An unresolved problem in physics is how the thermodynamic arrow of time
arises from an underlying time reversible dynamics. We contribute to this issue
by developing a measure of time-symmetry breaking, and by using the work
fluctuation relations, we determine the time asymmetry of recent single
molecule RNA unfolding experiments. We define time asymmetry as the
Jensen-Shannon divergence between trajectory probability distributions of an
experiment and its time-reversed conjugate. Among other interesting properties,
the length of time's arrow bounds the average dissipation and determines the
difficulty of accurately estimating free energy differences in nonequilibrium
experiments

A. S. Eddington

Edward H. Feng

Gavin E. Crooks

H. Jeffreys

H. Price

I. J. Taneja

T. M. Cover

English

arXiv

An unresolved problem in physics is how the thermodynamic arrow of time arises from an underlying time reversible dynamics. We contribute to this issue by developing a measure of time-symmetry breaking, and by using the work fluctuation relations, we determine the time asymmetry of recent single molecule RNA unfolding experiments. We define time asymmetry as the Jensen-Shannon divergencebetween trajectory probability distributions of an experiment and its time-reversed conjugate. Among other interesting properties, the length of time's arrow bounds the average dissipation and determines the difficulty of accurately estimating free energy differences in nonequilibrium experiments

Feng, Edward H.

Crooks, Gavin E.

UNT (University of North Texas) Digital Library

arXiv:0809.0025v1  [cond-mat.stat-mech]  29 Aug 2008The length of time’s arrowEdward H. FengCollege of Chemistry, University of California, Berkeley, Berkeley, California, 94720, USAGavin E. CrooksPhysical Biosciences Division, Lawrence Berkeley National Laboratory, Berkeley, California, 94720, USA(Dated: August 29, 2008)An unresolved problem in physics is how the thermodynamic arrow of time arises from an un-derlying time reversible dynamics. We contribute to this issue by developing a measure of timesymmetry breaking, and by using the work fluctuation relations, we determine the time asym-metry of recent single molecule RNA unfolding experiments. We define time asymmetry as theJensen-Shannon divergence between trajectory probability distributions of an experiment and itstime-reversed conjugate. Among other interesting properties, the length of time’s arrow bounds theaverage dissipation and determines the di!culty of accurately estimating free energy di"erences innon-equilibrium experiments.PACS numbers: 05.70.Ln, 05.40.-aIn our everyday lives we have the sense that time flowsinexorably from the past into the future; water flowsdownhill; mountains erode; we are born, grow old anddie; we anticipate the future but remember the past.Yet almost all of the fundamental theories of physics –classical mechanics, electrodynamics, quantum mechan-ics, general relativity and so on – are symmetric withrespect to time reversal. The only fundamental theorythat picks out a preferred direction of time is the secondlaw of thermodynamics, which asserts that the entropy ofthe universe increases as time flows towards the future [1].This provides an orientation, or arrow of time, and it isgenerally believed that all other time asymmetries, suchas our sense that future and past are di!erent, are a di-rect consequence of this thermodynamic arrow [2, 3].When the dissipation, or the total increase in entropy,is large, the orientation of time’s arrow is self evident. Ifwe watch a movie in which shards of pottery jump o! thefloor, assemble themselves into a cup, and land on a table,then clearly someone has threaded the film through theprojector backwards. On the other hand, if the dissipa-tion is microscopic, then the distinction between past andfuture becomes nebulous. This is because a more generalstatement the second law claims the dissipation is pos-itive on average, !"Stotal" # 0 [8, 9]. If we repeat thesame experiment many times, the entropy might increaseor decrease on di!erent occasions. Only the average dis-sipation must be positive. Thus, if we view a movie of amicroscopic system undergoing a dissipative transforma-tion, we cannot determine with certainty whether timemoves forward or backwards.Here, we seek a quantitative measure of time asym-metry in a driven microscopic system such as the singlemolecule RNA pulling experiments explained in Fig. 1.Naively, one might use the average dissipation for thisquantification, but we will show that a large average dis-sipation can arise for dynamics that are essentially timePizeoelectic   ActuatorLaser TrapTrap BeadActuator BeadRNAHairpin0100100 105 110 115MediumFastSlowObservations                Work (k  T)BUnfoldingRefolding (negative work)(a) (b)FIG. 1: In this Letter, we discuss the definition and mea-surement of time-asymmetry in microscopic systems. (a) Asa concrete example, we analyze the time asymmetry of a singlemolecule experiment in which an RNA molecule is attachedbetween two beads [4]. (a) One bead is captured in an opticallaser trap that can measure the applied force on the bead.The other bead is fixed to a piezoelectric actuator. The con-trollable parameter ! is the distance between the fixed beadand the center of the laser trap. For the forward protocol, theRNA hairpin is initially in thermal equilibrium in the foldedstate with extension !(a). The extension is then increasedto !(b), unfolding the RNA. In the conjugate, time reversedprotocol, the RNA is initially in thermal equilibrium in theunfolded state, and the extension is lowered from !(b) back to!(a), allowing the RNA to refold [4, 5, 6, 7]. (b) Histograms ofwork measurements for folding and unfolding an RNA hairpinat three di"erent rates. Observations are binned into integerscentered at 1 kBT intervals. Note that Eq. (6) predicts thatthe folding and unfolding work distributions cross at the freeenergy change.symmetric. Instead, we develop a measure based on theJensen-Shannon divergence between the forward and re-verse probability distributions of trajectories for a micro-scopic system. We find that this measure of time’s arrowhas intuitive physical and information theoretic interpre-tations and constrains the minimum average dissipation.Moreover, recent advances in far-from-equilibrium sta-tistical physics allow one to measure time’s arrow in real2world experiments.We consider a physical system driven from thermalequilibrium by an external perturbation. For such anexperimental protocol, # denotes a set of controllableparameters !(t) for t $ [a, b] which describe how the sys-tem is driven from the initial equilibrium at !(a). Weare also interested in the conjugate time reversed proto-col #̃ in which the system begins in thermal equilibriumat !(b) and the controllable parameters retrace the sameseries of changes, in reverse, back to !(a). In the singlemolecule experiments of Fig. 1, the distance between thecenter of the laser trap and the fixed bead plays the roleof !(t). For each realization of the forward protocol #,the system travels along a trajectory x which representsthe states x(t) for t $ [a, b]. We define a conjugate timereversed trajectory x̃ such that x̃(t) = x(t) for t $ [b, a].We quantify the intrinsic time asymmetry of a drivensystem as the distinguishability of conjugate forward andreverse experiments. Given a microscopic trajectory x,can we tell if it was generated by the protocol #, orwhether it is the time reversal of a trajectory generatedby the reverse protocol #̃? Specifically, we define thetime asymmetry A asA[#] % JS!P [x|#] ; P [x̃|#̃]"(1)in which P [x|#] and P [x̃|#̃] are the probabilities of tra-jectories during the forward and reverse protocol respec-tively, and JS is the Jensen-Shannon divergence betweentwo probability distributions [12, 13, 14, 15, 16]JS(p; q) = 12#ipi lnpi12 (pi + qi)+ 12#iqi lnqi12 (pi + qi).(2)Each of the two summands is the relative entropy (orKullback-Leibler divergence) between one of the distri-butions and the mean of the two distributions. Hence,JS(p; q) # 0, and is equal to zero only if the two distri-butions are identical: pi = qi for all i [17]. The Jensen-Shannon divergence reaches its maximum value of ln 2nats [i.e. 1 bit], if the two distributions do not overlap,piqi = 0 for all i, and therefore are perfectly distinguish-able.The Jensen-Shannon divergence has a direct interpre-tation in terms of a Bayesian inference problem [14]. Sup-pose we are given a sample k taken from one of two prob-ability distributions, p or q. With no other way to dis-tinguish between the distributions, the prior probabilityfor the distributions is P (s) = { 12 ,12} in which s repre-sents either p or q. The prior distribution of outcome kis therefore P (k) = 12pk +12qk while the posterior distri-bution isP (s|k) =$ 12pk12pk +12qk,12qk12pk +12qk%. (3)The information gained about s from observing state kis the relative entropy between the posterior and priordistribution. This information averaged over the priordistribution of outcomes is!"I" =#kP (k)"I =#kP (k)#sP (s|k) lnP (s|k)P (s).(4)Some elementary algebra reveals that this average in-formation gain is equal to the Jensen-Shannon diver-gence, !"I" = JS(p; q). Hence, the time asymmetryA[#] is the average gain in information about the ori-entation of time’s arrow obtained from one realization ofthe experiment. Moreover, the square root of the Jensen-Shannon divergence is a metric between probability dis-tributions [14, 15]. Consequently, the square root of thetime asymmetry&A[#] measures the distance betweenthe forward and reverse protocols in trajectory space, lit-erally the length of time’s arrow.In addition to its information theoretic interpretation,the time asymmetry can be measured in experimentsdue to recent advances in far-from-equilibrium statisti-cal physics. In particular, the ratio of the probabilityof a trajectory during the forward protocol P [x|#] andthe probability of its conjugate trajectory on the reverseprotocol P [x̃|#̃] is [18]P [x|#]P [x̃|#̃]= e!W [x|!]!!"F [!] , (5)in which " = 1/kBT , T is the temperature of the en-vironment in natural units (kB is the Boltzmann con-stant), and W [x|#] is the work performed on the sys-tem during the forward protocol # [9, 18, 19], and"F [#] = F"(b) &F"(a) is the di!erence in Helmholtz freeenergy between the initial and final ensembles. Eq. (5)is a direct consequence of the time reversal symmetry ofthe underlying dynamics[18] and implies the work fluc-tuation theoremP (+W |#)P (&W |#)= e!W!!"F (6)in which "F % "F [#]. Moreover, Eq. (5) gives that thetime asymmetry isA[#] = 12'ln21 + exp(&"W [x|#] + ""F )(!+ 12'ln21 + exp(&"W [x̃|#̃] & ""F )(!̃, (7)a non-linear average of the forward and reverse dissipa-tion.Time asymmetry is also closely related to the e$ciencywith which the free energy can be estimated from non-equilibrium measurements of the work. To determine"Ffrom experimental realizations of the forward and reverseprotocols, Bennett’s method gives the log likelihood of32 4 60.10.20.30.40.50.61 3 5 7 h : Average dissipation (kT)      A : Time asymmetry( Squared length of time’s arrow)Linear responseSlow unfolding/refolding rateFast unfolding/refolding rateMedium unfolding/refolding ratedisallowed regionln 2XLow time asymmetry, high dissipation regimeFIG. 2: The squared length of times arrow A [Eq. (1)] versus the hysteresis h [Eq. (10)], the dissipation averaged acrossa conjugate pair of forward and reversed experiments. The experiment is explained in Fig. 1. We equalize the number ofdata points between conjugate experiments, estimate the free energy from the data, obtain error bars by applying a Bayesianbootstrap [10], and apply a correction for experimental errors, as described in [11]. The slower the experiment is performed, thecloser to thermodynamic reversibility, the lower the dissipation, and the lower the time-asymmetry. The slowest experimentsare known to contain the greatest experimental error [11], which may explain the deviation of the slowest data from the linearresponse trend.the free energy di!erence as [11, 20, 21, 22]#("F ) =K#i=1ln11 + e!!W [xi|!]+!"F+K#j=1ln11 + e!!W [x̃j |!̃]!!"F. (8)Here, W [xi|#] and W [x̃j |#̃] are the work measured duringthe i and j realizations of the forward and reverse pro-tocols respectively. The maximum likelihood estimate"F̂ = argmax #("F ) has the minimum variance amongall estimators of "F [11, 21, 22]. Comparing Eqs. (7)and (8), the time asymmetry can be estimated with theBennett likelihood in the large sample limit, [23, 24]A[#] '12K#("F ) + ln 2 . (9)Thus, we can simultaneously estimate "F and the timeasymmetry by maximizing A[#] with respect to "F .It is enlightening to contrast the length of time’s arrowwith the hysteresis, the average dissipation of the forwardand reverse protocols,h[#] = 12"!W [x|#]"! +12"!W [x̃|#̃]"!̃ . (10)Because of Eq. (5), the hysteresis is also a divergencebetween the forward and reverse trajectory distribu-tions [25, 26, 27, 28, 29]h[#] = 12 Je!reys!P [x|#] ; P [x̃|#̃]"(11)in whichJe!reys(p; q) =#ipi logpiqi+#iqi logqipi(12)is the Je!reys J-divergence (or symmetrized Kullback-Leibler divergence) [30, 31]. In Fig. 2, we plot time asym-metry against the hysteresis for single molecule RNApulling experiments at three di!erent rates (see Fig. 1).Both the hysteresis and length of time’s arrow increasesas the pulling rate increases; however, we will show thatthis need not always be the case. For comparison, we alsodisplay the time asymmetry A[#] in the linear responseregime. The work distributions are normal with variancetwice the average dissipation [9].The relative values of time asymmetry and hystere-sis are bounded by several inequalities. Taneja demon-strated that A ( h/4 using convexity arguments [32].Figure (2) shows that this bound is obeyed by the lin-ear response calculation and the experimental data. Forlarge values of the hysteresis, we can derive a tighterbound than Taneja. Since the function f(x) = ln(1+e!x)is convex, Jensen’s inequality [17] implies that !ln(1 +e!x)" # ln(1 + e!"x#). Thus,JS(p; q) = 12#ipi ln21 + elnqipi+ 12#iqi ln21 + elnpiqi( 12 ln21 + exp)& D(p)q)* + 12 ln21 + exp)& D(q)p)*( ln21 + exp)& 12 Je!reys(p; q)* (13)in which last line follows by a second application ofJensen’s inequality. Hence, A ( ln[2/(1 + e!h)] whichwe show in Fig. 2 for large h.However, there is no lower bound to the time asym-metry given the hysteresis. A system can be almost timesymmetric, but exhibit a large average dissipation. To il-lustrate this situation, imagine that occasionally, whilegently unfolding an RNA hairpin, the RNA becomesstuck in a tangled configuration that resists being pulledapart by force. While most repetitions of the experiment4give a work measurement very close to the free energychange W ' "F , for the rare instances of entanglementthe work is very large W * "F . The normalized dis-tribution of dissipation D = W & "F for the forwardprocess may well be approximated asP (D|#) = (1 & p)N(D̄0, $20) + pN(D̄1, $21) (14)in which N(µ, $2) denotes a normal distribution withmean µ and variance $2. The second term correspondsto the rare events, so p + 1 and D̄1 * 1. The workfluctuation theorem, Eq. (6) implies that the dissipationof the reverse process isP (D|#̃) = (1&q)N($20&D̄0, $20)+qN($21&D̄1, $21) (15)in which q = p exp(&D̄1 + $21/2) and normalization re-quires that(1&p) exp(&D̄0 + $20/2)+p exp(&D̄1 + $21/2) = 1 . (16)The linear response regime corresponds to p = 0 withD̄0 = $20/2. For small p and relatively small variance, wealmost never see rare trajectories on the reverse protocolwith a negative dissipation &D̄1 since q will be exponen-tially smaller than p + 1. The values p = 0.1, D̄1 = 150,D̄0 = 0.2 and $20 = 1 give h = 7.8 and A = 0.1 (Marked inFig 2), so a system with small time asymmetry can havea large hysteresis. While Eq. (5) shows that dissipationmeasures the time symmetry breaking of individual tra-jectories [25, 29, 33], the average dissipation is sensitiveto unusual events and is not a reliable measure of timeasymmetry for the entire system.One interpretation of relative entropy, and thereforeJe!reys divergence, is that it represents an encodingcost [17]. If we encode messages using an optimal code forthe message probability distribution qi, but the messagesactually arrive with probabilities pi, then each message,on average, will require an additional D(p)q) bits to en-code compared to the optimal encoding. Analogously,the hysteresis represents a cost, the entropy lost to dissi-pation. Thus, the time asymmetry A measures the extentof time-symmetry breaking and the average dissipationmeasures the price paid.The interrelation between time asymmetry and dissi-pation may be important for molecular motors and othermacromolecular biological machinery. One of the cen-tral imperatives of any life form is to make tomorrowlook di!erent from today. On the molecular level, thisrequires rectifying the ever present thermal fluctuations.Since the horizontal-axis in Fig. 2 represents dissipationin units of kBT , it takes about 4-8 kBT of free energy percycle to ensure that a machine mostly advances forwardin time, assuming it stays in the linear regime. This is asubstantial fraction of the energy budget available fromthe hydrolysis of an ATP molecule, about 20 kBT .This research was supported by the U.S. Dept. ofEnergy, under contracts DE-AC02-05CH11231. E.H.F.thanks the Miller Institute for Basic Research in Sciencefor financial support. We thank Felix Ritort for providingthe experimental data used in our analysis.[1] R. Clausius, Annalen der Physik und Chemie 201, 353(1865).[2] A. S. Eddington, The nature of the physical world (Cam-bridge University Press, Cambridge, 1928).[3] H. Price, Time’s arrow and Archimedes’ point (OxfordUniversity Press, New York, 1996).[4] D. Collin, F. Ritort, C. Jarzynski, S. B. Smith,I. Tinoco Jr., and C. Bustamante, Nature 437, 231(2005).[5] J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco Jr., andC. Bustamante, Science 296, 1832 (2002).[6] C. Bustamante, J. Liphardt, and F. Ritort, Phys. Today58, 43 (2005).[7] F. Ritort, J. Phys.: Condens. Matter 18, R531 (2006).[8] D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Phys.Rev. Lett. 71, 3616 (1993), Errata 71(21):2401.[9] C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997).[10] D. B. Rubin, Ann. Stat. 9, 130 (1981).[11] P. Maragakis, F. Ritort, M. Karplus, C. Bustamante, andG. E. Crooks, J. Chem. Phys. 129, 024102 (2008).[12] J. Lin, IEEE Trans. Info. Theory 37, 145 (1991).[13] F. Topsøe, IEEE Trans. Info. Theory 46, 1602 (2000).[14] D. M. Endres and J. E. Schindelin, IEEE Trans. Info.Theory 49, 1858 (2003).[15] F. Österreicher and I. Vajda, Ann. Inst. Statist. Math.55, 639 (2003).[16] A. Majtey, P. W. Lamberti, M. T. Martin, and A. Plas-tino, Eur. Phys. J. D 32, 413 (2005).[17] T. M. Cover and J. A. Thomas, Elements of InformationTheory (Wiley, New York, 1991).[18] G. E. Crooks, J. Stat. Phys. 90, 1481 (1998).[19] L. Peliti, J. Stat. Mech.: Theor. Exp. p. P05002 (2008).[20] C. H. Bennett, J. Comput. Phys. 22, 245 (1976).[21] M. R. Shirts, E. Bair, G. Hooker, and V. S. Pande, Phys.Rev. Lett. 91, 140601 (2003).[22] P. Maragakis, M. Spichty, and M. Karplus, Phys. Rev.Lett. 96, 100602 (2006).[23] G. E. Crooks, Phys. Rev. Lett. 99, 100602 (4) (2007).[24] E. Feng and G. E. Crooks (2008), arXiv:0807.0621.[25] P. Gaspard, J. Chem. Phys. 120, 8898 (2004).[26] C. Jarzynski, Phys. Rev. E 73, 046105 (2006).[27] R. Kawai, J. M. R. Parrondo, and C. Van den Broeck,Phys. Rev. Lett. 98, 080602 (2007).[28] D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier,S. Joubaud, and A. Petrosyan, J. Stat. Mech.: Theor.Exp. p. P01002 (2008).[29] A. Gomez-Marin, J. M. R. Parrondo, and C. Van denBroeck (2008), arXiv:0805.4703v1.[30] H. Je"reys, Theory of probability (Clarendon Press,1948), 2nd ed.[31] S. Kullback and R. A. Leibler, Ann. Math. Stat. 22, 79(1951).[32] I. J. Taneja, Aust. J. Math. Anal. Appl. 2, 8 (2005).[33] C. Maes and K. Netǒcný, J. Stat. Phys. 110, 269 (2003).

The Length of Time's Arrow

Crossref

Length of Time’s Arrow

The length of time's arrow

The length of time's arrow

Abstract

Similar works

Full text

Available Versions

UNT (University of North Texas) Digital Library

Crossref