Transport properties in disordered ratchet potentials
F. Marchesoni
Department of Physics, University of Illinois, 1110 West Green Street, Urbana, Illinois 61801
and Istituto Nazionale di Fisica della Materia, Universita` di Camerino, I-62032 Camerino, Italy
Received 10 January 1997
The role of disorder in one-dimensional ratchet potentials is investigated by introducing the following: i
impurities, where a certain fraction of the asymmetric unit cells are replaced by unit cells with opposite
asymmetry, and ii randomness, where all unit cells have the same asymmetry, but random size. The relevant
color-induced currents are determined and compared with the current of the ideal periodic ratchet potential.
Altogether disorder is shown to quench the effectiveness of thermal ratchets, while remarkable transport
properties become detectable. S1063-651X9709708-0
PACS numbers: 05.40.+j
Thermal ratchets are remarkable devices capable of recti-
fying a zero-mean noisy signal 1–9. The simplest example
of a thermal ratchet is described by the stochastic process
x
˙
=-V ' x + t , 1
where ( t ) denotes a Gaussian, zero-mean, stationary noise
and V ( x ) is a periodic potential with unit-cell length L , that
is, V ( x +L ) =V ( x ) . A stationary nonzero average probability
current j ( ) (1/L )
0
L
j ( x ; ) dx may result from the com-
bined action of the spatial asymmetry of the drift
V ( x ) V ( -x ) and the noise finite correlation with time .
The process 1 represents a nonequilibrium dynamics, so
that no violation of the second law of thermodynamics is
implied 10. Ratchets have been proposed to rectify periodic
signals too rocked ratchets 11.
The purpose of the present paper is to investigate the role
of disorder in the transport properties of a thermal ratchet.
Two instances of disorder are discussed in some detail: i
impurities, for which a certain fraction of the asymmetric
unit cells or teeth of the potential V ( x ) are replaced at
random by unit cells with opposite asymmetry Fig. 1, and
ii randomness, for which the potential teeth all have the
same asymmetry, but their size is randomly distributed. A
variety of disordered ratchets may be construed by combin-
ing disorder of types i and ii. We conclude that disorder
quenches the rectifying power of both thermal and rocked
ratchets; most notably, a number of ratchet-related transport
properties become detectable in the presence of disorder. Our
results allow us to extend the notion of thermal ratchet to the
model of transport in polymers 12 and biological macro-
molecules 3,9, on random surfaces interfaces13, in ar-
rays of Josephson junctions 14, of vortex lines in two-
dimensional superconductors 15, and of point defects and
dislocations in polycrystalline media 16, to quote but a few
examples.
Let us specialize the ratchet model 1 by choosing the
autocorrelation function
t 0 =
2
exp -| t | / , 2
with
2
=D / for the noise source ( t ) and the shape
V x =
V
i
m
+A
1
x -x
i
, 0 x -x
i
l
1,i
V
i +1
m
-A
2
x -x
i +1
, -l
2,i
x -x
i +1
0
3
for the i th ratchet tooth see the inset of Fig. 1. The two
adjacent potential minima V
i
m
=V ( x
i
) and V
i +1
m
=V ( x
i +1
)
are separated by a distance x
i +1
-x
i
=l
1,i
+l
2,i
and a poten-
tial maximum V
i
M
=V
i
m
+A
1
l
1,i
=V
i +1
m
+A
2
l
2,i
. The overall
potential function V ( x ) is fully determined when the se-
quence of the minima V
i
m
or, equivalently, the sequence of
the lengths l
1,i
and l
2,i
for a given choice of V
0
m
– is assigned.
The ideal periodic ratchet potential 4–6 corresponds to set-
ting V
i
m
=0 for i =0,1,2, . . . , so that l
1,i
=l
1
and l
2,i
=l
2
with l
1
+l
2
=L and V
i
M
=V
M
with V
M
=A
1
l
1
=A
2
l
2
.
The directionality of the ratchets in Fig. 1 is determined
by the choice of A
1
and A
2
; here A
1
and A
2
are independent
of the tooth index i and A
2
A
1
. Thus we define the dimen-
sionless rectifying factor
, D =2
+
, D -
-
, D
+
, D +
-
, D
, 4
FIG. 1. Ratchet potentials V ( x ): a periodic, made of + cells
i.e., A
2
A
1
); b and c with impurities represented by - cells.
Impurities in b and c are topologically different. The potential in
c can be obtained from the one in b through the discrete trans-
formation V ( x ) →-V ( -x ). A +, - interface divides a + cell se-
quence on the right-hand side from a - cell sequence on the left-
hand side vice versa for a -, + interface. d Periodic pattern of
cells. The extended unit cell is made here of two + and two -
cells. Inset: the ratchet tooth of Eq. 3.
PHYSICAL REVIEW E SEPTEMBER 1997 VOLUME 56, NUMBER 3
56 1063-651X/97/563/24924/$10.00 2492 © 1997 The American Physical Society