Environmental Engineering Reference
Figure 3.28. Example of periodic asymmetric potential V (
). In particular, this
example is obtained with Eq. ( 3.68 ).
Now, let us introduce an additive, time-independent, and uncorrelated Gaussian
gn , with zero mean and constant intensity s gn . The corresponding stochastic
d t =−
+ ξ gn .
Even though the potential exhibits an asymmetry, it is possible to demonstrate that
the so-called average current in the long-time limit,
( t )
is zero, i.e., no preferential direction (or net transport) emerges in the randomBrownian
motion. At first glance, this behavior may seem to be strange because of the asymmetry
of the potential. However, it is a consequence of the second law of thermodynamics
that establishes zero current for a Brownian motor under the influence of a thermal-
equilibrium heat bath, i.e., a random forcing with steady properties ( Reimann , 2002 ).
Completely different dynamics can emerge if we let the noise be time dependent.
For example, we can assume that the noise intensity is periodically modulated:
gn ( t )
gn ( t
2 s gn T ( t )
where T ( t )
T ( t
) is a periodic modulation function with period
. A typical
case of temporal modulation of T ( t )is
A sign sin 2
T ( t )