Agreat challenge for theburgeoning field of

nanotechnology is the de-sign and construction of mi-croscopic motors that canuse input energy to drive di-rected motion in the face ofinescapable thermal andother noise. Driving suchmotion is what protein mo-tors—perfected over thecourse of millions of years by evolution—do in every cell inour bodies.1

To put the magnitude of the thermal noise in per-spective, consider that the chemical power available to atypical molecular motor, which consumes around100–1000 molecules of adenosine triphosphate (ATP) persecond, is 10⊗16 to 10⊗17 W. In comparison, a molecularmotor moving through water exchanges about 4 × 10⊗21 J(the thermal energy kT at room temperature) with its en-vironment in a thermal relaxation time of order 10⊗13 s.Thus, a thermal noise power of about 10⊗8 W continuallywashes back and forth over the molecule. That power,which, according to the second law of thermodynamics can-not be harnessed to perform work, is 8–9 orders of magni-tude greater than the power available to drive directed mo-tion. For molecules, moving in a straight line would seemto be as difficult as walking in a hurricane is for us.Nonetheless, molecular motors are able to move, and withalmost deterministic precision.

Inspired by the fascinating mechanism by which pro-teins move in the face of thermal noise, many physicistsare working to understand molecular motors at a meso-scopic scale. An important insight from this work is that,in some cases, thermal noise can assist directed motion byproviding a mechanism for overcoming energy barriers. Inthose cases, one speaks of “Brownian motors.”2 In this ar-ticle, we focus on several examples that bring out someprominent underlying physical concepts that haveemerged. But first we note that poets, too, have been fas-cinated by noise; see box 1.

Bivalves, bacteria, and biomotorsBacteria live in a world in which they are subject to vis-cous forces large enough that the inertial term mv� in New-ton’s equation of motion can be safely ignored. The motionof the bacteria, governed by those viscous forces, is verydifferent than the inertia-dominated motion that we knowfrom everyday experience. Edward Purcell, in his classicarticle “Life at Low Reynolds Number,” highlighted thatdifference by formulating what has come to be known asthe scallop theorem.3

A scallop is a bivalve (a mollusk with a hinged shell)

that could, in principle,move by slowly opening itsshell and then rapidly clos-ing it. (In fact, the scallop’smethod of locomotion issomewhat different.) Duringthe rapid closing, the scallopwould expel water and de-velop momentum, allowingit to glide along due to iner-tia. A typical scallop has a

body length a of about a centimeter and propels itself at aspeed v of several cm/s, that is, at several times its lengthper second. Thus, the Reynolds number, R ⊂ avr/h (a di-mensionless parameter that compares the effect of inertialand viscous forces), is about 100, where r is the density ofthe fluid (for water, the density is 1 g/cm3) and h is thefluid’s viscosity (for water, about 10⊗2 g/(cm�s)).

For organisms a few thousand times smaller than ascallop, also moving at several body lengths per second, theReynolds number is much less than one. In that case, theglide distance is negligible. Because the motion generatedby opening the shell cancels that produced on closing theshell, a tiny “scallop” cannot move. The mathematical rea-son is that motion at low Reynolds number is governed bythe Navier–Stokes equation without the inertial terms, ⊗∇p ⊕ h∇2v ⊂ 0. Because time does not enter explicitly intothe equation, the trajectory depends only on the sequenceof configurations, and not on how slowly or rapidly any partof the motion is executed. Hence, any sequence that retracesitself to complete a cycle—and that is the only type of se-quence possible for a system such as a “scallop” with justone degree of freedom—results in no net motion.

With typical lengths of about 10⊗5 m and typicalspeeds of some 10⊗5 m/s, bacteria live in a regime in whichthe Reynolds number is quite low, about 10⊗4. Thus, bac-teria must move by a different mechanism than that usedby a “scallop.” Our purpose is not to investigate how ac-tual bacteria move (see the article “Motile Behavior of Bac-teria” by Howard C. Berg in PHYSICS TODAY, January2000, page 24) but to examine generic mechanisms bywhich locomotion at low Reynolds number is possible, withan ultimate focus on molecule-size Brownian motors.

Purcell described several locomotion mechanisms, allof which pertain to motion induced by cyclic shape changesin which, unlike the scallop’s cycle, the sequence of con-figurations in one half of the cycle does not simply retracethe sequence of configurations in the other half. Here, weconsider the two mechanisms shown in figure 1, thecorkscrew and the flexible oar.

The two mechanisms shown have different symme-tries. The corkscrew mechanism avoids retracing its stepsby its chirality. At low frequency, the bacterium moves afixed distance for each complete rotation of the chiral screwabout its axis. Thus, the velocity is proportional to the fre-quency. Reversing the sense of corkscrew rotation reversesthe bacterium’s motion. Slow enough rotation produces mo-

© 2002 American Institute of Physics, S-0031-9228-0211-010-6 NOVEMBER 2002 PHYSICS TODAY 33

DEAN ASTUMIAN is a professor of physics at the University of Maine inOrono. PETER HÄNGGI is a professor of theoretical physics at the Uni-versity of Augsburg, Germany.

BROWNIANMOTORS

Thermal motion combined with inputenergy gives rise to a channeling ofchance that can be used to exercise control over microscopic systems.

R. Dean Astumian and Peter Hänggi

tion with essentially no dissipation, sowe call the corkscrew an adiabaticmechanism.

On the other hand, the flexible oarrelies on the internal relaxation of theoar curvature to escape the scalloptheorem. At low frequency, the ampli-tude of the bending of the oar is pro-portional to the frequency and the ve-locity is thus proportional to thesquare of the frequency. Because, inthis case, relaxation and dissipationare essential, the flexible oar is an ex-ample of a nonadiabatic mechanism.4

The role of noiseThe mechanisms in figure 1 illustratehow self propulsion at low Reynoldsnumber is possible. A new problemarises, however, when particles havelengths characteristic of molecular di-mensions, 10⊗8 m or so.

In that case, diffusion caused bythermal noise (Brownian motion) com-petes with self-propelled motion. Thetime to move a body length a at a self-propulsion velocity v is a/v, while thetime to diffuse that same distance is ofthe order a2/D. Here, the diffusion co-efficient D is given in terms of particlesize, solution viscosity, and thermal energy by theStokes–Einstein relation D ⊂ kT/(6pha). At room temper-ature, and in a medium whose viscosity is about that ofwater, a bacterium needs more time to diffuse a bodylength than it does to “swim” that distance. For smallermolecular-sized particles, however, a body length is cov-ered much faster by diffusion. For molecular motors, un-like bacteria, the diffusive motion overwhelms the directedmotion of swimming.

A solution widely adapted in biology is to have themotor on a track that constrains the motion to essentiallyone dimension along a periodic sequence of wells and bar-riers.5 The energy barriers significantly restrict the diffu-sion. Thermal noise plays a promi-nent constructive role by providinga mechanism, thermal activation,by which motors can escape overthe barriers.6 (See also the article“Tuning in to Noise” by Adi R. Bul-sara and Luca Gammaitoni inPHYSICS TODAY, March 1996, page39.) The energy necessary for di-rected motion is provided by ap-propriately raising and loweringthe barriers and wells, either viaan external time-dependent modu-lation or by energy input from anonequilibrium source such as achemical reaction.

A simple Brownian motorFigure 2 shows a simple exampleof a Brownian motor, in which amolecule-sized particle moves onan asymmetric sawtooth poten-tial. Such an asymmetric profile isoften called a ratchet after thebeautiful example given byRichard Feynman in his Lectures

on Physics, volume I, chapter 46 (Addison-Wesley, 1963).Feynman used his ratchet to show how structuralanisotropy never leads to directed motion in an equilib-rium system. But in the nonequilibrium system depictedin figure 2, the potential’s cycling—which provides the en-ergy input—combines with structural asymmetry and dif-fusion to allow directed motion of a particle, even againstan opposing force. (An excellent simulation of the Brown-ian motor is at the Web site http://monet.physik.unibas.ch/~elmer/bm/.)

For biological motors on a track, one might expect thelength a and track period L to be of molecular size and theviscous drag coefficient to be around 10–10 kg/s, somewhat

higher than that in water due tofriction between the motor andtrack. If the potential is switchedon and off with a frequency of103 Hz, consistent with the rate ofATP hydrolysis by many biologi-cal motors, the induced velocity isabout 10⊗6 m/s and the force nec-essary to stop the motion is ap-proximately 10⊗11 N: The velocityand force estimates are both con-sistent with values obtained fromsingle-molecule experiments onbiological motors.1 (See also thearticle “The Manipulation of Sin-gle Biomolecules” by TerenceStrick, Jean-François Allemand,Vincent Croquette, and DavidBensimon in PHYSICS TODAY, Oc-tober 2001, page 46).

An effect analogous to the di-rected motion caused by cyclicallyturning a potential on and off canbe demonstrated for particles mov-ing on a fixed asymmetric track,such as a sawtooth etched on a

Corkscrew

Flexible oar

FIGURE 1. SELF-PROPULSION at low Reynolds number can occur through a numberof mechanisms, two of which are shown here. In the top diagram, a bacterium is pro-pelled by a rotating corkscrew. At low frequency, the resulting velocity is propor-tional to the frequency. In the bottom illustration, a bacterium propels itself by wav-ing its flagellum up and down in an undulatory motion. Because the flagellum isflexible, it acquires a curvature whose concavity depends on the direction of its mo-tion—concave down while the flagellum is moving upward, concave up when mov-ing downward. The degree of curvature depends on the frequency with which theflagellum is waved up and down. Thus, the velocity that results from the flexible oarmechanism is proportional to the square of the frequency.

Box 1. The Place of the SolitairesLet the place of the solitaires Be a place of perpetual undulation.

Whether it be in mid-seaOn the dark, green water-wheelOr on the beaches.There must be no cessation Of motion, or of the noise of motion, The renewal of noise And manifold continuation;

And, most, of the motion of thought And its restless iteration,

In the place of the solitaires, Which is to be a place of perpetual undulation.

Wallace Stevens (1879–1955)

From The Collected Poems of Wallace Stevens byWallace Stevens, copyright 1954 by WallaceStevens and renewed 1982 by Holly Stevens. Usedby permission of Alfred A. Knopf, a division ofRandom House, Inc.

34 NOVEMBER 2002 PHYSICS TODAY http://www.physicstoday.org

glass slide.7 The energy input driving the directed motion isprovided by cyclically varying the temperature betweenhigh and low values. At high temperatures the particles dif-fuse, but when the temperature is low the particles arepinned in the potential wells. Because of the asymmetry ofthe track, the fluctuations over time between hot and coldcause the particles to move, on average, over the steeplysloped, shorter face of the etched sawtooth.

In the scheme depicted in figure 2, the fuel is the en-ergy supplied by turning the potential on and off. The track,or substrate, is the lattice on which the particle moves. Theparticle is the motor—the element that consumes fuel andundergoes directional translation. The model illustrates thetwo main ingredients necessary for self-propelled motion atlow Reynolds number: symmetry breaking and energyinput. The particle in the illustrated scheme is a trueBrownian motor, because without thermal noise to causeBrownian motion, the mechanism fails.8

The ratchet in figure 2 mimics aBrownian motor first proposed in 1992by Armand Ajdari and Jacques Prostworking at the Ecole Supérieure dePhysique et de Chimie Industrielles inParis.9 They envisioned a situation inwhich turning on and off an asymmet-ric electric potential would provide ameans for separating particles based ondiffusion.

Several groups explored that pos-sibility in various ways during themid-to-late 1990s.10 In 1994, Ajdariand Prost, along with colleagues Juli-ette Rousselet and Laurence Salome,constructed a device for moving smalllatex beads unidirectionally in a non-homogeneous electric potential thatwas turned on and off cyclically. Abouta year later, Albert Libchaber and col-leagues at Princeton University and atNEC Research Institute Inc, made anoptical ratchet. By modulating theheight of the barrier on an asymmet-ric sawtooth fashioned from light, theycould drive a single latex bead arounda circle. Most recently, Joel Bader andcolleagues at CuraGen Corp con-structed a device for efficient separa-

tion of DNA molecules using interlocking combed elec-trodes with an asymmetric spacing between the positiveand negative electrodes.

General descriptionWhen we considered the scallop theorem and the devicesbacteria use to evade it, we focused on physical changes—the opening and closing of a scallop’s shell, the turning ofa corkscrew, and the waving of a flexible flagellum. In ageneral mathematical description of motion at lowReynolds number, a time-dependent potential correspondsto the changes in shape that we discussed earlier.

Imagine a particle constrained to move on a line, witha spatially periodic time-modulated potential V(x, t). Theparticle is governed by the equation of motion

mv� ⊕ V� (x, t) ⊂ ⊗gy ⊕ =(2kTg)j(t),

On

Off

On

EN

ER

GY

POSITION

FIGURE 2. SWITCHING A SAWTOOTH POTENTIAL on and offcan do work against an external force. In the illustration, the

red gaussians indicate how the probability distribution of aparticle evolves as a sawtooth potential is turned off and then

on again. When the potential is off, a particle moves to the leftbecause of the force, but it also diffuses with equal probability

to the left and right. After some time, the potential is turnedon and the particle is trapped at the bottom of a well—morelikely the well to the right than the well to the left of where

the particle started. Were the sawtooth potential to remain ei-ther on or off, the net velocity would be to the left. But in the

illustrated system, the asymmetry of the potential combineswith diffusion and the cycling of the potential to allow di-

rected motion to the right, even against an external force. Theillustrated mechanism works even if, in imitation of the effectof a chemical reaction, the potential is turned on and off with

random, Poisson-distributed lifetimes. (See R. D. Astumian, M. Bier, Biophys. J. 70, 637, 1996.)

a b

v p= /2 v=0

y y

x x

FIGURE 3. TWO-DIMENSIONAL POTENTIALS, in combination with homogeneousforces, can serve as particle separators. The figure shows contour graphs of the 2Dpotential functions V(x, y) ⊂ V0 cos(4px/Lx) ⊕ u(y) cos(2px/Lx) ⊕ e(y) sin(2px/Lx),with u(y) ⊂ u0 cos(2py/Ly), e(y) ⊂ e0 cos(2py/Ly ⊕ v), and v ⊂ (0, p/2). The super-imposed triangles and rectangles emphasize the symmetry of the functions. (a) Themodulating function with v ⊂ p/2, is the spatial equivalent of a traveling-wave tem-poral modulation. For the resulting potential, switching the component of force inthe y direction switches the resulting component of particle velocity in the x direc-tion. (b) The modulating function with v ⊂ 0 is the spatial equivalent of a standing-wave temporal modulation. The resulting potential produces a drift in the ⊕x direc-tion regardless of the sign of the y component of force.

http://www.physicstoday.org NOVEMBER 2002 PHYSICS TODAY 35

where the prime denotes a spatial derivative and g ⊂ 6phais the viscous drag coefficient. The left-hand side of theequation of motion describes the deterministic, conserva-tive part of the dynamics, and the right-hand side accountsfor the effects of the thermal environment—viscous damp-ing and a fluctuating force modeled by thermal noise j(t).If both inertia and noise are negligible, the equation of mo-tion may be approximated as V�(x, t) ⊂ ⊗gv: Explicit timedependence enters only through V�(x, t). Thus, the patternof motion is independent of whether the modulations occurrapidly or slowly. If, as is the case for a standing-wavemodulation, the changes are such that the path in the sec-ond half of the cycle retraces those of the first, then the ve-locity must also retrace its steps. That retracing is irre-spective of the amplitude, waveform, or frequency of themodulation.

For Brownian motors, however, there is ineluctableand significant thermal noise, which changes the situationdramatically. Because noise provides a mechanism for re-laxation—an internal response of the system to a change

in the external parameters—a single external degree offreedom can combine with the internal dynamics to escapethe scallop theorem and yield directed motion, as in thescheme depicted in figure 2. In boxes 2 and 3, we haveworked out a second illustrative scheme in detail and dis-cussed the relationship of that scheme with the principleof detailed balance.

As illustrated in the first of those boxes, the net cur-rent (an appropriately normalized average velocity) due tomodulation of the potential can be broken into two contri-butions. One is a purely geometric term corresponding tomotion similar to that induced by a slow traveling-wavemodulation. That term depends only on the two externalparameters that define the modulation and describes thereversible part of the transport. Similar in spirit to theBerry, or geometric, phase in quantum mechanics (see thearticle “Anticipations of the Geometric Phase” by MichaelBerry in PHYSICS TODAY, December 1990, page 34), thatfirst term corresponds to the adiabatic transport mecha-nism described by David Thouless and recently used by

36 NOVEMBER 2002 PHYSICS TODAY http://www.physicstoday.org

The illustration below summarizes the way in which parti-cles that are subject to a time-modulated, spatially periodic

one-dimensional potential can exhibit net current. The upperpanel (a) shows the time-independent part of the model poten-tial, V0 cos(4px/L), which reflects the underlying lattice onwhich the particle moves. Superimposed on that potential is thetime-dependent modulation u(t) cos(2px/L) ⊕ e(t) sin(2px/L),which may arise from stochastic or deterministic changes in theenvironment.

If the modulation is not too large or too fast, and if the po-tential amplitude is several times larger than the thermal energykT, then the effect of the modulation can be viewed as a per-turbation on the escape rates for the underlying symmetric po-tential. In that case, one can think of u(t) as governing the rela-tive barrier heights and e(t) as independently regulating therelative well energies. In the following, we drop explicit nota-tion of the time dependence of parameters and take all energiesto be dimensionless quantities, measured relative to kT.

The modulation distinguishes two types of wells—states—designated A and B in panel (b) of the figure. Transitions be-tween the A and B wells may be described by a two-state ki-netic model with, for this model, time-dependent transition co-efficients of the form kesc exp(�u�e).12 For example, the

coefficient with both signs positive in the exponent corre-sponds to a transition from a B well to an A well over the bar-rier to the left of the A well. In the figure, the coefficient is de-noted k

OBA.

Because of the modulation of the relative well energies(characterized by e), the probability Q for a particle to be in anA well changes as particles flow back and forth between the Aand B wells. The rate of probability change is given by

dQ/dt ⊂ I1 ⊕ I2,

where the currents I1 and I2 are defined in panel (a) of the fig-ure. The probability for the particle to be in a B well is 1 ⊗ Q,which allows expression of the currents in the form

I1,2 ⊂ kescexp(e�u)q/f(e).

Here the plus sign applies for I1 and the minus sign for I2, andq ⊂ f (e) ⊗ Q is the deviation of the probability from its instan-taneous equilibrium value given by the Fermi distributionfunction f (e) ⊂ {1 ⊕ exp(⊗2e)}⊗1. Note that each individualcurrent goes to zero as equilibrium, q ⊂ 0, is approached, butthe fraction F of the current over the left-hand barrier

which depends only on u, does not. Once any transients havedecayed, the net current is the time average of F dQ/dt, and canbe written as the sum of two integrals:

Inet ⊂ w(�F df ⊗ �F dq),

where w is the frequency of the modulation. The first integral is independent of the frequency and de-

scribes the adiabatic contribution to the current. The first inte-gral’s dependence on time arises only through u (via F ) and e(via f ). It is nonzero only if u and e are out of phase with oneanother.

The second integral, which depends on the deviation fromequilibrium q, describes the nonadiabatic contribution to thecurrent. To lowest order, its value depends linearly on fre-quency. Because of the dissipation inherent in the relaxation ofthe system to equilibrium, that integral can be nonzero even ifu and e vary together and at random times. Panels (c), (d), and(e) in the figure show a situation in which randomly switchingthe signs of both u and e together drives net transport: The par-ticle moves on average almost one period per switching cycle.

F ⊂I1

I I1 2⊕[1 tanh( )],⊕ u⊂

12

Box 2. Adiabatic and Nonadiabatic Transport

I1

kBA

kAB kBA

kAB

I2

e

e

eu

u

Q

. . . . . .A A AB B

( , )=e u (.5 , .5 )V V0 0

( , )=e u (.5 , .5 )V V0 0

( , )=e u (–.5 , –.5 )V V0 0

a

b

c

d

e

Charles Marcus and his colleagues to pump electronsthrough a quantum dot with essentially no energy dissi-pation (see PHYSICS TODAY June 1999, page 19).11

The second term corresponds to the dissipative part ofthe transport and is the term typically associated with theratchet effect: In the on–off ratchet shown in figure 2, theaverage of the adiabatic term is zero—all of the net trans-port is described by the irreversible term. The dissipativenature of the mechanism corresponding to the irreversibleterm means that even random fluctuations such as thosethat might arise from a simple two-state nonequilibriumchemical reaction can drive transport.12

Two-dimensional ratchetsThe Brownian motors we have considered so far have beenconfined to one spatial dimension and subject to time-vary-ing potentials. One can develop a sharper intuition forBrownian ratchets by mapping time-modulated potentialsinto static, 2D potentials: (x/L, wt) O (x/Lx, y/Ly). The mod-ulation, instead of being characterized by functions oftime, is then characterized by functions of the coordinatey.13,14 The nonequilibrium features implicit in the originalexternal temporal modulation are introduced by externalforces in the x or y directions.

The resulting 2D potentials yield two classes of de-vices distinguished by symmetry. In one class, proposed byTom Duke and Bob Austin, symmetry is broken in both co-ordinates in the sense that changing the sign of either x ory changes the potential. A member of this symmetry classis illustrated in figure 3a, which also shows the responseof particles to forces in the up and down directions. Notethat changing the direction of the force changes the direc-tion of the resulting velocity. In general, for potentials withbroken symmetry in both coordinates, to leading order, thex-component of current resulting from a force in the y di-rection is the same as the y-component of current result-ing from the same magnitude force in the x direction. Gen-

erally, if the force in the, say, x direction is zero, current inboth the x and y directions is proportional to the force inthe y direction.

In the second class of devices, suggested by ImreDerényi and Dean Astumian and by Axel Lorke and col-leagues, symmetry is broken in only one coordinate. Fig-ure 3b shows a member of this class, with broken symme-try in the x coordinate. The figure also shows that a forceup or down induces flow to the right; by symmetry, a force inthe x direction induces no net flow in the y direction. Analgebraic manifestation of those responses to force is that,to lowest order, the particle current in the x direction isproportional to the square of the y component of force.

An advantage of the second class is that an oscillatingforce in the y direction can drive unidirectional motion inthe x direction, thus allowing devices to be much smaller.One can achieve good lateral separation without particlestraveling very far vertically. Combining systems with dif-ferent symmetry properties may make it possible to tailor apotential for the most effective separation in a given system.

So far, only the first symmetry class has been appliedexperimentally for particle separation,13 but the secondhas been realized for ratcheting electrons that move bal-listically through a maze of antidots.14 Electrons in a 2Dsquare array of triangular antidots move in a potentialsimilar to that shown in figure 3b. When irradiated by far-infrared light, the electrons are shaken and crash againstthe antidots rather like balls hitting the obstacles on apinball table. The electrons are then funneled into the nar-row gaps between the antidots, thereby yielding a well-di-rected beam. Indeed, one observes a net photovoltage be-tween source and drain—merely the expected ratcheteffect that turns an AC source into a DC one.

Two-dimensional ratchets open a doorway to wirelesselectronics on the nanoscale. For example, different orien-tations of asymmetric block structures may allow for theguiding of several electron beams across each other. As

http://www.physicstoday.org NOVEMBER 2002 PHYSICS TODAY 37

Directed motion driven by modulation of relative barrierheights and well energies seems to violate a principle dis-

cussed by Lars Onsager in his famous 1931 paper on reciprocalrelations in irreversible processes. In that paper, Onsager re-marked on the idea that, at thermodynamic equilibrium, eachforward transition in any chemical pathway is, on average, bal-anced by an identical transition in the reverse direction. Thatrequirement, known as detailed balance, is closely related to theprinciple of microscopic reversibility.

The system discussed in box 2 is at steady state—the proba-bility Q is constant—whenever the flow into well A equals theflow out of A, that is, when the two currents satisfy the rela-tion I1 ⊂ ⊗I2. (See box 2 for definitions of the terms used in thisbox.) The principle of detailed balance, however, asserts that, atthe special steady state known as thermodynamic equilibrium,each transition is independently balanced:

I1 ⊂ kO

BA (1 ⊗ Q) ⊗ kI

ABQ = 0,

I2 ⊂ kI

BA (1 ⊗ Q) ⊗ kO

ABQ = 0.

Thus, a corollary of detailed balance that holds even away fromequilibrium is

At first glance, it appears that as long as the above corollaryrelation holds, there can be no net current. As the example in

box 2 shows, however, that conclusion need not be true if therate constants are time-dependent due to fluctuations of thebarrier heights and well energies. When the rate constants de-rived in box 2 are inserted into the corollary relation, the time-dependent functions u and e drop out: The corollary holds atall times. Yet, for the system described in box 2—indeed, undera wide range of circumstances—barrier-height and well-energyfluctuations consistent with the corollary relation give rise tonet current, that is, detailed balance is broken.12

Systems at thermodynamic equilibrium experience energyfluctuations that may be described in terms of u and e. Howcan those fluctuations be consistent with the second law ofthermodynamics, which prohibits directed motion at equilib-rium? The figure in box 2 provides a key to the explanation. Itshows that an increase in e implies an increase in the potentialenergy if the particle is in a B well and a decrease if it is in anA well. Thus, an increase in e must, at equilibrium, be expo-nentially less likely when the particle is in a B well than whenit is in an A well; equilibrium fluctuations do not drive di-rected motion.

External fluctuations, or fluctuations driven by a nonequi-librium source, such as a chemical reaction, are not subject tothe energy constraint imposed on equilibrium fluctuations, andso they can drive directed motion. In the limit of strong driv-ing, the fluctuations of u and e are independent of the positionof the particle. That is the case shown in the lower panels of thefigure in box 2.

⊂ 1.k kBA AB

k kAB BA

O O

O O

Box 3. Detailed Balance

suggested by Franco Nori and his collaborators at the Uni-versity of Michigan, specially tailored 2D potentials, com-bined with a source of thermal or quantum noise, can pro-vide a lens for focusing or defocusing electrons, much asoptical lenses manipulate light.15

Ratchets in the quantum world Symmetry breaking and the use of noise to allow randomlyinput energy to drive directed motion can also be exploitedwhen quantum effects play a prominent role. An especiallyappealing possible application is the pumping and shut-tling of quantum objects such as electrons along previouslyselected pathways without the explicit use of directed wirenetworks or the like.

One of the most important features of quantum trans-port not present in the classical regime is quantum tun-neling. Tunneling provides a second mechanism—the firstbeing the thermal activation exploited for classical ratch-ets—for a particle to move among energy wells. HeinerLinke and colleagues took brilliant advantage of the twomechanisms in designing a quantum ratchet showing acurrent reversal as a function of temperature (see figure4).16 Peter Hänggi and coworkers applied quantum dissi-pation theory to a slowly rocked ratchet device to theoret-ically anticipate such a current reversal.17

Using electron beam lithography, Linke and cowork-ers constructed an asymmetric electron waveguide withina 2D sheet of electrons parallel to the surface of an alu-minum-doped gallium arsenide/gallium aluminum ar-senide heterostructure. The device comprised a string offunnel-shaped constrictions, each of which forms an asym-metric energy barrier for electrons traveling along thewaveguide. A slow (192 Hz), zero-average, periodic,square-wave voltage was applied along the channel to rockthe ratchet potential. In other words, electrons travelingthrough the waveguide were subjected to a uniform per-turbing force that periodically switched direction.

At low temperature, tunneling predominates. Thebarriers are narrowed when the force is to the right andwidened when the force is to the left, thus inducing elec-tron current to the right. Because of the asymmetry, therocking produces a net current. At high temperatures,where thermal activation predominates, electrons movepreferentially over the gentle slope of the potential. Thatmovement leads to an electron current to the left. More-over, the device functions as a heat pump even when thetemperature is set to the value that produces no net elec-trical current: The thermally activated current is predom-inantly due to electrons in higher-energy states whereasthe tunneling current is mainly due to electrons in thelower-energy states.17

Quantum ratchets are potentially useful in any num-ber of tools such as novel rectifiers, pumps, molecularswitches, and transistors. Some day, devices built withquantum ratchets may find their way into practical applications.

Perspective and overviewA Brownian motor is remarkably simple: The essentialstructure consists of two reservoirs, A and B, with twopathways between them. (In the example of box 2, the twopathways are defined by whether the potential barrierovercome is to the left or right of well A.) By periodicallyor stochastically altering both the relative energies of thereservoirs and the “conductances” of the pathways be-tween them, with a fixed relationship between the twomodulations, one can arrange that transport from A to Bis predominately via one pathway and transport from B toA is via the other pathway. Depending on the topology, theinduced particle motion could be realized as directed trans-port along a circle or a line, transfer of microscopic cargoor electrons between two reservoirs, or coupled transportin two dimensions.

In micro- and nanoscale materials such as polymersor mesoscopic conductors, thermal activation and quan-tum mechanical tunneling are mechanisms for overcom-ing energy barriers (incidentally introducing nonlinear-ity).6 In addition, the multiple time scales inherent incomplex systems allow the necessary correlations betweenthe fluctuations of the conductances and energies toemerge spontaneously from the dynamics of the system. Inthose cases, noise plays an essential or even dominatingrole: It cannot be switched off easily and, moreover, inmany situations, not even the direction of noise-inducedtransport is obvious! Because the direction and speed oftransport depend on different externally controllable pa-rameters—temperature, pressure, light, and the phase,frequency, and amplitude of the external modulation—as

38 NOVEMBER 2002 PHYSICS TODAY http://www.physicstoday.org

0.1

0

–0.10 1 2 3

T (K)

CU

RR

EN

T(n

A)

FIGURE 4. IN A QUANTUM RATCHET, tunneling can con-tribute to electron current. The two contributions to the time-

averaged net current—thermal activation over, and tunnelingthrough, the barriers—flow in opposite directions. Because therelative strengths of the two contributions depend on the elec-trons’ energy distribution, the direction of the net current can

be controlled by tuning the temperature, as shown in thegraph. Below the graph is a scanning electron micrograph of

the quantum ratchet discussed in ref. 16. (Courtesy of HeinerLinke, University of Oregon.)

well as on the characteristics of the potential and on theinternal degrees of freedom of the motor itself, syntheticBrownian molecular motors can be remarkably versatile.18

In the microscopic world, “There must be no cessa-tion / Of motion, or of the noise of motion” (box 1). Ratherthan fighting it, Brownian motors take advantage of theceaseless noise to move particles efficiently and reliably.

R.D.A. thanks Anita Goel for many stimulating discussionsand Ray Goldstein for an inspiring series of lectures onbiopolymers. We thank our colleagues for their help and com-ments, particularly Howard Berg, Hans von Baeyer, ImreDerényi, Igor Goychuk, Dudley Herschbach, Gert-Ludwig Ingold, Heiner Linke, Manuel Morillo, Peter Reimann, PeterTalkner, and Tian Tsong.

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