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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