Noise-induced phenomena in zero-dimensional systems - Noise-induced Phenomena in the Environmental Sciences - page 96

0.03

0.02

0.01

F

0.04 0.03 0.02 0.01 0.01 0.02

0.01

˙

Figure 3.31. Dependence of the drift φ on static deterministic force F . The pa-

T

rameter values are k = L = T = 1 , = 0 . 5 , A = 0 . 8 , V 0 = 1 / 2 π , and the variables

˙

are made dimensionless according to Reimann ( 2002 ). Note that φ = 0 also when

F = 0.

to stochastic dynamics ( 3.77 ) with a time-dependent random component, this result
demonstrates that particles may perform work against the load force F ; such work is
performed by white noise ( t )( Reimann , 2002 ).

Moreover, in general, the effect of the static force F is shown in Fig. 3.31 : The case
with F

ξ

0 corresponds in fact to dynamics ( 3.72 ) with periodic noise ( 3.74 ) (see
subsequent discussion). We observe that noise modulation, along with an asymmetric
potential, is able to induce a positive drift.
Several variations of the “basic” scheme of the Brownian motors presented in this
section were proposed. In particular, a number of shapes of the potential and the
temporal modulation of noise have turned out to be able to give rise to net transport.
In particular, it was demonstrated that noise modulation can also be random instead
of deterministic and that the noise term can be substituted with a chaotic forcing
( Reimann , 2002 ). The key mechanism remains the same: Net transport may arise
from the cooperation between an asymmetric deterministic potential and a random
forcing that alternates (periodically or stochastically) between phases of greater and
lower intensity. Thus the dynamics repeatedly switch between phases dominated by
noise and phases dominated by the deterministic terms. In this system, noise plays a
fundamental role in determining net transport. When the rate of noise modulation is
either relatively slow or fast (with respect to the intrawell deterministic dynamics),
the net transport tends to zero.
The type of Brownian motors described in this section is usually called thermal
ratchet because noise and its modulation are generally introduced to model thermal
fluctuations. However, we can obtain the same drift effect also by modulating the
deterministic potential instead of noise, i.e., when the dynamics are expressed as
d

=

d t =− d V

d

[1 + f ( t )] + ξ , (3.78) gn

φ