Environmental Engineering Reference

In-Depth Information

V

φ

1

0.5

φ

2

1

1

2

0.5

1

Figure 3.28. Example of periodic asymmetric potential
V
(

φ

). In particular, this

example is obtained with Eq. (
3.68
).

Now, let us introduce an additive, time-independent, and uncorrelated Gaussian

noise

gn
, with zero mean and constant intensity
s
gn
. The corresponding stochastic

dynamics become

ξ

d

d
t
=−

d
V

d

+
ξ
gn
.

(3.72)

φ

Even though the potential exhibits an asymmetry, it is possible to demonstrate that

the so-called average current in the long-time limit,

lim

t

φ

(
t
)

−
φ

(0)

˙

φ
=

,

(3.73)

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:

ξ

ξ

+
τ

=

δ

τ

,

gn
(
t
)

gn
(
t

)

2
s
gn
T
(
t
)

(

)

(3.74)

where
T
(
t
)

=

T
(
t

+
T

) is a periodic modulation function with period

T

. A typical

case of temporal modulation of
T
(
t
)is

T
1

A
sign
sin
2

π

t

T
(
t
)

=

+

,

(3.75)

T

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