However, in some cases transitions in the modes are not reflected by qualitative changes of the moments, and vice
versa (e.g., Garcia-Ojalvo and Sancho , 1999 ).