Environmental Engineering Reference
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a preferential direction of motion, i.e., a drift in the oscillations of the state variable.
This process is known under different names (e.g.,
Brownian motor
,
ratchet effect
,
Brownian ratchet
,or
stochastic ratchet
), depending on the context in which the phe-
nomenon is reported or investigated. The first studies on noise-induced transport go
back to work by
Smoluchowski
(
1912
)and
Feynman et al.
(
1963
) in the context of
intracellular transport processes (the so-called
molecular motors
; see the review by
Howard
,
1997
), and to some research by physicists in the 1970s (see
Reimann
,
2002
).
Some key contributions in the early 1990s (e.g.,
Ajdai and Prost
,
1992
;and
Magnasco
,
1993
) gave a new impetus to studies on noise-induced transport, with an increasing
number of experimental and theoretical results along with several technological ap-
plications (
Reimann
,
2002
;
Astunian and Hanggi
,
2002
). We use the same approach
as in the previous two sections and describe the basic aspects of Brownian motors;
we refer the reader to
Reimann
(
2002
)and
Reimann and Hanggi
(
2002
) for a more
detailed discussion.
Brownian motors have three basic ingredients: (i) an asymmetric periodic potential,
(ii) a noise source, and (iii) a temporal modulation, either of the potential or of the
noise intensity. Here we focus on the case in which the noise intensity is modulated,
though a similar mechanism is common to all Brownian motors, including those
relying on a modulation of the potential.
Let us first consider the deterministic dynamics driven by an asymmetric peri-
odic potential
V
(
φ
). An example of a potential satisfying these two properties (see
Reimann
,
2002
),
V
0
sin
2
4
sin
4
πφ
L
1
πφ
L
V
(
φ
)
=
+
,
(3.68)
is shown in Fig.
3.28
(
L
is the coefficient determining the period of the periodic
potential). As a consequence, in the deterministic dynamics described by
d
d
t
=−
d
V
d
φ
,
(3.69)
φ
(
t
) tends to one of the minima of the potential [i.e., one of the steady states of (
3.69
)],
depending on the initial condition. In particular, in example (
3.68
) the minima are at
arccos
1
+
√
3
2
L
π
φ
=
+
−
,
Ln
(3.70)
m
where
n
is an integer number. The corresponding attraction basins are
Ln
arccos
1
arccos
1
+
√
3
2
+
√
3
2
L
π
L
π
+
,
L
(
n
+
1)
+
.
(3.71)
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