Papers in Physics, vol. 8, art. 080004 (2016)

Received: 30 October 2015, Accepted: 4 February 2016
Edited by: G. Mart́ınez Mekler
Reviewed by: J. Mateos, Departamento de Sistemas Complejos, Instituto de F́ısica,

Universidad Nacional Autónoma de México, México.
Licence: Creative Commons Attribution 3.0
DOI: http://dx.doi.org/10.4279/PIP.080004

www.papersinphysics.org

ISSN 1852-4249

Invited review: Fluctuation-induced transport. From the very small to
the very large scales

G. P. Suárez,1 M. Hoyuelos,1∗ D. R. Chialvo2

The study of fluctuation-induced transport is concerned with the directed motion of par-
ticles on a substrate when subjected to a fluctuating external field. Work over the last
two decades provides now precise clues on how the average transport depends on three
fundamental aspects: the shape of the substrate, the correlations of the fluctuations and
the mass, geometry, interaction and density of the particles. These three aspects, reviewed
here, acquire additional relevance because the same notions apply to a bewildering variety
of problems at very different scales, from the small nano or micro-scale, where thermal
fluctuations effects dominate, up to very large scales including ubiquitous cooperative
phenomena in granular materials.

I. Introduction

Much of the efforts devoted to particle transport
were triggered by the famous challenge at very
small scales presented by Feynman in 1959: “A bi-
ological system can be exceedingly small. Many of
the cells are very tiny, but they are very active.
(...) Consider the possibility that we too can make
a thing very small, which does what we want —
that we can manufacture an object that maneuvers
at that level!” [1].

At the scales discussed by Feynman, our most
usual notions of work, energy and transport seem to
break down, including some counterintuitive obser-

∗E-mail: hoyuelos@mdp.edu.ar

1 Instituto de Investigaciones F́ısicas de Mar del Plata
(IFIMAR - CONICET) and Departamento de F́ısica,
Facultad de Ciencias Exactas y Naturales, Universidad
Nacional de Mar del Plata, Deán Funes 3350, 7600 Mar
del Plata, Argentina.

2 Consejo Nacional de Investigaciones Cient́ıficas y
Técnicas (CONICET), Godoy Cruz 2290, Buenos Aires,
Argentina.

vations. As discussed in these notes, these findings
are not restricted to small scales since work in the
last decades shows similar dynamics arising any-
time there is a peculiar interplay of fluctuations,
nonlinearity and correlations resulting in various
classes of fluctuation-induced transport.

To visualize the problem, consider a gedankenex-
periment involving, for the sake of discussion, our
desk. Elementary physics explains how all the ob-
jects at the desk stay in place and/or which forces
are needed to displace them. Now, consider the
imaginary case in which we progressively shrink all
the objects up to a size of a few nanometers. It
will be noticed that while at the natural scale ob-
jects remain steady without any energy expendi-
ture, at the nanometers scale things move around,
our “nano cell phone” which was quiet at the nat-
ural scale desk, moves and falls off the “nano desk.
This exercise reminds us that at the Brownian do-
main, energy would be required even to stay quiet
since the basic macroscopic methods of controlling
energy flow no longer remain valid. This nonin-
tuitive phenomenon in the function of molecular
machines was described by Astumian as follows [2]:

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Papers in Physics, vol. 8, art. 080004 (2016) / G. P. Suárez et al.

“any microscopic machine must either work with
Brownian motion or fight against it, and the for-
mer seems to be the preferable choice”. Analogous
observations, with some additional caveats due to
inertial forces, can be made if instead of shrinking
the mass, we apply an increasingly large external
fluctuating field, making now our real size desk to
shake around.

This brief review is dedicated to discuss the
essence of three elementary results of fluctuation-
induced transport including the potential shape,
the correlations of the fluctuations and the parti-
cle interactions and how they work, calling atten-
tion to some common lessons that can be borrowed
from problems in apparently far apart scales and
fields, from cellular biology to technological appli-
cations and applied physics. It should be noted
that it is not our intention to cover the extent of
the field, this is neither a fair, nor historically cor-
rect, exhaustive or updated review of the relevant
literature; it only encompasses some interesting re-
sults which, in our opinion, warrant further explo-
ration. The reader will find comprehensive reviews
covering specific topics, including those on Brow-
nian motors in [3–7], on the more general subject
of molecular motors in [2, 8–12], on a more biologi-
cal perspective of molecular springs and ratchets in
[13], or on a systematic analysis of the space-time
symmetries of the equations in [14].

The paper is organized as follows. The next sec-
tion revisits pioneer works on these types of prob-
lems, carried on a hundred years ago. Next, we
discuss the three fundamental aspects of the prob-
lem, including the substrate, the correlations of the
fluctuations and the particle interactions. We start
by briefly introducing the different realizations of
fluctuation-induced transport as popularized two
decades ago, i.e., in the so-called correlation ratch-
ets. After that, the two elementary ways to break
the symmetry are reviewed, either in the temporal
or in the spatial aspects of the system, to conclude
introducing yet another way to affect transport, the
correlations born out of many particle interactions.
The review closes with a discussion of some appli-
cations and new directions.

Figure 1: Feynman’s imaginary microscopic
ratchet, comprised by vanes, a pawl with a spring,
two thermal baths at temperatures T1 > T2, an
axle and wheel, and a load m.

II. Smoluchowski-Feynman’s
ratchet as a heat engine

Feynman famous lectures [15] include an imaginary
microscopic ratchet device to illustrate the second
law of thermodynamics. The basic idea belongs to
Smoluchowski who discussed it during a conference
talk in Münster in 1912 (published as proceedings-
article in Ref. [16]). As seen in Fig. 1, it consists
of a ratchet, a paw and a spring, vanes, two ther-
mal baths at temperatures T1 > T2, an axle and
wheel, and a load. The ratchet is free to rotate
in one direction, but rotation in the opposite di-
rection is prevented by the pawl. The system is as-
sumed small so that molecules of the gas at temper-
ature T1 that collide with the vanes produce large
fluctuations in the rotation of the axle. Fluctua-
tions are rectified by the pawl. The net effect is
a continuous rotation of the axle that can be used
to produce work by, for example, lifting a weight
against gravity. The pawl becomes a materializa-
tion of Maxwell’s demon, a small agent able to ma-
nipulate fluctuations at a microscopic level in order
to violate the second law of thermodynamics, since
in this case a given amount of heat is completely
transformed into work. A closer inspection shows
that such violation does not really take place. Feyn-
man demonstrated that, if T1 = T2, no net rotation
of the axle is produced. The reason is that the pawl
has its own thermal fluctuations that, from time to

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Papers in Physics, vol. 8, art. 080004 (2016) / G. P. Suárez et al.

U(x)

x

λ
Q	
  

Figure 2: Typical ratchet potential U(x).

time, allow a tooth of the ratchet to slip in the op-
posite direction. Not even demons are free from
thermal fluctuations. In order for the machine to
work as intended, the pawl should be colder than
the vanes, T2 < T1. But in this case, there is a heat
flux between thermal baths. The mechanical link
between vanes and ratchet through the axle implies
that the baths are not thermally isolated [17], even
when the materials are perfect insulators. The sys-
tem performs as a heat engine: some work is gen-
erated while some heat is transferred from a cold
reservoir to a hot reservoir.

In summary, Feynmann’s ratchet —and Brown-
ian motors— actually work, but without violating
the laws of thermodynamics.

III. Breaking the symmetry: time,
space and interactions

Feynman’s deep thinking motivated an entire gen-
eration of models around the same idea. The model
ratchet is a fluctuations-driven overdamped nonlin-
ear dynamical system described by

ẋ = −U′(x) + f(t),
U(x) = U(x + λ), 〈f(t)〉 = 0,

(1)

where U(x) is a periodic potential, such as the one
illustrated in Fig. 2, and f(t) is zero-mean fluctua-
tion of some type. In general, the initial theoretical
problem is to find the stationary current density
j = 〈ẋ(t)〉 in the ratchet given the statistical prop-
erties of the fluctuation f(t) and the shape of U(x),
and to be able to determine the most efficient con-
ditions for the transformation of fluctuations into
a net current.

Multiple variations and extensions of the same
problem were studied in the 90s resulting on a jar-
gon of names such as on-off ratchets [8], fluctuat-
ing potential ratchets [18,19], temperature ratchets
[20, 21], chiral ratchets [22–24], and so on. In any
case, three elements are always present: a parti-
cle which eventually will execute some motion and
two forces, one coming from the external applied
field and another given by the particular shape of
the potential (i.e., the substrate where the parti-
cle resides). Thus, an isolated particle “feels” two
forces, but while such information is available to
an observer, it is important to realize that the par-
ticle has no way to distinguish or separate these
sources. Thus, the break of symmetry resulting
in average directed motion of a particle could come
from either spatial or temporal sources. Yet, a third
force needs to be considered in the cases in which
the concentration of particles becomes relevant and
then particles mutual interactions are not negligi-
ble anymore, an aspect crucial to understand flow
in channels. We will consider all these cases in the
following sections.

i. Asymmetries in the substrate

Figure 3 summarizes the two basic ways in which
asymmetries in space (or some other degree of free-
dom of the system, such as phase [25]) contribute to
noise-induced transport. The common situation in-
volves an asymmetric periodic potential that breaks
the spatial inversion symmetry combined with a
temporal, zero mean, forcing periodicity. In Panel
A, the case of turning on and off the asymmetric
potential is depicted and Panel B shows the case in
which a tilting force is added.

The first important result was due to Magnasco
in [19] who considered the case of the piecewise lin-
ear potential U(x) shown in Fig. 4A, which is ex-
actly solvable [26] for slow fluctuation f(t); it has a
characteristic time much larger than the ratchet’s
relaxation time. The potential is periodic and ex-
tends to infinity in both directions. λ measures the
spacing of the wells, λ1 and λ2 the inverse steep-
nesses of the potential in opposite directions out
of the wells, and Q the well depths. The parti-
cle undergoes overdamped Brownian motion due
to its coupling with a thermal bath of temperature
T , and an external driving F(t) which represents
the forces. These two ingredients compose what

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Papers in Physics, vol. 8, art. 080004 (2016) / G. P. Suárez et al.

λ1 λ2

QON

ON

A

Q-Fλ2

Q-Fλ1

+F

–F

λ1 λ2
B

OFF

Figure 3: Spatial asymmetry. Panel A: The so-
called flashing ratchet is a type of ratchet in which
an asymmetric potential is periodically switched off
and on. Particles (green circles) diffuse evenly dur-
ing the off period while the asymmetric potential
favors the drift in one direction, producing a net
transport to the left. Panel B: In a rocked ratchet,
an asymmetric potential is tilted periodically deter-
mining a (right directed) transport of the particle
trough the relative lowest Q−Fλ2 value of the right
potential barrier.

we called the fluctuation f(t). The expression for
the current in the adiabatic limit, which measures
the work done by the ratchet was shown to be

J(F) =
P22 sinh(λF/2kT)

kT
(
λ
Q

)2
P3 −

(
λ
Q

)
P1P2 sinh

(
λF
2kT

), (2)
P1 = δ +

λ2 − δ2
4

F

Q
, P2 =

[
1 − δF

2Q

]2
−
(
λF

2Q

)2
,

P3 = cosh[(Q− δF/2)/kT] − cosh(λF/2kT),

where λ = λ1 + λ2 and δ = λ1 −λ2. The average
current, the quantity of primary interest, is given
by

j = 〈J〉 = 1
τ

∫ τ
0

J(F(t)) dt, (3)

where τ is the period of the driving force F(t),
which is assumed longer than any other time scale
of the system in this adiabatic limit. The current is
maximized for a given value of the periodic forcing
amplitude. Interestingly, numerical computations
showed robustness in the results when the forcing
is not periodic. The key feature is that it should
have a long time correlation.

According to Magnasco, “all that is needed to
generate motion and forces in the Brownian domain
is loss of symmetry and substantially long time cor-
relations” [19]. Indeed, if the forcing is white noise,
the system is at thermal equilibrium and j = 0.
However, if the fluctuation auto-correlations are
non-vanishing, i.e., for colored noise, the system
is no longer in thermal equilibrium, and in general
j 6= 0. Since onset of a current means breaking the
“right-left” symmetry, currents may only arise, in
the case of additive noise, if the potential U(x) is
asymmetric with respect to its extrema. It could
be argued [27] that the emergence of current can
be viewed as an example of “temporal order com-
ing out of disorder”, since the current is apparently
time-irreversible, whereas stationary noise does not
distinguish “future” from the “past”; we notice,
however, that Eq. (1) implies relaxation and is thus
time-irreversible itself.

The flashing or pulsating ratchet depicted in
Panel A of Fig. 3 was introduced in [28] and re-
introduced in a more general theoretical context
in [29]. Despite the huge structural complexity of
biological Brownian motors, the majority of the
models are compatible with a simplified descrip-
tion based on the flashing ratchet. The description

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Papers in Physics, vol. 8, art. 080004 (2016) / G. P. Suárez et al.

is in terms of only one variable x that may rep-
resent, for example, the position of a molecule or
the coordinate of a complex reaction with many in-
termediate steps. The environment, composed by
some aqueous solution acts, on one hand, as a heat
bath and, on the other hand, as a source or sink of
ATP, ADP and Pi molecules of the chemical reac-
tion cycle that provides energy to the motor. In this
simplified model, a periodic asymmetric potential
is periodically turned on and off, as shown in Fig.
3A. The situation is generalized to stochastic vari-
ations of the potential with a characteristic corre-
lation time. As happens for the rocked ratchet, the
current vanishes for zero correlation time, or fast
pulsating limit (white noise). It also disappears in
the slow pulsating limit, i.e., when the potential is
left on or off for a diverging time. There is an opti-
mum value of the correlation time that maximizes
the current.

A recent example of an experimental realization
of a rocked ratchet in a mesoscopic scale can be
found in [30], where dielectric particles suspended
in water are affected by a ratchet potential given
by a periodic and asymmetric light pattern.

ii. Temporal asymmetries

Figure 4B summarizes one type of ratchet in which
the higher order statistics of the driving force can
be responsible for the transport. Indeed, the work
of Millonas [25, 31, 32] and others [33] showed that
directed motion can be induced with an unbiased
driving force, deterministic or stochastic, as long
as it has asymmetric correlations: non zero odd
correlation of order higher than one [27].

The case analyzed in the seminal work of Mag-
nasco [19] only considered F(t) symmetric in time
F(t) = F(nτ−t). Instead, the work of Millonas [31]
considered the same setting but studying a more
general case in which the driving force still is non
biased zero mean, 〈F(t)〉 = 0, but which is asym-
metric in time,

F(t) =

{ (
1+�
1−�

)
A 0 ≤ t < τ(1−�)

2
, mod τ

−A τ(1−�)
2

< t ≤ τ, mod τ
(4)

as shown in Fig. 4B. In this case, the time averaged

A

B

U(x)

A(1+ℇ)/(1-ℇ)

τ(1+ℇ)/2

τ

x

Q

0

F(t)
τ(1-ℇ)/2

t

A	
  

λ

λ1 λ2

Figure 4: Panel A: The simplest piecewise ratchet
potential, where the spatial degree of asymmetry is
given by the parameter δ = λ1−λ2. Panel B: Fluc-
tuation’s temporal asymmetry. The driving force
F(t) preserves the zero mean 〈F(t)〉 = 0. The tem-
poral asymmetry is given by the parameter �.

current can be easily calculated,

〈J〉 = 1
2

(1 + �)J(−A)

+
1

2
(1 − �)J((1 + �)A/(1 − �))

(5)

Solving for different values of parameters, it was
shown that the current is a peaked function both
of kT (see Fig. 5A) and of the amplitude A of
the driving. As expected, the driving, the poten-
tial, and the thermal noise in fact play cooperative
roles. For low temperatures, any transport depends
on very large A values, while for large noise the fea-
tures of the potential and of the driving are washed
out.

The most striking results are concerning the com-
petition between the temporal asymmetry and the
spatial asymmetry, as pictured in Fig. 5B, result-
ing on the switching of the direction of the current
as the asymmetry factor � is varied. This reversal
represents the competition of the spatial asymme-
try, which dominates for small � an the temporal
asymmetry, which dominates for large �.

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Papers in Physics, vol. 8, art. 080004 (2016) / G. P. Suárez et al.

〈J〉

〈J〉

Figure 5: Temporally asymmetric fluctuations with
mean zero can optimize or reverse the current in
the ratchet of Fig. 4 (with Q = 1,λ = 1). Panel
A corresponds to the case of a symmetric poten-
tial (i.e., δ = 0) which shows a peaked function of
the net current 〈J〉 as a function of temperature
kT for � = 1 and three values of the driving am-
plitude A = 1,A = 0.8,A = 0.5 labeled (a), (b)
and (c), respectively. Panel B shows 〈J〉 versus the
temporal asymmetry parameter � for three asym-
metries in the shapes of the potential. Curve (a) is
for a symmetric potential, (i.e., δ = 0) and those
labeled (b) and (c) for two cases of asymmetric po-
tentials δ = −0.3 and δ = −0.7, respectively (with
kT = 0.01, and A = 2.1)

Temporal asymmetry and spatial asymmetry re-
late to the problem of nonequilibrium transport in
precisely the same way. In both cases, a net effect
arises due to an interplay between the strength of
a fluctuation, the time it acts, and the underly-

ing dynamics. In the case of a spatial asymme-
try, a fluctuation to the right with a given strength
which lasts a given time will tend to take the system
over the right-hand barrier while the same fluctu-
ation with sign reversed does not lift it over the
left-hand barrier. In the case of temporal asym-
metry, the probabilities of the fluctuations to the
right or to left are different, so the net effect arises
in the absence of spatial asymmetries. What both
of them show is that even a subtle asymmetry in
the shape of the potential or in the shape of the
spectral properties of the noise will give rise to an
effect even when the net force due to each vanishes.

The time asymmetry of the mean zero fluctu-
ations discussed above can be cast in several dif-
ferent ways. Dichotomic noise (a type of “Kubo-
Anderson” process) was used to demonstrate phase
transport in a pair of Josephson junctions [25].
There are also types of continuous noise exhibit-
ing similar asymmetry, including shot noise (com-
mon in quantum electronics) which are of this type.
Mean zero shot noise, which is temporally asym-
metric, can be produced if the frequency and ampli-
tude distribution are slightly different for positive
and negative fundamental pulses. Another trivial
example of temporally asymmetric driving force is
a simple bi-harmonic signal which constitutes a cu-
riosity since it results from adding two (zero mean
symmetric) periodic process of harmonic frequen-
cies.

iii. Particle interactions

The previous discussions were limited to the cases
in which an isolated or a few particles were present
in the potential. As the concentration is increased,
interaction among particles becomes relevant, and
it can be the cause of a reduction, and even of a
reversal [35–38] of the current. We present two ex-
amples.

a. Vortex current in a 2D array of Josephson
junctions

Current reversal has been experimentally observed
in a two dimensional array of Josephson junctions
[39]. It was numerically analyzed in [40]. A ratchet
potential for vortices is generated by modulating
the gap between superconducting islands. The den-
sity of vortices is controlled by an external magnetic

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Papers in Physics, vol. 8, art. 080004 (2016) / G. P. Suárez et al.

Figure 6: Diffusion in a periodic channel with asymmetric cavities when a force to the right is applied;
A(x) is the witdth of the channel. Parameters: total average concentration c = 0.5; force β aF = 0.5,
where a is the lattice spacing; Lx = Ly = 100 a.

field. There is a repulsive vortex-vortex interaction.
The results show a preferred direction of the vortex
motion, parallel to the ratchet modulation, when
an alternating force is applied. But as the vortex
concentration is increased, this direction is reversed
for appropriate values of the periodic forcing inten-
sity. (Vortex current reversal is also observed for a
fixed value of the concentration when the periodic
forcing, or AC current amplitude, is varied, see Fig.
4 in [39] or Fig. 2 in [40]).

The vortex current reversal produced by the in-
crease of concentration is a consequence of the fol-
lowing symmetry [39]. Let us consider that the ex-
ternal magnetic field is such that positive vortices
are produced. For small concentration (frustration
parameter between 0 and 1/2), we have a small
discrete number of positive vortices. For large con-
centration (frustration between 1/2 and 1), we can
consider that there is a background of positive vor-
tices in which some negative vortices move. But
the movement of this negative vortices is in the op-
posite direction. For them, the ratchet potential is
inverted so the rectification effect of the ratchet is
inverted too.

b. Particle diffusion in a channel with asymmetric
cavities

The same effect is observed in a different con-
text. In the next paragraphs, we refer to the hard
core interaction between particles that diffuse in
a channel with a transverse section A(x) that has

a ratchet shape, see Fig. 6. An external periodic
forcing is applied in the direction of the channel.
There is a particle-hole symmetry. But before go-
ing into the interaction effects, let us consider the
low concentration regime, where interactions can
be neglected. Several interesting experiments have
been performed with particles suspended in a liq-
uid and contained in a channel qualitatively as the
one shown in Fig. 6. There are basically two ways
to apply the periodic external forcing. In one case,
a periodic variation of the pressure is used: parti-
cles are drifted back and forth by the movement of
the liquid; see [41, 42] and the critical report [43]
(cavities of order 5 µm). In the other case, the liq-
uid remains still and the force is directly applied on
the particles by an external field as, for example,
an electric field on charged particles [44] (cavities
of order 50 µm).

Such a system has been proposed for separa-
tion of particles of different size [45]. The idea
is based on the difference between rectification ef-
fects for different size particles. When a periodic
—unbiased— forcing is applied, particles move in
the forward direction because of the ratchet; but,
in general, larger particles move faster than smaller
ones. Now we apply a bias, a constant force in
the backward direction that reduces the velocity of
the larger particles and reverses the velocity of the
smaller ones. Then we have that larger particles
end up in one extreme of the channel and smaller
particles in the opposite one, with an estimated pu-
rity of 99.997 % according to the authors of [45].

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Papers in Physics, vol. 8, art. 080004 (2016) / G. P. Suárez et al.

Figure 7: Rectified particle current ∆I against av-
erage particle concentration c for different force am-
plitudes. Units of ∆I: D/a2, where D is the diffu-
sion coefficient and a is the lattice spacing; units of
F: (βa)−1. Lower left inset: scheme of the channel
composed by an array of triangular cavities.

The Fick-Jacobs equation [46] gives an appropri-
ate description of the particle density in the channel
as long as its dependence on the transverse direc-
tion, y, can be neglected, i.e., n(x,y,t) ' n(x,t).
Let us consider the transverse integral of the con-
centration: ρ(x,t) =

∫
dy n(x,y,t) ' A(x)n(x,t)

(the two-dimensional channel can be easily ex-
tended to a three-dimensional tube). If D is the
diffusion coefficient and F(t) is the total applied
force, the Fick-Jacobs equation is

∂ρ(x,t)

∂t
=

∂

∂x

[
D

(
∂ρ

∂x
+ β

∂H

∂x
ρ

)]
, (6)

where ∂H
∂x

= −F(t)−β−1 d
dx

ln A(x) and β−1 = kT .
The expression β−1 ln A(x) is called entropic po-
tential due to the similarity with the thermody-
namic relation among energy, free energy, temper-
ature and entropy: H = U−TS, with ∂U

∂x
= −F(t).

In a first approximation, the diffusion coefficient is
constant; a further refinement considers a depen-
dence on A′(x), see [47].

Now, let us consider the hard core interaction be-
tween particles, of the same size, diffusing in a lat-
tice. A jump of a particle to the right is equivalent
to a jump of a hole to the left. A concentration c of

Figure 8: Particle concentration against longitudi-
nal position x for one cavity of the channel depicted
in Fig. 6, in a stationary state (cavity length nor-
malized to 1). Dots correspond to Monte Carlo
simulations. The curve is obtained from numeri-
cal integration of (7). Concetration c = 0.5, more
details in [49].

particles subjected to a force F is equivalent to a
concentration 1−c of holes subjected to a force −F.
This symmetry is the cause of the shape of Fig. 7,
where Monte Carlo results of the rectified current
∆I against the average concentration c for different
values of the forcing amplitude is plotted [48]. A
square wave in the limit of low frequency was used
for the applied force; in this limit, the rectified cur-
rent is equal to the difference between the current
for the force in the positive phase and the current
for the force in the negative phase. The particle-
hole symmetry is evident in the figure: changing
c → 1 − c and ∆I → −∆I (a consequence of the
change F → −F), we recover the same curves.
Le us note the current reversal for large concen-
tration. It is the same effect that was mentioned
in the previous section for vortex current in 2D ar-
rays of Josephson junctions. A description based on
the Fick-Jacobs equation is also possible for parti-
cles with hard-core [49]. Its derivation starts from
the non-linear Fokker-Planck equation for fermions
[50], where Pauli exclusion principle plays the role
of the hard core interaction. Following the same
steps used for derivation of the linear Fick-Jacobs
equation [47], we can arrive at the following non

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Papers in Physics, vol. 8, art. 080004 (2016) / G. P. Suárez et al.

linear version:

∂n(x,t)

∂t
=

1

A

∂

∂x
DA

(
∂n

∂x
−βFn (1 −n)

)
. (7)

The non linear term, n (1 − n), is the responsible
of the interaction. Fig. 8 shows numerical integra-
tion of (7) in good agreement with Monte Carlo
simulation results.

IV. Engineers knew it...

The principles discussed above ruling the corre-
lation ratchets have, in a macro scale, important
technological applications. Of course, some of them
were applied even before a detailed statistical un-
derstanding was available. Vibratory conveyors, or
vibratory bowl feeders, are regularly used in many
branches of industry such as food processing, syn-
thetic materials or small-parts assembly mechanics,
to mention just a few [51]. The conveying speed of
these devices was theoretically and experimentally
studied, for example, in [52] and references cited
therein.

There are many parameters involved in the oper-
ation of a vibratory conveyor: amplitude, frequency
and mode of the vibrations, inclination angle and
friction coefficient are only some of them. A clas-
sification in terms of the vibratory modes is as fol-
lows: sliding (linear horizontal vibration), ratchet-
ing (linear vertical vibration) or throwing (circular,
elliptical or linear tilted vibration).

For throw conveyors, the material being trans-
ported loses contact with the through during part
of the cycle, see Fig. 9A. Appropriate for granu-
lar materials o small objects, particles are forced
to perform repeated short flights with a preferred
direction, combined with rest and slide phases.

The other two types: sliding and ratcheting in-
volve temporal and spatial asymmetries, respec-
tively.

The sliding type of vibratory conveyors allows
transport over a deck that vibrates back and forth
with asymmetric motion (see Fig. 5A) in the hori-
zontal direction. The particle or object moves rela-
tive to the deck due to alternate stick and slip steps
driven by the asymmetric oscillations, as shown in
Fig. 9B.

B

C

slowfast

A

Figure 9: Three types of vibratory conveyors which
share some of the principles of small scale ratch-
ets. Panel A: throwing conveyor with linear tilted
vibration. Panel B: sliding conveyor; transport is
induced by asymmetric horizontal oscillations with
zero mean, of the kind shown in Fig. 4B. Panel C:
ratchet conveyor with vertical oscillations; similar
to the flashing ratchet of Fig. 3A.

The ratchet conveyor achieves transport of gran-
ular material using vertical vibrations [38, 53]. Di-
rected motion is caused by the broken space sym-
metry of the deck’s surface, given by a sawtooth-
shaped profile, see Fig. 9C. The ratchet conveyor
shares qualitative features with the flash or pulsat-
ing ratchet depicted in Fig. 3A. One difference is
that it includes a ballistic flight phase.

V. Conclusions

Spatial or temporal asymmetries, or both, are able
to generate directed motion in the presence of fluc-
tuations. In addition to a thermal bath, fluctua-
tions with large correlation time, compared to the
characteristic relaxation time of the system, should
be included. During the last decades, simple mod-
els based on these ideas provided a deeper under-

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Papers in Physics, vol. 8, art. 080004 (2016) / G. P. Suárez et al.

standing of the complex biological machinery at the
nano scale. This success stimulated the study of
ratchets in a wide variety of contexts, and in larger
scales. Interactions among transported particles
are relevant for high concentration; most notice-
able, they may produce an inversion of the pur-
ported motion direction.

Vibrations inducing directed motion are used in
industry for the transport of small —macroscopic—
objects since, at least, around 1950. Vibratory con-
veyors applied the qualitative features of ratchets,
with space or time asymmetries, before a detailed
theoretical understanding was available. Half a
century ago, Feynman called attention to the fact
that in his view, “there’s plenty of room at the bot-
tom” [1]. We can safely conclude that, even today,
there is plenty of room at the top as well.

Acknowledgements - This work was partially
supported by Consejo Nacional de Investigaciones
Cient́ıficas y Técnicas (CONICET, Argentina)

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