String ratchets: ac driven asymmetric kinks
G. Costantini,
1
F. Marchesoni,
1,2,3
and M. Borromeo
3,4
1
Istituto Nazionale di Fisica della Materia, Universita` di Camerino, I-62032 Camerino, Italy
2
Michigan Center for Theoretical Physics, University of Michigan, Ann Arbor, Michigan48109-1120
3
Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, I-06123 Perugia, Italy
4
Dipartimento di Fisica, Universita` di Perugia, I-06123 Perugia, Italy
Received 18 January 2002; published 26 April 2002
We simulated numerically the time evolution of a one-kink bearing, damped elastic string sitting on noise-
less periodic substrates of two types: I asymmetric, time independent, II symmetric, periodically deform-
able. An asymmetric kink subjected to an ac drive is shown to drift steadily with finite average speed inde-
pendent of its initial kinetic conditions. In the overdamped regime the resulting net kink transport can be
attributed to the rectification of the Brownian motion of a pointlike particle with oscillating mass. For inter-
mediate to low damping completely different features show up, due to the finite size of the objects being
transported; in particular, the kink current hits a maximum for an optimal value of the damping constant,
resonates at the kink internal-mode frequency and, finally, reverses sign within a certain range of the drive
parameters.
DOI: 10.1103/PhysRevE.65.051103 PACS numbers: 05.60.Cd, 05.50.+q, 11.27.+d
I. INTRODUCTION
A transport mechanism of potential relevance both to ap-
plied physics and nanobiology is the so-called ratchet effect
1. In its simplest instance a ratchet device can be assimi-
lated to a Brownian particle with coordinate x ( t ) moving in
an asymmetric periodic potential V ( x ), with V ( x +a )
=V ( x ), subjected to viscous damping and ac drive rocked
ratchet 2: The natural direction and the intensity of the
ratchet current x
˙
results from a rather intricate interplay of
particle inertia, spatial asymmetry, and time correlation of
the forcing terms including fluctuations when present.A
similar mechanism is expected to operate also when instead
of a pointlike Brownian particle one considers a fluctuating
one-dimensional elastic string ( x , t ) 3 and replaces, ac-
cordingly, V ( x ) with a periodic substrate potential V
,
such that V
+a =V
. The kinks and antikinks born by
the string as it connects adjacent substrate valleys tend to
glide apart, so that the string center of mass advances effec-
tively in the natural ratchet direction 4. In rocked ratchets
noise is required either to aid the escape of the Brownian
particles over the potential barriers or to nucleate kink-
antikink pairs along the string, directed parallel to the sub-
strate valleys. In both cases a sufficiently large drive ampli-
tude can activate a net ratchet current even in the absence of
fluctuations.
In the present paper we address the ratchet dynamics of an
elastic string diffusing on a periodically tilted substrate. In
order to catch the essence of the mechanism at work we
ignore the spatiotemporal fluctuations responsible for the
thermalization of unperturbed string 3; instead, we impose
that the string always bears at least one kink, that is
( , t ) - ( - , t ) =a . The initial problem is thus reduced
to the question as how each individual damped kink anti-
kink responds to an external periodic drive. In the case of a
symmetric substrate ( V - =V
, like in the sine-
Gordon theory and a sinusoidal tilt, it has been noticed 5
that an isolated frictionless kink may travel in either direc-
tion depending on the tilt parameters and, most importantly,
on its initial conditions momentum modulus and phase.
Such an effect follows from the spontaneous symmetry
breaking induced by the external tilt; indeed, on averaging
over all initial conditions the kink current vanishes, as one
would expect on the ground of simple symmetry arguments.
Furthermore, adding damping, no matter how small, makes
this effect vanish completely.
As stated in the earlier ratchet literature 1, the onset of a
kink ratchet current requires a sufficient amount of asymme-
try in the system. In this paper we address two classes of
asymmetric kink dynamics: I Ratchet Potential (RP): V
is intrinsically asymmetric and time independent; the kink is
asymmetric also in the absence of a tilt unperturbed kink
due to the asymmetry of the barrier separating any two ad-
jacent potential valleys; II Deformable Potentials (DP):
V
is symmetric at all times; its shape is modulated so that
its valleys broaden and shrink with constant height periodi-
cally in time 6; the tilt is phase locked to the substrate
modulation, thus providing for an effective cycle asymmetry.
To avoid further complications, we assume that the strings of
both classes bear one species of kink, only; this implies that
all V
valleys are degenerate and have the same curvature.
Entropic rectification effects like those described in Refs.
7,8 are thus ruled out.
The main conclusion of our study is that strings of classes
I and II possess sufficient asymmetric coupling to the ac
drive to sustain steady kink transport even in the presence of
finite damping. The rather complicated dependence of the
kink ratchet current on the damping constant and the tilt
parameters makes a detailed analysis of the system at hand
worthy our extensive numerical simulation effort reported on
here. Our presentation is organized as follows. In Sec. II we
introduce an example for each class of asymmetric strings
RP and DP; we then estimate the effect of a small periodic
tilt on the relevant kink dynamics in the adiabatic limit, thus
explaining why we expect a finite kink current. In Sec. III we
PHYSICAL REVIEW E, VOLUME 65, 051103
1063-651X/2002/655/0511037/$20.00 ©2002 The American Physical Society 65 051103-1