Hänggi, Peter

Jung, Peter

Kissner, Jürgen G.

OPUS Augsburg

VOLUME 76, NUMBER 18 P HY S I CA L REV I EW LE T T ER S 29 APRIL 19963436Regular and Chaotic Transport in Asymmetric Periodic Potentials: Inertia RatchetsP. JungCenter For Complex Systems Research, Beckman Institute, and Department of Physics, University of Illinois,Urbana, Illinois 61801J. G. Kissner and P. HänggiInstitut für Physik, Universität Augsburg, Memminger Strasse 6, D-86135 Augsburg, Germany(Received 29 November 1995)Motivated by recent work on stochastic ratchets, we consider the effect of finite inertia ontothe directed motion in a deterministically rocked, periodic potential lacking reflection symmetry.Characterizing the motion by cumulants of the contracted, time-dependent solution of the Liouvilleequation, we can distinguish regular from chaotic transport. The first cumulant describes a stationarycurrent that exhibits multiple reversals versus increasing driving strength, whereas the secondcumulant yields a measure for its variance. Chaotic transport exhibits universal (Gaussian) scalingbehavior. [S0031-9007(96)00064-6]PACS numbers: 82.20.Mj, 05.40.+j, 05.45.+bIt is generally appreciated that—in accordance with thesecond law of thermodynamics—usable work cannot beextracted from equilibrium fluctuations. Devices and phe-nomena that are only in apparent contradiction with thissecond law have been discussed by Feynman, Leighton,and Sands [1] and even much earlier in the heyday ofBrownian motion by Smoluchowski [2]. In the presenceof nonequilibrium forces the situation changes drastically.Then, directed transport of Brownian particles in asym-metric periodic potentials (ratchets) can be induced by theapplication of nonthermal forces [3–8] or with the helpof deterministic, periodic coherent forces [4,9,10]. Thesenonequilibrium models recently gained much interest inview of their role in describing the physics of molecu-lar motors [5,6] and their potential for novel technologi-cal applications on nanoscales and microscales [11]. Theprevious literature on these schemes is characterized bythe limitation of using an overdamped ratchet dynamicsthroughout. A particular challenge thus presents the studyof finite inertia for the ratcheting mechanism in the absenceof thermal and nonthermal forces. With finite inertia, thedynamics is allowed to become more complex, exhibitingboth regular and chaotic behavior. Using the model of aperiodically rocked ratchet, we shall investigate the mu-tual interplay of regular and chaotic dynamics for directedtransport, in particular, to what extent deterministically in-duced chaos mimics the role of noise. In doing so, we usenovel diagnostic tools, such as the behavior of cumulantaverages for the phase-space probability, whose time evo-lution is governed in the deterministic limit by a dissipa-tive, nonautonomous Liouvillian. Nonstationary effects,such as the broadening of the phase-space probability—a measure for the fluctuations of the net-flux (on a coarsegrained scale)—are taken into account by not applying pe-riodic boundary conditions.As a working model, we use an underdamped particle,periodically driven in an asymmetric periodic potential,0031-9007y96y76(18)y3436(4)$10.00eẍstd 1 Ùxstd ­ 2ddxVmsxd 1 A sinsVtd , (1)with the potential shown in Fig. 1 (see also [10])Vmsxd ­ 2sinsxd 2 m sins2xd . (2)All variables in (1) and (2) are scaled dimensionless andmis set to m ­ 1y4 throughout this work.We first briefly discuss the case of massless particles(e ­ 0), i.e., overdamped motion. The motion is boundfor driving amplitudes A , A0sVd; i.e., the solution ofEq. (1) approaches a function which is periodic in time forlarge times. For A0sVd , A, the motion can becomeunrestricted. The average velocity assumes an asymptoticvalue (that is independent of the initial conditions) of theformynm ­xst 1 nT d 2 xstdnT­mnV , (3)with integers m and n. A direct consequence of theoverdamped motion is that the net flux is directed towardsthe shallow side of the ratchet (positive current), i.e.,m, n . 0 in Eq. (3). For finite mass e, the situation isFIG. 1. The ratchet potential Vmsxd ­ 2 sinsxd 2 m sins2xdfor m ­ 1y4, as used in the text.© 1996 The American Physical SocietyVOLUME 76, NUMBER 18 P HY S I CA L REV I EW LE T T ER S 29 APRIL 1996different. The trajectories described by (1) can becomechaotic; i.e., the trajectories are erratic on a time scale largeagainst the largest Liapunov coefficient and the net flux cango in both directions. This raises the key question studiedin this paper: How does intrinsic chaotic motion resemblenoise driven motion in ratchets?Equation (1) has a variety of solutions which can beobtained by numerical integration. We are using theirtransport properties as a classification criterion. Reg-ular trajectories are described by xst 1 nT d ­ xstd 12pmwith n [ N andm [ Z . Regular, nontransportingtrajectories are being characterized by tuples sn, m ­ 0d.Regular, transporting solutions are characterized by thetuples sn, m fi 0d. These trajectories are periodic withperiod n (subharmonic for n . 1) on a toroidal phasespace obtained by using periodic boundary conditions;i.e., x is identified with x 1 2p . In contrast to theoverdamped case the transport can now be due to the inertiain either direction; i.e., the average velocity can assume thevalues given in Eq. (3) with m ­ 0, 61, 62, . . . . Givena fixed set of parameters, it is important to note that—depending on the initial conditions—different classes oftrajectories have been observed by numerical simulation.For example, at A ­ 0.624, V ­ 0.1, e ­ 20, asymptotictrajectories are either periodic, (1,0), or current carrying,s2, 21d. This situation calls for interesting technologicalapplications, since it allows for the separation of particlesstarting with different initial conditions (a detailed discus-sion will be published elsewhere [12]).A third class is chaotic trajectories. Their behavior hasbeen studied by numerically simulating the time evolutionof an initial probability density in sx, Ùxd space (see belowfor details). For V ­ 0.1, e ­ 20, and those values of Ain 0 , A , 10, where the system is living on a chaoticattractor, there was always a net drift to the right or tothe left; i.e., particles move on average either to the leftor to the right. The chaotic attractor can be observed ina stroboscopic plot in sx, Ùxd space after mapping the un-bounded dynamics in space onto one period of the po-tential, i.e., xstd ! xstd smod2pd. We did not observecoexisting chaotic attractors in our working model (1). Forchaotic maps with periodic forces, the onset of diffusivebehavior caused by the memory loss due to chaotic dy-namics has been studied some time ago [13].By integrating Eq. (1) numerically, one realizes thatthe drift of a single chaotic trajectory exhibits strongfluctuations. An appropriate tool for describing transportis therefore a time-dependent probability measure. Giventhe initial probability density r0sx, Ùx, td, its time evolutionis given byrsx, Ùx, td ­Zdx0Zd Ùx0 dsx 2 xdsx0, Ùx0, tdd3ds Ùx 2 Ùxdsx0, Ùx0, tdd r0sx0, Ùx0d . (4)Here xdsx0, Ùx0, td is the solution of (1) with initial con-ditions xdst ­ 0d ­ x0 and Ùxdst ­ 0d ­ Ùx0. In contrastto the conventional treatment, we do not apply periodicboundary conditions. This allows us to study the broad-ening of the probability density as a function of time [12].As an example for the chaotic regime, we depict inFig. 2(a) the time evolution of the contracted probabil-ity density r̄sx, td ­Rd Ùx0 rsx, Ùx0, td for A ­ 0.79, V ­0.1, e ­ 20 and m ­ 1y4. The x axis has been partitionedso that one partition ranges over one period of the po-tential. The initial probability density was chosen Gauss-ian with a width of 1y2 in space and Ùxst ­ 0d ­ 0. Ona short time scale the probability density becomes morecomplicated, but approaches asymptotically a Gaussianshape; cf. Fig. 2(b). Because we are not interested in timescales of a single period, where one observes the oscilla-tory contribution of the driving force, we switch to a coarsegrained description by looking at the system stroboscopi-cally st ­ nT ­ 2pnyVd. Time t will therefore denotethese stroboscopic times. In Fig. 3, we show the first andsecond cumulants of the probability density as a functionof time. Apart from initial transients, we observe a linearmean and a linear variance—typical for Brownian motion.The third cumulant and higher cumulants contain more ofthe detailed structure of the density, and are therefore moreFIG. 2. Snapshots of the probability density, obtained bysimulating 50 000 trajectories, are shown in (a) for A ­0.79, V ­ 0.1, and e ­ 20. The universal, normalized scalingfunction p̄sx̃, td, Eq. (7), obtained from the same densities isshown in (b).3437VOLUME 76, NUMBER 18 P HY S I CA L REV I EW LE T T ER S 29 APRIL 1996FIG. 3. The first two cumulants C1std and C2std for A ­0.79, V ­ 0.1, and e ­ 20 are shown stroboscopically as afunction of time (tn ­ nT).complicated functions of time. As a common feature wefind that their modulus increases slower than t3y2.Based on the observations above, we will show inthe following that the probability density will becomeeffectively Gaussian. Thus the first and second cumulantsare sufficient to describe the time evolution and transport.The characteristic function Fssd ­ kexpf2isxstdgl ofthe contracted probability density r̄sx, td is written interms of its cumulants Ckstd asFssd ­ exp√2isC1std 212s2C2std 1X̀k­3s2isdkk!Ckstd!.(5)Making use of the numerically observed behavior forlarge times, i.e., C1std ­ yt, and C2std ­ s2t as t ! `,we can analyze the long time behavior of the probabilitydensity. First, we introduce the integration variable u ­spt, and define the relative position x̃ ­ x 2 yt. Thescaled probability density p̄sx̃, td ­ r̄sx, td can then bewritten asp̄sx̃, td ­1ptZ `2`exp"iux̃pt212s2u2#3 exp"X̀k­3s2iudkk!t2ky2Ckstd#du . (6)In the scaling limit t ! ` with x̃ypt kept constant, theterms Ck$3std in the second exponential become irrelevant(this, of course, relies on the observation that for k $ 3Ckytky2 ! 0 as t ! `).The first two terms, which are relevant in the scalinglimit, describe a Gaussian density with the scaling formt21y2fsx̃yptd, while the other terms describe correctionsto it. With the knowledge of all coefficients, thosecorrections to scaling can be obtained by expanding thesecond exponential, leading to an asymptotic expansionof p̄,p̄sx̃, td ­1ptf√x̃pt! "1 1 t23y2C3stdhµx̃pt∂1 · · ·#,(7)where h is an analytic function describing the leading cor-rection to scaling, and fsyd ­ exps2y2y2s2d. Therefore,3438for large times, the probability density approaches a shapewhich allows rescaling to a (universal) Gaussian distri-bution; see Fig. 2(b). The long time evolution does notdepend on the choice of the initial probability. It is inter-esting to compare p̄sx̃, td to the time-dependent probabil-ity density of a Brownian particle in a symmetric periodicpotential (see, e.g., in [14]). For large times it shows,apart from the vanishing drift, the scaling form (7).In case of a single regular attractor, i.e., all trajecto-ries are regular and obey xst 1 nT d ­ xstd 1 2pm withthe same tuple sn, md, the probability density for largetimes approaches a positive normalized function of theform r̄sx, t ! `d ­ fsx 2 ytd with y ­ mVyn. Thecharacteristic function of fsx 2 ytd, given by fsy, sd ­exps2iytsdf0ssd, where f0ssd is time independent, im-plies a first cumulant C1std ­ yt and constant higher cu-mulants Ck$2 (see Fig. 4). For nontransporting regularattractors, i.e., y ­ 0, all cumulants are time independent,corresponding to a stationary probability density. We re-mind the reader that the time scale t is actually a coarsegrained time scale, on which periodic variations of theprobability density within nT are not visible.In the case of coexisting regular attractors differ-ent trajectories, corresponding to different sets sn1, m1dand sn2, m2d, are drifting with different average veloci-ties y1 ­ m1Vyn1 and y2 ­ m2Vyn2, respectively. Theprobability density consists of several moving peakswith constant widths each (for large times), but gener-ally different velocities, y2 fi y1. The cumulant generat-ing function, gssd ­ lnfssd ­ lnfexps2iy1tsdfs1d0 ssd 1exps2iy2tsdfs2d0 ssdg yields cumulants of the form Ckstd ~tk . The first cumulant is again linear in time, i.e., C1std ­at ­ sAy1 1 By2dt, where A and B are the relativeweights of the two coexisting attractors. The second cu-mulant C2std is, in contrast to the cases above, quadraticin time. Such a case is realized, for example, with theparameter values A ­ 0.624, V ­ 0.1, and e ­ 20.With the above developed tools, we will now analyzethe transport properties in the asymmetric periodic po-tential. In Fig. 5, we depict the current y as a func-tion of the driving amplitude at e ­ 20. In the range0 , A , 1.00, where the overdamped system (e ­ 0)FIG. 4. The first two cumulants C1std and C2std for A ­1.1, V ­ 0.1, and e ­ 20 are shown stroboscopically as afunction of time (tn ­ nT ).VOLUME 76, NUMBER 18 P HY S I CA L REV I EW LE T T ER S 29 APRIL 1996always exhibits either zero or positive current, we ob-serve a variety of transport phenomena: in situations withzero transport, i.e., y ­ 0, the system has a single reg-ular attractor with period n ­ 1. The transport in theinterval 0.61 , A , 0.64 is regular for most parts. Inthis region we find a negative current. Applying our di-agnostic tools above, we conclude from the dependenceC2std ~ t2 that there are coexisting regular attractors [15].As a check, a stroboscopic plot yields a regular behaviorthat does not match with any of the values ynm ­ mVyn,because the transport is regular with contributions fromseveral coexisting regular attractors. For A . 0.7, themotion is typically chaotic. As an example we showFig. 3 (A ­ 0.79). The linear dependence of the first twocumulants as a function of time implies chaotic motion.This is confirmed by a stroboscopic plot. The transport isnegative, which is towards the steep side of the potential(cf. Fig. 1), for all trajectories. At some parameters (e.g.,A ­ 0.615), we observe behavior which resembles regu-lar transport for small times, with a clear-cut transitionto diffusive chaotic transport for large times (see [12] fora detailed discussion). For larger amplitudes A . 1, onefinds regular regimes and chaotic regimes. The currentsin the chaotic regimes are directed in either the positiveor the negative direction. The linear time dependence ofthe first and second cumulants, the observation that thehigher cumulants Cn$3std increase slower than t3y2, andthe resulting scaling laws, as presented above, apply in alltested cases in A , 10.In summary, we have presented novel results for aninertia ratchet. The system can exhibit a current flow ineither direction (multiple current reversals). The directioncan be controlled by adjusting the amplitude of theexternal, periodic driving (cf. Fig. 5); furthermore, itdepends sensitively on the strength of the inertia andFIG. 5. The transport coefficient y ­ ÙC1st ! `d is shown asa function of the driving amplitude A for e ­ 20 and V ­ 0.1.friction [16]. This makes it especially interesting fortechnological applications. The system’s dynamics ischaracterized by the time dependence of cumulants ofthe particle distribution. In the case of directed chaotictransport, the second cumulant is a measure for theintrinsic current fluctuations, and thus for the reliabilityof the ratchet mechanism. Our diagnostic tools allow adistinction between different forms of regular and chaotictransport; these enable a systematic description of chaos-induced currents in terms of a universal Gaussian scaling.P.J. thanks the Deutsche Forschungsgemeinschaft forfinancial support within the Heisenberg program.[1] R. P. Feynman, R. B. Leighton, and M. Sands, TheFeynman Lectures on Physics (Addison-Wesley, Reading,MA, 1966), Vol. 1, Chap. 46.[2] M. v. Smoluchowski, Phys. Z. XIII, 1069 (1912).[3] A. Ajdari and J. Prost, C. R. Acad. Sci. Ser. II 315, 1635(1992).[4] M.O. Magnasco, Phys. Rev. Lett. 71, 1477 (1993).[5] R. D. Astumian and M. Bier, Phys. Rev. Lett. 72, 1766(1994); R.D. Astumian and M. Bier, Biophys. J. 70, 637(1996).[6] C. S. Peskin, B. Ermentrout, and G. Oster, Cell Mechanicsand Cellular Engineering, edited by V. C. 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Risken, The Fokker-Planck Equation, Springer Se-ries in Synergetics (Springer-Verlag, Berlin, 1984), Sec.11.7.2.[15] Coexisting strange attractors would yield the same depen-dence to leading order, but we have not observed this casein our simulations.[16] Setting the friction to zero, we obtain a nonergodicbehavior. In particular, a stationary, nonzero current doesno longer emerge. Hence, a finite, stationary currentrequires also a finite dissipation, see also [1].3439

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Regular and chaotic transport in asymmetric periodic potentials: inertia ratchets

P. Jung

J. G. Kissner

P. Hänggi

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Physical Review Letters

Regular and Chaotic Transport in Asymmetric Periodic Potentials: Inertia Ratchets

Regular and Chaotic Transport in Asymmetric Periodic Potentials: Inertia Ratchets

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