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Fluctuations and Irreversibility: An Experimental Demonstration of a Second-Law-LikeTheorem Using a Colloidal Particle Held in an Optical Trap

D. M. Ca rberr y,1 J. C. Reid,1 G. M. Wang,1 E. M. Sevick,1 Debra J. Searles,2 and Denis J. Evans1

1 Research School of Chemist ry, The Australia n National University, Canber ra ACT 0200 Australia2School of Science, Griffith University, Brisbane QLD 4111 Australia

(Received 9 September 2003; published 6 April 2004)

The puzzle of how time-reversible microscopic equations of mechanics lead to the time-irreversiblemacroscopic equations of thermodynamics has been a paradox since the days of Boltzmann. Boltzmannsimply sidestepped this enigma by stating ‘‘as soon as one looks at bodies of such small dimension thatthey contain only very few molecules, the validity of this theorem [the second law of thermodynamicsand its description of irreversibility] must cease.’’ Today we can state that the transient fluctuationtheorem (TFT) of Evans and Searles is a generalized, second-law-like theorem that bridges themicroscopic and macroscopic domains and links the time-reversible and irreversible descriptions. Weapply th is theorem to a colloidal par ticle in an optical trap. For the first time, we demonstrate the TFT inan experiment and show quantitative agreement with Langevin dynamics.

DOI: 10.1103/ PhysRevLett.92.140601 PACS numbers: 05.70.Ln, 05.40.– a

The second law of thermodynamics [1,2] states that forsystems in the thermodynamic limit, the entropy produc-

tion must be greater than or equal to zero. Although theunderlying equations of motion are time reversible, thesecond law predicts an irreversible entropy production.This law applies to systems that are of infinite size andpersist over long times. Despite this strict l imitation, t hesecond law is often treated as being universal in applica-tion because the size of most systems can be consideredinfinite when compared to atomic length and time scales.However, several systems of current scientific interest,such as nanomachines and protein motors, operate atlength and time scales where the system cannot be con-sidered infinite. At the nano- and microscales the thermalenergy available per degree of freedom can be compa-rable to the work performed by the system. Classicalthermodynamics does not apply to these small systems.

In 1993 the first quantitative description of entropyproduction in finite systems was given by the fluctuationtheorem (FT) of Evans et al. [3]. In its most general form,the theorem provides an analytic expression for theprobability that a dissipative flux flows in the directionopposite to that required by the second law of thermody-namics. For thermostated dissipative systems [4] thetheorem relates the probability of observing a process of duration t with entropy production t A dA, Pt A, to that of a process with the same magnitude of

entropy change, but where the entropy is consumed,rather than produced:

Pt A

Pt A exp A: (1)

Since entropy production is extensive and the total en-tropy production increases with time, the FT predicts thatpositive entropy production will be overwhelminglylikely as either the system size or the observation time

increases. In th is way t he FT can be viewed as a general-ization of the second law since the FT applies to finite

systems observed over finite time and trivially recoversthe second law in the thermodynamic limit.When appliedto the t ransient response of a system, the t heorem isreferred to as the transient fluctuation theorem or TFT [2].

The FT has been verified in a number of moleculardynamics (MD) simulations, and it is possible to demon-strate the predictions of the theorem in an experimentinvolving a colloidal particle and an optical t rap. Anoptical trap is formed when a transparent, micron-sizedparticle, whose index of refraction is greater than that of the surrounding medium, is located within a focusedlaser beam. The refracted rays differ in intensity overthe volume of the sphere and exert a sub-picoNewtonforce (pN 1012 N) on the par ticle, drawing it t owardsthe region of highest light intensity. The optical trap isharmonic near the focal point; i.e., the potential energy of the system is 1

2kr r, where r is the vector position of the

particle relative to the origin-centered trap, and k isthe trapping constant which can be tuned by adjustingthe laser power.Wang et al. [5] recorded the t rajectories of a colloidal particle localized in a translating optical trapand evaluated the entropy production for a large numberof trajectories. They were unable to demonstrate the TF T,but their results agreed quantitatively with a n integratedform of the TFT (IT FT). This ITF T provides a prediction

of the ratio of the frequency of entropy-consuming trajectories to that of entropy-generati ng trajectories:

Pt < 0

Pt > 0 hexptit>0; (2)

where the angular brackets on the right-hand side (rhs)denote an average accumulated over entropy-producingtrajectories.

In this Letter, we demonstrate the TF T, E q. (1), directlyby observing the time-dependent relaxation of a colloidal

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particle under a step change in the strength of a stationaryoptical trap. For this experiment, a particle is localized ina stationary trap of strength k0 over a sufficiently longtime so that its position is described by an equilibriumdistribution. At t 0 the optical trap strength is increaseddiscontinuously from k0 to k1 so that we more tightlyconfine or ‘‘capture’’ the particle. The particle’s positionis recorded as it relaxes to its new equilibrium distribu-

tion, and we evaluate the function t over an ensemble of experiments. We demonstrate for the first time that t,derived consistently from Langevin dynamics, evolves intime according to the TFT, as well as the ITFT, Eqs. (1)and (2). In addition, we show for the first time that theexperimental results and TFT predictions are in quanti-tative agreement with Langevin dynamics.

The TFT is a simple but general relation that is derivedexactly f rom the laws of mechanics. For thermostateddissipative systems such as that of Wang et al. [5], theargument of the theorem, t, is identified with the en-tropy production and its average, hti, is identical to theirreversible entropy production [6]. The relationship be-

tween the fluctuation theorem and entropy has been ex-amined by several researchers [7–9]. For more generalsystems, as in the experiment described here, t is iden-tified as the dissipation function, having properties thatare strikingly similar to the entropy production. Motionof a system which is consistent with the second law isassociated with a positive dissipation function, analogousto entropy generation. A negative dissipation function isindicative of motion contrary to the second law and issimilar to entropy consumption. And finally, like entropyproduction, the average of the dissipation function is apositive quantity. Regardless of whether t is a general-ized dissipation function or the entropy production, tquantifies the irreversibility of the system. The TFT is arelation describing how t evolves in time, and in thisway the theorem describes the transition from time-reversible, microscopic equations of motion to irrevers-ible macroscopic behavior. The puzzle of how thetime-reversible microscopic equations of quantum andclassical mechanics lead to the time-irreversible macro-scopic equations of thermodynamics and hydrodynamicshas been a paradox since the days of Boltzmann. Boltz-mann simply sidestepped this enigma by stating ‘‘as soonas one looks at bodies of such small di mension that theycontain only very few molecules, the validity of thistheorem [the second law and its description of irreversi-bility] must cease’’ [1]. Today we can state that the TFT isa generalized, second-law-like theorem that bridges themicroscopic and macroscopic domains and links thetime-reversible and irreversible descriptions.

A system’s trajectory can be expressed in terms of thecoordinates q, and momenta, p, of all constituent mole-cules, including solvent, as well as the thermostat em-ployed t o describe the system’s heat exchange with itssurroundings. Figure 1 illustrates these sets of trajector iesin the q;p space. A system’s t rajectory t from an

initial state q0;p0 at t 0 to a state qt;pt some time tlater is a solution of Newton’s equations of motion. AsLoschmidt [10] pointed out in 1876, since these equationsare time reversible, for every trajectory t that satisfiesthe equations of motion, there is a time-reversed trajec-tory or antitrajectory which is also a solution to theequations. This antitrajectory, represented by t,evolves from an initial state qt;pt to a final state

q0;p0. Thus, t and

t represent a pair of conjugate trajectories that are related by time reversal.Consider a set of trajectories that start inside an infini-tesimal volume V q0p0 about an initial stateq0;p0, and the set of conjugate trajectories that startinside the volume element V qtpt about qt;pt.Every trajectory initiated within V has a conjugatetrajectory initiated in V . The ratio of the probabilityof observing trajectories initiated within V;PV , tothose withi n V is the dissipation function,

tq0;p0 ln

PV

PV

: (3)

If the probability of observing a trajectory in V is equalto the probability of observing one in V , then thesystem is thermodynamically reversible and t 0. Arecent review [11] rigorously derives the probabilitydensities of trajectories and the resulting dissipationfunctions from reversible equations of motion under vari-ous statistical ensembles.

FIG. 1 (color online). An illustration of a set of neighboringtrajectories initiated in a volume element V (top tube) and thecorresponding set of antitrajectories initiated in V (lowertube) in coordinate momentum q;p and time, t, space. Theratio of the probability of observing trajectories initiatedwithin V to those in V is a measure of the system’s ir-reversibility. Every trajectory in the q;p space corresponds toa solution of Newton’s time-reversible equations of motion.Thus, for every trajectory t that starts at q0; p0 and endsat qt;pt some time t later, there exists its time-reversed orantitrajectory t. This antitrajectory starts at qt;pt andends at q0;p0 at time t.


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provides the data necessary for analysis of the TFT. Therelaxation time of the particle in the strong trap, =k1, is 37.5 ms, where the friction coefficient was de-termined by Stokes drag as 1:05 107 Ns=m.

Figure 2 is a histogram of the values of the dissipationfunction, t, evaluated over 3300 experimental trajecto-ries where t he trap strength is increased from k0 to k1 att 0. The distribution of t is strongly peaked at t 0and is clearly non-Gaussian. At early times, the distribu-tion of t is roughly symmetric, showing an e qual num-ber of trajectories with positive and negative t. As timeincreases, the distribution of t becomes skewed towardspositive values, showing that the system is increasingly

characterized as irreversible. The average value of thedissipation function is always greater than zero and atlong times reaches a plateau value.

This histogram data can be recast to directly demon-strate the TFT, Eq. (1). Let N i be the number of experi-mental trajectories with dissipation function evaluated tobe between t;i =2 and t;i =2, where is thesize of the histogram bin in dimensionless units of t andt;i i. T he experimental analog of t he left-hand side(lhs) of the TFT then becomes lnN i=N i, where theargument of the logarithm is the ratio of the number of trajectories in a pair of histogram bins. Figure 3(a) is t helogarithm of the ratio of populations of pairs of bins,

lnN i=N i, versus the value the dissipation function forthe ith bin, t;i, evaluated for trajectories of duration t 26 ms. Also shown in the figure is a line of slope unity,representing the T FT prediction. The previous experi-mental work [5] only confirmed the integrated form of the TFT.

The ITFT, Eq. (2), is also demonstrated from theexperiments and the result is shown in Fig. 3(b). Over

the complete 10 s duration of the trajectories, the experi-mental values of the lhs and rhs of the ITF T are nearlyequivalent, thereby demonstrating the ITFT [20].Both show that the relaxation of hti is roughly on thetime scale of the motion of the colloidal particle; i.e., 37:5 ms.

The TFT quantitatively describes how irreversiblemacroscopic behavior evolves from time-reversible mi-

croscopic dynamics as either the system size or the ob-servation time increases. This experiment confirms, forthe first time, the predictions of this theorem. Further-more, these experiments demonstrate that the effectspredicted occur over colloidal length and time scalesand, consequently, show that t he TF T is relevant to nano-technological applications.

The authors thank Russell Koehne and Deshan Lu of the RSC Workshop for equipment modification and ac-knowledge financial support through a Discovery Grantfrom the Australian Research Council (ARC).

[1] E. Broda, Ludwig Boltzmann: Man—Physicist— Philosopher (Ox Bow Press, Woodbridge, 1983).

[2] D. J. Evans and D. J. Searles, Phys. Rev. E 50, 1645(1994).

[3] D. J. Evans, E. G. D. Cohen, and G. P. Mor ris, Phys. Rev.Lett. 71, 2401 (1993).

[4] A thermostated dissipative system is a system in whichwork is turned into heat, in the presence of a heat baththat maintains the system at a constant temperature.

[5] G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, andD. J. Evans, Phys. Rev. Lett. 89, 050601 (2002).

[6] S. R. de Groot and P. Mazur, Non-equilibrium Thermo-dynamics (Dover Publications, Mineola, NY, 1984).

[7] G. Crooks, Phys. Rev. E 60, 2721 (1999).[8] C. Maes, F. Redig, and A. V. Moffaert, J. Math. Phys.

(N.Y.) 41, 1528 (2000).[9] C. Maes and F. Redig, J. Stat. Phys. 101, 3 (2000).

[10] J. Loschmidt, J. Sitzungsber. der kais. Akad. d. W. Math.Naturw. II 73, 128 (1876).

[11] D. J. Evans and D. J. Searles, Adv. Phys. 51, 1529 (2002).[12] In this system the inertia time scale is less t han 1 ms.

This is smaller t han our sampling rate and the inertialessLangevin equation can therefore be used.

[13] J. L. Lebowitz and H. Spohn, J. Stat. Phys. 95, 333 (1999).[14] G. Crooks, Phys. Rev. E 61, 2361 (2000).[15] C. Maes, J. Stat. Phys. 95, 367 (1999).[16] J. Kurchan, J. Phys. A 31, 3719 (1998).[17] R. van Zon and E. G. D. Cohen, Phys. Rev. E 67, 046102

(2003).[18] S. Tasaki , I. Terasaki , and T. Monnai, cond-mat/0208154.[19] M. Doi and S. F. Edwards, The Theory of Polymer

Dynam ics (Oxford University Press, Oxford, 1988).[20] J. Reid, D. M. Carberr y, G. M. Wang, E. M. Sevick, D. J.

Searles, and D. J. Evans (to be published).

(a) (b)

FIG. 3 (color online). (a) Logarithm of the ratio of number of

trajectories in histogram bins, lnN i=N i, versus the value of the dissipation function associated with the ith bin, t;i,evaluated for trajectories of time duration t 26 ms. T heline of best fit has a slope of 0:98 0:04 which agrees withthe TFT predicted slope of 1 (shown as a l ine). This is the firstdirect demonstration of the TFT. (b) The lhs and rhs of the

ITFT, N i0 (4) and hexptit>0 ( ) versus ti me, t.


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