Noise-induced phenomena in zero-dimensional systems - Noise-induced Phenomena in the Environmental Sciences - page 94

t 1

(a)

t 2

(b)

t 3

(c)
Figure 3.29. Schematic representation of noise-induced transport. At time t 1 the
noise is weak and particles are concentrated in the well A. When the noise becomes
dominant (time t 2 ), particles spread across the potential barriers. Finally, when the
noise is again reduced (time t 3 ), the asymmetry of the potential means that particles
tend to accumulate more in the well located to the right of well A. Thus a net positive
current

˙

φ occurs.

where A < 1 and sign( · )isthe signum function [i.e., sign( z ) = 1if z > 0; sign( z ) =

− 0]. Equation ( 3.75 ) expresses a steplike behavior, with noise intensity

jumping between 2 s gn T (1

1if z <

A )and2 s gn T (1 /2. It is possible

to demonstrate ( Reimann , 2002 ) that, in this case, for a large set of parameters,

+ − A ) every half-period T

˙

φ = .

0 (3.76)

These conditions are associated with the occurrence of net transport.
To understand the physical mechanisms causing net transport, we consider the case
in which (i) T (1

T (1

V is the height of the potential
barrier (defined as in Fig. 3.17 ), and (ii) the noise intensity has a period

− A ) V + A ), where

greater than
the time scale of the intrawell deterministic dynamics (see Subsection 3.3.1 ). Under
these conditions, the dynamics of

T

can be schematized as an alternation of the three
phases, sketched in Fig. 3.29 : When the noise level is low, the deterministic component
of the dynamics prevails. Thus the values of

φ

tend to concentrate in correspondence
to the minima of the potential [see Fig. 3.29 (a)]. Conversely, during the phase with
high noise levels, the dynamics are dominated by the random component, whereas
the shape of the potential has only a weak effect on the process. In these conditions

φ