-
Agreat challenge for theburgeoning field ofnanotechnology 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
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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
periodicone-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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