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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