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Artificial Brownian motors: Controlling transport on the nanoscale Peter Hänggi * Institut für Physik, Universität Augsburg, Universitätsstrasse 1, D-86135 Augsburg, Germany and Department of Physics and Centre for Computational Science and Engineering, National University of Singapore, Singapore 117542, Singapore Fabio Marchesoni Dipartimento di Fisica, Università di Camerino, I-62032 Camerino, Italy and School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea Published 30 March 2009 In systems possessing spatial or dynamical symmetry breaking, Brownian motion combined with unbiased external input signals, deterministic and random alike, can assist directed motion of particles at submicron scales. In such cases, one speaks of “Brownian motors.” In this review the constructive role of Brownian motion is exemplified for various physical and technological setups, which are inspired by the cellular molecular machinery: the working principles and characteristics of stylized devices are discussed to show how fluctuations, either thermal or extrinsic, can be used to control diffusive particle transport. Recent experimental demonstrations of this concept are surveyed with particular attention to transport in artificial, i.e., nonbiological, nanopores, lithographic tracks, and optical traps, where single-particle currents were first measured. Much emphasis is given to two- and three-dimensional devices containing many interacting particles of one or more species; for this class of artificial motors, noise rectification results also from the interplay of particle Brownian motion and geometric constraints. Recently, selective control and optimization of the transport of interacting colloidal particles and magnetic vortices have been successfully achieved, thus leading to the new generation of microfluidic and superconducting devices presented here. The field has recently been enriched with impressive experimental achievements in building artificial Brownian motor devices that even operate within the quantum domain by harvesting quantum Brownian motion. Sundry akin topics include activities aimed at noise-assisted shuttling other degrees of freedom such as charge, spin, or even heat and the assembly of chemical synthetic molecular motors. This review ends with a perspective for future pathways and potential new applications. DOI: 10.1103/RevModPhys.81.387 PACS numbers: 05.60.k, 47.61.k, 81.07.b, 85.25.j CONTENTS I. Introduction 388 A. Artificial nanodevices 389 B. Brownian motors 389 II. Single-Particle Transport 390 A. Symmetric substrates 391 1. dc drive 391 2. ac drive 392 3. Diffusion peak 393 4. Single-file diffusion 393 B. Rectification of asymmetric processes 393 C. Nonlinear mechanisms 394 1. Harmonic mixing 394 2. Gating 395 3. Noise induced transport 395 a. Noise mixing 395 b. Noise recycling 396 D. Brownian motors 396 1. Rocked ratchets 397 2. Pulsated ratchets 399 3. Correlation ratchets 400 4. Further asymmetry effects 400 a. Current offsets 400 b. Asymmetry induced mixing 401 E. Efficiency and control issues 402 1. Optimization 402 2. Vibrated ratchets 403 III. Transport in Nanopores 404 A. Ion pumps 404 B. Artificial nanopores 405 C. Chain translocation 405 D. Toward a next generation of mass rectifiers 406 1. Zeolites 406 2. Nanotubes 407 IV. Cold Atoms in Optical Lattices 408 A. Biharmonic driving 408 B. Multifrequency driving 409 C. More cold atom devices 410 V. Collective Transport 410 A. Asymmetric 1D geometries 411 1. Boundary effects 411 2. Asymmetric patterns of symmetric traps 411 B. 2D lattices of asymmetric traps 412 C. Binary mixtures 413 * [email protected] [email protected] REVIEWS OF MODERN PHYSICS, VOLUME 81, JANUARY–MARCH 2009 0034-6861/2009/811/38756 ©2009 The American Physical Society 387

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Artificial Brownian motors: Controlling transport on the nanoscale

Peter Hänggi*

Institut für Physik, Universität Augsburg, Universitätsstrasse 1, D-86135 Augsburg,Germanyand Department of Physics and Centre for Computational Science and Engineering,National University of Singapore, Singapore 117542, Singapore

Fabio Marchesoni†

Dipartimento di Fisica, Università di Camerino, I-62032 Camerino, Italyand School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea

Published 30 March 2009

In systems possessing spatial or dynamical symmetry breaking, Brownian motion combined withunbiased external input signals, deterministic and random alike, can assist directed motion of particlesat submicron scales. In such cases, one speaks of “Brownian motors.” In this review the constructiverole of Brownian motion is exemplified for various physical and technological setups, which areinspired by the cellular molecular machinery: the working principles and characteristics of stylizeddevices are discussed to show how fluctuations, either thermal or extrinsic, can be used to controldiffusive particle transport. Recent experimental demonstrations of this concept are surveyed withparticular attention to transport in artificial, i.e., nonbiological, nanopores, lithographic tracks, andoptical traps, where single-particle currents were first measured. Much emphasis is given to two- andthree-dimensional devices containing many interacting particles of one or more species; for this classof artificial motors, noise rectification results also from the interplay of particle Brownian motion andgeometric constraints. Recently, selective control and optimization of the transport of interactingcolloidal particles and magnetic vortices have been successfully achieved, thus leading to the newgeneration of microfluidic and superconducting devices presented here. The field has recently beenenriched with impressive experimental achievements in building artificial Brownian motor devicesthat even operate within the quantum domain by harvesting quantum Brownian motion. Sundry akintopics include activities aimed at noise-assisted shuttling other degrees of freedom such as charge,spin, or even heat and the assembly of chemical synthetic molecular motors. This review ends with aperspective for future pathways and potential new applications.

DOI: 10.1103/RevModPhys.81.387 PACS numbers: 05.60.k, 47.61.k, 81.07.b, 85.25.j

CONTENTS

I. Introduction 388

A. Artificial nanodevices 389

B. Brownian motors 389

II. Single-Particle Transport 390

A. Symmetric substrates 391

1. dc drive 391

2. ac drive 392

3. Diffusion peak 393

4. Single-file diffusion 393

B. Rectification of asymmetric processes 393

C. Nonlinear mechanisms 394

1. Harmonic mixing 394

2. Gating 395

3. Noise induced transport 395

a. Noise mixing 395

b. Noise recycling 396

D. Brownian motors 396

1. Rocked ratchets 397

2. Pulsated ratchets 399

3. Correlation ratchets 400

4. Further asymmetry effects 400

a. Current offsets 400

b. Asymmetry induced mixing 401

E. Efficiency and control issues 402

1. Optimization 402

2. Vibrated ratchets 403

III. Transport in Nanopores 404

A. Ion pumps 404

B. Artificial nanopores 405

C. Chain translocation 405

D. Toward a next generation of mass rectifiers 406

1. Zeolites 406

2. Nanotubes 407

IV. Cold Atoms in Optical Lattices 408

A. Biharmonic driving 408

B. Multifrequency driving 409

C. More cold atom devices 410

V. Collective Transport 410

A. Asymmetric 1D geometries 411

1. Boundary effects 411

2. Asymmetric patterns of symmetric traps 411

B. 2D lattices of asymmetric traps 412

C. Binary mixtures 413*[email protected][email protected]

REVIEWS OF MODERN PHYSICS, VOLUME 81, JANUARY–MARCH 2009

0034-6861/2009/811/38756 ©2009 The American Physical Society387

VI. Microfluidics 414A. Transporting colloids 415B. Transporting condensed phases 416C. Granular flows 418

VII. Superconducting Devices 419A. Fluxon channels 420B. Fluxon rectification in 2D arrays 422

1. Symmetric arrays of asymmetric traps 4222. Asymmetric arrays of symmetric traps 423

C. Anisotropic fluxon rectifiers 423VIII. Quantum Devices 424

A. Quantum dissipative Brownian transport 424B. Josephson Brownian motors 425

1. Classical regime 4262. Quantum regime 426

C. Quantum dot ratchets 426D. Coherent quantum ratchets 428

1. Quantum ratchets from molecular wires 4292. Hamiltonian quantum ratchet for cold

atoms 429IX. Sundry Topics 430

A. Pumping of charge, spin, and heat 430B. Synthetic molecular motors and machines 430

X. Concluding Remarks 432Acknowledgments 432References 433

I. INTRODUCTION

Over the past two decades advances in microscopyand microscale control have allowed scientists and engi-neers to delve into the workings of biological matter.One century after Kelvin’s death, today’s researchersaim to explain how the engines of life actually work bystretching thermodynamics beyond its 19th-century lim-its Haw, 2007.

Carnot realized that all engines transform energyfrom one form into another with a maximum possibleefficiency that does not depend on the technology devel-oped or on the fuel utilized, but rather on fundamentalquantities such as heat and temperature. Kelvin andClausius came up with two “rules of the engine,” laterknown as the first two laws of thermodynamics. The firstlaw states that energy cannot be destroyed or createdbut only transformed; the second law sets fundamentallimitations on the practical achievements of energytransformation. Just as the first law was centered on thenotion of energy from the Greek for “work capability”,the second law revolved around the new concept of en-tropy a term coined by Clausius, also from Greek, for“change capability”.1 When expressed in such terms,the second law states that entropy cannot decrease dur-

ing any spontaneous or natural process. Notably, withinthe whole virtual factory of all natural processes, thefirst law takes on the role of an account clerk, keepingtrack of all energy changes, while the second law takeson the role of the director, determining the direction andaction of all processes.

The fathers of thermodynamics developed their lawshaving in mind macroscopic systems that they could de-scribe in terms of state i.e., average quantities such aspressure and temperature, a reasonable assumptionwhen dealing with the monster steam engines of Victo-rian industry. A typical protein, however, is a few na-nometers in size and consists of just a few tens of thou-sands of atoms. As a consequence, the movements of aprotein engine are swamped by fluctuations resultingfrom the Brownian motion of its surroundings, whichcauses the energy of any of its parts to fluctuate continu-ally in units of kT, with k denoting the Boltzmann con-stant and T the temperature. The effects of such energyfluctuations were demonstrated by Yanagida and co-workers Nishiyama et al., 2001, 2002, who observed ki-nesin molecules climbing the cytoskeletal track in a jud-dering motion made up of random hesitations, jumps,and even backward steps. Similar results have been re-ported for a host of protein engines. The key question inmodern thermodynamics is therefore how far energyfluctuations drive microengines and nanoengines beyondthe limits of the macroscopic laws.

A revealing experiment was performed by Busta-mante et al. 2005, who first stretched a single RNAmolecule by optically tugging at a tiny plastic bead, at-tached to one end, and then released the bead, to studythe effect of random energy fluctuations on the moleculerecovery process. By repeating many identical stretchingcycles, they found that the molecule “relaxation path”was different every time. In fact, the bead was drawinguseful energy from the thermal motion of the suspensionfluid and transforming it into motion. However, by aver-aging over increasingly longer bead trajectories, that is,approaching a macroscopic situation, Bustamante et al.2005 were able to reconcile their findings with the sec-ond law. These results led to the conclusion that thestraightforward extension of the second law to micro-scopic systems is unwarranted; individual small systemsdo evolve under inherently nonequilibrium conditions.

However, a decade ago Jarzynski 1997 showed thatthe demarcation line between equilibrium and nonequi-librium processes is not always as clear-cut as we used tothink. Imagining a microscopic single-molecule process,Jarzynski evaluated not the simple average of thechange of the random work of the underlying per-turbed nanosystem, as it was pulled away from equilib-rium according to an arbitrary protocol of forcing, butrather the average of the exponential of that tailorednonequilibrium work. This quantity turned out to be thesame as for an adiabatically slow version of the sameprocess, and, most remarkably, equals the exponential ofthe system’s equilibrium free energy change. This result,also experimentally demonstrated by Bustamante et al.2005, came as a great surprise because it meant that

1Rudolf Julius Emanuel Clausius throughout his life allegedlypreferred the German word Verwandlungswert rather than en-tropy, although his colleagues suggested that he choose a namefor his new thermodynamic quantity S his chosen symbol forlabeling entropy, possibly in honor of Sadi Carnot thatsounded as close as possible to the word energy.

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information about macroscopic equilibrium was some-how buried inside individual, randomly fluctuating mi-croscopic systems far from equilibrium see also Gal-lavotti and Cohen, 1995; Crooks, 1999; Jarzynski, 2007;Talkner et al., 2007, 2008.

There is an additional limitation of 19th-century ther-modynamics that is potentially even more significant inthe design and operation of engines at submicron scale.Kelvin’s thermodynamics was based on the simplifyingnotion of an isolated system. The laws of macroscopicthermodynamics therefore apply only to systems thatare either separated from their environment or coupledto it under controlled operating conditions, that is, mea-sured in terms of the state variables of the system itself.However, unlike cylinders inside steam engines, proteinengines do not cannot work in isolation.

Following very much in the footsteps of 19th-centuryscientists and engineers, modern experimenters have in-dividually probed the proteins that play a crucial role inthe cell, feeding them with energy by injecting somechemical fuel “by hand” e.g., ATP molecules or exert-ing mechanical action of some sort Astumian, 1997;Bustamante et al., 2005. In their natural setting, how-ever, life engines are just parts of a closely intercon-nected functional web that keeps a cell alive. The greatchallenge of systems biology is therefore to put our un-derstanding of isolated life engines back into the realworld of the cell.

A. Artificial nanodevices

Nanotechnology has been intricately linked with bio-logical systems since its inception. Fascinated by thecomplexity and smallness of the cell, Feynman 1960challenged the scientific community to “make a thingvery small which does what we want.” In his visionaryresponse, Drexler 1992 proposed to focus on proteinsynthesis as a pathway for creating nanoscale devices.Both Feynman’s and Drexler’s propositions were metwith much skepticism, as accurate manipulations at thenanoscale were deemed impossible. However, in view ofthe recent advances in systems biology Gross, 1999,cellular mechanisms are now being cited as the keyproof of the nanotechnological viability of devices withatomic precision. In spite of their established comple-mentarity, a fundamental difference between systems bi-ology and nanotechnology is their ultimate goal. Sys-tems biology aims to uncover the fundamentaloperations of the cell in an effort to predict the exactresponse to specific stimuli and genetic variations,whereas nanotechnology is chiefly concerned with usefuldesign.

Manufacture of nanodevices through positional as-sembly and self-assembly of biological componentsavailable at the cellular level is the goal of the so-calledbiomimetic approach—as opposed to the inorganic ap-proach aimed at fabricating nanomechanical devices inhard, inorganic materials e.g., using modern litho-graphic techniques, atomic force, and scanning tunnelingmicroscopy, etc.. Nature has already proven that it is

possible to engineer complex machines on the nano-scale; there is an existing framework of working compo-nents manufactured by nature that can be used as aguide to develop our own biology-inspired nanodevices.It is also true that the molecular machinery still outper-forms by many orders of magnitude anything that can beartificially manufactured. Nevertheless, inorganic nan-odevices are attracting growing interest as a viable op-tion due to their potential simplicity and robustness; inaddition, inorganic nanodevices may provide additionalexperimental access to the molecular machinery itself.

In this review we intend to pursue further the inor-ganic approach to nanodevices, based on three main as-sumptions: i In view of the most recent developmentsin nonequilibrium thermodynamics, the science of nano-devices, regardless of the fabrication technique, is in-separable from the thermodynamics of microscopic en-gines Hänggi et al., 2005. ii Fabrication techniques onthe nanoscale will perform better and better followingthe trend of the last two decades. iii A better under-standing of the molecular machinery can help us to de-vise and implement new transport and control mecha-nisms for biology-inspired nanodevices. In other words,we bet on a two-way cross fertilization between the bio-mimetic and inorganic approaches.

B. Brownian motors

Nature has provided microorganisms with characteris-tic sizes of about 10−5 m with a variety of self-propulsionmechanisms, all of which pertain to motion induced bycyclic shape changes. During one such cycle the configu-ration changes follow an asymmetric sequence, whereone half cycle does not simply retrace the other half, inorder to circumvent the absence of hydrodynamic iner-tia at the microscale, i.e., for low Reynolds numbersPurcell, 1977. A typical example is motile bacteria that“swim” in a suspension fluid by rotating or waving theirflagella Astumian and Hänggi, 2002; Astumian, 2007.As anticipated above, a further complication ariseswhen the moving parts of a submicron-sized enginehave molecular dimensions of 10−8 m or so. In that case,diffusion caused by Brownian motion competes withself-propelled motion. For example, a molecular motormechanism becomes efficient if, at room temperatureand in a medium with viscosity close to that of water, abacterium needs more time to diffuse a body length thanit does to swim the same distance.

A solution common to most cell machinery is to havemolecular motors operating on a track that constrainsthe motion to essentially one dimension along a periodicsequence of wells and barriers. The energy barriers sig-nificantly suppress diffusion, while thermal noise plays aconstructive role by providing a mechanism, thermal ac-tivation Hänggi et al., 1990, by which motors can es-cape over the barriers. The energy necessary for di-rected motion is supplied by asymmetrically raising andlowering the barriers and wells, either via an externaltime-dependent modulation e.g., due to the couplingwith other motors or by energy input from a nonequi-

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librium source such as a chemical reaction, like ATPhydrolysis. Thus, in agreement with the reasoning under-lying the analysis2 of the Smoluchowski-Feynman styl-ized ratchet engine Smoluchowski, 1912; Feynman etal., 1963, under appropriate nonequilibrium conditionsstructural anisotropy can sustain directed motion. Such adevice does not violate the second law of thermodynam-ics because the very presence of nonequilibrium rendersinoperative those limiting thermal equilibrium restric-tions.

In the case of a bacterium, as for any ordinary heatengine, the relevant state variables, namely, its positionand the phase of the flagellum stroke, always cyclethrough one and the same periodic time sequence; thetwo variables are tightly coupled and almost synchro-nized. In clear contrast to this familiar scenario, the statevariables of artificial motors are often loosely coupleddue to the prominent action of fluctuations, a salientfeature captured by Hänggi, who coined the termBrownian motors in this context Bartussek and Hänggi,1995.3

Important hallmarks of any genuine Brownian motorare as follows Astumian and Hänggi, 2002; Hänggi etal., 2005: i The presence of some amount of not nec-essarily thermal noise. The intricate interplay amongnonlinearity, noise-activated escape dynamics, and non-equilibrium driving implies that, generally, not even thedirection of transport is predictable a priori. ii Somesort of symmetry breaking supplemented by temporalperiodicity typically via an unbiased, nonequilibriumforcing, if a cyclically operating device is involved.Therefore not every small ratchet device falls in the cat-egory of Brownian motors. This holds true especially ifthe governing transport principle is deterministic, as inmechanical ratchet devices of macroscopic or mesos-copic size.

The following prescriptions should be observed whendesigning an efficient Brownian motor: a Spatial andtemporal periodicity critically affect rectification. b Allacting forces and gradients must vanish after averagingover space, time, and statistical ensembles. c Randomforces of thermal, nonthermal, or even deterministicorigin assume a prominent role. d Detailed balancesymmetry, ruling thermal equilibrium dynamics, must bebroken by operating the device away from thermal equi-librium. e A symmetry-breaking mechanism must ap-ply. There exist several possibilities to induce symmetrybreaking. First, the spatial inversion symmetry of theperiodic system itself may be broken intrinsically, that is,already, in the absence of nonequilibrium perturbations.

This is the most common situation and typically involvesa type of periodic, asymmetric ratchet potential. A sec-ond option consists of the use of an unbiased drivingforce either deterministic or stochastic possessing non-vanishing, higher-order odd time correlations. Yet athird possibility arises via collective effects in coupled,perfectly symmetric nonequilibrium systems, namely, inthe form of spontaneous symmetry breaking. Note thatin the latter two cases we speak of Brownian motor dy-namics even though a ratchet potential is not necessarilyinvolved.

The idea of using unbiased thermal fluctuationsand/or unbiased nonequilibrium perturbations to drivedirected motion of particles and the like has seen severalrediscoveries since the visionary works by von Smolu-chowski 1912 and Feynman et al. 1963. From a his-torical perspective, the theme of directed transport andBrownian motors was put into the limelight of statisticaland biological physics research with the appearance ofseveral ground-breaking works, both theoretical and ex-perimental, which appeared during the period 1992–1994. Most notably, Ajdari and Prost 1992, Astumianand Bier 1994, Bartussek et al. 1994, Chauwin et al.1994, Doering et al. 1994, Magnasco 1993, Millonasand Dykman 1994, Prost et al. 1994, and Rousselet etal. 1994 helped ignite tumultuous interest in this topi-cal area which kept growing until the present day. Read-ers may deepen their historical insight by consulting ear-lier introductory reports such as those published byHänggi and Bartussek 1996, Astumian 1997, Jülicheret al. 1997, Astumian and Hänggi 2002, Reimann2002, Reimann and Hänggi 2002, and Hänggi et al.2005.

This review focuses on recent advances in the scienceof nonbiological, artificial Brownian motors. In contrastto those reviews and articles mentioned above, whichcover the rich variety of possible Brownian motor sce-narios and working principles, our focus is on nonbio-logical, artificial, mostly solid-state-based Brownian mo-tors. In this spirit we have attempted to present acomprehensive overview of today’s status of this field,including the newest theory developments, most com-pelling experimental demonstrations, and first successfultechnological applications. Some closely related topics,such as the engineering of synthetic molecular motorsand nanomachines based on chemical species, are onlybriefly discussed because detailed, up-to-date reviewshave been published by groups active in those areasKottas et al., 2005; Balzani et al., 2006; Kay et al., 2007.

II. SINGLE-PARTICLE TRANSPORT

Signal rectification schemes in the absence of noisehave been known for a long time, especially in the elec-trical engineering literature. However, rectification innanodevices cannot ignore fluctuations and Brownianmotion, in particular. New experiments on both biologi-cal and artificial devices showed how noise rectificationcan actually be utilized to effectively control particletransport on the small scale. By now, noise rectification

2Note also the examination of Feynman’s analysis by Par-rondo and Español 1996.

3The notion of a molecular motor is reserved for motorsspecifying biological, intracelluar transport. Likewise, the no-tion of a Brownian or thermal ratchet is reserved for the op-erating principle of protein translocation processes. The latterterm was introduced by Simon et al. 1992 to describe isother-mal trapping of Brownian particles to drive protein transloca-tion; see also Wang and Oster 2002.

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has become one of the most promising techniques forpowering microdevices and nanodevices.

In order to set the stage, in the next section we firstconsider the case of systems where rectification cannotoccur. In the subsequent sections we single out all ingre-dients that do make rectification possible.

Consider a Brownian particle with mass m, coordinatext, and friction coefficient in one dimension, sub-jected to an external static force F and thermal noiset. The corresponding stochastic dynamics is describedby the inertial Langevin equation

mx = − Vx − mx + F + t , 1

where Vx is a periodic potential with period L, namely,Vx+L=Vx, the prime indicates differentiation withrespect to x, and the overdot differentiation with respectto time t. Thermal fluctuations are modeled by a station-ary Gaussian noise of vanishing mean, t=0, satisfy-ing the fluctuation-dissipation relation

t0 = 2Dpt , 2

where the momentum-diffusion strength reads Dp=mkT, with k denoting the Boltzmann constant and Tthe temperature of an equilibrium heat bath.

In extremely small systems, particle dynamics andfluctuations occurring in biological and liquid environ-ments are often well described by the overdamped limitof Eq. 1—for a discussion see Purcell 1977—in termsof a massless Langevin equation, which is driven by po-sition diffusion Dx=kT /mD, i.e.,

x = − Vx + F + t , 3

with

t0 = 2Dt . 4

In this overdamped regime, the inertia term mx can bedropped altogether with respect to the friction term−mx the Smoluchowski approximation. In the follow-ing, m has been scaled to unity for convenience, i.e.,DkT.

In the absence of an external bias, i.e., F=0, the equi-librium stochastic mechanism in Eq. 1 cannot sustain anonzero stationary current, i.e., xt=0, no matter whatVx. This can be readily proven by solving the corre-sponding Fokker-Planck equation with periodic bound-ary conditions Risken, 1984; Mel’nikov, 1991.

A. Symmetric substrates

We consider first the case when a periodic substratewith potential Vx is symmetric under reflection, that is,Vx−x0=V−x+x0 for certain x0 with 0x0L. Themost studied example is the symmetric washboard po-tential Risken, 1984

Vx = − V0 sin2x/L , 5

displayed in the presence of a static tilt force F in Fig.1a. The particle mobility Fx /F is symmetric for

F→−F, namely, F=−F. For this reason we restrictourselves to F0.

1. dc drive

The Brownian motion, Eq. 1, in the washboard po-tential, Eq. 5, has been detailed in the Risken 1984textbook. To make contact with Risken’s notation onemust rewrite Eq. 1 in terms of the rescaled quantitiesx→ 2 /Lx, F→ 2 /LF /m, and T→ 2 /L2T /m, sothat

x = − x + 02 cos x + F + t . 6

The angular frequency 0= 2 /LV0 /m characterizesthe oscillating motion of the particle at the bottom ofthe potential wells. To reconcile Eq. 6 with Eq. 1 forthe potential in Eq. 5, it suffices to scale m=1 and L=2, as assumed throughout unless stated otherwise.

The dynamics of Eq. 6 is characterized by randomswitches occurring either between two locked states,dwelling in a well minimum large friction , or be-tween a locked state with zero average velocity and adownhill running state with a finite, averageasymptotic velocity x=F / small friction . In termsof mobility, locked and running states correspond to=0 and 1, respectively.

In the overdamped regime, Eq. 3, the particle istrapped most of the time at a local minimum of the tiltedsubstrate as long as FF3=0

2; for FF3 there exist nosuch minima and the particle runs in the F direction withaverage speed approaching F /. This behavior is de-scribed quantitatively by the mobility formula Risken,1984

5 15x

-1

1

V(x

)-

xF

0

1

F1

F2

F3 F

γ µ

(a)

(b)

FIG. 1. Color online Transport in a tilted washboard poten-tial. a Tilted periodic potential of Eq. 5, Vx−xF, with F=0.1. b Locked-to-running transitions. The thresholds F1 andF3 of the hysteretic loop dashed curves and the zero tempera-ture step at F2 solid curve are marked explicitly. Parametervalues: m=V0=1, i.e., F3=1, and =0.03.

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F = L/tL,FF , 7

where tL ,F, the mean first-passage time of the par-ticle across a substrate unit cell in the F direction, can becomputed explicitly for any choice of Vx Hänggi et al.,1990.

In the underdamped regime 0 the locked-to-running transition depends crucially on the presence ofnoise, no matter how weak. In the noiseless case, t0, the average speed of a Brownian particle with co-ordinate xt depends on the initial conditions accordingto the hysteretic cycle illustrated in Fig. 1b: The tran-sition from the locked to the running state occurs onraising F above F3 depinning threshold, while the op-posite transition takes place on lowering F below F1= 4/0 repinning threshold. Of course, for suffi-ciently large values of , say, in the damped regime with0= /40, the distinction between F1 and F3 be-comes meaningless; the locked-to-running transition oc-curs at F=F3, no matter what the initial conditions.

At zero temperature, T=0+, we have a totally differ-ent scenario, as the stationary dynamics of xt is con-trolled by one threshold F2=3.3576. . .0 only Risken,1984: For FF2 the Brownian particle remains trappedin one potential well; for FF2 it falls down the tiltedwashboard potential with speed close to F /. At thethreshold F2 the quantity jumps from 0 to very closeto 1, stepwise Fig. 1b. Note that, unlike F3, F2 indi-cates a dynamical transition occurring in the presence ofa relatively small tilt. Nevertheless, at low damping,switches between locked and running states correspondto long forward particle jumps, which can span manysubstrate unit cells L; the distributions of the relevantjump lengths exhibit persistent nonexponential tailsPollak et al., 1993; Ferrando et al., 1995; Costantini andMarchesoni, 1999; Borromeo and Marchesoni, 2000;Shushin, 2002.

For finite but low temperatures, kTV0, the transi-tion from the locked to the running state is continuousbut still confined within a narrow neighborhood of F2;the relevant locked-to-running transition threshold Fth isnumerically identifiable with high accuracy as a convexfunction of ; F2 and F3 are, respectively, the →0 and→ asymptotes of the numerical curve Fth Risken,1984.

2. ac drive

Suppose now that the external force Ft acting on theunit-mass Brownian particle is periodic in time. The sim-plest case possible is represented by a noiseless particle,t0, moving on a sinusoidal potential, Eq. 5, underthe action of a harmonic force

Ft = A1 cos1t + 1 . 8

The wide class of devices thus modeled may be assimi-lated to a damped-driven pendulum operated at zeronoise level, a chaotic system investigated in depth in the1980s Baker and Gollub, 1990. The dynamics of a mas-sive particle in a sinusoidal potential was reproduced in

terms of a climbing-sine map Geisel and Nierwetberg,1982; Grossmann and Fujisaka, 1982: Running orbits,leading to a spontaneous symmetry breaking, can be ei-ther periodic or diffusive, depending on the value of themap control parameter viz., the amplitude of the sineterm. The phase-space portrait of the actual damped-driven pendulum was computed by Huberman et al.1980, who revealed the existence of delocalized strangeattractors with an intricate structure on all scales, laterrecognized to be fractal objects Baker and Gollub,1990. This means that, despite the global reflection sym-metry of the dynamics in Eqs. 1 and 8, for sufficientlysmall the particle drifts either to the right or to the left,depending on the initial conditions, but with equal prob-ability in phase space.

Coupling the particle to a heat bath, no matter howlow the temperature, changes this scenario completely.The action of the noise source t amounts to scram-bling the initial conditions, which therefore must be av-eraged over. As a consequence, trajectories to the rightand to the left compensate one another and the symme-try is restored: No Brownian drift is expected in thezero-temperature limit.

The system symmetry can be broken by adding a dccomponent A0 to the external force,

Ft = A0 + A1 cos1t + 1 . 9

The most evident effect of such a periodic drive is theappearance of hysteresis loops Borromeo et al., 1990 inthe parametric curves of the mobility t vs Ft. For10 and A0Fth, the mobility hysteresis loop is cen-tered on the static mobility curve A0 and traversedcounterclockwise; with decreasing 1, its major axis ap-proaches the tangent to the curve A0. Hysteresisloops have been observed even for 1 much smallerthan the relevant Kramers rate, the smallest relaxationrate in the unperturbed stationary process of Eq. 6with F=0. The area encircled by the hysteretic loops ismaximum for A0Fth, that is, close to the transitionjump. Even more interestingly, it exhibits a resonant de-pendence on both the forcing frequency and the tem-perature, thus pointing to a dynamical stochastic reso-nance Gammaitoni et al., 1998 mechanism betweenlocked and running states Borromeo and Marchesoni,2000.

Finally, Machura, Kostur, et al. 2007 showed that,under special conditions, involving small Brownian mo-tion at temperature T and appropriate relaxation con-stants 10, the damped process in Eq. 6 occa-sionally exhibits the phenomenon of absolute negativemobility: The ac cycle-averaged drift velocity x may beoriented against the dc bias A0, as the result of a delicateinterplay of chaotic and stochastic dynamics. This obser-vation was corroborated theoretically by Speer et al.2007 and Kostur et al. 2008 and experimentally byNagel et al. 2008. Yet another artificial Brownian motorsystem where this phenomenon can likely be validatedexperimentally is a system of cold atoms dwelling in pe-riodic optical lattices.

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3. Diffusion peak

Diffusive transport of particles or, more generally,small objects, is a ubiquitous feature of physical andchemical reaction systems Burada et al., 2009. DirectedBrownian motor transport is typically controlled by boththe fluctuation statistics of the jittering objects and thephase space available to their dynamics. For example, asthe particle in Eq. 6 drifts with average speed x in thedirection of the external force F, the random switchesbetween locked and running states cause a spatial dis-persion of the particle around its average position. Thecorresponding normal diffusion constant

DF = limt→

xt2 − xt2/2t 10

was computed numerically as a function of F at constanttemperature by Costantini and Marchesoni 1999. Apeak in the curves of D vs F is detectable in the vicinityof the transition threshold Fth, irrespective of the valueof . In particular, at low damping, where FthF2, andlow temperature, kT0

2, the peak of D at F=F2 is verypronounced; as the damping is increased, the diffusionpeak eventually shrinks down to a bump correspondingto the threshold FthF3. In any case, in an appropriateneighborhood of Fth, the diffusion constant can growlarger than Einstein’s diffusion constant for the freeBrownian motion in one dimension, D0=kT /. On in-creasing T the diffusion peak eventually disappears, nomatter what the value of .

A refined analytical formula for the diffusion peakwas obtained by Reimann and collaborators, who re-garded the locked-to-running transition in the over-damped regime as a renewal process, that is Reimann etal., 2001, 2002,

DF =L2

2t2L,F − tL,F2

tL,F3 , 11

where the nth moments of the first-passage timetnL ,F can be computed explicitly. For F→0, Eq. 11reproduces the zero-bias identity D /D0= Festa andd’Agliano, 1978.

4. Single-file diffusion

When a gas of particles is confined to a linear path, anindividual particle cannot diffuse past its neighbors. Thisconstrained one-dimensional 1D geometry is oftencalled a “single-file” or Jepsen gas Jepsen, 1965; Harris,1974. Consider a file of N indistinguishable, unit-massBrownian particles moving on the tilted sinusoidal sub-strate of length LS. If the particle interaction is hardcore zero radius, the file constituents can be labeledaccording to a given ordered sequence, and the long-time diffusion of an individual particle is strongly sup-pressed. In early studies Jepsen, 1965; Lebowitz andPercus, 1967; Levitt, 1973; Harris, 1974; Marchesoni andTaloni, 2006, the mean square displacement of a singleparticle in the thermodynamic limit LS ,N→ with con-stant density =N /LS was calculated analytically. Those

results were generalized to the diffusion of a single fileof driven Brownian particles on a periodic substrate byTaloni and Marchesoni 2006, who derived the subdif-fusive law

limt→

xt2 − xt2 =2

DFt

, 12

with DF given in Eq. 11. This result applies to anychoice of the substrate potential Vx Burada et al.,2009 and to the transport of composite objects Hein-salu et al., 2008, as well. Excess diffusion peaks havebeen obtained experimentally in the context of particletransport in quasi-1D systems Sec. III.

B. Rectification of asymmetric processes

Stochastic transport across a device can be induced byapplying a macroscopic gradient, like a force or a tem-perature difference. However, under many practical cir-cumstances this is not a viable option: i currents in-duced by macroscopic gradients are rarely selective; iia target particle that carries no charge or dipole canhardly be manipulated by means of an external field offorce; iii external controls, including powering, criti-cally overburden the design of a small device. Ideally,the optimal solution would be a self-propelled devicethat operates by rectifying environmental signals. In thequest for rectification mechanisms of easy implementa-tion, we start from the symmetric dynamics of Eq. 1and add the minimal ingredients needed to induce andcontrol particle transport.

If Vx is symmetric under reflection, the only way toinduce a drift of the Brownian particle consists in drivingit by means of a nonsymmetric force Ft, either deter-ministic or random Luczka et al., 1995; Hänggi et al.,1996; Chialvo et al., 1997; Astumian and Hänggi, 2002;Reimann and Hänggi, 2002. Here symmetric means thatall force moments are invariant under sign reversal, F→−F. Note that the condition of a vanishing dc compo-nent, limt→ 1/ t0

t Fsds=0, would not be sufficient. Forinstance, a biharmonic signal with commensurate fre-quencies and arbitrary phase constants, although zero-mean valued, is in general nonsymmetric.

On the contrary, particles in an asymmetric potentialcan drift on average in one direction even when all per-turbing forces or gradients are symmetric. However, aspointed out in Sec. II.D, to achieve directed transport insuch a class of devices the external perturbation is re-quired to be time correlated, as in the presence of anon-Markovian noise source correlation ratchets or atime-periodic drive rocked and pulsated ratchets.

The interplay of time and space asymmetry in the op-eration of a Brownian motor has been established math-ematically by Reimann 2001 and Yevtushenko et al.2001. We slightly generalize the overdamped dynamicsin Eq. 3 to incorporate the case of time-dependent sub-strates, that is,

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x = − Vx,F2t + F1t + t . 13

The potential Vx ,F2t is termed supersymmetricMarchesoni et al., 1988; Jung and Hänggi, 1991 if, foran appropriate choice of the x and t origins,

− Vx,F2t = Vx + L/2,F2− t . 14

Analogously, the additive drive F1t is supersymmetricif for an appropriate t origin

− F1t = F1− t . 15

Should Fit, with i=1,2, be stationary noises, clearly norestriction can be set on the t origin; the equalities inEqs. 14 and 15 must then hold in a statistical sense,meaning that the two terms of each equality must bestatistically indistinguishable.

Consider now the time-reversed process zt=x−t+L /2. By definition, z=−x, whereas, on simul-taneously imposing the supersymmetry conditions 14and 15, the Langevin equation 13 yields z= x;hence x=0. As a consequence, a nonzero rectificationcurrent requires that either the substrate or the additivedrive or both is nonsupersymmetric.

We remark that the above theorem has been provenonly for zero-mass particles, that is, when the Smolu-chowski regime mx=0 applies. In the presence of iner-tia, instead, rectification may occur, under special condi-tions, also in fully supersymmetric devices, as shownrecently by Machura et al. 2007 for a rocked cosinepotential.

C. Nonlinear mechanisms

In this section we review transport on symmetric sub-strates driven by asymmetric forces. The rectificationmechanisms outlined below can be traced back to thenonlinear nature of the substrate; for this reason, unlikeBrownian motors, they work also, if not more effec-tively, in the absence of noise.

We remind the reader that these mechanisms havebeen introduced and demonstrated experimentally inthe most diverse fields of physics and engineering. Di-rect applications to various categories of artificial nan-odevices will be discussed in the subsequent sections.

1. Harmonic mixing

A charged particle spatially confined by a nonlinearforce is capable of mixing two alternating input electricfields of angular frequencies 1 and 2, its response con-taining all possible higher harmonics of 1 and 2. Forcommensurate input frequencies, i.e., 1 /2=n /m withn and m coprime integer numbers, the output contains adc component too Schneider and Seeger, 1966; har-monic mixing thus induces a rectification effect of then+mth order in the dynamical parameters of the sys-tem Marchesoni, 1986; Goychuk and Hänggi, 1998.

Consider the overdamped stochastic dynamics of Eq.3 in the potential of Eq. 5, driven by the biharmonicforce

Ft = A1 cos1t + 1 + A2 cos2t + 2 . 16

Let the two harmonic components of Ft be commen-surate with one another, meaning that 1 and 2 areinteger-valued multiples of a fundamental frequency 0,i.e., 1=n0 and 2=m0. For small amplitudes andlow frequencies of the drive in Eq. 16, a simple expan-sion of the mobility function Eq. 7 in powers of Ftyields, after time averaging, the following approximateexpression for the nonvanishing dc component of theparticle velocity:

x = 2m + n

m!n!A1

2mA2

2n

cosm,n , 17

where is the positive n+m−1th derivative of F atF=0, m,n=n2−m1, and m+n is an odd number. Har-monic mixing currents j= x /L clearly result from aspontaneous symmetry-breaking mechanism: indeed, av-eraging j over 1 or 2 would eliminate the effect com-pletely. This is true under any regime of temperatureand forcing, as proven by means of standard perturba-tion techniques Wonneberger and Breymayer, 1981;Marchesoni, 1986. In particular, rectification of twosmall commensurate driving signals, A1 ,A20

2, is anoise-assisted process and therefore is strongly sup-pressed for kT0

2, when decays exponentially tozero Risken, 1984. Moreover, we anticipate that ac-counting for finite inertia effects requires introducing inEq. 17 an additional damping and frequency-dependent phase lag, as discussed in Sec. IV.A.

In all calculations of x, including Eq. 17, the reflec-tion symmetry of Vx plays no significant role; rectifica-tion via harmonic mixing is caused solely by the nonlin-earity of the substrate. However, it must be noted that asymmetric nonlinear device cannot mix rectangularwave forms, like

Ft = A1 sgncos1t + 1 + A2 sgncos2t + 2 ,

18

with A1 ,A20 and sgn¯ denoting the sign of ¯Savel’ev, Marchesoni, Hänggi, et al., 2004a, 2004b. Asshown in Sec. II.D.4, asymmetric devices do not exhibitthis peculiarity. Moreover, we underscore that for in-commensurate frequencies, where 1 /2 is not a ratio-nal number, harmonic mixing takes place only in thepresence of spatial asymmetry, as reported in Sec. II.D.4.

For practical applications, any commensuration condi-tion, like 1 /2=n /m for harmonic mixing but see alsoSecs. II.C.2 and II.D.4, is affected by an uncertaintythat is inversely proportional to the observation time,i.e., the code running time for numerical simulations orthe data recording time for real experiments. As suchtime is necessarily finite, only a limited number of com-mensuration spikes can be actually detected.

We conclude by recalling that the notion of harmonicmixing has been introduced long ago, e.g., to interpretthe output of certain charge wave density experimentsSchneider and Seeger, 1966 and to design laser ringgyroscopes Chow et al., 1985 and annular Josephson

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junctions Ustinov et al., 2004; applications in the con-text of nanoparticle transport are more recent see Secs.II.C.3 and IV.

2. Gating

A periodically driven Brownian motion can also berectified by modulating the amplitude of the substratepotential Vx. We specialize the overdamped dynamicsin Eq. 13 as follows:

x = − Vx1 + F2t + F1t + t . 19

To avoid interference with possible harmonic mixing ef-fects we follow the prescription of Sec. II.C.1, namely,we take Fit=Ai sgncosit+i, i=1,2, with A10and 0A21. Mixing of the additive F1t and the mul-tiplicative signal F2t provides a control mechanism ofpotential interest in device design.

In the adiabatic limit, the Brownian particle movescyclically back and forth subjected to opposite dc forceswith amplitude A1; the substrate potential Vx , t=Vx1+F2t switches, in turn, periodically betweenthe two symmetric configurations V±x=Vx1±A2.The relevant mobilities ±A1 can be easily related tothe static mobility function A for the tilted potentialVx studied in Fig. 1, namely Savel’ev, Marchesoni,Hänggi, et al., 2004a, 2004b,

±A1 = 1 ± A2A1/1 ± A2 20

with T→T / 1±A2.In particular, for any pair of commensurate frequen-

cies 1 and 2 such that 2 /1= 2m−1 / 2n−1 withm ,n positive integers, the net particle velocity mediatedover an integer number of cycles of both F1t and F2tcan be cast in the form

x = −− 1m+nA1

2m − 12n − 1A1,A2pn,m , 21

where A1 ,A2= −A1−+A1 /2 and pn,m −n,m /−0.5 is a modulation factor with n,m= 2n−12− 2m−11, mod 2. Note that for anyother choice of the ratio 2 /1, no induced drift is pre-dicted see Fig. 2.

As a consequence, a relatively small modulation ofthe sinusoidal potential amplitude at low temperaturesresults in a net transport current as illustrated in Fig. 2.Consider the simplest possible case, 1=2 and 1=2:As the ac drive points to the right, the amplitude ofVx , t is set at its maximum value 0

21+A2; at low tem-peratures the Brownian particle can hardly overcomethe substrate barriers within a half ac-drive period /1.In the subsequent half period F1t switches to the left,while the potential amplitude drops to its minimumvalue 0

21−A2: The particle has a better chance to es-cape a potential well to the left than to the right, thusinducing a negative current with maximum intensity for1 of the order of the Kramers rate resonant activationBorromeo and Marchesoni, 2004. Of course, the am-

plitude and sign of the net current may be controlled viathe modulation parameters A2 and 2 too see the insetof Fig. 2.

3. Noise induced transport

Induced transport in the symmetric dynamics can beachieved also by employing two correlated noisy signals.

a. Noise mixing

Borromeo and Marchesoni 2004 showed that thegating process in Eq. 19 can also be driven by twostationary, zero-mean-valued Gaussian noises Fit=it with i=1,2. The two random drives may be crosscorrelated and autocorrelated with

itj0 = QiQjij/ijexp− t /ij, i,j = 1,2.

22

Without loss of generality, one sets 11=22=1 and 12=21=, and to avoid technical complications ij. Ofcourse →0 corresponds to taking the white noise limitof it, Eq. 4. The parameter characterizes the crosscorrelation of the two signals; in particular, =0, inde-pendent signals; =1, identical signals, 2t1t; =−1, signals with reversed sign, 2t−1t. Two suchsignals may have been generated by a unique noisesource and then partially decorrelated through differenttransmission or coupling mechanisms.

In the white noise limit, →0, the Fokker-Planckequation associated with the process of Eqs. 19 and22 contains a stationary solution in closed form. Non-vanishing values of x for 0 are the signature of astochastic symmetry breaking due to nonlinear noise

0 0.5 1 1.5 2 2.5D

-0.3

-0.2

-0.1

0

⟨x⟩__A

.

0 0.5 φ2 / 2π

-0.1

0

⟨x⟩__A

.

A = 0.5

A = 1.0

A = 1.5

FIG. 2. Gating mechanism. Equation 19 has been simulatednumerically for 1=2=0.01, 1=2=0, A1=A2=A, andVx=0

21−cos x with 0=1; the net velocity x has beenplotted vs D=kT for different amplitudes A. For a comparisonwe show also x vs D for 1=0.01, 2 /1=2, and A=0.5crosses. Inset: x vs 2 for 1=0, A=0.5, and D=1. Note theresonant behavior of x for subthreshold drives, A1, wherea low noise level may enhance rectification; increasing D forA1 degrades the rectification effect. From Borromeo andMarchesoni, 2005b.

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mixing; the sign of , similarly to the relative phase m,nin the gating setup of Sec. II.C.2, determines the direc-tion of the particle drift. The interpretation of this resultis straightforward. Consider, for instance, the case of thesymmetric potential in Eq. 5, rocked and pulsated bythe same signal, i.e., 1t=2t: For =1, when pushedto the left i0, the Brownian particle encounterslower substrate barriers than when pushed to the righti0, hence the negative average drift current de-tected by means of numerical simulation Borromeo andMarchesoni, 2005b; Borromeo et al., 2006.

The magnitude of the induced current is controlled bythe correlation time Borromeo and Marchesoni,2005b. While one can easily estimate x for 0 and=0 white noise Risken, 1984 or → strongly cor-related noise Hänggi et al., 1989, the intermediate values are accessible solely through numerical simula-tion. In Fig. 3a we show x vs for different noiseintensities: the two sets of curves at Q and Q / fixed

illustrate the noise mixing effect for →0 and → , re-spectively.

b. Noise recycling

If the noises it are generated by the same sourceand then coupled to the diffusing particle through differ-ent paths, it may well happen that they are simply de-layed in time. Under stationary conditions, we can as-sume that 2t=1t−d, with it given by Eq. 22and, for simplicity, =0. In the notation of Borromeo etal. 2006 and Borromeo and Marchesoni 2007b, 1trepresents the primary noise source and 2t a recyclednoise to be used as a control signal.

By the same argument as for noise mixing, we expectthat the Brownian dynamics becomes rectified to the leftwith negative velocity xd. Note that x−d= xdand xd→−xd, upon changing the relative signsof i. The dependence of the characteristic curve xdon the time constants d and displayed in Fig. 3b isimportant in view of practical applications. Indeed, inmany circumstances, it would be extremely difficult torecycle a control signal 2t so that d; stated other-wise, measurement of x0 requires a certain degree ofexperimental sophistication. On the contrary, if we agreeto work on the resonant tail of its response curve xd , a noise-controlled rectification device can beoperated with less effort; its net output current may benot the highest for d but is still appreciable, and,more importantly, stable against the accidental floatingof the time constant d.

In this sense both schemes discussed are a simple at-tempt at implementing the operation of a Maxwell’s de-mon: The ideal device we set up is intended to gauge theprimary random signal 1 at the sampling time t and,depending on the sign of each reading, to lower or raisethe gate barriers accordingly at a later time t+d, i.e., toopen or close the trap door. The rectifying power of sucha demon is far from optimal; lacking the dexterity ofMaxwell’s “gate keeper” Leff and Rex, 2003;Maruyama et al., 2008 it works only on average like anautomaton.

D. Brownian motors

As detailed in Sec. II.B, a necessary condition for therectification of symmetric signals, whether random orperiodic in time, is the spatial asymmetry of the sub-strate. Rectification devices involving asymmetric sub-strates are termed ratchets. In most such devices, how-ever, noise no matter what its source, i.e., stochastic,chaotic, or thermal plays a non-negligible or even domi-nant role. Under such conditions one speaks of Brown-ian motors Bartussek and Hänggi, 1995; Hänggi andBartussek, 1996; Reimann et al., 1996; Astumian, 1997;Astumian and Hänggi, 2002; Hänggi et al., 2002, 2005;Linke, 2002; Kay et al., 2007. The label “Brownian mo-tor” should not be abused to refer to all small ratchet-like devices. For instance, the rocked ratchets of Sec.II.D.1 work quite differently in the presence or absence

0 2 4 6 8 10τ

-0.3

-0.2

-0.1

0

Q / τ = 0.25

Q / τ = 0.5

Q = 0.5

Q = 1.0

0 0.5 1 1.5 2 2.5τ

d

10-2

10-1

0.0125

0.025

0.050.1

0.20.0

0 0.5

0.2

0.4

τc

=

(b)

(a)

⟩ ⟩x

⟩ ⟩ x

FIG. 3. Color online Numerical simulation of Eq. 19, withFit=it, i=1,2, and Vx=−sin x. The noises it, Eq. 22,have the same strength, Q1=Q2=Q. a Nonlinear noise mix-ing: Characteristic curve x- for different intensities Q of thenoises and =1, D=0.1. b Noise recycling: Characteristiccurve x-d, where d is the relative time delay of i see text.Data are for different and Q=1, D=0; see Borromeo et al.,2006. From Borromeo and Marchesoni, 2005b.

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of noise, whereas the pulsated and correlation ratchetssometimes also referred to as “stochastic” ratchets ofSecs. II.D.2 and II.D.3 work only in the presence ofnoise.

The hallmarks of genuine Brownian motors are listedin Sec. I. In this section we discuss in detail noise recti-fication and directed transport on asymmetric periodicsubstrates and potentials. We point out to the reader thefollowing cautions: i Strict periodicity is not a require-ment for the operation of a Brownian motor. Theratchet system may contain small amounts of disorderHarms and Lipowsky, 1997; Popescu et al., 2000; Kafriand Nelson, 2005; Martinez and Chacon, 2008 or evenbe nonperiodic Marchesoni, 1997. ii Spatial asymme-try can also result as a collective effect, for instance, inextended systems consisting of interacting, symmetricdynamical components, introduced in Sec. V.

The archetypal model of ratchet substrates in one-dimension is the double-sine potential proposed by Bar-tussek et al. 1994,

Vx = − V0sin2x/L + 14 sin4x/L , 23

or, in the rescaled units of Eq. 6, Vx=−02sin x

+1/4 sin2x. In Fig. 4, the barriers are skewed to theright with ratchet length l+ l−. A Brownian particlewith Langevin equation 1 moving on such a substrate ischaracterized by an asymmetric mobility function F−F; as the particle mobility depends now not onlyon the amplitude, but also on the orientation of thedrive F, symmetric substrate perturbations are expectedto induce a net current in either direction.

Broadly speaking, ratchet devices fall into three cat-egories depending on how the applied perturbationcouples to the substrate asymmetry.

1. Rocked ratchets

Consider first, for simplicity, the Langevin equation3 with the sinusoidal drive of Eq. 8. When applied tothe reflection-symmetric sine potential of Eq. 5, Ftbreaks instantaneously the symmetry of the potential bytilting it to the right for Ft0, and to the left for Ft0. However, due to the spatiotemporal symmetry ofthe process, Vx=V−x and Ft=F−t, over one drivecycle T1=2 /1 the drifts to the right and left compen-sate one another: the net current is null. When placed inthe double-sine potential of Eq. 23, instead, an over-damped particle is more easily moved to the right thanto the left. Indeed, depinning occurs at F=F3R and F=−F3L for Ft oriented, respectively, to the right andleft; moreover, from the asymmetry condition l+ l− theinequality F3RF3L follows immediately. The depinningthresholds for the potential in Eq. 23 in the rescaledunits of Eq. 6 are F3R=3/4 and F3L=3/2.

If the forcing frequency 1 is taken much smaller thanall intrawell relaxation constants, 0

2T11 and T1 tL ,0, then the net transport current can be com-puted in the adiabatic approximation by averaging theinstantaneous velocity xt=FtFt see Eq. 7,over one forcing cycle, that is Bartussek et al., 1994;Borromeo et al., 2002,

-1

0

1

V(x

)

-1 0 1x

0

2

4

6

p(x,

t)

t1

t2

t3

-1 0 1 20

2

4

6

p(x,

t)

(a)

(c)

(b)

l- l

+

∆V

L

x-

x0 x

+

FIG. 4. Color online Ratchet mechanism. a Sketch of thepotential in Eq. 23 with L=1 and V0=L /2. The three con-secutive extremal points x−=−0.19, x0=0.19, and x+=0.81 de-fine a potential well; the barrier height is V=Vx±−Vx00.35 and its asymmetry is quantified by the difference l+− l−with l±= x±−x0 . The curvature of the potential at the bottomof the well is 0

2= 33/20210.1. b Rocket ratchet. Prob-

ability density px , t for a Brownian particle initially centeredaround x=0 and then driven for t=10 by a dc force A0=−0.5and 0.5. The backward displacement is strongly suppressed,hence the positive natural orientation of this ratchet. c Pul-sated ratchet. The ratchet potential is switched “on” and “off”periodically with ton=1, toff=3, and T2= ton+ toff see text. Theparticle, initially set at x=x0, relaxes first in the starting wellduring ton curve 1, t1= ton, then diffuses symmetrically drivenonly by noise for toff curve 2, t2=T2, and finally gets retrappedin the neighboring wells as the next cycle begins curve 3, t3=T2+ ton. As the left peak of px , t3 is more pronounced thanthe right peak, the natural orientation of this ratchet is nega-tive. In the simulations of b and c the ratchet potential Vxis the same as in a also drawn to guide the eye and theintensity of the noise in Eq. 4 is D=0.1V. Courtesy of Mar-cello Borromeo.

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j xL

=1

T1

0

T1 sgnFstL,Fs

ds . 24

The adiabatic current in Eq. 24 is positive definite, asthe natural ratchet direction is defined by the choiceF3RF3L. However, its range of validity is restricted toextremely low temperatures T and drive frequencies 1.On raising 1, the rectification current develops a morecomplicated dependence on the noise intensity D andthe drive amplitude A1, as proven by the numerical

simulations reported in Figs. 5a–5c. This is because,for forcing periods T1 shorter than the mean first-passage time tL ,0, the driven particle oscillationsspan fewer substrate cells. A few properties of therocked ratchet current in Fig. 5 are remarkable.

i At finite temperature, the maximum rectificationeffect occurs right in the adiabatic regime, 1→0, andwith natural orientation, panels; Figs. 5a and 5b.

ii For 10, the drive and interwell oscillationsmay combine to reverse the current orientation. Suchcurrent inversions are restricted to selected D and A1ranges, but never in the absence of noise; Figs. 5a and5b.

iii In the noiseless regime, T0, the incommensura-tion between forced oscillation amplitude and substrateperiodicity causes the quantized locking structure of Fig.5c. This structure disappears either in the limit 1→0because the step height, i.e., L1 / 2, is proportionalto 1 or in the presence of noise, when due to random-ness the finer steps for A1F3L merge into the broadoscillations of Fig. 5b.

iv As the temperature vanishes, the adiabatic cur-rent in Eq. 24 is suppressed for A1F3R and enhancedelsewhere see Fig. 8a. For F3RA1F3L the particlecan only move to the right, so that j grows almost lin-early with A1, while for A1F3L the particle can drift inboth directions, thus causing j to decrease.

Rectifiers, like rocked ratchets, can work against anexternal dc drive. On adding a dc component A0 to thesinusoidal force, namely, on applying the drive 9 in-stead of Eq. 8, we can determine the value of A0,termed the stopping force, such that for a certain choiceof the other perturbation parameters A1, 1, and D thenet current vanishes.

Rocked ratchets with massive particles exhibit stronginertial effects also capable of reversing their current. Anoiseless inertial rocked ratchet is naturally subject todeveloping chaotic dynamics; indeed, its current turnsout to be extremely sensitive to the initial conditionsJung et al., 1996; Mateos, 2000, 2003; Borromeo et al.,2002; Machura, Kostur, et al., 2007. However, an appro-priate noise level suffices to stabilize the rectificationproperties of an inertial ratchet and makes it a usefuldevice with potential applications in science and tech-nology Machura, Kostur, Talkner, et al., 2004;Marchesoni et al., 2006a.

In the notation of Sec. II.A.1, slightly generalized toaccount for the asymmetry of the potential in Eq. 23,we define two additional pairs of thresholds, besides thedepinning thresholds F3R,L: the repinning thresholdsF1R,L and the underdamped transition thresholds F2R,L,with FiRFiL for i=1,2 ,3 the subscript R ,L denotingthe orientation of the drive. Under the adiabatic condi-tion 1 ,0, the following approximations have beenobtained by Borromeo et al. 2002 and Marchesoni et al.2006a:

i In the zero-temperature limit, T→0,

(b)

(a)

(c)

FIG. 5. Rocked ratchet. a Rectification current j vs the noiseintensity D see Eq. 4 and b, c vs ac modulation strengthAA1, for the ratchet potential in Eq. 23 with L=1. Therelevant simulation parameters are a A1=0.5 and 1=0.01, 1,2.5, 4, and 7 top to bottom; b D=0.1 and 1=0.01, 1, 4, 7,and 10 top to bottom. In a and b the solid line for 1=0.01 coincides with the adiabatic limit in Eq. 24. c Deter-ministic regime with D=0, and 1=0.01 dashed line and 1=0.25 solid line. The inset depicts the width of the nth pla-teau step vs n for A7.8 with error bars. This lockingmechansim for the plateau width is governed by a devil’s stair-case. From Bartussek et al., 1994.

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jA1 = jRA1 , F2R A1 F2L,

jRA1 − jLA1 , A1 F2L, 25

and jA1=0 for A1F2R. The right and left currentcomponents are jR,LA1=2A1

2−F2R,L2 /L.

ii In the noiseless regime, T0,

jA1 = jRA1 , F3R A1 F3L,

jRA1 − jLA1 , A1 F3L, 26

and jA1=0 for A1F3R. Here jR,LA1= A12−F1R,L

2

+A12−F3R,L

2 /L. Note that jA1 in Eq. 26 is discon-tinuous at the depinning thresholds F3R,L as expected forthe response curve of any underdamped rocked ratchetoperated in the deterministic regime see Sec. II.A.1.

2. Pulsated ratchets

Consider now the overdamped process of Eq. 19with F1t=0 and F2t=A2 sgncos2t+2. As dis-cussed in Sec. II.C.2, modulation of the amplitude of asymmetric potential Vx in the presence of an uncorre-lated, time-symmetric perturbation t does not sufficeto induce rectification. This state of affairs changes whenF2t couples instead to an asymmetric substrate, likeour reference ratchet potential 23. For the sake of sim-plicity, we set A2=1, so that the effective substrate po-tential 1+F2tVx appears to switch on and off at ev-ery half period T2 /2= /2. The interplay between timemodulation and spatial asymmetry generates a nonzerodrift current, which, at low but not too low frequencies,is oriented in the positive direction. Devices operatedunder similar conditions are termed pulsated ratchets.

Rectification by a pulsated ratchet can be explainedqualitatively by looking at Fig. 4. The probability densityof a particle initially placed at the bottom of a potentialwell top is mostly confined to that well, as long as T2 /2is not exceedingly longer than the escape time tL ,0,which is the case at low temperatures. When, during the

following half cycle, the substrate is switched offmiddle, the particle, still subject to noise, diffusesfreely with Einstein’s constant D and, for T2L2 /D, itsprobability density approaches a Gaussian spanningmany unit cells. By switching on again, the substrate cutssuch a probability density into smaller peaks of differentsizes, one for each well surrounding the initial one. Forpotential barriers skewed as in the figure, l+ l−, wellson the right are expected to be less because of thelarger diffusion length l+ populated than wells on theleft, hence there is a negative net current. As a conse-quence, the potential in Eq. 23 has positive naturalorientation when rocked, and negative orientation whenpulsated. The mechanism illustrated in Fig. 6 can workeven in the presence of a dc force F1t=A0 pointing inthe opposite direction, A00. Such an upward-directedmotion is clearly powered by the thermal fluctuationst.

The directed particle current is bound to vanish in theadiabatic limit, 2→0, when thermal equilibrium is ap-proached. A similar conclusion holds true for fast sub-strate modulations, 20

2 in rescaled units. In pertur-bation analysis Savel’ev, Marchesoni, and Nori, 2004b,one finds the noteworthy result that the current decaysto zero in both asymptotic regimes remarkably fast,namely, as 2

2 in the slow-modulation limit and 2−2 in

the fast-modulation limit. Moreover, for intermediatemodulation frequencies, the current in a pulsatedratchet is generally not oriented in its natural direction;current inversions are possible when the forcing fre-quency matches some intrinsic relaxation rate of theprocess.

Finally, pulsated ratchets can be operated under verygeneral amplitude modulations F2t as well. Thisratchet effect proved robust with respect to the follow-ing: i Modifications of the potential shape Reimann,2002. ii Changes in the switching sequence. A genericdiscrete modulation signal F2t is characterized by dif-ferent residence times, namely, ton in the on state, +A2,and toff in the off state, −A2. For periodic modulations,ton+ toff=T2 defines the so-called duty cycle of the deviceBug and Berne, 1987; Ajdari and Prost, 1992. For ran-dom modulations, instead, ton and toff must be inter-preted as the average residence times in the correspond-ing on or off state Astumian and Bier, 1994; Faucheuxet al., 1995. In both cases the modulation is asymmetricfor ton toff. iii Replacement of pulsated with flashingsubstrates. The substrate is made to switch, either peri-odically or randomly in time, among two or more dis-crete configurations, which do not result from the ampli-tude modulation of a unique substrate profile Gormanet al., 1996; Chen, 1997; Borromeo and Marchesoni,1998; Lee et al., 2004. The rectification process is con-trolled by the spatial asymmetry of the single-substrateconfigurations, by the temporal asymmetry of theswitching sequence, or by a combination of both. ivModulation of the temperature Bao, 1999; Reimann etal., 1996. Modulation of the intensity of the ambientnoise t or its coupling to the device corresponds to

0.1 1 10Ω

2

10-4

10-3

10-2

j

1 / Ω2

Ω2

D = 0.5

FIG. 6. Color online Pulsated ratchet. Rectification currentj=−x vs 2 dimensionless units, for the ratchet potential ofEq. 23 with L=2 and V0=1. Other simulation parametersare A2=0.8 and D=0.5 The low-frequency 2

2 and high-frequency 2

−2 asymptotic limits are denoted by straight lines.Courtesy of Marcello Borromeo.

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introduction of the time-dependent temperature Tt= 1+F2tT for an appropriate control signal F2t withamplitude smaller than 1. Pulsation of the potential am-plitude V0 or the temperature T yields similar rectifica-tion effects, since the net current of a pulsated ratchet iscontrolled by the barrier-to-noise ratio. v The additionof constant or spatially modulated damping termsLuchsinger 2000; Suzuki and Munakata 2003; seealso Reimann 2002. Inertial effects make the rectifica-tion mechanism sensitive to the particle mass, so thatselective transport and segregation of mixed species be-comes possible Derènyi and Vicsek, 1995; Derènyi andAjdari, 1996.

3. Correlation ratchets

To better understand the role of asymmetry in therectification of a nonequilibrium process, we considernow an overdamped Brownian particle with Langevinequation 3 diffusing in the ratchet potential of Eq. 23subject to a zero-mean, colored Gaussian noise Ft,with

t0 = Q/exp− t / . 27

The substrate asymmetry is capable per se of rectifyingthe correlated fluctuation t, even in the absence ofexternal modulations or thermal noise. The early predic-tion of this transport effect, based on simple perturba-tion arguments Magnasco, 1993; Bartussek et al., 1994;Doering et al., 1994; Millonas and Dykman, 1994; Luc-zka et al., 1995, kindled a widespread interest in thethermodynamics of molecular motors. A more sophisti-cated path-integral analysis Bartussek et al., 1996 ledlater to the following low-noise estimates of the correla-tion ratchet current:

i In the weak-color limit, 021,

j = j+ − j− , 28

where j±=rK exp−2Qc±/ D+Q2 and rK is theequilibrium Kramers rate for the particle to exit apotential well through one side, rK

= 0± exp−V / D+Q / 2. Here x0 and x± are theextremal points of Vx defined in Fig. 4a; 0

2=Vx0,±

2 = Vx± , and c±=x0

x±VxVxdx.ii In the strong-color limit, → ,

j = j01 − exp− QLl+ − l−/2D2 , 29

where j0 is a positive definite constant. Equation 28 canbe regarded as the difference of two exit currents j±,respectively, through x+ forward and x− backward, inthe presence of colored noise Millonas and Dykman,1994.

The orientation of the current of a correlation ratchetis extremely sensitive to the substrate geometry. In thestrong-color limit, the condition l+− l−0 between thetwo ratchet lengths guarantees that the current is posi-tive, no matter what the choice of Vx; correlationratchets work in the natural direction of the correspond-ing rocked ratchet.

In the weak-color limit, the current in Eq. 28 has thesame sign as the difference c−−c+=0

LVxVxdx,which in turn depends on the detailed profile of Vx,regardless of sgnl+− l−. Numerical simulation of corre-lation ratchets with the potential given by Eq. 23 con-firmed that, consistently with the inequality c−−c+0,j is positive definite, as illustrated in Fig. 7; asj→0= j→ =0, optimal rectification is achieved foran intermediate noise correlation time, 0

21. Modifi-cation of the potential profile, e.g., by adding an appro-priate higher-order Fourier component without chang-ing l+− l−, suffices to reverse the sign of c−−c+ and thusintroduce at least one current inversion Bartussek et al.,1996.

Correlation ratchets have the potential for technologi-cal applications to “noise harvesting,” rather than tonanoparticle transport. The low current output and itsextreme sensitivity to the substrate geometry and theparticle mass Marchesoni, 1998; Lindner et al., 1999makes the design and operation of correlation ratchetsas mass rectifiers questionable. However, theasymmetry-induced rectification of nonequilibrium fluc-tuations, no matter how efficient and hard to control,can be exploited by a small device to extract from itsenvironment and store the power it needs to operate.

4. Further asymmetry effects

The rectification mechanisms introduced in Sec. II.Capply to any periodic substrate, independently of its spa-tial symmetry. In the presence of spatial asymmetry, therelevant drift currents get modified as follows.

a. Current offsets

At variance with Secs. II.C.1 and II.C.2, two ac drivesapplied to an asymmetric device are expected to inducea net current jav= xav also for incommensurate frequen-cies 1 and 2. This is a ratchet effect that in most ex-periments is handled as a simple current offset. Thisconclusion is apparent, for instance, in the case of low-frequency rectangular input waves, where the device

FIG. 7. Correlation ratchet. Rectification current j vs and Qfor D=0.1. The parameters of the ratchet potential, Eq. 23,are as in Fig. 4. More data can be found in Bartussek et al.,1996.

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output current can be expressed as a linear combinationof two known ratchet currents Savel’ev, Marchesoni,Hänggi, et al., 2004a, 2004b:

i Harmonic mixing current Sec. II.C.1. When thedrive is a double rectangular wave, Eq. 18, jav can beregarded as the incoherent superposition of currentsfrom two rocked ratchets driven by rectangular waveswith amplitudes A1+A2 and A1−A2 , respectively, i.e.,

jav = 12 j0 A1 − A2 + j0A1 + A2 . 30

ii Gating current Sec. II.C.2. The rocked-pulsated de-vice of Eq. 19 can be regarded as the incoherent super-position of two ratchets with potentials V±x= 1±A2Vx, respectively, both rocked with amplitudeA1; hence

jav = 12 j+A1 + j−A1 . 31

In Eqs. 30 and 31 j0,±A are the net currents of Eq.24 for a rocked ratchet driven by a rectangular wavewith amplitude A and vanishing frequency, the sub-scripts 0, referring to the regular, high- and low-amplitude potential configurations, Vx and V±x.

b. Asymmetry induced mixing

As anticipated in Sec. II.C.1, a double rectangularwave is not capable of rectifying a Brownian particle in asymmetric potential, that is, x=0 also under harmonicmixing conditions, 2 /1=m /n with m ,n coprime inte-gers and m+n odd. This is no longer the case if Vx isasymmetric. The nonzero odd moments of the processxt, determined by the substrate asymmetry, generateadditional harmonic couplings of the “odd” harmonicsof the drive, i.e., for 2 /1= 2m−1 / 2n−1 with 2m

FIG. 8. Color online Net current in a piecewise linear potential with amplitude V, driven by two rectangular wave forms, Eq.18: a one-frequency rocked ratchet; b,c harmonic mixing of Sec. II.C.1; d gating mechanism, Eq. 19. The ratchet potentialis Vx=xV /L2 for −L2x0 and Vx=xV /L1 for 0xL1, with L1=0.9 and L=1; V=1 in a–c and V=2 in d. aResponse curve jAx /L to a low-frequency rectangular force with amplitude A at zero temperature D=0 dashed curve, andlow temperature D /V=0.05 solid curve. b Numerical simulations for a doubly rocked ratchet with 1=2= and 1=0.01open circles vs adiabatic approximation, Eqs. 21 and 30 line and crosses. The offset jav, Eq. 30, is indicated by the line; thespikes of Eq. 21 at some selected integer-valued odd harmonics are marked with crosses . c Adiabatic approximation for1=2=3 /2 main panel and 1=3 /2, 2= /2 inset. In both cases A1=3, A2=2, D=0.6. d Numerical simulations vsadiabatic approximation, Eqs. 21 and 31, for a rocked-pulsated ratchet with A1=4, A2=0.5, and 1=0.01; noise level D=0.4.Main panel: 1=2= adiabatic approximation; inset: simulation open circles vs the fully adiabatic approximation for1= and 2=0. From Borromeo and Marchesoni, 2005b.

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−1 and 2n−1 coprime, as shown in Fig. 8. The totalrectification current, including both the incommensurateterm of Eq. 30 and this new harmonic mixing spike,was calculated analytically by Savel’ev, Marchesoni,Hänggi, et al. 2004a, 2004b:

j = jav −− 1m+npn,m2m − 12n − 1

jodd, 32

where pn,m is given by Eq. 21 and jodd= 12 j0 A1

−A2 − j0A1+A2.The competition between current offset, item a, and

asymmetry-induced spikes, item b, may cause surpris-ing current reversals for special values of the drivingfrequencies, as observed in several experimental setupscf. Secs. III, IV, and VII.B. Moreover, replacement ofthe rectangular wave form, Eq. 18, with a more con-ventional linear superposition of two sinusoids of thesame frequency, Eq. 16, leads to an even more compli-cated interplay of nonlinearity and asymmetry-inducedharmonic mixing Savel’ev, Marchesoni, Hänggi, et al.,2004a. A special limit of the biharmonically drivenrocked ratchet is discussed in Sec. II.E.2.

A demonstration of the combination of harmonic mix-ing and asymmetry effects in the context of particletransport at the nanoscale has been reported by Kalmanet al. 2007, who drove dilute ions through conical nano-pores by applying a biharmonic rectangular voltage.Harmonic mixing and gating currents could be sepa-rated, as predicted by the theory, and the resulting com-mensuration effects are displayed in Fig. 9.

E. Efficiency and control issues

The analysis of Sec. II.D suggests that the output of areal ratchet device is hard to control experimentally, letalone to predict. As a matter of fact, if we use a ratchetto rectify an assigned signal, the only tunable variablesare the temperature T, the damping constant or themass m, and the substrate profile Vx. Under certainexperimental circumstances these quantities may provenot directly accessible for this purpose or inconvenient

to change. This is why control of particle transport insmall devices sometimes requires the introduction ofauxiliary signals or auxiliary particle species Sec. V.C,or even the use of a feedback control scheme to opti-mize the Brownian motor current Feito and Cao, 2007;Craig, Kuwada, et al., 2008; Craig, Long, et al., 2008; Sonet al., 2008.

1. Optimization

The most common definition of efficiency for a loadedBrownian motor is

0 = xA0/Pin, 33

where xA0 is the average mechanical work done perunit of time against a working load A0 and Pin is theaverage net power pumped into the system by the exter-nal drives, no matter how applied. The quantity 0 hasbeen interpreted in terms of macroscopic thermodynam-ics by Sekimoto 1998 for further developments on is-sues of energetics and efficiency, see also Parrondo andde Cisneros 2002, and references therein. In theirscheme, a ratchet operates like a Carnot cycle, wherethe lower temperature is determined by the thermalnoise, Eqs. 2 and 4, and the higher temperature isrelated to the magnitude of the external modulation.Since the Brownian motor operates far from thermalequilibrium, the efficiency is typically smaller than thelimiting efficiency of a reversible Carnot cycle.

The ensuing question as to whether a ratchet can beoperated at the maximal Carnot efficiency spurred anintense debate for a review, see Sec. 6.9 in Reimann2002. The issue has been clarified by an analysis per-formed within the framework of linear irreversible ther-modynamics by Van den Broeck 2007: The key to ob-taining maximal Carnot efficiency for a Brownian motoris zero overall entropy production. This can be achievedby the use of architectural constraints for which the lin-ear Onsager matrix has a determinant equal to zero,implying vanishing linear irreversible heat fluxes.Maximizing efficiency subject to “maximum-outputpower” Van den Broeck, 2005 also leads to the samecondition of a vanishing determinant of the Onsager ma-trix i.e., perfect coupling, yielding the Curzon-Ahlborn1975 limit, which in turn at small temperature differ-ence yields half the Carnot efficiency. For the archetypeSmoluchowski-Feynman Brownian motor deviceSmoluchowski, 1912; Feynman et al., 1963 the effi-ciency at maximum power has been evaluated by Tu2008 for more general coupling schemes: Interestinglyenough, the typical upper bound set by Curzon and Ahl-born 1975 may well be surpassed.

In spite of the combined efforts of theorists and ex-perimenters, as of today the question remains unan-swered. At present, ratchet devices operating under con-trollable experimental conditions hardly achieve anefficiency 0 larger than a few percent.

On the other hand, the definition 33 of efficiency isnot always adequate to determine the optimal perfor-mance of a Brownian motor. First, 0 assumes that work

FIG. 9. Color online Net current through a single conicalnanopore for the sum of two voltage signals applied across thepore, 0.5 sgnsin1t+0.5 sgnsin2t+, with different2 /1. The measurements were taken in a 0.1M KCl solutionat pH 8.0. For further details, see Kalman et al., 2007.

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is being done against a load A0, which is not always thecase. For example, one clearly finds a vanishing effi-ciency whenever no load A0 is present. Second, the onlytransport quantifier used in Eq. 33 is the drift velocityv, whereas the fluctuations of xt=vt, i.e., the vari-ance v

2 = v2− v2, are also of practical importance.Note that we use vt to denote the particle velocity inEq. 6 as a proper stochastic process. If v v, andeven more so if v v, the Brownian motor can possi-bly move for some time against its drift direction v.

A load-independent rectification efficiency r hasbeen introduced by Suzuki and Munakata 2003 andthen generalized by Machura, Kostur, Talkner, et al.2004. They computed r as the ratio of the dissipatedpower associated with the directed motion of the motoragainst both the friction and the load, and the ‘‘total’’input power in the presence of the time-periodic forcing.The result assumes the explicit form

r = vA0 + v/ v2 + vA0 − D0 , 34

where D0 is the free diffusion coefficient of Sec. II.A.3.This definition holds evidently also for A0=0, while nu-merical evidence indicates that v2D0, consistentlywith the inequality r1. Note that the definition 34assumes that xt is a damped process with finite nomatter how large. In this way r accounts explicitly forthe power dissipated as velocity fluctuations during therectification process; in particular, it increases upon de-creasing v. Machura, Kostur, Talkner, et al. 2004 no-ticed that, for a rocked ratchet, v exhibits pronouncedpeaks in correspondence with the rectification thresh-olds F3R,L large damping and F2R,L small damping,where the diffusion coefficient also has a maximum Sec.II.A.3. That led to the general conclusion that transportby a Brownian motor can be optimized by operatingaway from activation thresholds, in regimes of large netcurrents, where velocity fluctuations are intrinsicallysmall.

2. Vibrated ratchets

Besides the exceptions presented in Sec. IV, transportcontrol in a rocked ratchet cannot be obtained experi-mentally by deforming the substrate potential “on de-mand.” As shown by Borromeo and Marchesoni2005a, this goal can more easily be achieved by meansof an external tunable signal. Let Vx be assigned afixed profile, say, the standard double sine in Eq. 23.The ratchet current can be still varied by injecting anadditional control signal with frequency 2, that is, byreplacing the harmonic drive in Eq. 8 with the bihar-monic drive in Eq. 16; in the adiabatic regime 21 the system response is very different than in Sec.II.D.4. A rocked ratchet operated under such conditionsis termed vibrational ratchet.

Following the perturbation approach of Blechman2000, Landa and McClintock 2000, and Baltanás etal., 2003 the variable xt in Eq. 3 can be separated asxt→xt+t: in the remainder of this section xt will

represent a slowly time-modulated stochastic processand t the particle free spatial oscillation t=0 sin2t+2, with 0=A2 /2. On averaging out tover time, the Langevin equation for the slow reducedspatial variable xt can be written as

x = − Vx + A1 cos1t + 1 + t , 35

where

Vx = − V0J00sin x + 14J020sin 2x 36

and J0x is the Bessel function of zeroth order. Thesymmetry of the effective vibrated potential 35 is re-stored if one of its Fourier components vanishes,namely, for A2 /2= 1

2 j1 , j1 , 12 j2 , 1

2 j3 , j2 , 12 j4 , 1

2 j5 , j3 , . . ., wherej1=2.405, j2=5.520, j3=8.654, j4=11.79, j5=14.93. . ., arethe ordered zeros of the function J0x; correspondingly,the ratchet is expected to vanish, as confirmed by thesimulation data displayed in Fig. 10.

Not all zeros of this sequence mark an inversion of theratchet current. For instance, for A2 /2

12 j1 the current

is oriented in the natural direction of the effective po-tential, Eq. 36; for 1

2 j1A2 /2 j1, the coefficient ofsin 2x changes sign and so does the ratchet natural ori-entation, or polarity; on further increasing A2 /2 largerthan j1, the sign of both Fourier coefficients in Eq. 36become reversed: this is equivalent to turning Vx up-side down besides slightly remodulating its profile, so

that the polarity of Vx stays negative. Following thisline of reasoning one predicts double zeros i.e., no cur-rent inversions at 0= j1 , j2 , j3 , j4 , . . ..

This control technique can be easily applied to thetransport of massive Brownian particles in both symmet-

0 2 4 6 8 10 12 14A

2/ Ω

2

-0.1

0

0.1

µ

0.06

0.12

0.36

0.60

0 0.5 1D

0.02

1 2

-0.03

0

0.03

FIG. 10. Color online Mobility vs A2 /2 for a vibratedratchet with A1=0.5, 1=0.01, 1=2=0, and different valuesof the noise intensity D see legend, main panel. All simula-tion data have been obtained for 2=10, but the black crosseswhere we set D=0.12 and 2=20. Bottom inset: simulationdata for as in the main panel with an additional curve atD=0.6. Top inset: vs D for A2=0, A1=0.5, and 1=0.01;circles: simulation data; solid curve: adiabatic formula 11.44in Risken 1984. From Borromeo and Marchesoni, 2005a.

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ric Borromeo and Marchesoni, 2007a—see inset of Fig.10—and asymmetric devices Borromeo andMarchesoni, 2007. Likewise, the use of a delay in afeedback control signal Feito and Cao, 2007, 2008:Craig, Kuwada, et al., 2008; Craig, Long, et al., 2008; Sonet al., 2008 can, via synchronization mechanisms, effi-ciently improve the performance of Brownian motorcurrents; a scheme that can also be implemented readilyin experimental flashing ratchets Craig, Kuwada, et al.,2008. In fact, an experimental implementation of afeedback controlled flashing ratchet mechanism hasbeen realized with an optical line trap: It has been ob-served that the use of feedback increases the ratchetvelocity by up to an order of magnitude Lopez, Ku-wada, et al., 2008, in agreement with theory.

III. TRANSPORT IN NANOPORES

Membranes in biology encase cells and their or-ganelles and allow the compartmentalization of cellularprocesses, thereby operating far from thermal equilib-rium, a condition that is essential to life. Membranesseparate two phases by creating an active or passive bar-rier to the transport of matter between them. As a firstclassification, membranes can be divided into biologicaland artificial membranes. The latter term is applied toall membranes made by man with natural, possiblymodified, materials and with synthetic materials syn-thetic membranes. Synthetic membranes can be furtherdivided into organic made with polymers and inorganicmembranes made with alumina, metals, etc..

Transport across a membrane occurs through channelsor pores Berezhkovskii et al., 2003; Kolomeisky, 2007.In several cases, the membrane transport propertiesmust be regarded as a collective effect, where the func-tion of an individual channel is influenced by the pres-ence of possibly diverse neighboring channels. This isthe case, for instance, of ion pumps Läuger, 1991; Imand Roux, 2002 in cellular membranes Sec. III.A orfor coupled ion channels that experience a commontransmembrane voltage. Single-molecule techniques,however, allow for the study and characterization of rec-tification properties of individual entities, such as single-ion channels. Nowadays, synthetic membranes with as-signed pore density and patterns are made commerciallyavailable. Moreover, by means of increasingly sophisti-cated growth methods, irradiation Fleischer et al., 1975,and nanofabrication techniques Li et al., 2001; Storm etal., 2003; Dekker, 2007; Healy et al., 2007, the cross sec-tions of membrane pores can be modulated along theiraxis. As a result, transport in artificial ion channels fab-ricated from asymmetric single pores of the most diversegeometries have become accessible to experimentsSiwy and Fulinski, 2004; Healy et al., 2007; see also Sec.III.B.

In this section we focus on devices where the size ofthe transported particles is comparable to the pore crosssection. Transport in these devices can be analyzed interms of the one-particle 1D mechanisms illustrated inSec. II. Larger pores, commonly used to channel fluids

liquid or gaseous or colloidal particles suspended in afluid, are considered in Sec. VI in the context of micro-fluidic devices.

A. Ion pumps

As remarked, biology teaches us useful lessons thatcan guide us in the design of artificial Brownian motors.Biological membranes are known to form lipophilic bar-riers and have embedded a diverse range of units whichfacilitate the selective movement of various ionic andpolar segments or the pumping of protons and electronsacross channel-like membrane openings. These biologi-cal nanodevices typically make use of electrochemicalgradients that enable them to pump a species against itsconcentration gradient at the expense of yet anothergradient. Ultimately, such devices convert nondirec-tional chemical energy, say, from the hydrolysis ofadenosine triphosphate ATP, into directed transport ofcharged species against electrochemical gradients.

Although many details of the underlying mechanismare far from being understood, these machines appar-ently make use of mechanisms that characterize thephysics of Brownian motors Astumian, 2007. A char-acteristic feature is that changes in the binding affinity atselective sites in the transmembrane region must becoupled to conformational changes which, in turn, con-trol motion into the desired direction. Thus, in contrastto rocked or pulsated ratchets, where the particle-potential interaction acts globally along the whole peri-odic substrate landscape, the transport mechanism atwork here can be better described as an “information”ratchet Astumian and Derényi, 1998; Parrondo and deCisneros, 2002. Indeed, the effective potential bottle-necks to transduction of Brownian motion becomemodified locally according to the actual location of thetransported unit; as a result, information is transferredfrom the unit to the potential landscape. For example,this scheme can be used to model the pumping of Ca2+

ions in Ca2+-ATPase Xu et al., 2002. Related schemeshave also been invoked in the theoretical and experi-mental demonstration of pumping of Na+ and K+ ionsvia pulsed electric field fluctuations in Na- andK-ATPase Xie et al., 1994; Freund and Schimansky-Geier, 1999; Tsong, 2002 or for the operation of a cata-lytic wheel with help of a ratchetlike, electroconforma-tional coupling model Rozenbaum et al., 2004.

Yet another mechanism can be utilized to pump elec-trons in biomolecules. It involves nonadiabatic electrontunneling in combination with asymmetric, but nonbi-ased, nonequilibrium fluctuations, as proposed by Goy-chuk 2006. The nonequilibrium fluctuations originatefrom either random binding of negatively charged ATPor externally applied asymmetric, but nonbiased, electricstochastic fields. Likewise, unbiased nonequilibrum two-state fluctuations telegraphic noise can induce direc-tional motion across an asymmetric biological nanopore,as numerically investigated for an aquaglyceroporinchannel, where water and glycerol are transported by

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means of a rocked ratchet mechanism Kosztin andSchulten, 2004.

Finally, we stress that in such realistic complex bio-logical channels the physical implications of an exter-nally applied control are often difficult to predict. This isdue to the multifaceted consequences any control actionmay have in terms of chemical variations, conforma-tional changes, polarization effects, and the like. More-over, unlike single-molecule-type experiments, single-channel recordings are not easily accessible whendealing with biological molecules. Nanopores of lessercomplexity are thus synthetic nanopores which can befabricated by the use of bottom-down nanosciencetechniques—the theme we review next.

B. Artificial nanopores

With the recent advances of track-etching Fleischer etal., 1975 and silicon technologies Li et al., 2001; Stormet al., 2003; Dekker, 2007, charge transport in a singlenanopore became experimentally accessible. This is asubstantial leap forward with respect to ion pump andzeolite transport experiments, where experimental dataare taken over a relatively high channel density. Fabri-cated nanopores in polymer films and silicon materialsare investigated in view of their potential applications asbiomimetic systems, that is, for modeling biologicalchannels, and as biosensors.

Siwy and co-workers Siwy and Fulinski, 2002; Siwy etal., 2005; Constantin and Siwy, 2007; Vlassiouk and Siwy,2007 have prepared a nanofluidic diode, which hadbeen predicted to rectify ion current as a bipolar semi-conductor diode rectifies electron current Daiguji et al.,2005. This diode is based on a single conically shapednanopore track-etched in a polymer film with openingsof several nanometer and 1 m, respectively see Fig.11. The surface charge of the pore is patterned so thattwo regions of the pore with positive and negative sur-face charges create a sharp barrier called the transitionzone. This nanofluidic diode is bipolar in character sinceboth positively and negatively charged ions contribute tothe measured current. Majumdar and co-workersKarnik et al., 2007 fabricated a similar nanofluidic di-ode with a sharp barrier between a positively chargedand a neutral side of the pore. The presence of only onetype of surface charge causes the latter device to be uni-polar.

Ion rectification was achieved by applying a longitudi-nal ac voltage Fig. 11; the system thus operates as aone-cell 1D rocked ratchet Sec. II.D.1, where the spa-tial asymmetry is determined by the interaction of asingle ion with the inhomogeneous charge distributionon the pore walls. The rectification power of the pore isdefined as the ratio of the ionic currents recorded forpositive and negative driving voltages, i.e., V= IV / I−V . Of course, due to their asymmetric ge-ometry, conical pores can rectify diffusing ions also foruniform wall-charge distributions Siwy et al., 2005;however, the corresponding factor would be at leastone order of magnitude smaller than reported here.

As a serious limitation of the present design, it is notpossible to control the rectification power of a givenconical nanopore without introducing changes to itsbuilt-in electrochemical potential. An alternate ap-proach has been proposed by Kalman et al. 2007,where two superposed rectangular voltage signals ofzero mean were used to control the net ion currentthrough a nanopore of preassigned geometry. By chang-ing the amplitude, frequency, and relative phase of thesesignals they used the gating effect of Sec. II.C.2 to gaincontrol over the orientation and the magnitude of ionflow through the pore. Their experimental data werefound in excellent agreement with the theoretical pre-diction of Eq. 32. The magnitude of ion current varia-tion achieved by asymmetric signal mixing was compa-rable with the incommensurate current offset Eq. 30,which means that the nanopore diode could be operatedwithout regard for the details of its intrinsic rectificationpower.

We conclude this section on artificial nanopores byanticipating that asymmetric micropores etched in sili-con membranes also work as microfluidic ratchet pumpsfor suspended colloidal particles Kettner et al., 2000;Müller, 2003. For instance, entropic effects on the rec-tification efficiency of the conical nanopores of Siwy andco-workers have been analyzed by Kosinska et al. 2008.However, in-pore diffusion in a liquid suspension re-quires a fully 3D analysis of the pumping mechanism,which sets the basis for the fabrication of more compli-cated ionic devices Stein et al., 2004; van der Heyden etal., 2004. This category of device is reviewed in Sec. VI.

C. Chain translocation

The opposite limit of composite objects passingthrough a much narrower opening is often called “trans-

FIG. 11. Color online Ion current recorded experimentallyby Constantin and Siwy 2007 at 0.1M KCl, pH=5.5, througha single conical nanopore with end diameters 5 and 1000 nm,respectively. The pore rectification power is =217 at 5 V. In-set: Geometry of a conical nanopore with schematic represen-tation of surface charge distribution creating a bipolar nano-fluidic diode. Figure provided by Zuzanna Siwy.

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location.” Using the experimental setup in Fig. 12a,Kasianowicz et al. 1996 first measured the blockagecurrents of single-stranded RNA and DNA electro-phoretically driven through a transmembrane pore. Re-cent developments of this technique demonstratedsingle-nucleotide resolution for DNA hairpins Vercou-tere et al., 2003; Gerland et al., 2004; Ashkenasy et al.,2005, thus raising the prospect of creation of nanoporesensors capable of reading the nucleotide sequence di-rectly from a DNA or RNA strand see Healy 2007;Movileanu 2008; Zwolak and Di Ventra 2008. Nowa-days, translocation mechanisms are investigated in bothprotein channels mainly the bacterial -hemolysin poreKasianowicz et al., 1996 and synthetic pores. Boththese approaches have advantages and disadvantagesDekker, 2007; Healy, 2007. Protein channels can be en-gineered with almost angstrom precision, but the lipidmembrane in which they may be incorporated is verymechanically unstable. Synthetic pores, on the otherhand, offer robustness to the system, which allows us tobetter characterize the physical aspects of translocationphenomena.

As a main difference with ion transport, the translo-cation of a long polymer molecule in a 1D device in-volves entropic effects, which become important whenthe opening cross section grows comparable with the ra-dius of gyration R0 of the polymer. These effects werepredicted by Arvanitidou and Hoagland 1991 to ac-count for the conformation changes of a chain in movingpast a conduit constriction, and finally observed by Hanet al. 1991 and Han and Craighead 2000 in an artifi-cial channel. As a model pore-constriction system, theyfabricated a channel consisting of a periodic sequence ofregions of two different depths, as shown in Fig. 12b.The thick regions were 1 mm deep, i.e., comparable withthe R0 of the double-stranded DNA molecules they usedin their experiment, whereas the depth of the thin re-gion, 90 nm, was much smaller than R0.

The thick regions act like “entropic traps,” as theDNA molecules are entropically prevented from enter-ing the thin regions. For the same reason, a chain caughtbetween two traps tends to fall back into the trap thatcontains most of it. In the presence of an external elec-tric field, escape from a trap is initiated by the introduc-tion of a small portion of DNA into an adjacent thinregion, just enough to overcome the activation barrierfor escape. This initiation process is local in nature, andthe energy barrier does not depend on the total lengthof the trapped DNA molecule. Once a DNA molecule isin the transition state once a proper length of “beach-head” is formed, it readily escapes the entropic trap,regardless of the length of the remaining molecule in thetrap. Quite counterintuitively, Han and co-workersfound that escape of DNA occours faster in longer en-tropic traps than in shorter ones.

A theoretical interpretation of these results was givenby Park and Sung 1999, who treated the dynamics of aflexible polymer surmounting a 1D potential barrier as amultidimensional Kramers activation process Hänggi etal., 1990. To determine the activation free energy, Park

and Sung computed the free energy of the polymer atthe transition state. For a small-curvature barrier, thepolymer retains its random coil conformation during thewhole translocation process, giving rise to essentially thesame dynamics as that of a Brownian particle. For alarge-curvature barrier, on the other hand, a conforma-tional transition coil-stretch transition occurs at the on-set of the barrier crossing, which significantly lowers theactivation free energy and so enhances the barrier cross-ing rate. As the chain length varies, the rate shows aminimum at a certain chain length due to the competi-tion between the potential barrier and the free energydecrease by chain stretching. Synthetic nanopores canthus be used as Coulter counterdevices to selectively de-tect single DNA molecules, resolving their length anddiameter.

Spatial asymmetry can rectify translocation throughsynthetic and biological pores alike. This is the case, forinstance, of a hydrophobic polymer translocating acrossa curved bilayer membrane. Extensive simulationBaumgärtner and Skolnick, 1995 showed that the poly-mer crosses spontaneously and almost irreversibly fromthe side of lower to that of higher curvature, so as tomaximize its conformational entropy “entropy” ratch-ets; Sec. V.A.1. Moreover, in protein translocationthrough a biomembrane, unlike that through the chemi-cal artificial channels, structure of the pores can comeinto play by helping rectify the thermal fluctuations ofthe stretched molecule: When specific predeterminedsegments of the protein cross the membrane, chemicalsacting as chaperons bind on the segments to preventtheir backward diffusion Hartl, 1996; Jülicher et al.,1997; Nigg, 1997. The ensuing chemical asymmetry thencompetes with the entropic asymmetry to determine thetranslocation current.

D. Toward a next generation of mass rectifiers

1. Zeolites

Zeolites are three-dimensional, nanoporous, crystal-line solids either natural or synthetic with well-definedstructures that contain aluminum, silicon, and oxygen intheir regular framework. The silicon and aluminum at-oms are tetrahedrally connected to each other throughshared oxygen atoms; this defines a regular frameworkof voids and channels of discrete size, which is accessiblethrough nanopores of well-defined molecular dimen-sions. The negative electric charge of the zeolite frame-work is compensated by inorganic or even organic cat-ions or by protons in the acidic form of the zeolites.The ions are not a part of the zeolite framework, andthey stand in the channels. The combination of manyproperties—such as the uniform cross section of theirpores, the ion exchange properties, the ability to de-velop internal acidity, high thermal stability, high inter-nal surface area—makes zeolites unique among inor-ganic oxides and also leads to unique activity andselectivity. As a result, zeolites can separate moleculesbased on size, shape, polarity, and degree of unsatura-

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tion, among others properties Kärger and Ruthven,1992; Kärger, 2008b.

Nuclear magnetic resonance NMR provides directaccess to the density and mobility of molecules in zeo-lites. By implementing a time-dependent NMR tech-nique, termed pulsed field gradient NMR, Kärger andco-workers see Kärger 2008a investigated the prob-lem of particle diffusion in a narrow zeolite nanopore.For instance, they observed that, owing to the con-strained pore geometry 0.73 nm across, the meansquare displacement of CF4 molecules 0.47 nm in diam-eter diffusing in the zeolite AlPO4-5 increases linearlywith the square root of the observation time rather thanwith the observation time itself. That was an early ex-perimental demonstration of the single-file diffusionmechanism introduced in Sec. II.A.4. In the meantime,however, real zeolite crystals have been repeatedlyfound to deviate notably from their ideal textbook struc-ture Schemmert et al., 1999, so that their pore crosssections ought to be regarded as longitudinally corru-gated. To what extent this may affect particle diffusion is

still a matter of ongoing research Kärger et al., 2005;Taloni and Marchesoni, 2006.

Most zeolite structure types exhibit 3D pore net-works. The most studied example is a synthetic zeoliteof type MFI Zeolite Socony Mobil-five. Their pore net-work is formed by mutually intersecting straight in thecrystallographic y direction and sinusoidal in the crys-tallographic x direction channels. Although there is nocorresponding third channel system, molecular propaga-tion has also been observed in the z direction. Experi-mental data are consistent with a simple law of corre-lated diffusion anisotropy Fenzke and Kärger, 1993az

2 /Dz=ax2 /Dx+ay

2 /Dy, where ai and Di, with i=x ,y ,z,denote, respectively, the lattice and the diffusion con-stants in the i direction. This means that the molecular“memory” is shorter than the mean travel time betweentwo adjacent network intersections.

With the advent of synthetic zeolites and zeoliticmembranes, researchers are fascinated by the optionthat the existence of different channel types within oneand the same material may be used for an enhancementof the performance of catalytic chemical reactions. Asthe diffusion streams of the reactant and product mol-ecules tend to interfere with each other, rerouting themthrough different channels may notably speed up a cata-lytic reactor; hence, the idea of reactivity enhancementby “molecular traffic control” Derouane and Gabelica,1980; Neugebauer et al., 2000.

2. Nanotubes

Another interesting category of artificial nanopore isthe carbon nanotubes Dresselhaus et al., 1996. Single-walled carbon nanotubes are cylindrical molecules ofabout 1 nm in diameter and 1–100 m in length. Theyconsist of carbon atoms only, and can essentially bethought of as a layer of graphite rolled up into a cylin-der. Multiple layers of graphite rolled in on themselvesare said to form a multiwall carbon nanotube. The elec-tronic properties of nanotubes depend strongly on thetube diameter as well as on the helicity of the hexagonalcarbon lattice along the tube chirality. For example, aslight change in the pitch of the helicity can transformthe tube from a metal into a large-gap semiconductor,hence their potential use as quantum wires in nanocir-cuits Dekker, 1999; Collins and Avouris, 2000. An evenwider range of geometries and applications becameavailable recently with the synthesis of various oxidenanotubes Rêmskar, 2004. With their hollow cores andlarge aspect ratios, nanotubes are excellent conduits fornanoscale amounts of material. Depending on the fillingmaterial, experimenters have thus realized nanoscalemagnets, hydrogen accumulators, thermometers, andswitches.

Nanotubes also provide an artificial substrate for con-trollable, reversible, atomic-scale mass transport. Reganet al. 2004 attached indium nanocrystals to a multiwallcarbon nanotube and placed it between electrodes setup in the sample chamber of an electron microscope.Applying a voltage to the tubes, they observed that the

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FIG. 12. Color online DNA translocation. a Electric detec-tion of individual DNA molecules with a nanopore. A constantvoltage bias induces a steady-state ionic current through asingle nanopore left panel, from simulation; adding DNA tothe negatively biased compartment causes transient reductionsof the ionic current right panel, from experiment. This re-duced conductance is associated with the translocation ofDNA through the pore, which partially blocks the ionic cur-rent. From Aksimentiev et al., 2004. b Schematic diagram ofthe entropic trap in Han et al., 1999.

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metallic particles at one end of the tube gradually disap-peared, while the number at the other end grew. Whilethe details of the underlying driving mechanism ther-momigration versus electromigration remain unclear,they concluded that the voltage dictates the directional-ity of the nanoscale mass conveyor. Experimenters evensucceeded in synthesizing carbon nanotubes encapsulat-ing metallic atoms and characterizing the electrome-chanical properties of such nanochannels see, e.g., Gaoand Bando 2002. In the near future, carbon nanotubeswill also be combined to form molecular “gears,” whosefeasibility has been proven so far only by simulationDrexler, 1992. A conceptual example is provided by adouble-walled carbon nanotube consisting of two co-axial single-walled nanotubes with different chiralities,immersed in an isothermal bath. In the presence of avarying axial electrical voltage, this system would exhibita unidirectional ratchetlike average rotation as a func-tion of the chirality difference between its constituentsMarchesoni, 1996.

Nanotubes have also been used to realize prototypesof thermal diodes for phonon transport Chang et al.,2006. With this concept in mind, one can further devisea phonon Brownian motor aimed to ratchet a net heatflux from cold to hot, as numerically demonstrated by Liet al. 2008. This class of device has the potential toallow an efficient control of heat fluxes at the nanos-cales.

In spite of the recent advances in nanotechnology, in-ner or outer transport along nanotubes is still con-trolled by external gradients. Nevertheless, nanotube-based ratchets, though not immediately available, arelikely to represent one of the next frontiers in artificialBrownian motor research. Moreover, synthetic nano-tubes are potential building blocks for nanofluidic de-vices also Holt et al., 2006, as discussed in Sec. VI.

IV. COLD ATOMS IN OPTICAL LATTICES

Optical lattices are periodic potentials for atoms cre-ated by the interference of two or more laser fields Jes-sen and Deutsch, 1996; Grynberg and Robilliard, 2001.In near-resonant optical lattices the laser fields producesimultaneously a periodic potential for the atoms and acooling mechanism. The optical potential for an atom inan optical lattice is given by the light shift, or ac Starkshift, of the atomic energy level that acts as the groundstate of an assigned optical transition. The simplest caseis represented by the Jg=1/2→Je=3/2 atomic transitionin a 1D configuration.

Consider two counterpropagating laser fields, detunedbelow the atomic resonance, with orthogonal linear po-larizations and the same intensity and wavelength .Their interference results in a spatial gradient with po-larization ellipticity of period /2. This in turn producesa periodic potential for the atom. For instance, theatomic hyperfine ground states g , ± = Jg=1/2 ,M= ±1/2 experience periodic potentials in phase opposi-tion along the direction x of light propagation, namely,

V±x = V0− 2 ± cos kx/2, 37

where k=2 / and the depth of the potential wells V0scales as IL /, with IL the total laser intensity and thedetuning from atomic resonance. As the laser fields arenear resonance with the atomic transition Jg=1/2→Je=3/2, the interaction with the light fields also leads tostochastic transitions between the Zeeman sublevels g , ± of the ground state. The rate of this transition canbe quantified by the photon-atom scattering rate which scales as IL /2. It is therefore possible to varyindependently the optical lattice depth V0 and bychanging simultaneously IL and .

The stochastic transitions between ground states alsolead to damping and fluctuations. The damping mecha-nism, named Sisyphus cooling, originates from the com-bined action of light shifts and optical pumping, whichtransfers, through cycles of absorption and spontaneousemission involving the excited state Je, atoms from oneground state sublevel to the other one. Moreover, thestochastic transitions between the two potentials V±xalso generate fluctuations of the instantaneous force ex-perienced by the atom. In conclusion, the equilibriumbetween cooling and heating mechanisms determines astationary diffusive dynamics of the atoms, a processconfined to the symmetric ground-state optical latticeV±x.

Random amplitude pulsations of one symmetric po-tential, Eq. 37, do not suffice to produce rectification.The only missing element needed to reproduce therocked setups of Secs. II.C and II.D is the additive acforce Ft. In order to generate a time-dependent homo-geneous force, one of the lattice beams is phase modu-lated, and we indicate by t its time-dependent phase.In the laboratory reference frame the phase modulationof one of the lattice beams results in the generation of amoving optical lattice V±„x−t /2k…, reminiscent of theperiodic sieves by Borromeo and Marchesoni 2007a. Inthe accelerated reference frame x→x−t /2k, the op-tical potential would be stationary; however, an atom ofmass m experiences also an inertial force in the z direc-tion proportional to the acceleration of the movingframe, namely, Ft= m /2kt. This is the homoge-neous ac drive needed to rock a cold atom ratchet.

The harmonic mixing and rocking ratchet setups de-scribed above were used by Renzoni and co-workersSchiavoni et al., 2003; Gommers et al., 2005, 2006, 2007,2008 to investigate experimentally the relationship be-tween symmetry and transport in 1D atom traps. Wediscuss here separately the experimental results for twodifferent cases: i a biharmonic driving including twoharmonics at frequencies 0 and 20, and ii a multifre-quency driving obtained by combining signals at threedifferent frequencies.

A. Biharmonic driving

Schiavoni et al. 2003 and Gommers et al. 2005 gen-erated a biharmonic drive, Eq. 16, with 2=21 and

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1=0, and searched for a harmonic mixing current pro-portional to sin2−0. At variance with Eq. 17, anadditional phase lag 0 was introduced to account forthe finite atom dissipation. Indeed, by generalizing ourargument of Sec. II.C.1, one can easily prove that in-crease in the damping makes 0 drop from 0 down to− /2. In the experiment, the atom dissipation could betuned continuously, without changing the optical poten-tial constant V0, by varying simultaneously IL and at aconstant IL / ratio; thus harmonic mixing was experi-mentally accessible in limiting regimes of both zero andinfinite damping.

The experimental results of Fig. 13 for cesium atomscooled in the microkelvin range clearly demonstrate themechanism of harmonic mixing at work. In fact, the netatom velocity plotted in panel a is well fitted by v /vr=A sin2−0, where the dependence of 0 on is asin panel b, and A is a characteristic function of thesystem Gommers et al., 2005. For the smallest scatter-ing rate examined in the experiment, no current wasgenerated at 2= l, with l integer, as expected from Eq.17. On the other hand, the magnitude of the phaseshift 0 increases with increasing scattering rate, thuscausing current generation for 2= l. Besides confirm-ing the notion of harmonic mixing, the experimental re-sults of Schiavoni et al. 2003 and Gommers et al. 2005suggest that symmetry breaking can be controlled bydissipation. Closely related results have been reportedby Ustinov et al. 2004 in their attempt to control fluxon

ratcheting in Josephson junctions by means of harmonicmixing.

B. Multifrequency driving

Recent experiments with multifrequency drivingGommers et al., 2006, 2007 aimed to investigate thetransition from periodic to quasiperiodic driving, and toexamine how the analysis of Sec. II.C is modified in thistransition. The multifrequency driving was obtained byadding a sinusoidal component to the ac drive employedin the previous section, i.e.,

Ft = A1 cos1t + A2 cos21 + 2

+ A3 cos3t + 3 . 38

For irrational 3 /1 the driving is quasiperiodic.Clearly, in a real experiment 3 /1 is always a rationalnumber, which can be written as p /q, with p and q twocoprime positive integers. However, as the duration ofthe experiment is finite, by choosing p and q sufficientlylarge it is possible to obtain a driving that is effectivelyquasiperiodic on the time scale of the experiment.

Consider first the case of periodic driving, with ratio-nal 3 /1. Adding a third harmonic with phase constant30 in Eq. 38 breaks the time symmetry of Ft sothat directed transport is allowed for 0=0 and 2= l.In other words, for 30, the third driving componentleads to an additional phase shift of the current as afunction of 2. As a result, the current will retain itssinusoidal phase dependence sin2−0 as above, withthe difference that now 0 accounts for the phase shiftsproduced both by dissipation and by the added drivingcomponent. Following the discussion in Sec. II.D.4, oneconcludes that in the quasiperiodic regime the third-order harmonic at frequency 3 is not relevant to char-acterize the time symmetry of the forcing signal, which isentirely determined by the biharmonic terms at fre-quency 1 and 22.

In later experiments Gommers et al., 2006, 2007, thetransition to quasiperiodicity was investigated by study-ing the atomic current as a function of 2 for differentp /q. By increasing p and q the driving was made moreand more quasiperiodic on the finite duration of the ex-periment, with the quantity pq an easily accessible mea-sure of the degree of quasiperiodicity. The data for theaverage atomic current were fitted with the functionv /vmax=sin2−0 and the resulting values for 0 wereplotted as a function of pq. For small values of pq, i.e.,for periodic driving, the added component at frequency3 leads to a shift which strongly depends on the actualvalue of pq. For larger values of pq, i.e., approachingquasiperiodicity, the phase shift 0 tends to a constantvalue independent of 3, which coincides with the purelydissipative phase shift measured in the biharmonic driv-ing case of Sec. IV.A. The experimental results confirmthat, in the quasiperiodic limit, the only relevant symme-tries are those determined by the periodic biharmonicdriving and dissipation.

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FIG. 13. Cold atoms in an optical lattice. a Average atomicvelocity as a function of the phase 2 for different values ofs= v / 22 / a quantity proportional to the scatteringrate . The data are labeled by the lattice detuning , as thevibrational frequency at the bottom of the wells was kept con-stant, v /2=170 kHz; the forcing frequency is 1 /2=100 kHz. The lines are the best fit of the data with the func-tion v /vr=A sin−0, where vr=k /m is the so-called atomrecoil velocity. b Experimental results for the phase shift 0as a function of s. From Gommers et al., 2005.

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C. More cold atom devices

Cold atoms in optical traps proved to be a playgroundfor rectification experiments.

(i) Gating effect. For instance, Gommers et al. 2008modified the experimental setup described above todemonstrate experimentally a gating ratchet with coldrubidium atoms in a driven near-resonant optical lattice.As suggested in Sec. II.C.2, a single-harmonic periodicmodulation of the optical potential depth with frequency2 was applied, together with a single-harmonic rockingforce with frequency 1. The modulation of the opticalpotential depth V0 was obtained by modulating the in-tensity IL of the laser beams. This also resulted in anunavoidable modulation of the optical pumping rate,which affected only the fitting phase shift 0. Directedmotion was observed for rational values of 2 /1, a re-sult due to the breaking of the symmetries of the system.

(ii) Pulsated ratchets. Although a bona fide rockedratchet for cold atoms in an optical lattice has not beenrealized see, e.g., Ritt et al. 2006, a variation of the1D optical lattice described above allowed an early dem-onstration of a randomly pulsated ratchet. Mennerat-Robilliard et al. 1999 set the polarizations of the laserbeams at an angle /2 and applied a weak Zeemanmagnetic field orthogonal to the optical lattice. The ef-fect of the magnetic field consisted in removing the de-generacy of the ground states by adding a /2 wave-length component to both potentials in Eq. 37, whichthus acquired different asymmetric profiles. As a conse-quence, the random optical transitions between themodified potentials V±x turned out to propel trappedcold rubidium atoms in the vertical direction, with a signthat depended on and the orientation of the magneticfield.

In Sec. II.D.2 we reported that rectification can occuron symmetric 1D substrates that shift instantaneouslyback and forth in space with a fixed amplitude, namely,on substrates that are subjected to a time-discrete phasemodulation periodic or random. In this case, breakingthe supersymmetry condition 14 requires appropriateasymmetric space-dependent switching rates Gorman etal., 1996. Based on this rectification scheme, Sjölund etal. 2006 realized a simple flashing ratchet for cold at-oms. It consisted again of a millikelvin cold gas of ce-sium atoms switching between two symmetric ground-state optical lattices, coupled via optical pumping. In thepresence of induced friction, although small, and for ap-propriate laser detunings and intensities of the laserbeams, the degeneracy of the ground states was re-moved by making one ground state, say V+, long livedand the other one, V−, short lived. If we further considerthat the two optical lattices were shifted one relative tothe other, V−x=V+x−x0, then we conclude that theswitching rates between potentials were state and posi-tion dependent with k−→+xk+→−x. In this setup, theatoms execute stationary time-asymmetric sequences ofrandom jumps between V+ and V−. During the timespent in the short-lived lattice V−, they experience a po-

tential with an incline that depends on x0. Thus theirdiffusion is strongly enhanced in one specific direction,and correspondingly reduced in the opposite direction.In the experiment, the spatial shift x0 and the transitionrates between the two optical lattices could be adjustedat will. A directed motion with constant velocity wasobserved in the absence of additional forcing terms, i.e.,for Ft=0, except for specific system parameters, wheresymmetry was restored. Moreover, they showed thattheir device, when operated in three dimensions Ell-mann et al., 2003, can generate directed motion in anydirection.

V. COLLECTIVE TRANSPORT

The studies on rectification mechanisms reviewed inSec. II were conducted mostly for a single particle; how-ever, often systems contain many identical particles, andthe collective interactions between them may signifi-cantly influence the transport of particle aggregatesAghababaie et al., 1999. For example, interactionsamong coupled Brownian motors can give rise to novelcooperative phenomena like anomalous hysteresis andzero-bias absolute negative resistance Reimann et al.,1999. The interactions among individual motors plays aparticular important role for the occurrence of unidirec-tional transport in biological systems; see, e.g., Sec. IX inthe review by Reimann 2002.

In this section we restrict ourselves to the propertiesof 1D and 2D systems of pointlike interacting particles,although an extension to 3D systems is straightforward.Let the pair interaction potential Wr be a function ofthe pair relative coordinates r= x ,y, characterized byan appropriate parameter g, quantifying the strength ofthe repelling g0 or attracting force g0, and by aconstant , defining the pair interaction length. An ap-propriate choice of Wr, with g0, can be used tomodel hard-core particles with diameter Savel’ev etal., 2003, 2004a. We term the pair interaction long rangeif is larger than the average interparticle distance, andshort range in the opposite regime. In the absence ofperturbations due to the substrate or the external drives,long-range repelling particles rearrange themselves toform a triangular Abrikosov 1957 lattice. When placedon a disordered substrate or forced through a con-strained geometry, such an ideal lattice breaks up intolattice fragments punctuated by pointlike defects anddislocations and separated by faulty boundaries. In thepresence of a drive, such boundaries act like 1D easy-flow paths, or rivers, for the movable particles. This pro-cess, often referred to as “plastic” flow, has been ana-lyzed numerically by Reichhardt et al. 1998.

Rectification of interacting particles occurs as a com-bined effect of the configuration of the device and thegeometry of its microscopic constituents. In this contextthe dimensionality of the underlying dynamics is alsoimportant: i transport in some 2D geometries can oftenbe reduced to the 1D mechanisms illustrated in Sec. IIreducible 2D geometries; ii under certain circum-

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stances, however, a bona fide 2D rectification may occurat a nonzero angle with the driving force irreducible 2Dgeometries. Finally, we remark that particle interactioncan be exploited to rectify particles of one species byacting on particles of another species, alone, by either anappropriate drive or a specially tailored substrate geom-etry.

A. Asymmetric 1D geometries

We consider now two examples of 2D reducible geom-etries where the transport of massless repelling particles,subjected to an external ac drive, occurs as a collectiveeffect, namely, under conditions incompatible with therectification of a single particle. In particular, we showthat collective effects make transport of interacting par-ticles possible even in the absence of an ad hoc ratchetsubstrate.

1. Boundary effects

Consider the asymmetric channel in Fig. 14a filledwith n repelling particles per unit cell. The correspond-ing particle density is =n /al, where al is the area of thechannel unit cell. The walls are rigid and particle-wallcollisions are taken elastic and relatively short range in-teraction length not much larger than . The profiles±yx of the upper and lower walls are modeledby an appropriate double-sine function; 2d and 2a de-note, respectively, the width of the channel and of itsbottlenecks. Due to the repulsive interactions, the par-ticles are pressed against the walls which corresponds toan effective asymmetric spatial modulation. As a conse-quence, when driven by an ac force, the particles aremore likely to flow to the left than to the right. Theensuing rectification mechanism is reminiscent of a 1Drocked ratchet with an inverted potential 23. Numeri-cal simulations, besides supporting this prediction,clearly indicate an optimal or resonant, temperature re-gime in which the particle drift is maximized. This ob-

servation can be explained by noticing that, at low tem-peratures, the ac drive causes a moving particle tomigrate to the center of the channel where it no longerinteracts with the boundaries; while at high tempera-tures the driving force becomes irrelevant and thus theparticle is no longer pushed through the channel bottle-necks periodically in time. The first simulation evidenceof this phenomenon was produced by Wambaugh et al.1999. Their results obtained for the special case ofmagnetic vortex channeling are discussed in Sec. VII.A.

At high enough particle densities , the net current inthe channel is expected to be suppressed, as the repel-ling particles end up clogging the bottlenecks and thushampering collective longitudinal oscillations. In the op-posite limit of, say, n=1, the arguments above seem torule out rectification, as a single oscillating particlewould be confined in the inner channel of radius a. How-ever, this conclusion holds only in the absence of ther-mal fluctuations, T=0. At finite T, reducing the diffusionin a 2D channel to a 1D process implies defining aneffective diffusion constant Sec. II.A.3 Dx /D0= 1+yx2−1/3 equivalent to a periodic asymmetric modu-lation of the temperature Reguera et al., 2006 a See-beck ratchet in the notation of Reimann 2002. A par-ticle crossing a channel bottleneck perceives a lowereffective temperature on the left, where the wall issteeper, than on the right, so that it gets sucked forward;the opposite happens in correspondence to the largestchannel cross sections. As long as dx /Dx0, the os-cillating motion of a single particle can indeed be recti-fied, with a sign that depends on the details of the wallprofile Ai and Liu, 2006. No matter how weak, such amechanism, termed an entropic ratchet Slater et al.,1997, supports the conclusion of Wambaugh et al. 1999that rectification in a channel occurs only at finite T, as acertain amount of noise is needed for the particle toexplore the asymmetric geometric of the device. More-over, the argument above hints at the occurrence of anoptimal channel density , as detected in real supercon-ducting devices Sec. VII.A.

2. Asymmetric patterns of symmetric traps

Olson et al. 2001 proposed a new type of 2D ratchetsystem which utilizes gradients of pointlike disorder,rather than a uniformly varying substrate potential.Consider a 2D sample containing a periodically gradu-ated density of point defects, as sketched in Fig. 14b.Each defect is depicted as a circular microhole, whichacts as a symmetric short-range particle trap of finitedepth. In real experiments, such defects can actually becreated by either controlled irradiation techniques ordirect-write electron-beam lithography Kwok et al.,2002. The defect density lx was chosen to be uniformalong the vertical axis, and to follow the typical double-sine asymmetric profile of Eq. 23 along the horizontalaxis. We now inject into the sample repelling masslessparticles with average density and interaction length .The defect radius controls particle pinning by defectsand was taken much smaller than . For a sufficiently

x

2a2d

L(a)

(b)y

FIG. 14. 1D reducible asymmetric geometries. a Channelwith standard asymmetric profile yx=y0−y1sin2x /L+ 1/4sin4x /L; the parameters y1 and y0 control the width2d and the bottleneck 2a. b Disordered pattern of circulardefects generated with periodic distribution yx along the xaxis.

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high particle-to-defect density ratio, the particles fillmost of the pinning sites and create an effective repul-sive potential. If we further assume long-range particlepair interactions Wr, such a 2D potential becomes in-sensitive to the details of the defect distribution; it re-tains only the periodic horizontal modulation of lx,thus resulting in a mean-field ratchet potential Vx, asin Eq. 23. The effective amplitude V0 is a function of atleast three length scales: the interaction constant, theaverage particle distance, and the average defect spac-ing. A certain fraction of particles does not becomepinned at individual defects but, subjected to an appliedac drive, can move in the interstitial regions betweenpinning sites. Although the moving interstitials do notdirectly interact with the short-ranged defects, they feelthe long-range interaction of the particles trapped at thepinning sites and described by the mean-field potentialVx. As a consequence, a horizontally applied ac drivecan induce a longitudinal particle transport, as provenby Olson et al. 2001 by means of numerical simulation.

Two conditions are instrumental to the onset of a rec-tification current: i a finite temperature, T0, and iia defect filling fraction close to or larger than 1, bothconditions required for interstitial particles to get andstay unpinned. As in Sec. V.A.1, condition i implies theexistence of an optimal rectification temperature. Thedependence on the forcing frequency and amplitude areas discussed in Sec. II.D.1. These ideas have been imple-mented to control transport of colloids and charge car-riers in experimental setups where point defect gradientscould be engineered at will cf. Secs. VI.A, VII.B, andVIII.C.

B. 2D lattices of asymmetric traps

Examples of substrates sustaining transverse rectifica-tion are asymmetric potential barriers, height q0, orwells, depth q0, either isolated or arranged into 1Dchains and 2D lattices. Similar lateral displacement de-vices, also known as bumper arrays, have been proposedto separate particles by exploiting their mass and sizedispersion Savel’ev et al., 2005; Heller and Bruus, 2008.Consider, for instance, pyramidal potential barriers orwells with isosceles triangular cross section. This geom-etry, sketched in Fig. 15, is a generalization of the experi-mental setup by Villegas et al. 2003.

The substrates discussed here combine two types ofasymmetry: the triangular shape of their building blocksand the asymmetry associated with the pyramidal struc-ture of each block. The latter asymmetry affects the mo-tion of the particles only if the drive is strong enough topush them across the barriers or wells. Savel’ev et al.2005 numerically simulated the dynamics of a gas ofrepelling massive particles driven across 1D or 2D lat-tices of barriers or wells for different parameters of thesubstrate lattice Fig. 15, bottom panel, the drive Ft,and the particle pair potential Wr. Although Savel’evet al. 2005 showed that inertial effects often enhancetransverse rectification, we restrict our presentation to

the case of massless particles, certainly the most relevantfor technological applications. We consider next two dis-tinct operating regimes.

(i) Deterministic setups. An ac force applied along they axis, i.e., parallel to the symmetry axis of the pyramidcross section, is known to induce a longitudinal particledrift in the drive direction for more detail see Zhu et al.2004 and Sec. VII. Indeed, driving a distorted latticeof repelling particles along the crystallographic axis ofthe substrate, oriented parallel to the symmetry axis ofthe substrate blocks, makes the system reducible to amean-field 1D dynamics in that direction, along the lineof Sec. V.A. Due to the pyramidal shape of its buildingblock, the reduced 1D substrate is spatially asymmetric,which explains the reported longitudinal particle flows.

(ii) Diffusive setups. Under appropriate conditions dcor ac forces applied perpendicularly to the symmetryaxis may induce a transverse particle drift in the y direc-tion. For all geometries considered, the net velocity of agas of noninteracting overdamped particles driven per-pendicularly to the symmetry axis vanishes for T→0.Indeed, sooner or later each particle gets captured in ahorizontal lane between two triangle rows and thenkeeps oscillating back and forth in it forever. At finitetemperature fluctuations tend to push the particle out ofits lane, thus inducing the net transverse currents re-ported in the earlier literature Duke and Austin, 1998;Bier et al., 2000; Huang et al., 2004. More remarkably,Savel’ev et al. 2005 reported that the interaction amongparticles not only contributes to transverse rectificationbut actually plays a dominant role if the interactionlength or the particle density is large enough. In par-ticular, the particle-particle interaction was shown tocontrol transverse rectification for both weak short-range and strong long-range interparticle forces see Fig.15a. Therefore transverse rectifications of interactingcolloidal particles short range and magnetic vorticeslong range are expected to differ appreciably see Secs.VI.A and VII.

To illustrate the key mechanism of transverse rectifi-cation, we consider only rectification by pyramidal bar-riers, q0, subjected to the dc force Ft=A0. For a de-tailed analysis, see Savel’ev et al. 2005. Consider firstthe geometry in Figs. 15b–15e bottom panel, wherethe pyramids are stacked up in a close row y=0. Theparticle interactions have a strong impact on the equilib-rium particle distribution; a transition from an orderedlatticelike to a disordered liquidlike phase is displayed inframes b–e. Nevertheless, the transverse currents vyshow a qualitatively similar A0 dependence for bothweak short-range and strong long-range interactions;only the decaying tail is longer for the latter. The regionof the linear growth of vy for the case of weak short-range interaction corresponds to the regime when a con-stant fraction of particles less than 1/2 because of thegeometry of the system is rectified into the y direction.This regime applies for increasing A0 until particles startcrossing over the triangles frames d and e. Theseresults can be easily generalized to describe transverse

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rectification in any 2D lattice of triangular-shaped barri-ers or wells. Moreover, the orientation of the transversenet current is independent of the sign of A0, so that theabove description can be readily extended to low-frequency ac drives.

Savel’ev et al. 2005 also simulated the case of a 2Darray of pyramids top panel of Fig. 15, where a gapbetween triangles along the y axis, y0, turned out tomake the net current sensitive to the particle-particleinteraction length . In the limit of low T, if is smallerthan a certain threshold value cy, then the net cur-rent vanishes; if exceeds c, then the current rapidlyincreases. This effect has been advocated to separateparticles according to their interaction length. More pre-cisely, particles having an interaction length smaller thanc would pass through the array or sieve of barriersseparated by y. In contrast, particles with a longer in-teraction length c would be sifted sideways. On con-necting several such sieves with different gaps y, onecan construct a device capable of separating the differ-ent fractions of a particle mixture.

The overall conclusions of Savel’ev et al. 2005 do notchange when barriers are replaced with wells of thesame shape, nor for lattices of asymmetric pins of differ-ent aspect ratios and geometries Duke and Austin,1998; Ertas, 1998; Bier et al., 2000; Zau, Marchesoni,Moshchalkov, et al., 2003; Zau et al., 2004; Huang et al.,2004; Chepelianskii and Shepelyansky, 2005. This effecthad been anticipated to some extent by Lorke et al.1998, who investigated the magnetotransport proper-ties of a square lattice of triangular antidots of the typeshown in Fig. 15. Antidot lattices are 2D electron gaseswith appropriately placed submicron voids. They fabri-cated their lattices by electron beam lithography onshallow high-electron-mobility transistor structures,grown on semi-insulating gallium-arsenide substrates. Inparticular, they reported evidence that, under far-infrared irradiation, electron sloshing between the anti-dot rows lead to a net electric current along the symme-try axis of the antidots. Significant transverse effectswere observed in the presence of an orthogonal mag-netic field. Moreover, in most applications to the electro-phoresis of macromolecules, the particles movingthrough these sieves are suspended in the fluid wherethey diffuse, thus involving additional microfluidic ef-fects Sec. VI.

C. Binary mixtures

We address now the problem of induction and controlof the net transport of passive particles target or A par-ticles, namely, particles that are little sensitive to theapplied drives and/or substrates. Savel’ev et al. 2003,2004 proposed to employ auxiliary B particles that iinteract with the target species and ii are easy to driveby means of external forces. By driving the auxiliary par-ticles one can regulate the motion of otherwise passiveparticles through experimentally accessible means.

2l

EC

DB

A

l0

2l

ax

ay

y

x

Drivingforce

y

x(a)

0 2 4

0

g = 0.1, = 0.7

g = 1.0, = 1.2

<v

y>

A0

(a)

(b)

(c)

(d)

(e)

0.1

Short-range weakinteraction

Long-range stronginteraction

(b) (c)

(d) (e)

0.010

(b)

FIG. 15. Color online 2D sieves for magnetic vortices. Toppanel: asymmetric 2D arrays of potential energy barriers andwells, top view. The parameters l; l0; ax; ay; x; y define thegeometry of the array. The dashed and dash-dotted arrowsrepresent trajectories perturbed by thermal noise or particle-particle repulsion. Bottom panel: a Transverse net velocityvy vs A0 for massless interacting particles dc driven through agapless triangular chain: ay=2, ax=6, l=1, x=4, y=0, andT=0. The repulsive potential Wr is a wedge function with g=1 and half-width =0.05. Solid circles are for weak short-range interacting particles, while open circles are for stronglong-range interacting particles. Particle distributions forstrong long-range b, c and weak short-range d, e inter-actions for the A0 values indicated by arrows in a. The resultsin a–e coincide with those obtainable for low-frequency acdrives, as shown in Savel’ev et al. 2005.

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Savel’ev and co-workers considered a mixture of twospecies of pointlike overdamped Brownian particles Aand B at temperature T, diffusing on the 1D periodicsubstrates described by potentials VAx and VBx, re-spectively. Particles of type A interact pairwise with oneanother as well as with the B particles via the potentialsWAA and WAB =WBA, while WBB describes the inter-action of the B pairs. The pair interaction is quantifiedby the tunable strengths gij, with i , j=A ,B, whereas theinteraction constants ij play no significant role as longas they are conveniently small.

To illustrate the rectification mechanisms in a binarymixture, we review the case of a pulsated device withoscillating temperature, i.e., a temperature ratchet stud-ied originally by Reimann et al. 1996; see also Sec.II.D.2. The model can be made simpler with two as-sumptions: i One subsystem say, the auxiliary B par-ticles is subject to an asymmetric ratchet potentialVBx, while the other one the target A particles is not,VAx=0. This may happen, for instance, in a mixture ofneutral and charged particles with the ratchet potentialproduced by an electrical field. ii The interactionamong particles of the same type is repulsive, i.e., gAA0, gBB0.

Under these operating conditions, the B particles con-dense naturally at the minima of the asymmetric sub-strate potential see Figs. 16a and 16b, bottom pan-els. In addition, if the B particles repel the A particles,gAB0, then the latter will accumulate in regions wherethe density of the B particles is minimum, that is, nearthe maxima of the substrate potential VBx. Vice versa,for attractive AB interactions, gAB0, the B particlesconcentrate around the minima of VBx. Therefore thetarget A particles feel an effective potential VA

effxwhich has opposite spatial asymmetry with respect to

VBeffx for gAB0, and has the same asymmetry for

gAB0 see Figs. 16a and 16b, top panels. Note that,for low occupation numbers nA and nB, the bare poten-tial VBx is only marginally affected by particle interac-tion, i.e., VB

effxVBx. During the lower-T half cycle,all particles are confined more tightly around theminima of the relevant effective potential, while, duringthe higher-T half cycle, the particles of both species dif-fuse more easily out of the VA

effx and VBeffx potential

wells. As the asymmetry of the ratchet potentials VAeffx

and VBeffx for repelling A and B particles is opposite, so

is the orientation of their currents. On the contrary, twoattracting A and B species drift in the same direction.This implies that the transport of the two particle speciescan be effectively and separately controlled by regulat-ing their numbers nA and nB per unit cell, without theneed of tuning their substrate.

As gAA ,gBB0, the potential wells of VAeffx VB

effxtend to flatten out when the corresponding density nBnA increases. This results in the decay of the associatedratchet current vB vB. In contrast, the ratchet asym-metry and current of one species may be enhanced byincreasing the density of the other mixture component.For instance, in the case of AB attractive forces, the Aparticles tend to concentrate in the regions of higher Bdensities, that is, around the minima of VB, thus attract-ing even more B particles and making the VB

effx poten-tial wells deeper. In short, one can enhance the transportof the “target” particles by adding a certain amount ofauxiliary particles. All these properties have been repro-duced analytically in the framework of the nonlinearFokker-Planck formalism Savel’ev et al., 2003, 2004b.

The influence of the interspecies interaction on thetransport of a binary mixture has been investigated inSavel’ev et al. 2004a, where dc and ac external forceswere applied at constant temperature, to the particleseither of one species only or of both simultaneously.Their main results can be summarized as follows: i withincreasing intensity of an applied dc driving force, thereis a dynamical phase transition from a “clustered” mo-tion of A and B particles to a regime of weakly coupledmotion Marchesoni et al., 2006b; ii by applying atime-asymmetric zero-mean drive to the B species only,one can obtain a net current for both the A and B spe-cies, even in the absence of a substrate; iii when twosymmetric ac signals act independently on the A and Bparticles, and only the particles of one species feel anasymmetric substrate, then the two species can be deliv-ered in the same or opposite direction by tuning therelative signal phase for both attractive and repulsiveAB interactions. Superconducting devices based onthese two-species transport mechanisms have been real-ized experimentally in recent years Sec. VII.C.

VI. MICROFLUIDICS

Microfluidics Squires and Quake, 2005 is playing anever-growing role in controlling transport of particles, or

-0.2

-0.1

0.0

0.1nA = 0.005, nB = 0.003nA = 0.09, nB = 0.06

u

Beff

nA = 0.005, nB = 0.003nA = 0.08, nB = 0.05

b) Attracting A-B speciesa) Repelling A-B species

u

Aeff

u

Beff

u

Aeff

A

B

0.0 0.5 1.0 1.5 2.00.0

0.4

0.8

1.2

x/l

nA = 0.5, nB = 0.4nA = 5.2, nB = 5

nA = 0.32, nB = 0.2nA = 1.1, nB = 0.68

A

0.0

0.2

0.4

0.6

B

0.0 0.5 1.0 1.5 2.00.0

0.4

0.8

1.2

x/l

FIG. 16. Color online Spatial dependence of the effectivepotentials VA

effx and VBeffx at different densities of the A and

B particles. In both panels, particles of the same type repel oneanother; the interaction between particles of different speciesis repulsive in a and attractive in b. There is no substrate forthe A particles, VAx=0, whereas the ratchet potential VBxis piecewise linear: VB0x l+=x / l+, VBl+x l= 1−x / l−, with l−=0.2 and l+=0.8. The other coupling param-eters are gAA=gBB= gAB =1 and T=1. From Savel’ev et al.,2005.

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even whole extended phases, on the microscale andnanoscale. The ability to manipulate the dynamics of liq-uids is crucial in various applications such as for lab-on-a-chip technology. The Brownian motor concept has re-cently been invoked in different contexts to face thischallenge. In this regard we remark that the inertialforces of small suspended particles are typically quitesmall Purcell, 1977. Pointlike particles can then be con-sidered to be advectively transported by the fluid veloc-ity field at the particle’s actual position. Then, for anincompressible liquid, the particle dynamics is volumeconserving, and consequently no dynamical attractorsemerge Kostur et al., 2006. However, for extended par-ticles the local velocity of a surface point need not coin-cide with the fluid velocity which would act at this pointin the absence of the particle. Notably, for extended ob-jects with internal degrees of freedom the volume of thestate space is no longer conserved by the dynamics, thusgiving rise to attractors for stationary flow fields Kosturet al., 2006. Of course, Brownian diffusion provides anadditional transport mechanism that must also be takeninto account.

We remark here that, in spite of the intrinsic hydro-dynamical effects, microfluidic devices do not fall intothe category of collective ratchets as defined in Sec. V,because here transported objects are not required to in-teract with one another. The suspension fluid still plays acentral role: a it powers particle transport, and b itdetermines how the particle dynamics is coupled to theasymmetric geometry of the substrate. Genuine collec-tive ratchets are discussed in Sec. VII.

A. Transporting colloids

Many physical examples and technological applica-tions involve particles or molecules in solution that un-dergo a directed net motion in response to the action ofa ratchet. There, the ratchet does not induce a meanflow of the solvent itself. For instance, colloidal particlesor macromolecules, suspended in solution, move whenexposed to a sawtooth electric potential that is succes-sively turned on and off Rousselet et al., 1994. Electro-lytic effects can be avoided by shuttling microsizedBrownian polystyrene particles by optically trappingthem with repeatedly applied on-off cycles in an opticaltweezer, thus mimicking a flashing Brownian motorFaucheux and Libchaber, 1995; Faucheux et al., 1995;Marquet et al., 2002. As a function of the cycle fre-quency one can even detect flux reversal of diffusingcolloidal spheres in an optical three-state thermalBrownian motor Lee et al., 2004. This modus operandiof a Brownian motor can therefore be put to work topump or separate charged species such as fragments ofDNA. A micromachined silicon-chip device that trans-ports rhodamine-labeled fragments of DNA in water hasbeen demonstrated with a flashing on-off Brownian mo-tor scheme by Bader et al. 1999. Yet other devices arebased on ideas and experimental realizations of entropicratchets Slater et al., 1997; Duke and Austin, 1998; Er-tas, 1998; Chou et al., 1999; van Oudenaarden and

Boxer, 1999; Tessier and Slater, 2002 which make use ofasymmetry within geometric sieve devices to transportand separate polyelectrolytes.

A similar concept is employed to selectively filtermesoscopic particles through a microfabricatedmacroporous silicon membrane Fig. 17, containing aparallel array of etched asymmetrical bottlenecklikepores Kettner et al., 2000; Müller et al., 2000; Matthiasand Müller, 2003. The working principle and predictedparticle flow for this microfluidic Browinan motor device

(a)

(b)

FIG. 17. Artificial porous sieves. a Scanning-electron-microscope picture of a single pore with ratchet longitudinalprofile. The length of one period is 8.4 m. The maximumpore diameter is 4.8 m and the minimum pore diameter is2.5 m. b Scanning-electron-microscope picture of a siliconwafer pierced by practically identical pores about 1.5 mm apartand 1 mm in diameter. This illustrates the enormous potentialfor particle separation of a parallel 3D ratchet architecture.Figure provided by Frank Müller.

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are shown in Fig. 18. A fluid such as water containingimmersed, suspended polystyrene particles is pumpedback and forth with no net bias through the 3D array ofasymmetric pores of Fig. 17. Such an artificial Brownianmotor is thus kept far away from equilibrium by theperiodically varying pressure across the membrane. Dueto the asymmetry of the pores, the fluid develops asym-metric flow patterns Kettner et al., 2000, thus providinga ratchetlike 3D force profile in which a Brownian par-ticle of finite size can i undergo Brownian diffusioninto liquid layers of differing speed, and/or ii becomereflected asymmetrically from the pore walls. Bothmechanisms will then result in a driven nonequilibriumnet flow of particles. Note that the direction of the netflow cannot be easily guessed a priori. Indeed, the direc-tion of the Brownian motor current is determined by theinterplay of the Navier-Stokes flow in this engineered

geometry and hydrodynamic thermal fluctuations.A detailed quantitative interpretation of the experi-

mental data is plagued by several complications such asthe influence of hydrodynamic interactions and, possibly,electric response effects due to residual charge accumu-lation near the boundaries. A most striking feature ofthis setup, however, is the distinct dependence of currentreversals on particle size. The sharply peaked current-size characteristics curves of this directed flow, that is,the theoretical flow versus size prediction in Fig. 18 andthe experimental current-pressure characteristics in Fig.19, suggest a highly selective particle separation effi-ciency Kettner et al., 2000. This microfluidic artificialBrownian device has been implemented in experimentswith suspended polystyrene spheres of well-defined di-ameters 100, 320, 530, and 1000 nm by Matthias andMüller 2003: Their experimental findings are in goodqualitative agrement with the theory as shown in Fig. 19.

Remarkably, this device has advantageous 3D scalingproperties: A massively parallel architecture composedof about 1.7106 pores illustrated in Fig. 17 is capableof directing and separating suspended microparticlesvery efficiently. For this reason, this type of device hasclear potential for biomedical separation applicationsand therapy use.

The separation and sorting of cellular or colloidal par-ticles is definitely a topic attracting wide interest in dif-ferent areas of biology, physical chemistry and soft mat-ter physics. The powerful toolbox of opticalmanipulation Grier, 2003; Dholakia et al., 2007 uses theoptical forces exerted on colloids by focused laser beamsto move and control objects ranging in size from tens ofnanometers to microns. If the optical forces are suffi-ciently strong to rule transport, the stochastic Brownianforces play only a minor role, so that transport occurs asa deterministic process with an efficiency close to unity,a circumstance known as optical peristalsis Koss andGrier, 2003; Bleil et al., 2007. Such a scheme can then beimplemented to experimentally realize ratchet cellularautomata capable of performing logical operations withinteracting objects. All together, in combination withthis new optical technology, colloids provide ideal modelsystems to investigate statistical problems near and faraway from thermal equilibrium Babic and Bechinger,2005; Babic et al. 2005.

B. Transporting condensed phases

Most present applications use ratchet concepts totransport or filter discrete objects, like colloidal particlesor macromolecules. However, ratchets may also serve toinduce macroscopic transport of a continuous phase us-ing local gradients only. One such realization, stronglyrelated to the above cases of particle transport in a“resting” liquid phase, is the Brownian motion of mag-netic particles in ferrofluids subjected to an oscillatingmagnetic field Engel and Reimann, 2004. In contrast tothe cases reported above, here the motion is also trans-mitted to the solvent by viscous coupling.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Particle diameter [µm]

−1.8

−1.5

−1.2

−0.9

−0.6

−0.3

0.0

0.3

0.6

v e[µ

m/s

]

40 Hertz, νR=1.0

40 Hertz, νR=0.5

100 Hertz, νR=0.5

(b)

(a)

FIG. 18. Microfluidic drift ratchet of Kettner et al. 2000. Toppanel: A macroporous silicon wafer is connected at both endsto basins. The pores with their ratchet-shaped profile aresketched in dark gray. The basins and pores are filled withliquid; microparticles of two different species are represented.The fluid is pumped back and forth by the piston on the left.Figure provided by Christiane Kettner. Bottom panel: Theo-retically predicted net particle current ve vs the particle diam-eter for different driving frequencies and viscosities relative towater R. Note in particular the very sharp velocity reversalaround 0.5 m. The results are for a pore of infinite lengthconsisting of periodic units with a pore length of L=6 m, amaximum pore diameter of 4 m, and a minimum pore diam-eter of 1.6 m; see Fig. 17a. For further details see Kettner etal. 2000.

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Another class of system does not require colloidalparticles to drive the motion of the liquid phase. In afirst example, a secondary large scale mean flow is trig-gered in Marangoni-Bénard convection over a solid sub-strate with asymmetric grooves Stroock et al., 2003.The direction and strength of the mean flow can be con-trolled by changing the thickness of the liquid layer andthe temperature gradient across the layer.

Yet another intriguing situation involves self-propelled Leidenfrost drops placed on a hot surface alsowith a ratchetlike grooved profile Fig. 20. The locallyasymmetric geometry of the support induces a directeddrop motion. This effect has been observed for manyliquids and in wide temperature ranges within the filmboiling Leidenfrost regime Linke et al., 2006; Quéréand Ajdari, 2006.

Moreover, microdrops confined to the gap betweenasymmetrically structured plates move when drop shapeor wetting properties are changed periodically, for in-stance by vibrating the substrate, applying an on-offelectric field across the gap or a low-frequency zero-mean electric field along the gap Gorre et al., 1996; Bu-guin et al., 2002. In a related experiment, the motion ofdrops on a global wettability gradient was strongly en-hanced when a periodic force was applied Daniel et al.,2004. In that work the ratchet aspect was determined bythe intrinsically asymmetric shape of the drops and thehysteresis of their contact angle on the gradient sub-strate.

The theory for particle transport operated by Brown-ian motors is presently well developed Hänggi and Bar-

tussek, 1996; Astumian, 1997; Jülicher et al., 1997; As-tumian and Hänggi, 2002; Reimann and Hänggi, 2002;Hänggi et al., 2005. In clear contrast, models for ratchet-driven transport of a continuous phase are rather scarce,although the concept is thought to be important foremerging microfluidic applications Squires and Quake,2005.

FIG. 19. Operating a microfluidic drift ratchet. Parallel Brownian motors in a macroporous silicon membrane containing approxi-mately 1.7106 asymmetric pores with 17 etched modulations each see Fig. 17, top panel. a When the pressure oscillations ofthe water are switched on, the measured photoluminescence signal and thus the number of particles in the basin located to theright see top panel of Fig. 18 increases linearly in time, corresponding to a finite transport velocity. In contrast, for symmetriccylindrical-shaped pores no net drift is observed. b The experimentally observed net transport behavior strongly depends on theamplitude of the applied pressure and qualitatively shows the theoretically predicted current inversion Kettner et al., 2000. Thepressure oscillations are toggled on and off each 60 s. Other experimental parameters are as follows: the suspended luminescentpolystyrene spheres in water are 0.32 m across, the frequency of the pressure oscillations is 40 Hz, and the root mean squarerms of the applied pressure during the “on” phase is 2000 Pa. A control experiment using straight cylindrical pores with adiameter of 2.4 m showed no directed Brownian motor transport. Image: Max-Planck-Institute of Microstructure Physics, Halle,Germany.

FIG. 20. Self-propelled droplets. a Video sequence time in-terval 32 ms demonstrating droplet motion perpendicular to athermal gradient. A droplet of the refrigerant tetrafluorethaneR134a; boiling point Tb=26.1 °C was placed on a room-temperature, horizontally leveled, brass surface with a ratchet-like grooved profile d=0.3 mm, s=1.5 mm. b Droplet of liq-uid nitrogen on a flat surface moving with a small initialvelocity toward a piece of tape shaded. Figure provided byHeiner Linke.

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In recent work, a model for the flow of two immisciblelayered liquids confined between two walls and drivenby a flashing wettability ratchet has been put forward byJohn and Thiele 2007 and John et al. 2008. Here oneemploys the general concept of wettability comprisingall effective interactions between the liquid-liquid freeinterface and the solid walls. Notably, any interactionthat is applicable in a time-periodic, spatially periodicbut locally asymmetric fashion is capable of driving aflow, i.e., will result in directed transport. There existseveral possibilities for experimental realizations of spa-tially inhomogeneous and time-switching interactions. Apractical setup consists of thin films of dielectric liquidsin a capacitor with a periodic, locally asymmetric voltageprofile, that can be periodically switched on and off. Aspatially homogeneous electric field would destabilizethe interface between two dielectric liquids. Thereforean on-off ratchet with a constant lateral force can resultin the dewetting-spreading cycle sketched in Fig. 21. Theprocess underlying this motor scheme may be termed“electro dewetting.”

C. Granular flows

The concept of ratchet physics found an early applica-tion in the field of granular matter Jaeger et al., 1996.Indeed, to explain the vertical grain size segregation un-der vibration, the so-called Brazil-nut effect Rosato etal., 1987, Bug and Berne 1987 initially investigated di-rected diffusion in sawtoothlike inclined grooves. In themeantime, this problem spurred several computer simu-

lations aimed at exploring granular ratchets. Anotherunusual phenomenon, observed both experimentallyand in computer simulations, is horizontal size segrega-tion. It involves the occurrence of horizontal flows ofgranular layers that are vibrated vertically on plateswhose surface profile consists of sawtoothlike groovesDerényi and Vicsek, 1995; Rapaport, 2002.

The spontaneous anelastic clustering of a vibrofluid-ized granular gas has been employed, for instance, togenerate phenomena such as granular fountains con-vection rolls and ratchet transport in compartmental-ized containers, where symmetry-breaking flow patternsemerge perpendicular to the direction of the energy in-put Eggers, 1999; Brey et al., 2001; Farkas et al., 2002;van der Meer et al., 2004. Granular systems exhibit arich scenario of intriguing collective transport phenom-ena, which have been tested and validated both compu-tationally and experimentally van der Meer et al., 2007.

All these systems are kept away from thermal equilib-rium by vertical vibrations, while the presence of ran-dom perturbations plays a role as a source of complexity,due to the intrinsically chaotic nature of granular dy-namics. Closer in spirit to the present review are, how-ever, those devised granular Brownian motors which aretruly capable of putting thermal Brownian motion atwork Cleuren and Van den Broeck, 2007; Costantini etal., 2007, 2008. In those setups, an asymmetric object ofmass m Fig. 22 is free to slide, without rotation, in agiven direction. Its motion with velocity v is induced bydissipative collisions with a dilute gas of surroundingparticles of average temperature T. On each collision,only a certain fraction called the restitution coefficientof the total kinetic energy of the object-particle system isconserved. At variance with the case of elastic collisions,here isotropy, implying v=0, and energy equipartition,implying v2=kT /m, do not apply: Dissipation is re-sponsible for breaking the time-reversal symmetry. As aresult one finds that the asymmetric object undergoes adirected random walk with nonzero average velocity,which can be modeled in terms of an effective biased

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FIG. 21. Color online Fluidic ratchet. a–c Illustration ofthe working principle of a fluidic ratchet based on dewetting-spreading cycles. d The spatial asymmetric periodic interac-tion profile !x responsible for the wettability pattern and ethe time dependence "t of the switching in relation to thedewetting and spreading phases a–c. The filled circle indi-cates a fluid element that gets transported during one ratchetcycle although the evolution of the interface profile is exactlytime periodic. For further details and resulting directed flowpattern, see John et al. 2008.

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FIG. 22. Color online Taming thermal motion. a A sche-matic representation of the Smoluchowski-Feynman rachet de-vice, which is able to rectify thermal Brownian motion of gasesheld at two different temperatures Feynman et al., 1963. Anidealized version suitable for computer studies is presented inb with the ratchet replaced by a triangular unit and the pawlreplaced by a rigidly joined blade residing in the lower gascompartment. c A 3D rotational construction. Figure pro-vided by C. Van den Broeck.

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Langevin dynamics for the velocity variable v. Such aLangevin dynamics assumes the form of an Ornstein-Uhlenbeck process Hänggi and Thomas, 1982; Risken,1984 governed by an effective Stokesian friction and aneffective external force, both depending on the restitu-tion coefficient Cleuren and Van den Broeck, 2007;Costantini et al., 2007. The directed motion of the ob-ject is accompanied by a continuous flow of heat fromthe gas to the object, in order to balance the dissipationof the collisional processes; as a consequence, the aver-age temperature of the asymmetric object is lower thanthe gas temperature.

Ever since the formulation of the second law of ther-modynamics there have been debates about its validityat the small scales. In this context the ratchet gadget bySmoluchowski 1912 and later popularized by Feynmanet al. 1963 points at the heart of a controversy, whichconcerns the very working principle of Brownian mo-tors: To be consistent with this law, no rectification ofthermal fluctuations can take place at thermal equilib-rium, at which all dynamics is intrinsically governed bythe principle of detailed balance. A steadily workingBrownian motor necessarily requires a combination ofasymmetry and nonequilibrium, such as a temperaturegradient. This point has been convincingly elucidated inrecent molecular dynamics studies by Van den Broeckand collaborators Van den Broeck et al., 2004, 2005;Van den Broeck and Kawai, 2006, who stylized a Max-well demon operating far away from equilibrium Fig.22. Several computer versions of Smoluchowski-Feynman ratchets operating in a dilute gas of collidinggranular particles have thus been confirmed to generatedirected transport in the presence of a finite tempera-ture difference. With the possibility of today’s observa-tion and manipulation of physical, chemical, and biologi-cal objects at the nanoscale and mesoscale, such devicesno longer represent a theorist’s thought experiment, butare rather becoming a physical reality. It is right on theselength scales that thermal fluctuations cannot be ig-nored, as they combine with nonequilibrium forces toyield constructive transport.

VII. SUPERCONDUCTING DEVICES

Magnetic vortices in type-II superconducting devicesprovide a flexible and experimentally accessible play-ground for testing our stochastic transport models. Inrecent years an impressive effort has been directed to-ward the design and operation of a new generation ofvortex devices with potential applications to terahertzTHz technology Hechtfischer et al., 1997; Zitzmann etal., 2002, fluxon circuitry Shalóm and Pastoriza, 2005,and quantum computation You and Nori, 2005, tomention but a few. We anticipate that magnetic vorticesare inherently quantum objects that, under most experi-mental conditions, behave like massless, pointlike clas-sical particles. Genuine quantum effects in the vortextransport are discussed in Sec. VIII.

A magnetic field applied to a type-II superconductingfilm, with intensity H comprised between the critical val-

ues Hc1 and Hc2, penetrates into the sample producingsupercurrent vortices, termed Abrikosov vortices Abri-kosov, 1957; Blatter et al., 1994. The supercurrent circu-lates around the normal i.e., nonsuperconducting coreof the vortex; the core has a size of the order of thesuperconducting coherence length parameter of aGinzburg-Landau theory; the supercurrents decay on adistance London penetration depth from the core. Inthe following we assume that . The circulating su-percurrents produce magnetic vortices, each character-ized by a total flux equal to a single flux quantum !0=hc /2e hence the name fluxon for magnetic vortex anda radially decaying magnetic field

Br = B0K0r/ , 39

where K0x is a modified Bessel function of zeroth or-der and B0=!0 / 22. Far from the vortex core, r,the field decays exponentially Br /B0= /2re−r/,whereas, approaching the vortex center, it diverges loga-rithmically for r, Br /B0=ln /r, and then satu-rates at B0=B0 ln / inside the core, r. As a con-sequence, the coupling between vortex supercurrentsand magnetic fields determines a long-range repulsivevortex-vortex interaction.

Due to mutual repulsion, fluxons form a lattice alsocalled the Abrikosov vortex lattice, usually triangular,possibly with defects and dislocations with average vor-tex density approximately equal to H /!0. A local den-sity I of either a transport current or a magnetizationcurrent or both exerts on each fluxon a Lorentz forceFL=!0IH /cH.

The distribution and microscopic properties of pin-ning centers can qualitatively influence the thermody-namic and vortex transport properties of the supercon-ducting sample. Pinning forces created by isolateddefects in the material oppose the motion of the fluxons,thus determining a critical or threshold current, belowwhich the vortex motion is suppressed. Many kinds ofartificial pinning centers have been proposed and devel-oped to increase the critical current, ranging from thedispersal of small nonsuperconducting second phases tocreation of defects by irradiation Altshuler and Jo-hansen, 2004. A novel approach to the problem camewith advances in lithography, which allowed for regularstructuring and modulation of the sample propertiesover a large surface area Harada et al., 1996. Long-range correlation in the position of the pinning centersresulted in the interplay between the length scales char-acterizing the pin lattice and the vortex lattice. Indeed,the magnitude of the commensuration effects is readilycontrolled by tuning the intensity of the magnetic fieldH: at the first matching field H1=!0 /al, pin and vortexlattices are exactly commensurate, with one fluxon perpin lattice cell of area al; at the nth matching fieldHn=nH1, each pin lattice cell is occupied by n=alfluxons; for irrational H /H1, pin and vortex lattices areincommensurate. Commensuration effects are respon-sible for a variety of dynamical superconducting states

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of relevance to the problem of vortex rectificationReichhardt et al., 1998.

As a first attempt to design a vortex ratchet, Lee et al.1999 showed that an ac electric current applied to asuperconductor patterned with the asymmetric pinningpotential shown in Fig. 23 can induce vortex transportwhose direction is determined only by the asymmetry ofthe pattern. The fluxons were treated as zero-masspointlike particles moving on a sawtooth potential alongthe x axis, according to the scheme of Sec. II.D.1. Theydemonstrated theoretically that, for an appropriatechoice of the pinning potential, such a rocked ratchetcan be used to manipulate single vortices in supercon-ducting samples under realistic conditions. The rectifica-tion mechanisms of an extended overdamped string, likea fluxon, on a ratchet potential had been anticipated byMarchesoni and co-workers Marchesoni, 1996; Costan-tini et al., 2002. Brownian motor ratcheting of singleoscillating fluxons have been observed experimentally inasymmetrically etched Nb strips Plourde et al., 2000, onplanar patterns of columnar defect Kwok et al. 2002,and in asymmetric linear arrays of underdamped Jo-sephson junctions Lee, 2003, and in an annular Joseph-son junction Ustinov et al., 2004, to mention just a fewrecent experiments.

A. Fluxon channels

A more viable approach to achieve controllable sto-chastic transport of fluxons in superconducting devicesconsists of exploiting collective boundary effects as sug-gested in Sec. V.A.1. Fluxon rectification in asymmetricchannels has been investigated by Wambaugh et al.1999 by means of a molecular dynamics MD code,originally developed to study magnetic systems with ran-dom and correlated pinning Reichhardt et al., 1998.They simulated a thin slice of a zero-field-cooled, type-IIsuperconducting slab, taken orthogonal to the magneticvortex lines generated by a tunable external magneticfield of intensity H. When regarded as fairly rigid fieldstructures, magnetic vortices formed a confined 2D gasof zero-mass repelling particles with density =H /!0.The simulated sample had very strong, effectively infi-nite pinning except for the “zero pinning” central

sawtooth-shaped channel, sketched in the inset of Fig.24. In the channel fluxons moved subject to fluxon-fluxon repulsions, an externally applied ac driving Lor-entz force, forces due to thermal fluctuations, and strongdamping. The fluxons outside the channel could notmove, thus impeding the movable fluxons in the middlefrom crossing the channel walls. As a consequence, theinteraction length of the fluxon-wall collisions was thesame as of the fluxon-fluxon collisions. The net fluxonvelocity v vs temperature shown in Fig. 24 is negativeand exhibits an apparent resonant behavior, vs both Tand H i.e., the fluxon density , as anticipated in Sec.V.A.1.

These conclusions have been corroborated experi-mentally by Togawa et al. 2005 and Yu et al. 2007.The more recent work fabricated triangular channelsfrom bilayer films of amorphous niobium germanium, anextremely weak-pinning superconductor, and niobiumnitride NbN, with relatively strong pinning. A reactiveion etching process removed NbN from regions as nar-row as 100 nm, defined with electron-beam lithography,to produce weak-pinning channels for vortices to movethrough easily. In contrast, vortices trapped in the NbNbanks outside of the channels remain strongly pinned.The vortex motion through such asymmetric channelsexhibited interesting asymmetries in both the static de-pinning and the dynamic flux flow. The vortex ratcheteffect thus revealed a even richer dependence on mag-netic field and driving force amplitude than anticipatedby the simplified model simulated by Wambaugh et al.1999.

Devices like that simulated in Fig. 24 serve as excel-lent fluxon rectifiers. By coupling two or more such de-vices, one can design fluxon lenses, to regulate thefluxon concentration in chosen regions of a sample, and

FIG. 23. Color online Diagram of a superconductor in thepresence of an external magnetic field H. A direct current withdensity I flowing along the y direction indicated by the largearrow induces a Lorentz force FL that moves the vortex in thex direction upper panel. The superconductor is patternedwith a pinning potential lower panel. The potential is peri-odic and asymmetric along the x direction, and is invariantunder translation along y; the potential Vx is an effectiveratchet potential see Sec. II.D.

FIG. 24. Color online Rectified average fluxon velocity vin units of / vs T for the sawtooth channel geometry in theinset width, 7, bottleneck, . The magnetic field H is di-rected out of the figure. A vertically applied alternating cur-rent square wave drives the fluxons back and forth horizontallyalong the channel with period =100 MD steps. The numberof fluxons per unit cell is 1 circles, 25 diamonds, 50squares, 75 triangles, 100 , 150 , 250 *. FromWambaugh et al., 1999.

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eventually channel networks or circuits for fluxons insuperconducting films.

Fluxon channeling was observed first experimentallyin an effective 1D vortex rectifier by Villegas et al.2003. Their device has the pedagogical merit of show-ing all collective transport effects summarized in Sec. Vat work. A 100-m-thick niobiun film was grown on anarray of nanoscale triangular pinning potentials orientedas in Fig. 25a, with bases lined up along the x axis and

vertices pointing in the y direction. MagnetoresistanceRH experiments were done with a magnetic field Happlied perpendicular to the substrate in a liquid heliumsystem. The dc magnetoresistance curves in the bottompanel of Fig. 25 exhibit commensurability effects inwhich dissipation minima develop as a consequence ofthe geometrical matching between the vortex lattice andthe underlying periodic structure. At these matchingfields, the vortex lattice motion slows down, and RHminima appear at the equally spaced values Hn=nH1,with H1=32 Oe. The RH minima are sharp at the n=1,2 ,3 matching fields, but shallow and not well definedfor higher n. This effect is a consequence of the appear-ance of interstitial vortices at increasing H beyond H3,three being the maximum number of vortices containedin each triangle.

In order to interpret the experimental results, oneseparates all vortices in two groups: i pinned vortices,which move from one triangular-shaped pinning trap toanother one and thus are directly affected by an effec-tive 1D rocked ratchet substrate with positive polarity inthe y direction; and ii interstitial vortices, which arechanneled in-between triangles and do not directly inter-act with the pinning traps Savel’ev et al., 2003; Zhu,Marchesoni, Moshchalkov, et al., 2003. However, as isapparent in the top panel of Fig. 25 and discussed in Sec.V.C, pinned vortices, being strongly coupled to the sub-strate potential, determine a weaker mean-field poten-tial that acts on the interstitials with opposite asymmetrySavel’ev et al., 2003. As a consequence, when all flux-ons are subjected to the same ac drive force, the fluxon-fluxon repulsion pushes the interstitials in the directionopposite to the pinned vortices for an animation seehttp://dml.riken.go.jp/vortex.

The effects of this mechanism are illustrated in Fig.26. At constant temperature close to Tc to avoid ran-dom pinning and H multiple of H1, an ac current den-sity It=Iac sint was injected along the x axis. Thisyields a sinusoidal Lorentz force with amplitude FL act-ing on the vortices along the y axis. Despite the zerotime average of such Lorentz force, a nonzero dc voltagedrop Vdc in the x direction was measured, thus provingthat the vortices actually drift in the y direction. Notethat, due to the peculiar substrate symmetry, an ac cur-rent It oscillating along the y axis can rectify the fluxonmotion only parallel to it transverse rectification, Sec.VII.B. The amplitude of the dc voltage signal decreaseswith increasing H because the effective pinning is sup-pressed by the intervortex repulsion. Moreover, whenn3 corresponding to more than three vortices perunit cell, a reversed Vdc signal begins to develop with amaximum occurring at a lower Lorentz force FL thanthe positive dc maxima: The interstitials, which feel aweak inverted ratchet potential with respect to thepinned vortices, dominate the rectification process. Thiscurrent reversal effect is enhanced when further increas-ing the magnetic field Figs. 26c and 26d and finally,at very high magnetic fields, close to the normal state,the voltage reversal, although suppressed in magnitude,

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FIG. 25. Color online Experimental implementation of a 2Dsieve for magnetic vortices. a Sketch of the positions of thevortices for several matching fields Hn, i.e., for n vortices perunit cell. Vortices pinned on the triangles no more than threeper unit cell and interstitial vortices are shown. The pinningsubstrate is a square lattice with constant 750 nm; the triangu-lar pinning sites have been grown on a Si support, are 40 nmthick with side about 600 nm long. b dc magnetoresistanceRH vs H. The temperature is T=0.98Tc and the injected dccurrent density is Idc=12.5 kA cm−2. The upper inset shows amicrograph of the measuring bridge, which is 40 m wide. Thedarker central area of the inset is the 9090-m2 array ofmagnetic triangles. The lower inset shows the characteristicscurves Vdc−Idc at the matching magnetic fields Hn=nH1 withH1=32 Oe and n=1, 3, 4, 6, 8, and 10 top to bottom. TheVIdc curves change abruptly at magnetic fields from n=3 to 4,because an Ohmic regime appears at low currents. This is aclear signature of the presence of interstitial vortices. FromVillegas et al., 2003.

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spans over the entire FL range Fig. 26f.Moshchalkov and co-workers Silva et al., 2006; de

Vondel et al., 2007 also demonstrated longitudinalfluxon rectification. Silva et al. 2006 observed multiplecurrent reversals in regular square lattices of asymmetricdouble-well traps periodically driven along an easy di-rection, as in the simulations by Zhu and co-workersZhu, Marchesoni, Moshchalkov, et al., 2003; Zhu,Marchesoni, and Nori, 2003. More interestingly, deVondel et al. 2007 detected inverted fluxon currents inlarge triangular arrays, similar to those reviewed here,but at magnetic fields or fluxon densities so high that acollective ratchet mechanism would not be plausible.They attributed their finding to a new intra-antidot rec-tification mechanism controlled by the magnetic fluxquantization condition synchronously satisfied at theedge of each asymmetric antidot. Finally, Silva et al.2007 detected vortex rectification also as a dipole-induced effect in reducible symmetric arrays of micro-magnets.

B. Fluxon rectification in 2D arrays

The rectification of a fluxon lattice, no matter howdistorted or disordered, is an inherently 2D process. Fol-lowing the theoretical and numerical investigations re-

viewed in Sec. V.B, a number of experimental groupsCrisan et al. 2005; Menghini et al. 2007; Van Look etal. 2002 to mention a few prototyped devices aimed atcontrolling vortex ratcheting in both directions parallellongitudinal rectifiers and perpendicular transverserectifiers to an applied ac Lorentz force.

Longitudinal transport in a 2D array of asymmetricpinning traps has been experimentally obtained, for in-stance, by de Vondel et al. 2005, who reported that,under appropriate operating conditions, fluxon rectifica-tion can be enhanced for FL amplitudes comprised be-tween two critical pinning forces of the underlying asym-metric substrate. The shape of the net dc voltage dropVdc as a function of the drive amplitude indicated thattheir vortex ratchet behaved differently from standardoverdamped models. Rather, as the repinning force nec-essary to stop vortex motion is considerably smaller thanthe depinning force, their device resembles the behaviorof the inertial ratchets of Sec. II.D.1. More recently,Shalóm and Pastoriza 2005 reported longitudinalfluxon rectification in square arrays of Josephson junc-tions, where the coupling energies had been periodicallymodulated along one symmetry axis. Similarly, longitu-dinal fluxon rectification has also been observed insquare arrays of magnetic dots with size-graded periodGillijns et al., 2007. Both are experimental implemen-tations of the asymmetric patterns of symmetric trapsintroduced in Sec. V.A.2. Finally, Ooi et al. 2007trapped symmetric fluxon lattices in a triangular dot lat-tice that they had photolithographed on a thinBi2Sr2CaCu2O8+ Bi2212 single-crystal film, by settingthe applied magnetic field at the lowest multiples of H1.By subjecting the fluxon lattice to a biharmonic Lorentzforce oriented along the crystallographic axes of the dotlattice, they obtained demonstration of harmonic mixingon a symmetric substrate. However, the results of theseexperiments can be easily explained also in terms ofsimple 1D models.

Transverse transport, instead, requires genuinely irre-ducible 2D geometries. When designing and operating adevice capable of transverse rectification, experimenterscan vary the orientation of the pinning lattice axes, thesymmetry axes of the individual traps if any, and thedirection of the injected current i.e., of the Lorentzforce. Many have explored by numerical simulation thegeometries best suited for the experimental implemen-tation of this concept Zhu et al., 2001; Reichhardt andReichhardt, 2005. The most recent realizations of trans-verse fluxon rectifiers fall into two main categories.

1. Symmetric arrays of asymmetric traps

Gonzalez et al. 2007 modified the experimental setupof Villegas et al. 2003 to investigate the rectificationmechanisms presented in Secs. V.A and V.B. They con-firmed the numerical simulations by Savel’ev et al.2005, who had predicted that the same device can ex-hibit either longitudinal or transverse output current de-pending on orientation of the ac drive with respect tothe pinning lattice axes. In the longitudinal ratchet con-

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figuration a sinusoidal driving current It was appliedperpendicular to the triangle reflection symmetry axis xaxis and the output voltage signal Vdc was recorded inthe same direction. We recall that the Lorentz-force-induced vortex motion, parallel to the triangle reflectionsymmetry axis y axis, corresponds to a voltage drop inthe direction of the injected current. To observe an op-timal transverse rectification effect, the axes of the cur-rent injection and the voltage drop were inverted. Theasymmetry of the traps with respect to the direction ofthe Lorentz force is responsible for the observed fluxondrift in the y direction.

The number n of vortices per lattice cell was con-trolled by varying the intensity of the magnetic field or-thogonal to the device, H=nH1. On increasing n, thevoltage associated with the longitudinal ratchet currentchanges sign and gets amplified, as is to be expected dueto the presence of the n−3 interstitials per unit cell Sec.VII.A. The H dependence of the transverse fluxon rec-tification is very different: i the transverse net currentshows no inversions; ii increase in the number of inter-stitials suppresses transverse rectification.

2. Asymmetric arrays of symmetric traps

Controlled transport of vortices through rows of anti-dots was measured by Wördenweber et al. 2004 viastandard four-probe Hall-type experiments Altshulerand Johansen, 2004. 100–150-nm-thick YBa2Cu3O2films were deposited on CeO2 buffered sapphire andthen covered with a 50-nm-thick Au layer. Asymmetriclattices of symmetric antidots in the shope of circularmicroholes were patterned via optical lithography andion beam etching. The inset of Fig. 27 shows a typicalantidot lattice. A tunable dc current Idc was then in-jected so that the corresponding Lorentz force FL was at

an angle with respect to the orientation of the antidotrows.

Reference measurements on samples without antidotsas well as temperature-dependent measurements of theHall resistance RH for different angles clearly indicatethat, for T83 K and low dc drive current, the Hallresistance is dominated by the directed vortex motionalong the antidot rows. Under these circumstances, RHquantifies the fluxon current in the direction of the dccurrent Idc with negative Hall voltage, or, equivalently,transverse to the drive force FL. In view of the symmetryof the traps, for currents perpendicular to the antidotrows, no transverse fluxon current was observed; asshown in Fig. 27, changing the sign of led to inversionof the transverse current. Most notably, the detecteddrift was not restricted by a current threshold, that is,fluxons seemed not to jump from antidot to antidot, likeindividual vortices in a defective superconductor sampleSec. V.A.2, but rather to obey an Ohmic behavior. Theabsence of an activation threshold was explained byWördenweber et al. 2004 by invoking a combination ofi finite-size effects, as the area where vortices can benucleated is severely restricted by the device geometry;and ii screening effects, as trapped fluxons induce spa-tially nonuniform current distributions around neighbor-ing antidots. Due to current screening, the antidot rowsserve as easy-flow i.e., thresholdless paths for thedriven vortices, which thus acquire a transverse velocityproportional to sin 2 see Fig. 27.

At higher temperatures and stronger drives, this effectbecomes negligible and no transverse rectification wasdetected Wördenweber et al., 2004. Moreover, on re-placing Idc with a symmetric ac drive Iact, no net cur-rent is expected, either longitudinal or transversal.

C. Anisotropic fluxon rectifiers

Transport control in a binary mixture can be achievedin samples with no ratchet substrate. As reported in Sec.V.C, when one component of the binary mixture isdriven, the moving particles drag along the other non-driven component, interacting with it. A time-asymmetric ac drive produces rectification of both com-ponents of the binary mixture, which can be tuned bymeans of the ac drive parameters. The device acts like aratchet, but it has no ratchet substrate.

Savel’ev and Nori 2002 proposed to implement thisnew ratchet concept in a layered superconductor. In thisclass of materials, a magnetic field, tilted away from thehigh-symmetry crystalline c axis, penetrates the sampleas two perpendicular vortex arrays Ooi et al., 1999;Grigorenko et al., 2001; Matsuda et al., 2001, known as“crossing” vortex lattices Koshelev, 1999. One vortexsublattice consists of stacks of pancake vortices PVsaligned along the c axis, whereas the other is formed byJosephson vortices JVs confined between CuO2 layersFig. 28. A weak attractive PV-JV interaction has beenexperimentally observed Grigorenko et al., 2001. TheJVs are usually very weakly pinned and can be easily

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10

15dV

Hall,

I=0/d

I

angle [deg]

FL

I0

||

Channel 1

FL

F Fguid L= cos

I0

F Fguid L= cos sin ||[m]

FIG. 27. Color online Angular dependence of RH measuredon a circular shaped 90-nm-thick sample at T=30 K, H=143 mT, and with dc current Idc=10 mA. Inset: Portion of a20 m10 m rectangular lattice of circular antidots with ra-dius of 1 m; Idc is horizontally oriented from left to rightcorresponding to a Lorentz force FL pointing upward. The an-gular dependence is fitted by the sinusoidal law sin 2, assketched in the inset. From Wördenweber et al., 2004.

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driven either by changing the in-plane magnetic field H

or by applying an electrical current Iz flowing along thec axis.

Time-asymmetrically-driven JVs can drag along thePVs, thus causing a net motion of the PVs, as illustratedin Fig. 28. The simplest possible operating mode consistsof slowly increasing the in-plane field H from 0 to H

max

for a fixed value of the out-of-plane magnetic field Hz.The increasing in-plane field slowly drives the JVs fromthe edges to the middle of the sample. In turn, the JVsdrag the PVs along with them toward the sample center.As a result, asymmetric cycling causes either pumpingfocusing or antipumping defocusing of the PVs at thecenter of the lens.

Cole et al. 2006 performed a vortex lensing experi-ment on an as-grown single Bi2212 crystal. The changesin magnetic induction, arising from PV lensing and anti-lensing, were detected using one centrally placed ele-ment of a micro-Hall probe array Altshuler and Jo-hansen, 2004. In order to realize the asymmetric acdriven mode for the vortex lens, the following stepssketched in Fig. 29 were carried out: i The sample wascooled in fixed Hz at H =0; ii a “conditioning” trian-gular wave was run for 4 min to equilibrate the PV sys-tem; iii a time-asymmetric, zero-mean ac drive wasswitched on for 4 min for an animation see http://dml.riken.go.jp. The magnetic induction Bz was thenmonitored in real time starting from step ii at the cen-trally located Hall element and then related to the localPV density. The measured efficiency of this vortex lens,displayed in Fig. 29, is quite low. However, the sameexperiment, when performed on a symmetric pinningsubstrate, yields a much higher efficiency, as proven byCole et al. 2006 for a film of Bi2212 patterned with asquare array of circular antidots.

The two main advantages of this class of vortex de-vices over the earlier proposals reviewed in Secs. VII.Aand VII.B are i the possibility to guide particles withno tailored spatial asymmetry and ii the tuning of thevortex motion by simply changing the parameters of theexternally applied drive. The first feature allows expen-sive and cumbersome nanofabrication processing to beavoided; the second property becomes important if the

transport properties of a device must be frequently var-ied, something which is generally hard to achieve instandard ratchet devices.

VIII. QUANTUM DEVICES

In the foregoing sections, we focused on directedtransport within the realm of classical dynamical lawsand classical statistical fluctuations. Totally new sce-narios open up when we try to consistently incorporatequantum mechanical laws in the operation of an artifi-cial Brownian motor. The quantum nature of fluctua-tions and quantum evolution laws, as governed by quan-tum statistical mechanics, allow for unexpectedtransport mechanisms. In particular, quantum mechanicsprovides the doorway to new features such as under-barrier -threshold tunneling, above-barrier reflection,and the interplay of coherent i.e., with oscillatory relax-ation and incoherent i.e., with overdamped relaxationquantum transport. Last but not least, quantum mechan-ics generates the possibility for nonclassical correlations,including entanglement among quantum states in thepresence of coupling.

A. Quantum dissipative Brownian transport

To set the stage and in order to elucidate the complex-ity involving directed quantum Brownian transport, weconsider a 1D quantum particle of coordinate q andmass m, moving in a typically time-dependent rachetlikepotential landscape VRq , t. The particle is bilinearlycoupled with strength ci to a set of N harmonic oscilla-

FIG. 28. Color online Vortex lensing mechanisms. Subjectedto a superposition of dc and time asymmetric ac in-plane mag-netic fields, Ht=H

dc+Hact, the JVs horizontal cylinders

are asymmetrically pushed in and out of the sample. If the accomponent of Ht increases slowly, the PV stacks pancakesremain trapped by the JVs, and both move together towardthe center of the lens. If, on the way back, H

act decreasesrapidly, the JVs leave the PVs behind them. Courtesy ofSergey Savel’ev.

0.38

0.405

0.43

0.455

0.48

0.505

0 100 200 300 400 500 600t (s)

Pumping

Anti-pumping

Conditioningf = 10 Hz H//c = 32 Oe

Nopumping

f = 3 Hz H//r = 32 Oe

T = 77KHz = 5.3 OeH//off= 16Oe

0

0.25

0.50

0.75

-0.25

-0.50

-0.75

ZeropumpingB

z/Bz0x10

-2

t (s)

FIG. 29. Color online Vortex lens operation. The curvesshow the measured percentage change of the magnetic induc-tion PV density at the sample center, when an initial condi-tioning signal is applied, followed by pumping and antipump-ing time-asymmetric drives. The conditioned PV densityincreases decreases during several cycles of the pumping an-tipumping drive and then saturates. As soon as the drive isswitched off, the PV density starts to relax from the nonequi-librium pumping antipumping state toward an equilibriumstate. Courtesy of Sergey Savel’ev.

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tors xi, with i=1, . . . ,N. The latter set of oscillators mod-els the heat bath, with the oscillators prepared in a ca-nonical state with density matrix bath, corresponding toan isolated, bare bath Hamiltonian, and a temperaturekT. Accordingly, the total dynamics is governed by theHamiltonian

H =p2

2m+ VRq,t +

i=1

N pi2

2mi+

mi

2i

2xi2 − qcixi

+ q2 ci2

2mii2 . 40

The last term, which depends only on the system coor-dinate, represents a potential renormalization termwhich is needed to ensure that the potential VRq , t co-incides with the bare ratchet potential in the presence ofcoupling to the bath degrees of freedom. This Hamil-tonian has been studied since the early 1960s for systemsthat are weakly coupled to their environment. Only inthe 1980s was it realized by Caldeira and Leggett 1983,1984 that this model is also applicable to stronglydamped systems and may be employed to describe, forexample, dissipative tunneling in solid state and chemi-cal physics Hänggi et al., 1990.

One may convince oneself that the Hamiltonian inEq. 40 indeed models dissipation. Making use of thesolution of the Heisenberg equations of motion for theexternal degrees of freedom, one derives a reduced sys-tem operator-valued equation of motion, the so-calledinertial generalized quantum Langevin equation, that is,

mqt + mt0

t

ds t − sqs +VRq,t

q

= t − mt − t0qt0 . 41

The friction kernel is given by

t = − t =1

mi=1

Nci

2

mii2 cosit , 42

and the quantum Brownian force operator reads

t = i=1

N

cixit0cosit − t0

+pit0mii

sinit − t0 . 43

The random force t is a stationary Gaussian opera-tor noise with correlation functions

tbath= 0, 44

St − s =12

ts + stbath

=

2 i=1

Nci

2

miicosit − scoth i

2kT . 45

Moreover, being an operator-valued noise, the com-mutators do not vanish, i.e.,

t,s = − ii=1

Nci

2

miisinit − s . 46

This complex structure for driven quantum Brownianmotion in a potential landscape follows the fact that aconsistent description of quantum dissipation necessarilyrequires the study of the dynamics in the full Hilbertspace of the system plus environment. This is in clearcontrast to the classical models, where the Langevin dy-namics is directly formulated in the system state spaceHänggi and Ingold, 2005. Caution applies in makingapproximations to this structure, even if done semiclas-sically. The interplay of quantum noise with the commu-tator structure is necessary to avoid inconsistencies withthe thermodynamic laws, such as a spurious finite di-rected current in an equilibrium quantum ratchet, i.e.,even in absence of rocking, i.e., for VRq , t=VRq Ma-chura, Kostur, et al., 2004; Hänggi and Ingold, 2005.Simplifications are possible only under specific circum-stances as in the case of very strong friction, when quan-tum corrections can consistently be accounted for by asemiclassical quantum Smoluchowski equation operat-ing on the state space of the classical system onlyAnkerhold et al., 2001; Machura, Kostur, et al., 2004;Machura, Luczka, et al., 2007. The situation is presentlyless settled for weakly damped quantum Brownian mo-tors Carlo et al., 2005; Denisov et al., 2009 and for sys-tems at high temperatures, where approximate reduceddescriptions are often in conflict with the laws of ther-modynamics; this is true in particular for the second law,central to this review Hänggi and Ingold, 2005; Zuecoand Garcia-Palacios, 2005.

An alternative approach to the quantum Langevin de-scription, which in addition allows powerful computa-tional methods, is based on the real-time path integraltechnique. Starting from the quantum statistical repre-sentation of the density operator evolution of the totaldynamics of a system coupled to its environments, onetraces over the bath degrees of freedom to end up with apath integral representation for the reduced density op-erator, the so-called influence functional, which consis-tently incorporates quantum dissipation see, for in-stance, Hänggi et al., 1990; Grifoni and Hänggi, 1998;Hänggi and Ingold, 2005; Kohler et al., 2005.

B. Josephson Brownian motors

In the following we consider the role of quantum ef-fects in artificial Brownian motors made of coupled Jo-sephson junctions.

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1. Classical regime

Our first case refers to the limit when quantum coher-ence and tunneling events, such as photon-assisted tun-neling, can safely be neglected, which is often the case atsufficiently high temperatures. In this regime the de-scription of Josephson quantum systems can well be ap-proximated by an effective Fokker-Planck dynamics inthe context of the Stewart-McCumber model Baroneand Paternò, 1982, which includes effective parameterscontaining . An artificial Brownian motor then can beexperimentally realized, for instance, with the asymmet-ric superconducting quantum interference deviceSQUID illustrated in Fig. 30 Zapata et al., 1996. Inthe overdamped regime, where capacitative effects canbe ignored, such a device maps precisely onto therocked ratchet of Sec. II.D.1. Under such operating con-ditions, the phase of the device is a classical variablewhich can be adequately described by the aforemen-tioned “resistively shunted junction” model. ThermalBrownian motion at temperature T is included by add-ing Nyquist noise. The behavior of the rachet voltage soinduced is displayed in Fig. 31 for a nonadiabaticallyrocked SQUID ratchet Zapata et al., 1996.

The results for this ratchet setup have been experi-mentally validated by use of high-temperature supercon-ducting dc SQUIDs by the Tübingen group Weiss et al.,2000; Sterch et al., 2002, 2005. In the meantime, severalvariants of this scheme have been studied, both theoreti-cally and experimentally, including fluxon ratchets ofvarious designs Zapata et al., 1998; Carapella and Cos-tabile, 2001; Falo et al., 2002; Lee, 2003; Berger, 2004;Beck et al., 2005; Shalóm and Pastoriza, 2005; cf. alsoSec. VII. Moreover, transport of fluxons in an extended,annular Josephson junction has also been demonstratedby Ustinov et al. 2004 as a harmonic mixing effect.

2. Quantum regime

To tackle true quantum effects, including quantumtunneling phenomena, we must resort to the theoreticalscheme outlined in Sec. VIII.A. A first theoretical studyof quantum ratchets was pioneered by Reimann et al.1997 see also Reimann and Hänggi, 1998. For a re-

gime with several quasibound states below a potentialbarrier one can evaluate the ratchet tunneling dynamicsin terms of an effective action for the extremal bouncesolution in combination with a fluctuation analysisaround this bounce solution to obtain the correspond-ing tunneling rates. In Fig. 32 the result of such a quan-tum calculation is compared with the classical result foran adiabatically rocked quantum particle of mass m andcoordinate q dwelling in a ratchet potential VRq. Notethat, within a quantum Brownian motor scheme, therole of mass m, i.e., the inertia, does enter the analysisexplicitly. The most distinctive quantum signature is astriking current reversal which emerges solely as a con-sequence of quantum tunneling under adiabatic rockingconditions. Moreover, in contrast to the classical result,the directed current no longer vanishes as T tends tozero and additional resonancelike features show up.

Such a dependence of the current versus decreasingtemperature in a quantum Brownian motor has beencorroborated experimentally for vortices moving in aquasi-1D Josephson junction array with ratchet poten-tial profile specially tailored so as to allow several bandsbelow the barrier Majer et al., 2003. The experimentalsetup of this quantum ratchet device is displayed in Fig.33. The experimental findings are in good agreementwith the theoretical analysis by Grifoni et al. 2002.

C. Quantum dot ratchets

Another ideal resource to observe the interplay of iquantum Brownian motion, ii quantum dissipation,and iii nonequilibrium driving are semiconductor engi-neered quantum rectifiers. Composed of arrays of asym-metric quantum rectifiers Linke et al., 1998, these de-

r

2

1

ext

a)

V

Jr sin( r)Rr

Lr

IrIl

Jl sin( l /2) Rl

Ll

I(t) = I0 + I1sin( t)b)

FIG. 30. SQUID rocked ratchet. a Schematic picture of anasymmetric SQUID with three junctions threaded by an exter-nal magnetic flux. b Equivalent electric circuit: the two junc-tions in series of the left branch behave like a single junctionwith # replaced by # /2. For further details, see Zapata et al.,1996.

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

2V0

/JlR

I0

/ Jl

-0.040 -0.035 -0.030

0.0

0.1

0.2

0.3

FIG. 31. Theoretically evaluated dc current-voltage Joseph-son relation V0= /2e#t characteristics for the SQUIDratchet of Fig. 30. Simulation parameters: nonadiabatic rock-ing frequency =2 /eRJl=0.3, ac drive amplitude A=I0 /Jl=1.7, and different noise levels D=ekBT /Jl=0 solid, 0.01dashed, and 0.5 dotted. Inset: Magnified plot showing char-acteristic steps at fractional values of for D=0. Here R and Jldenote the resistance and the critical current amplitude of thetwo identical Josephson junctions placed in the left branch ofthe device cf. Fig. 30.

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vices operate on geometric and dynamical length scalesranging between a few nanometers and microns. Thisclass of devices allowed the first experimental validationof the distinctive features of quantum rectifiers, namely,tunneling-enhanced rachet current and tunneling-induced current reversals, by Linke and collaboratorsLinke et al., 1999, 2002. Their central results are illus-trated in Fig. 34. Their quantum Brownian motor consisted of a 2D gas

of electrons moving at the interface of a fabricated,ratchet-tailored GaAs/AlGaAs heterostructure. Its op-erating regime was achieved by adiabatically switchingon and off a square-wave source-drain periodic voltageof frequency 190 Hz and amplitude of about 1 mV. Werecall that, for an adiabatically rocked classical ratchetof Sec. II.D.1, noise-induced transport exhibits no cur-rent reversals, but things change drastically when quan-tum tunneling enters into the dynamics. A true bench-mark for the quantum behavior of an adiabaticallyrocked Brownian motor is then the occurrence of atunneling-induced reversal at low temperaturesReimann et al., 1997. This characteristic feature wasfirst experimentally verified with an electron quantumrocked ratchet by Linke et al. 1999. Moreover, the cur-rent reversal reported in the top panel of Fig. 34 indi-cates the existence of parameter configurations wherethe current of a quantum Brownian motor vanishes. Incorrespondence to such configurations, one can imagineoperating the device as a quantum refrigerator to sepa-rate “cold” from “hot” electrons in the absence of cur-rents Linke et al., 2002. At higher temperatures this

FIG. 32. The classical Brownian motor current Icl occurring ina dichotomously rocked ratchet potential, Eq. 23, of periodL, compared with the corresponding quantum Brownian mo-tor current Iqm vs dimensionless inverse temperature T.The current is measured in units of L

0* with

0*

= 42V0 / L2m1/2 and the temperature is measured in unitsof the maximal crossover temperature T0

max occurring in therocked potential landscape. Notably, in contrast to an adiabati-cally rocked classical Brownian motor, the quantum currentchanges sign near T0

max/T=2.8 which is a manifestation of truequantum tunneling. For more details see Reimann and Hänggi,1998.

FIG. 33. Color online Asymmetric Josephson junction array.Top left: Scanning electron picture with enlargement on theright of a strongly asymmetric array of a long, narrow networkof Josephson junctions arranged in a rectangular lattice. Thisarrangement determines the potential shape felt by vortices.Bottom left: Schematic layout. The Josephson junctions arerepresented by crosses, each network cell being delimited byfour junctions. All arrays have a length of 303 cells and a widthof 5 cells, placed between solid superconducting electrodes.The vortices assume a lower energy in cells with larger areaand smaller junctions. Figure provided by Milena Grifoni.

FIG. 34. Color online In an experimental quantum Brownianmotor driven by an adiabatic square-wave zero-mean rockingvoltage, quantum tunneling can contribute to the electron cur-rent. Due to the underlying asymmetric potential structure, thetwo opposite components of the time-averaged-driven currentdiffer in magnitude, yielding a net quantum ratchet currentReimann et al., 1997; Linke et al., 1999. The magnitudes canbe individually controlled by tuning the temperature. This inturn causes a tunneling-induced current reversal occurringnear 1.5 K in the top graph that can be exploited to directelectrons along a priori designed routes Linke et al., 1999.Below the current vs T graph is a scanning electron micro-graph of the relevant quantum device. Figure provided by Hei-ner Linke.

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and other asymmetric quantum-dot arrays, when sub-jected to unbiased ac drives, exhibit incoherent quantumratchet currents. Such experimental realizations using2D electron systems with broken spatial inversion sym-metry have been reported by Lorke et al. 1998, Vidanet al. 2004, and Sassine et al. 2008. An even richerbehavior of the quantum current, including, for ex-ample, multiple current reversals, emerges when thisclass of devices is operated in the nonadiabatic ac re-gime, as revealed by recent theoretical studies Goychukand Hanggi, 1998, 2005; Goychuck et al., 1998a, 1998b;Grifoni et al., 2002; Kohler et al., 2005.

As a second example of an artificial quantum dotBrownian motor, we consider an experimental double-quantum-dot device where the two dots are indepen-dently coupled to a nonequilibrium energy source whichis given by a biased quantum point contact. Moreover,the two quantum dots are embedded in independentelectric circuits via a common central gate, marked by aC in Fig. 35. When its internal symmetry is broken bytuning the dot gate voltages, this double quantum dotacts as a quantum ratchet device Khrapai et al., 2006.For weak interdot tunneling, detuning of the quantumdot energy levels results in the localization of an elec-tron in one dot, so that elastic electron transfer to theother dot is energetically forbidden. To operate as aquantum ratchet, the device still requires a nonequilib-rium energy source. For this purpose, a strong tunablebias on the quantum point contact induces nonequilib-rium energy fluctuations across the dividing central gate,thus promoting inelastic interdot tunneling inside theCoulomb-blockaded double quantum dot. This in turnleads to a net quantum ratchet current flow, as plotted inFig. 36. In the insets of the same figure, the interdottunneling process is sketched for the right-to-left transi-tion m ,n+1→ m+1,n, with asymmetry energy Em+1,n−Em,n+1, and for the opposite left-to-right tran-sition with energy −. Notably, a finite ratchet current isobserved only if the electron energy states in the dots

are detuned asymmetrically, i.e., when 0. In contrast,a likely inelastic ionization of one dot toward its adja-cent lead, followed by recharging from the same lead,does not result in a net current. The nonequilibrium en-ergy fluctuations, carried by the quantum point contactelectrons and absorbed by the electrons in the doublequantum dot, could consist of acoustic phonons, long-wavelength photons, or plasmons Khrapai et al., 2006.

The technology available to generate 2D electrongases can be generalized to ratchet not only charge butalso spin carriers. The interplay of spatially periodic po-tentials, lateral confinement, spin-orbit or Zeeman-typeinteractions, and ac driving then gives rise to directedspin ratchet currents, as theoretically proposed in recentstudies Scheid, Bercioux, et al., 2007; Scheid, Pfund, etal., 2007. Of particular experimental relevance is thetheoretically predicted phenomenon of a spin currentwhich emerges via an unbiased ac charge current drivinga dc spin current in a nonmagnetic, dissipative spinquantum ratchet which is composed of an asymmetricperiodic structure with Rashba spin-orbit interactionsFlatte, 2008; Smirnov et al., 2008. Remarkably, this spincurrent occurs although no magnetic fields are present.

Likewise, substantial rectification of a spin current canalso be achieved by coupling impurities to spatiallyasymmetric Luttinger liquids under ac voltage rockingBraunecker et al., 2007. The transport mechanism inthose schemes is governed by coherent tunneling pro-cesses, which will be addressed in the following section.

D. Coherent quantum ratchets

The study of quantum Brownian motors is far frombeing complete. In fact, there is an urgent need for fur-

1

234

5

C

6

7 8 910

IQPC

IDQD

VQPC

VDQD

24

8

C

(a) (b)

FIG. 35. Color online Double-dot quantum ratchet. a AFMmicrograph of a double-quantum-dot device. Metal surfacegates have a light color, black squares mark source and drainregions. The black scale bar marks the length of 1 m. bSchematic diagram: A biased quantum point contact QPCprovides the nonequilibrium fluctuation source for driving atunneling current IDQD in the asymmetric double quantum dotDQD two dashed circles Khrapai et al., 2006. The asym-metry is induced via gate voltages at the plunger gates 2 and 4.Figure provided by Vadim S. Khrapai.

-1 0 1

asymmetry energy ∆ (meV)

-0.2

0.0

0.2

ratc

het

curr

ent

I DQ

D(p

A)

−∆

FIG. 36. Color online Experimental quantum ratchet currentIDQD measured through the double quantum dot as a functionof its asymmetry energy . The energy source is a nearbyquantum point contact biased with VQPC=−1.55 mV; see Fig.35. An elastic contribution to IDQD is subtracted Khrapai etal., 2006. The two insets sketch the corresponding inelastictunneling processes which drive the ratchet current. Notably,this quantum ratchet current vanishes when =0 i.e., noasymmetry. Figure provided by Stefan Ludwig.

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ther development. In many cases the roles of incoherenttunneling and coherent transport channels cannot beclearly separated. For quantum devices built bottom upwith engineered molecules or artificial molecules such asquantum dots, the role of coherent quantum transporttypically plays a dominant role, as other interactions likeelectron-phonon processes turn out to be negligible. Thekey transport process for quantum ratcheting is thendriven coherent quantum tunneling Grifoni andHänggi, 1998; Kohler et al., 2005.

1. Quantum ratchets from molecular wires

This is the case of quantum Brownian motors consist-ing of nanowires formed by asymmetric moleculargroups and subjected to infrared light sources Lehmannet al., 2002, or symmetric molecular wires with symme-try breaking now provided by irradiation of the wirewith harmonic mixing signals Lehmann et al., 2003.The theoretically predicted quantum ratchet currentthrough an asymmetric molecular wire irradiated by far-infrared light is displayed in Fig. 37.

In this class of quantum ratchets, transport proceedscoherently between two or more fermionic leads thatensure the dissipative mechanism for the transportedelectrons to relax toward equilibrium Fermi distribu-tions. Such a ratchet current can also be used to sensi-tively probe the role of the electron correlations in theleads, as is the case with a ratchet device coupled toLuttinger liquids rather than to Fermi liquids Komnikand Gogolin, 2003; Feldmann et al., 2005.

2. Hamiltonian quantum ratchet for cold atoms

Surprisingly, systems governed by strictly Hamiltoniandynamics are able to yield directed transport as well.Directed transport may then take place even though nodissipation mechanisms are present. Trivial examples ofthis process are provided by integrable regular dynamicssubjected to unbiased ac drives with fixed initial phase,i.e., not averaged over Yevtushenko et al., 2000; Goy-chuk and Hänggi, 2001. Much more intriguing are time-dependent driven Hamiltonian quantum Brownian mo-tors exhibiting a nontrivial, mixed classical phase spacestructure Flach et al., 2000; Denisov and Flach, 2001;Schanz et al., 2001, 2005. In these systems, the onset ofa directed flow requires, apart from the necessity ofbreaking time-reversal symmetry, an additional dynami-cal symmetry breaking. As it turns out, in the semiclas-sical limit the corresponding directed flow then obeys aremarkable sum rule: Directed currents occurring inregular regimes of the underlying phase space dynamicsare counterbalanced by directed flows occurring in cha-otic regimes in phase space Schanz et al., 2001, 2005.As a result, if those individual directed currents aresummed up over all disjoint regimes in semiclassicalmixed phase space, no net transport emerges.

The concept of Hamiltonian quantum Brownian mo-tors extends as well to fully quantum systems governedby a unitary time evolution Goychuk and Hänggi, 2000,2001; Schanz et al., 2001, 2005; Denisov et al., 2006, 2007;Gong et al., 2007. In fact, the issue of pure quantumcoherence in the directed transport of chaotic Hamil-tonian systems is presently a topic of active research.This in particular holds true for cold atoms loaded inoptical lattices: If properly detuned, the intrinsic quan-tum dynamics is then practically dissipation-free, thusproviding a paradigm for Hamiltonian quantum ratchettransport. Various experimental cold atom ratchets havebeen realized Sec. IV. Relevant to the current topic isalso the demonstration of sawtoothlike, asymmetric coldatom potentials by Salger et al. 2007. All these systemsare currently employed to effectively steer cold matteron a quantum scale.

In the context of quantum ratchets for cold atoms,one must distinguish the common case of “rectificationof velocity,” implying that the mean position of particlesis growing linearly in time, from the case with “rectifica-tion of force,” i.e., with mean momentum growing lin-early in time. The latter class is better referred to as“quantum ratchet accelerators” Gong and Brumer,2004, 2006. In the context of cold atom ratchets the firstsituation is obtained by rocking a cold atom gas withtemporally asymmetric driving forces or temporallyasymmetrically flashing, in combination with spatiallyasymmetric potential kicks Denisov et al., 2007. Theaccelerator case originates from the physics of quantum-kicked cold atoms displaying quantum chaos featuressuch as the quantum suppression of classical chaotic dif-fusion dynamical localization and the diametrically op-posite phenomenon of quantum resonance occurringwhen the kicking period is commensurate with the in-

Ω [∆/h]

I[1

0−3eΓ

]

V [∆/e]I

[10−

3eΓ

]

FIG. 37. Quantum Brownian motor from a molecular wire.Time-averaged current I measured in units of the lead cou-pling strength e as a function of the infrared drive angularfrequency measured in units of the tunneling matrixelement =0.1 eV between sites for a laser amplitude A=;Lehmann et al., 2002. The inset displays the dependence of Ion the static voltage drop V applied along the molecule. Notethat the current at zero voltage is finite, thereby indicating aratchet effect. The driving frequency corresponds to the verti-cal arrow in the main panel.

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verse recoil frequency. Under the quantum resonancecondition a linear quadratic increase of momentum ki-netic energy takes place. These ratchet acceleration fea-tures, theoretically predicted in suitably modified kickedrotor models Lundh and Wallin, 2005; Poletti, Carlo, etal., 2007; Kenfack et al., 2008, require tuning to exactresonance. In contrast, the accelerator models obtainedfrom a generalization of either a quantum kicked rotoror a kicked Harper model Gong and Brumer, 2004,2006; Wang and Gong, 2008 are generic rectifiers offorce in the sense that here no tuning to exact resonanceis necessary. Interestingly, these fully quantum ratchetaccelerators display unbounded linear growth of meanmomentum, while the underlying classical dynamics isfully chaotic, a situation where classical quantumBrownian motor transport necessarily vanishes accord-ing to the above-mentioned classical sum rule Schanz etal., 2001, 2005. Thus, within this full quantum regime,which carries no clear relationship with the dynamics inthe semiclassical regime, the quantum accelerator workby Gong and Brumer 2006 see also Wang and Gong2008 presents a generic quantum mechanical excep-tion to the classical sum rule.

Early quantum resonance experiments in quantumratchets have already been successfully carried out. Fora -kicked rotor model with time symmetry broken by atwo-period kicking cycle asymmetric temporal drive,directed growth of momentum has been detected byJones et al. 2007. For a phase-dependent initial prepa-ration of a Bose-Einstein condensate kicked at reso-nance, a momentum acceleration has been observed bySadgrove et al. 2007 at zero quasimomentum, while foran arbitrary quasimomentum directed quantum Brown-ian transport has been realized by Dana et al. 2008.The last experiment also showed that the intrinsic ex-perimentally nonavoidable finite width in quasimomen-tum causes a suppression of the acceleration, eventuallyleading to a saturation effect after short times.

This field of Hamiltonian quantum ratchets is pres-ently undergoing rapid development. For example, it ispossible to apply control schemes to the relative phasesfor the resulting single-particle quantum Brownian mo-tor currents by harvesting Landau-Zener transitionsMorales-Molina et al., 2008. Of particular theoreticaland experimental interest is the study of the role of non-linearity on the size of directed quantum currents in in-teracting cold gases, as described within mean-fieldtheory by a nonlinear quantum ratchet evolution of theGross-Pitaevskii type. There the interplay of time-dependent driving and nonlinear Floquet states yieldsnew features, such as lifting of accidental symmetriesPoletti, Benenti, et al., 2007 and a resonant enhance-ment of the directed ratchet currents Morales-Molinaand Flach, 2008.

IX. SUNDRY TOPICS

In the following we discuss some classes of nanosys-tems and devices which feature directed transport in thespirit of Brownian motors. In these systems, however,

rectification of Brownian motion does not constitute themain element for directed transport. We recall that anartificial Brownian motor is mainly noise controlled,meaning that such a motor operates in an unpredictablemanner. In contrast, we discuss next systems that exhibitdirected transport predominantly as the result of strong-coupling schemes. Typical examples are adiabatic pumpscenarios of the peristaltic type, or nanosystems whichare driven by unbiased but asymmetric mechanical orchemical causes that are tightly coupled to the resultingmotion.

A. Pumping of charge, spin, and heat

A first class of physical systems that comes to mind inrelation to the working principles of artificial Brownianmotors is nanoscale pump devices. Pumping is character-ized by the occurrence of a net flux of particles, charges,spins, and soon, in response to time-dependent externalmanipulations of an otherwise unbiased system. Thismechanism is well studied, and peristaltic pumps arewidely exploited in technological applications. Thesesystems do not require a periodic arrangement of com-ponents, nor is thermal Brownian motion an issue fortheir operation. In particular, adiabatic turnstiles andpumps for charge and other degrees of freedom haveattracted considerable interest both experimentallyKouwenhoven et al., 1991; Pothier et al., 1992; Switkeset al., 1999; Hohberger et al., 2001 and theoreticallyThouless, 1983; Spivak et al., 1995; Brouwer, 1998; Zhouet al., 1999; Shutenko et al., 2000; Vavilov et al., 2001;Brandes and Vorrath, 2002; Aono, 2003; Moskalets andBüttiker, 2004; Sinitsyn and Nemenman, 2007.

Realizations of artificial pumps on the nanoscale ofteninvolve coupled quantum dots or superlattices. Most no-tably, in such peristaltic devices the number of trans-ferred charges, or, more generally, the number of trans-porting units per cycle, is directly linked to the cycleperiod. As a result, the output current becomes propor-tional to the driving frequency. This observation leads tothe conclusion that high-frequency nonadiabatic pump-ing might become more effective. Indeed, nonadiabaticpumping of charge, spin Scheid, Bercioux, et al., 2007;Flatte, 2008; Smirnov et al., 2008, or heat Li et al., 2008;Van den Broek and Van den Broeck, 2008 does exhibita rich phenomenology, including resonances Staffordand Wingreen, 1996; Moskalets and Büttiker, 2002; Plat-ero and Aguado, 2004; Cota et al., 2005; Kohler et al.,2005; Arrachea et al., 2007; Rey et al., 2007 and otherpotentially useful noise-induced features Strass et al.,2005; Sanchez et al., 2008, hence yielding a close inter-relation between nonadiabatic pumping in the presenceof noise and the physics of Brownian motors.

B. Synthetic molecular motors and machines

As emphasized in the Introduction, the field ofBrownian motors has its roots in the study and applica-tions of intracelluar transport in terms of molecular mo-tors Jülicher et al., 1997; Astumian and Hänggi, 2002;

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Reimann, 2002; Reimann and Hänggi, 2002; Wang andOster, 2002; Lipowsky and Klumpp, 2005. These mo-lecular motors function in terms of structure and motil-ity by use of specialized proteins in living systems. Suchbiological devices are fueled by ATP hydrolysis and areable to efficiently perform mechanical work on thenanoscale inside the cell structures Howard, 2001;Schliwa, 2002.

Closely related to the topic of synthetic motors is thepossibility of devising DNA-fueled artificial motors: sev-eral settings render it possible to biologically engineer

nanomachines which move along tracks designed a pri-ori Yorke et al., 2000; Turberfield et al., 2003; Yin et al.,2004.

An offspring of this topic is the engineering of nano-machines based on interlocked molecular species. Thishas spurred a flurry of new investigations within thephysical biology and organic and physical chemistrycommunities, aimed at building bottom-up synthetic mo-lecular systems which carry out such diverse functions asmolecular switches, molecular rotors, and any other kindof molecular gear.

An example of this class of molecular motor is thelight-powered molecular machine developed by Hugel etal. 2002. Bistable photosensitive azobenzene moleculescan be synthesized into long chains, each containingmolecules in either their trans or cis form. The free en-ergy landscape corresponding to different molecularconformational states is sketched in Fig. 38 versus thereaction coordinate. Polymeric chains have the advan-tage of scaling up the length changes corresponding tothe two different conformations of the constituents. Thelength changes of an azobenzene polymeric chain canthen be transformed into mechanical energy by meansof the lever arm of an atomic force microscope AFM,which, in combination with the light sources, forms thecore of the light-driven molecular motor of Fig. 39.

Such a molecular motor can be operated according toan idealized thermodynamic cycle of the Stirling type:Individual azobenzene polymers are made to stretch andcontract by inducing optical trans-cis transitions of theirconstituents. This machine demonstrates optomechani-cal energy conversion in a single-molecule motor device.The analogy with a thermodynamic cycle, though,should not be taken too seriously. A thermal Stirlingengine operates between baths of differing tempera-tures; here the “expansion” and “compression” of the

FIG. 38. Color online Free energy landscape of a singleazobenzene molecule along the reaction coordinate of the con-formation coordinate given by the bond angle !NNC, whichvaries from about −60° to +60°. Transitions can be inducedoptically 1420 nm for the short cis form 0.6 nm and 2365 nm for the extended trans form 0.9 nm, respectivelyfrom the singlet ground state S0 into the excited singlet stateS1, from which the molecule subsequently relaxes fluctuationdriven into either the cis or trans form. Notably, the trans formis thermally favored. The insets depict the corresponding con-formations with the transition state located near !NNC0°.Figure provided by Thorsten Hugel.

IIIII

IV

F1

F0

Cantileverstiffness

Forc

e

Poly cis Poly trans

∆L = N∆l

Mechanicalwork byopticalcontraction

Extension

I

220

200

180

160

140

120

100

80

21.0 21.5 22.0 22.5Extension (nm)

Forc

e(p

N) II

III

IV

I

FIG. 39. Color online Operation of a light-powered molecular motor. a The schematic force extension cycle for the optom-echanical energy conversion cycle of a single poly-azobenzene. In analogy to a thermodynamic Stirling cycle, the polymer is firststretched by the AFM tip acting as the piston I→II. Then, the application of the first optical excitation with 2365 nm shortensthe polymer by inducing a polytrans–poly cis transition. This yields the first part of the work in bending the AFM cantilever,which is clearly not as stiff as a piston in a common Stirling motor. In analogy to the Stirling cycle, another amount of work is doneduring the relaxation of the polymer III→IV. Finally, a second optical excitation with 1420 nm is needed to reset themolecule into its starting polytrans state IV→I. The total work output of the system is the mechanical energy corresponding tothe contraction L=Nl of the entire polymer chain of N azobenzenes against the external load. The experimental realizationof a full single-molecule operating cycle Hugel et al., 2002 is depicted in b. Figure provided by Thorsten Hugel.

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molecular motor are performed at a fixed temperatureand, most notably, by means of nonequilibrium lightsources. This laser-operated molecular motor can alsobe used as a building block to devise a bioinspired mo-lecular locomotive which can be guided back and forthon a preassigned track via a laser-assisted protocolWang, 2004.

In the operation of these synthetic nanoscale devices,especially when powered by light or chemical reactiveadditives, thermal noise does not necessarily play a ma-jor role, i.e., the function of these synthetic molecularmotors is ruled predominantly by deterministic forces,forces that depend on the mechanical and chemicalproperties of the molecules Neuert et al., 2006. Yet thisarea of research is fascinating and we therefore refer theinterested reader to recent items and tutorials Porto etal., 2000; Balzani et al., 2006; Browne and Ferina, 2006;Astumian, 2007 and to consult the reviews by Kottas etal. 2005 and Kay et al. 2007 for more details.

X. CONCLUDING REMARKS

With this review we have taken a tour through themany intriguing and multifacetted applications thatBrownian motion can offer in the most diverse areas ofnanotechnology, when combined with spatiotemporalsymmetry breaking, nonlinearity, and, possibly, collec-tive interaction effects. The physics of classical andquantum Brownian motion is by now well established,by virtue of the breadth of the theoretical modeling andexperimental realizations produced over the last century.Many research activities are still developing in interdis-ciplinary fields encompassing chemistry, biological re-search, information sciences, and even extending intosocial sciences and economics. The main lesson to belearned from Robert Brown’s and Albert Einstein’swork is therefore as follows: Rather than fighting ther-mal motion, we should put it to work to our advantage.Brownian motors use these ceaseless noise source to ef-ficiently direct, separate, pump, and steer degrees offreedom of differing nature reliably and effectively.

In writing this overview we spared no efforts in cov-ering a wide range of interesting developments and po-tential achievements. In doing so we nevertheless had tomake some selective choices of topics and applications,which to some extent reflect our preferences and preju-dices. Closely related topics of ongoing research werenot reviewed due to space limits. For example, in Sec.IX.B we did not cover in sufficient detail the topic ofATP-driven molecular motors and DNA-fueled motors,for which we refer the reder to earlier reviews andbooks, such as Jülicher et al. 1997, Howard 2001,Schliwa 2002, and Lipowsky and Klumpp 2005. Thisis surely the case of the bottom-up design and operationof synthetic Brownian molecular devices, extensivelycovered by Kay et al. 2007. Another such topic is thequestion of so-called absolute negative mobility, whichoccurs via quantum tunneling events in quantum sys-tems Keay et al., 1995; Aguado and Platero, 1997; Gri-foni and Hänggi, 1998; Platero and Aguado, 2004 and

through nonequilibrium-driven diffusive dynamics inclassical systems Eichhorn et al., 2002a, 2002b; Ros etal., 2005; Machura, Kostur, et al., 2007; Kostur et al.,2008.

Our main focus was on noise-assisted directionaltransport, shuttling, and pumping of individual or collec-tive particle, charge, or matter degrees of freedom. Theconcept, however, extends as well to the transport ofother degrees of freedom such as energy heat and spinmodes see also Sec. IX.A. Both topics are experiencinga surge of interest with new exciting achievements cur-rently reported. The concept carries potential for yetother applications. Examples include the noise-assisteddirectional transport and transfer of informational de-grees of freedom such as probability or entropy and,within a quantum context, the shuttling of entanglementinformation. These issues immediately relate to the en-ergetics of artificial Brownian motors reviewed in Sec.II.E, including measures of open- and closed-loop con-trol scenarios and other optimization schemes.

We have attempted to indicate the potential ofBrownian motors in present nanotechnology by showinghow they can be put to work; we discussed how suchmotors can be constructed and characterized, and howdirectional, Brownian-motion-driven transport can becontrolled, measured, and optimized. Moreover, we areconfident that the interdisciplinary style will encouragereaders to develop new approaches and motivations inthis challenging and fast-growing research area.

ACKNOWLEDGMENTS

This review would not have emerged without continu-ous support from and insightful discussions with ourclose collaborators and colleagues. We first express ourgratitude to all members of our work groups amongwhich special credit should be given to M. Borromeo, J.Dunkel, I. Goychuk, G.L. Ingold, S. Kohler, G. Schmid,P. Talkner, and U. Thiele for their scientific contribu-tions and personal encouragement. A special thanksgoes to our many colleagues and friends who are alsoengaged in the field of Brownian motors and, in particu-lar, to R. D. Astumian, M. Bier, J. Casado, W. Ebeling, J.A. Freund, L. Gammaitoni, H. E. Gaub, E. Goldobin,M. Grifoni, D. Hennig, P. Jung, J. Kärger, I. Kosinska,M. Kostur, J. P. Kotthaus, B.-W. Li, H. Linke, J. Luczka,S. Ludwig, L. Machura, V. R. Misko, M. Morillo, F. Nori,J. Prost, P. Reimann, F. Renzoni, K. Richter, M. Rubí, S.Savel’ev, L. Schimansky-Geier, Z. Siwy, B. Spagnolo, C.Van den Broeck, and A. Vulpiani. In addition, we areindebted to M. Borromeo, M. Grifoni, T. Hugel, V. S.Khrapai, H. Linke, S. Ludwig, F. Müller, S. Savelev, U.Thiele, and C. Van den Broeck for providing us withoriginal figures and unpublished material. F.M. wishes tothank Professor Hunggyu Park for his kind hospitality atthe Korea Institute for Advanced Study and, likewise,the Physics Department of the National University ofSingapore for kind hospitality: At these two institutionswe were both given the opportunity to efficiently con-tinue working on a preliminary version of this review.

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P.H. acknowledges financial support by the DeutscheForschungsgemeinschaft via the Collaborative ResearchCentre SFB-486, Project No. A10, B 13 and by the Ger-man Excellence Cluster “Nanosystems Initiative Mu-nich” NIM. Both authors thank the Alexander vonHumboldt Stiftung, who made this joint project possibleby granting to one of us F.M. a Humboldt ResearchAward to visit the Universität Augsburg, where most ofthe work was done.

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