The NonequilibriumThermodynamics of Small SystemsThe interactions of tiny objects with their environments aredominated by thermai fiuctuations. Guided by theory andassisted by new micromanlpuiation toois, scientists havebegun to study such interactions in detail.

Carlos Bustamante, Jan Liphardt, and Felix Ritort

Small systems found throughout physics, chemistry, andbiology manifest striking properties as a result of their

tiny dimensions. Examples of such systems include mag-netic domains in ferromagnets, which are typically smallerthan 300 nm; quantum dots and biological molecular ma-chines tbat range in size from 2 to 100 nm; and solidlikeclusters tbat are important in the relaxation of glassy sys-tems and whose dimensions are a few nanometers. Scien-tists nowadays are interested in understanding tbe prop-erties of sucb small systems. For example, tbey arebeginning to investigate tbe dynamics of tbe biological mo-tors responsible for converting cbemical energy into use-ful work in tbe cell (see tbe article by Terence Strick, Jean-Franfois Allemand, Vincent Croquette, and DavidBensimon, PHYSICS TODAY October 2001, page 46). Thosemotors operate away from equilibrium, dissipate energycontinuously, and make transitions between steady states.

Until the early 1990s, researchers had lacked experi-mental methods to investigate such properties of smallsystems as bow they exchange beat and work with theirenvironments. The development of modern techniques ofmicroscopic manipulation has changed the experimentalsituation. In parallel, during the past decade, theoristsbave developed several results collectively known as fluc-tuation theorems (FTs), some of which have been experi-mentally tested. Tbe much-improved experimental accessto tbe energy fluctuations of small systems and the for-mulation of the principles tbat govern both energy ex-changes and tbeir statistical excursions are starting toshed light on the unique properties of microscopic systems.Ultimately, tbe knowledge pbysicists are gaining withtheir new experimental and theoretical tools may serve asthe basis for a tbeory of the nonequilihrium thermody-namics of small systems.

Molecuiar maciiinesThermodynamics describes energy exchange processes ofmacroscopic systems: Objects as varied as liquids, mag-nets, superconductors, and even hlack boles comply withits laws. In macroscopic systems, behavior is reproducibleand fluctuations (deviations from the typically observed,average hebavior) are small. It is only under some special

Carlos Bustamante is a Howard Hughes Medical Institute inves-tigator and a professor of molecular and cell biology, chemistry,and physics ai the University of California, Berkeley. JanLiphardt is an assistant professor of biophysios, also at the Uni-versity of California, Berkeley. Felix Ritort is a professor of statis-tical physics at the University of Barcelona in Spain.

conditions tbat thermal fluctuationsproduce readily detectable conse-quences in macroscopic systems. Well-known examples include the opales-cence of light in a fluid at its criticalpoint and the blue color of the sky,which is a result of light scattering.

As a system's dimensions de-crease, fluctuations away from equi-

librium begin to dominate its behavior. In particular, in anonequilibrium small system, thermal fluctuations canlead to observable and significant deviations from the sys-tem's average behavior. Therefore, sucb systems are notwell described by classical thermodynamics. Systems oftbis type abound in the laboratory, where scientists arebuilding motors witb dimensions of less than 100 nm (seefigure la), and in the cell, where the biological functionand efficiency of molecules sucb as the molecular motor ki-nesin are determined by molecular size (figure lb).

Kinesin is one of many molecular macbines. In thecell, those macbines use tbe energy of bond hydrolysis toperform useful work such as the replication, transcription,and repair of DNA and the translation of RNA. Kinesin'srole is to carry subcellular cargoes along microtubules. Onaverage, a kinesin motor takes one 8-nm step every 10-15milliseconds. A single adenosinetriphospbate (ATP) mole-cule is hydrolyzed per step, and the cbemical energy re-leased is tigbtly coupled to movement and force genera-tion. The kinesin is highly processive—that is, it takesmany steps before detacbing from its microtubule track.

How efficient is tbe kinesin motor and bow mucb en-ergy does it dissipate as it moves along tbe track? Thechemical energy released by ATP hydrolysis is about20kgT. (In the world of small systems, tbe product of Boltz-mann's constant and temperature is a convenient energyunit.) Tbe motor does about 12 fejjTof work witb eacb step.Tbus, tbe machine's efficiency is roughly 60% and it dissi-pates about 650 k^T per second into its environment.

Molecular machines are unlike macroscopic ones inthat they can harness thermal fluctuations and rectifythem using energy from chemical sources. Consider, for ex-ample, RNA polymerase, an enzyme tbat moves alongDNA to produce a newly syntbesized RNA strand—aprocess called transcription. Although it has not yet beenproven unequivocally, evidence suggests tbat during tran-scription tbe polymerase moves by extracting energy fromthe thermal bath and uses bond hydrolysis to ensure thatonly "forward" fluctuations are captured. That is, the en-zyme rectifies tbermal fluctuations to directed motion. Theamount of energy required to step from one DNA base toanotber, the shape of tbe enzyme, tbe structural rougbnessof DNA, and tbe information encoded in the steps (tbat is,the base sequence of the DNA helix) are all essential as-pects tbat are attributable to tbe smallness of tbe systemand that ultimately determine its dynamics.

As witb macroscopic systems, for small systems onecan distinguisb between two situations in wbich tbesystems' bebavior and properties do not change with

© 2005 American Institute ot Physics, S-0031 -9228-0507-020-1 July 2005 Physics Today 43

Figure 1. Nonequilibrium small systems, (a) In this scanning electron microscope image of an integrated synthetic actuator,the central metal-plate rotor is attached to a multiwalled carbon nanotube (MWNT) that acts as a support shaft. Electricalcontact to the rotor plate is made via the MWNT and its anchor pads. A synchronized electrostatic force can induce rotarymotion about the axis of the MWNT. (Adapted from ref. 16.) (b) This artist's rendition, based on crystallographic studies,shows a kinesin motor walking along a microtubule in a hand-over-hand (blue regions) fashion to carry organelles and othercargo tVom one part of the cell to another. Every step of the motor involves the hydrolytic conversion of chemical energyfrom adenosinetriphosphate into mechanical work. The reaction cycle is completed with the release of the hydrolysis prod-uct, adenosinediphosphate (ADP). Each of the motor's hands Is about 5 nm long.

time: equilibrium states and nonequilibrium steadystates. Systems in nonequilibrium steady states have netcurrents that flow across them, but their properties do notdisplay observable time dependence. Examples of non-equilibrium steady-state systems include an object in con-tact with two thermal sources at different temperatures,for which the current is a heat flux; a resistor with elec-tric current flowing across it; and the kinesin-microtubulesystem, for which kinesin motion is the current. Such sys-tems require a constant input of energy to maintain theirsteady state. Because the systems constantly dissipate netenergy, they operate away from equilibrium. Most biolog-ical systems, including molecular machines and evenwhole cells, are nonequilibrium steady states. In a non-steady-state system, the most general case, one or more ofthe system's properties change in time.

Figure 2 sbows various thermal systems classified ac-cording to tbeir size and typical dissipation energy rate,along witb a couple of macroscopic systems for compari-son. Most of the small systems are characterized by lengthscales in the nanometer-to-micrometer range and dissipa-tion rates of 10-1000 kj"!?..

The state of a small systemExternal variables sucb as temperature, pressure, andchemical potential specify the different ensembles in sta-tistical mechanics. All such ensembles yield the sameequation of state in the large-volume or thermodynamicHmit. For small systems, the equation of state and tbespectrum of fluctuations are fully determined by so-calledcontrol parameters. Figure 3 illustrates two control pa-rameters that may be used to define the state of a smallstretched polymer.

By giving direct access to control parameters of singlemicroscopic systems, micromanipulation technology hasopened up new opportunities to study nonequilibriumsmall systems. By varying such parameters, one can per-form controlled experiments in whicb tbe system is drivenaway from its initial state of equilibrium and its subse-quent response is observed.'

Consider, for example, the system illustrated infigure 3a, and imagine that a tethered polymer initially at

equilibrium is driven out of equilibrium by the action of anexternal perturbation that moves tbe two walls fartberapart. In that situation, the control parameter is tbe dis-tance between tbe two walls, and the nonequilibrium pro-tocol is fully specified by giving tbe wall-to-wall distanceas a function of time Xit). Since the system is small and isplaced in a thermal bath, its dynamics will be effectivelyrandom: Even if tbe nonequilibrium protocol is exactly re-peated, tbe trajectory followed by the system will be dif-f'erent. Each trajectory may be represented by the timeevolution of the positions of all atoms (:c,(f)|. As the controlparameter X evolves, tbe total energy of tbe system,U{\x},X), varies in a manner tbat has two distinctcontributions,

dW. (1)

The first term is tbe variation of the energy resultingfrom the cbange in the internal configuration—that is, acbange in beat Q—and the second term is the variation oftbe energy resulting from tbe perturbation applied whilevarying the control parameter—that is, a change in workW. If tbe control parameter changes from 0 to Xf, tbe totalwork done on the system is given by

FdX, (2)

where ((iU/fiX)^,^^ is the force applied to move the walls. Theheat Q exchanged in tbe nonequilibrium process may be ex-pressed as At/ - W, where A[/is the variation of the energy.

Random fluctuations dominate tbe tbermal behaviorin small systems. Since the force is a fluctuating quantity,W, Q, and AC/ will also fluctuate for different trajectories,and tbe amount of heat or work exchanged witb the bathwill fluctuate in magnitude and even sign. For a given non-equilibrium protocol, the work and heat probability dis-tributions PiW) and PiQ) cbaracterize tbe work and beatcollected over an infinite number of experiments. In gen-eral, tbose distributions will depend on the details of tbeexperimental protocol. Distributions sucb as PiW) andP{Q} are important for providing detailed information

44 July 2005 Physics Today http://www.physicstoday.org

about bow a system responds when subjected to a partic-ular experimental process.

Fluctuation theoremsNonequilibrium systems are characterized by irreversibleheat losses hetween the system and its environment, typ-ically a thermal bath. Fluctuation theorems embody recentdevelopments toward a unified treatment of arbitrarilylarge fluctuations in small systems.

In equilibrated, time-reversal-invariant systems, nonet beat is transferred from the system to the bath. There-fore the probability of absorbing a given amount of heatmust be identical to that of releasing it, and the ratioP(Q)IP{ -Q) equals 1. The probability ratio becomes differ-ent from 1 under nonequilibrium conditions. We assumetime-reversal invariance, but note tbat for equilibrated,noninvariant systems—if, for example, magnetic fields arepresent—it is the total probability for beat absorption thatis equated witb the total probability for beat release.

Tbe mid-1990s saw the introduction of two importantFTs. Denis Evans and Debra Searles derived an FT for sys-tems evolving from equilibrium toward a nonequilibriumsteady state, and Giovanni Gallavotti and Eddie Coben de-veloped an FT for steady-state systems.^ The two workswere based on numerical evidence obtained previously.^ Insteady-state systems, an external agent continuously pro-duces beat that is transferred to tbe bath. Tbe averageamount of beat <Q> so produced implies an increase in tbetotal average entropy of tbe system plus environmentequal to <S> - <Q>IT.

Tbe rate at which the system exchanges heat witb thebath is called the entropy production, a = QITt, where t istbe interval of time over wbich tbe system exchanges tbebeat Q. Associated with the entropy production is a time-dependent probability distribution P /a ) , Gallavotti andCohen established an explicit mathematical expressiontha t holds under very general conditions for the ratioP,{(T]/P,{-a) in steady states.

lim — lnI-co t

(3)

Although this expression involves a limit of infinite time,a similar expression without the limit should be valid, togood approximation, as long as t is much greater than thedecorrelation time, which is, roughly speaking, tbe recov-ery time of tbe steady state after it is slightly perturbed.

Equation 3 indi-cates tbat a steady-statesystem is more likely todeliver heat to tbe bathUr is positive) tban it isto absorb an equal quan-tity of beat from thebath (tr is negative).Nonequilibrium steady-s ta te systems alwaysdissipate heat on aver-age. For macroscopicsystems, the heat is anextensive quantity andtherefore the rat io ofprobabilities Pi(j)/P(-a)grows exponentiallywitb the system's size.That is to say, tbe prob-ability of heat absorp-tion by macroscopic sys-tems is insignificant.

10--

10"-

1 0 - -

10-12-

io-i«-

10-24.

1 k

• Steam engine

Car engine

E. coli bacterium swimming in water

1- m bead dragged \tbrough water at \ \a speed of 1 ^irci/s \ , ^ \

\ Smallest artiiicial motor

Single R N A Ihairpin 1

Kinesin \

100 lo-.T 10-6CHARACTERISTIC LENGTH (m)

10-9

Figure 2. Thermodynamic systems characterized accord-ing to their typical length scales and energy dissipationrates. The two systems set off by boxes have been used totest fluctuation theorems and tbe Jarzynski equality as de-scribed in the text.

Our bodies, for example, are maintained in a nonequilib-rium state by metabolic processes that dissipate beat allthe time. For small systems such as molecular motors tbatmove along a molecular track, however, the probability ofabsorbing heat can be significant. On average, molecularmotors produce heat, but it may be that tbey move by rec-tifying thermal fluctuations—^a process tbat would implythe occasional capture of beat from tbe bath.

Fluctuation theorems shed light on Loschmidt's para-dox. In 1876, Josef Loscbmidt raised an objection to Boltz-mann's derivation of tbe second law of tbermodynamicsfrom Newton's laws of motion. According to Loscbmidt,since tbe microscopic laws of mechanics are invariantunder time reversal, there must also exist entropy-decreasing evolutions tbat apparently violate tbe secondlaw. Tbe FTs show bow macroscopic irreversibility arisesfrom time-reversible microscopic equations of motion.Time-reversed trajectories do occur, but they become van-

Control parameter: end-to-end distance X Control parameter:external force

Figure 3. Control parameters for a stretched polymer, (a) The end-to-end distance X is the fixed control parameter. In experiments, one canvary X by moving the walls. When X is the control parameter, theforce acting on a bead that attaches the polymer to the wall is a fluc-tuating variable, (b) Here the polymer is fixed at one wall and thecontrol parameter is the force acting on a bead attached to the poly-mer at its free end. Experimentally, one can fix that force by using amagnetic bead with magnetic moment equal to fx and applying a uni-form external magnetic field gradient HBJciz. By changing the value ofthe gradient, one controls the force F acting on the bead. For a fixedF, the polymer's extension is a fluctuating variable. In general, the re-lation between force and extension for a small system whose re-sponse is not linear will depend on which variable is the control pa-rameter and which is allowed to fluctuate. (Adapted from ref. 17.)

http://www.physicstoday.org July 2005 Physics Today 45

Trap bead

Handle

RNAmolecule

Actuatorbead

Figure 4. Testing the Jarzynski equality. A molecule of RNAis attached to two beads and subjected to reversible and ir-reversible cycles of folding and unfolding. A piezoelectricactuator controls the position of the bottom bead, which,when moved, stretches the RNA. An optical trap formed bytwo opposing lasers captures the top bead, and the changein momentum of light that exits the two-beam trap deter-mines the force exerted on the molecule connecting the twobeads. The difference in positions of the bottom and topbeads gives the end-to-end length of the molecule. Theblowup shows how the RNA molecule (green) is coupledwith the two beads via molecular handles (blue). The han-dles end in chemical groups (red) that can be stuck to com-plementary groups (yellow) on the bead. The blowup is notto scale: The diameter of the beads is around 3000 nm,much greater than the 20-nm length of the RNA.

ishingly rare with increasing system size. For large sys-tems, the eonventional seeond law emerges.

The Jarzynski equalityThe various FTs that have been reported differ in the de-tails of such considerations as whether the system's dy-namics are stochastic or deterministic, whether the kineticenergy or some other variable is kept constant, andwhether the system is initially prepared in equilibrium orin a nonequilibrium steady state. A novel treatment of dis-sipative processes in nonequilibrium systems was intro-duced in 1997 when Christopher Jarzynski reported a non-equilibrium work relation,^ now called the Jarzynskiequality (JE). (See PHYSICS TODAY, September 2002, page

19.) The JE indicates a practical way to determine free-energy differences. Consider a system, kept in contact witha bath at temperature T, whose equilibrium state is de-termined by a control parameter x. Initially, the control pa-rameter is x^ and the system is in an equilibrium state A.The nonequilibrium process is obtained by changing :r ac-cording to a given protocol xit), from x^ to some final valuejCp. In general, the final state of the system will not be atequilibrium. It will equilibrate to a state B if it is allowedto further evolve with the control parameter fixed at x-^.The JE states that

^G= { exp (4)

where AG is the free-energy difference between the equilib-rium states A and B, and the angle brackets denote anaverage taken over an infinite number of nonequilibriumexperiments repeated under the protocol x(t). Frequently,the JE is recast in the form {expi-W^^^Vk^T) = 1,where W , W - AG is the dissipated work along a giventrajectory.

The exponential average appearing in the JE impliesthat (W) > AG or, equivalently, (W^J > 0, which, for macro-scopic systems, is the statement of the second law of ther-modynamics in terms of free energy and work. An impor-tant consequence of the JE is that, although on averageW^-^ > 0, the equality can only hold if there exist nonequi-librium trajectories with W j < 0. Those trajectories, some-times referred to as transient violations of the second law,represent work fluctuations that ensure the microscopicequations of motion are time-reversal invariant. The re-markable JE implies that one can determine the free-energy difference between initial and final equilibriumstates not just from a reversible or quasi-static process thatconnects those states, but also via a nonequilibrium, irre-versible process that connects them. The ability to bypassreversible paths is of great practical importance.

In 1999, Gavin Crooks related various FTs by deriv-ing a generalized theorem for stochastic microscopicallyreversible dynamics.^ The box below gives details.The past six years have seen further consolidation, andphysicists now understand that neither the details of justwhich quantities are maintained constant during the dy-namics nor the somewhat differing interpretations of en-tropy production, entropy production rate, dissipatedwork, exchanged heat, and so forth lead to fundamentallydistinct FTs.

The Crooks Fluctuation Theorem

Gavin Crooks provided a significant generalization of an important fluctuation theorem (FT) obtained earlier by ChristopherJarzynski. As described in the text, thelarzynski equality (|F) relates the change AG in freeenergy of two equilibrium states

to an appropriate work average calculated with an irreversible path. In the Jarzynski scenario, and also in Crooks's general-ized FT, the system is initially in thermal equilibrium but then driven out of equilibrium by the action of an external agent. LetXpls) denote a time-dependent nonequilibrium "forward" process for which the variable s runs from 0 to some final time t. Theforward process initially acts on an equilibrium state A and it and ends at a state B that is not at equilibrium. In the reverseprocess, the initial state B is allowed to reach equilibrium and the system evolves to a nonequilibrium state A. The nonequi-librium protocol for the reverse process x (5) is time-reversed with respect to the forward one, Xf^^s) = Xf(( - s), so that bothprocesses last for the same time t. Let Pf{W) and PR(IV) stand for the work probability distributions along the forward and re-versed processes respectively. Then the Crooks FT asserts

Pp(W) IW-AC— = exp

The Crooks FT can be manipulated to yield the JE. It also resembles the Gallavotti-Cohen FT (equation 3) derived forsteady-state systems if one identifies o-f with W^JT = {W - AQ/T. The main difference is that the Callavotti-Cohen relationis asymptotically valid, whereas the Crooks theorem holds for any finite time (.

46 July 2005 Physics Today http://www.physicstoday.org

Nonlinear regime

CONTROL PARAMETER, X

10

Ed

Gaussian"fit

-10W-W,

mp

Computer simulations have played an essential rolein the development of the FTs.' ' Indeed, the first paper onthe subject, the 1993 report by Evans, Cohen, and GaryMorriss,^ included molecular-dynamics simulations of atwo-dimensional gas of disks. Over suitably short times,their computer runs showed spontaneous ordering of thegas, in agreement with the expression tbe authors bad de-rived for the probability of fluctuations of a nonequilibriumsteady-state fluid's sbeer stress. Computer simulations ofnonequilibrium systems continue to be important prima-rily due to the difficulty of setting up and characterizingsuitable nonequilibrium small systems. Conversely, FTs,and especially the JE, can potentially be used to improvethe performance of molecular-dynamics simulations.

Experimental tests of fluctuation theoremsTheorists who consider small systems have greatly bene-fited from advances in micromanipulation that make itpossible to measure energy fluctuations in nonequilibriumsmall systems. With such measurements, experimenterscan test tbe validity of FTs and scrutinize some funda-mental assumptions of statistical mechanics. Sergio Cilib-erto and Claude Laroche, in their 1998 study ofRayleigb-Benard convection, performed the first experi-mental test of the Gallavotti-Cohen FT.' In 2002, Evans'sgroup verified an integrated form of equation 3 in an ex-periment that used an optical trap to repeatedly drag mi-croscopic beads through water." Tbey computed tbe en-tropy production for each head trajectory and found thatthe likelihood of entropy-consuming trajectories relative toentropy-producing trajectories was precisely as theory pre-

Figure 5. The nonlinear regime, (a) The v -ork required tostretch a short polymer is a fluctuating function. The plotshows three different nonequilibrium trajectories obtained asthe control parameter X varies from 0 to X,. The continuousblack line is the work averaged over all trajectories. Tbedashed line is the linear behavior described by the quadraticfunction W= (}/2)kX'^. The size of work fiuctuations andthe deviations of the average work from the dashed line aregreater in the nonlinear regime than in the linear regime, (b)In the nonlinear regime, the work probability distributionRW) has a Gaussian component plus long non-Gaussiantails describing rare processes. Such a mix of Gaussian andnon-Gaussian behavior bas been seen in the power fluctua-tions in resistors, the relaxation of glassy systems,Rayleigh-Bernard convection, turbulent flows, and energyfluctuations in granular media. The existence of linear andnonlinear processes occurring along different time scales isreminiscent of the phenomenon of intermittency in turbulentflows and suggests an interesting link between apparentlyunrelated physical phenomena. In the plot, the zero of thehorizontal axis is the most probable (mp) work and the en-ergy unit is k^J. The black dots represent an analytical solu-tion for a nonlinear two-state system, and the solid curve is anon-Gaussian fit. (Adapted from ref. 18.)

dieted. For short times in the millisecond range, the re-searchers readily observed entropy-consuming trajecto-ries. And as expected, tbe classical bulk behavior was re-covered for longer times on the order of seconds.

Gerbard Hummer and Attila Szabo noted the biophys-ical relevance of the JE and showed bow free energies couldbe extracted via single-molecule experiments carried outunder nonequilibrium conditions.^ Soon tbereafter, a groupled by one of us (Bustamante) tested the JE by mecbanicallystretcbing a single molecule of RNA, botb reversibly and ir-reversibly, between its folded and unfolded conformations.'**Figure 4 illustrates the group's experimental design. Whenthe RNA was unfolded slowly, tbe average forward and re-verse trajectories could be superimposed; that is, tbe reac-tion was reversible. When the RNA was unfolded more rap-idly, tbe mean unfolding force increased and the meanrefolding force decreased. Tbe folding-unfolding cycle wasthus hysteretic, an indication that work was dissipated.When Bustamante and coworkers applied the JE to the ir-reversible work trajectories, tbey recovered the free energyof the unfolding process to within kf^T/2 of its best inde-pendent estimate—tbe work needed to reversibly stretchtbe RNA. Their experimental test is an example of how theJE bridges tbe statistica] mechanics of equilibrium and non-equilibrium systems.

Experimenters have continued to progress in theirability to test FTs. Technical improvements have recentlyenabled Evans's group to test equation 3, rather than itsintegrated form." Nicolas Garnier and Ciliberto bave usedelectrical circuits as tbe driven, dissipative system.'•^ Tbeyinjected current to maintain an electrical dipole, composedof a resistor and a capacitor, in a nonequilibrium steadystate; collected the probability distributions of work andheat; and showed those distributions to be in very goodagreement with tbe appropriate FT.

Compared to tests involving trapped beads or stretchedpolymers, experiments with electrical circuits are less proneto drift and other systematic biases, and they permit muchgreater numbers of trajectories. Those advantages allow fortbe investigation of systems with larger dissipation rates.By recording several hours of fluctuation data from theirdriven electrical dipole, Garnier and Ciliberto investigatedtbe exchanged heat and work witb unusually higb resolu-tion, and detected the non-Gaussian tails in the beat

http://www.physicstoday.org July 2005 Physics Today 47

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distribution. Such tails, predicted by Ramses van Zon andCohen for the heat distribution in linear systems,'^ are alsocharacteristic of work distributions in systems with a non-linear response to change in a control parameter. For fur-ther details see figure 5.

Physicists have also deepened their understanding ofthe JE. Jarzynski's result—and Crooks's generalization aswell—applies to systems that start at equilibrium and arethen driven out of equilibrium by some external influence.For many systems of interest, however, including biologi-cal molecular machines and nanophotonics devices, the JEdoes not apply. Such systems often execute irreversibletransitions between nonequilibrium steady states. In1998, Yoshitsugu Oono and Marco Paniconi proposed ageneral phenomenologica! framework that encompassednonequihbrium steady states and transitions betweensuch states. Three years later, Takahiro Hatano and Shin-ichi Sasa built on that work and generalized the JE to ar-bitrary transitions between nonequilibrium steadystates.^'* Tbeir result was tested and confirmed last yearby a measurement of the dissipation and fluctuations ofmicrospheres optically driven through water.'"^ Those the-oretical and experimental advances represent steps to-ward a complete theory of steady-state thermodynamics.Such a theory would have a profound effect on how scien-tists describe nonequilibrium steady-state systems such asmolecular machines and cells.

Although we have not discussed them, several quan-tum versions of the classical FTs have appeared in thephysics literature. To date, no quantum FT has yet re-ceived experimental scrutiny, but such experiments migbtshow interesting surprises. For example, quantum coher-ence may allow large fluctuations to be observed and FTsto be tested in much larger systems than would be possi-ble in a classical world. That and many other exciting chal-lenges remain for scientists continuing to work with smallsystems, a fertile ground where physics, chemistry, and bi-ology converge.

References1. For a review, see F. Ritort, Seminaire Poincare 2, 193 (2003),

available at http://arXiv.org/abs/cond-maty0401311.2. D. J. Evans, D. J. Searles, Phy.s. Rev. E 50, 1645 (1994); G.

Gallavotti, E. G. D. Cohen, Phys. Rev. Lett. 74, 2694 (1995).For a review of fluctuation theorems, see D. J. Evans, D. J.Searles, Adv. Phys. 51, 1529 (2002).

3. D. J. Evans, E. G. D. Cohen, G. P. Morriss, Phys. Rev. Lett.71,2401(1993).

4. C. Jarzynski, Phys. Rev. Utt. 78, 2690 (1997).5. G. E. Crooks, Phys. Rev. E 60, 2721 (1999).6. S. Park, K. Schulten, J. Chem. Phys. 120, 5946 (2004).7. S. Ciliberto, C. Larocbe, J. Physique W 8, 215 (1998).8. G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, D. J.

Evans, Phys. Rev. Lett. 89, 050601 (2002).9. G. Hummer, A. Szabo, Proc. Nail. Acad. Sci. USA 98, 3658

(2001).10. J, Liphardt, S. Dumont, S. B. Smith, I. Tinoco Jr, C. Busta-

mante. Science 296, 1832 (2002).11. D. M. Carberry, J, C. Reid, G. M. Wang, E. M. Sevick, D. J.

Searles, D. J. Evans, Phys. Rev. Lett. 92, 140601 (2004).12. N. Gamier, S. Ciliberto, Phys. Rev. E 71, 060101 (2005).13. R. van Zon, E. G. D. Cohen, Phys. Rev. E 67, 046102 (2003),14. Y. Oono, M. Paniconi, Prog. Theor. Phys. Suppl. 130, 29

(1998); T. Hatano, S. Sasa, Phys. Rev. Lett. 86, 3463 (2001).15. E. H. Trepagnier, C. Jarzynski, F. Ritort, G. E. Crooks, C.

Bustamante, J. Liphardt, Proc. Natl. Acad. Sci. USA 101,15038(2004).

16. A. M. Fennimore, T. D. Yuzvinsky, W.-Q. Han, M. S. Fuhrer,J. Cumings, A. Zettl, Nature 424, 408 (2003),

17. D. Keller, D. Swigon, C. Bustamante, Biophys. J. 84, 733(2003).

18. F. Ritort, J. Stat. Mech. P10016 (2004), •

48 July 2005 Physics Today