Bartussek, Roland

Hänggi, Peter

Kissner, Jürgen G.

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Periodically Rocked Thermal Ratchets

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                                                                             Periodically Rocked Thermal Ratchets.R. BARTUSSEK, P. H ~ G G I  and J. G. KISSNERInstitut f i r  Theoretische Physik, Universitat AugsburgMemmingerstr, 6, 0-86135 Augsburg, G e m n y                                                                                                                                                                                                                                                                                    Abstract. - We consider overdamped Brownian particles in anisotropic, periodic structures(ratchets) that are rocked periodically. Together with the periodic forcing, white thermal noisecan generate a non-zero, macroscopic velocity. By tuning the parameters, the direction of thecurrent can be reversed. Additionally, the current as a function of the driving amplitude exhibitsseveral local maxima at  finite driving frequencies. For zero thermal noise, the deterministiccurrent assumes an intriguing structure, reflecting the complex dynamics of particle excursionsalong the ratchet.Biological applications relating to phenomena such as movement of muscles or theoperation of molecular combustion motors involve a transport of Brownian particles alongperiodic structures in the apparent absence of any external driving forces [l-61. Hence, thechallenge of seeking the physical mechanism(s) of directed transport without having to applymacroscopic bias forces or field gradients has stimulated the interest of many biologists,chemists and physicists (note the series of News and Views contributions in ref. [l]). Indeed,it could be demonstrated theoretically [l-51, as well as experimentally [6], that fluctuation-induced transport is possible in periodic structures with spatial asymmetry (no reflectionsymmetry). Such systems are termed thermal ratchets. Fluctuation-induced transportrequires the presence of non-equilibrium forces F(t): a finite current exists solely due to thefact that the second law of thermodynamics is no longer applicable to these non-equilibriumsystems.In this paper, we study a ratchet-like mesoscopic system, driven by thermal noise and aperiodic driving force with zero time average. Since most of the molecular transport occurs inthe overdamped regime[2-5], we can safely neglect inertial effects. Much to our ownsurprise, we find that a periodically driven ratchet exhibits a series of unexpected features,such as the occurrence of current reversal, and a complex, multipeaked current-drivingamplitude characteristics.Starting point of our analysis is the overdamped Langevin equationi = - a, {V(Z) - xF(t)} + E(t). (1)                         460                    The ratchet potential V(x)  = - k - l [ s in (kx )  + (1/4) sin(Zkx)] has the shape of a sawtoothwith smoothed edges (see right-hand side of fig. 2b)). In the following we choose k = 2x,which implies an intrawell relaxation frequency oo = 3.18. For the periodic force we takeF ( t )  = A sin(&). The thermal noise is modelled by a Gaussian white noise F(t) with zero meanand the correlation (t;(t)t;(t’)) = 2D6(t - t’). We mention that the spatial average of a,V(x),as well as the time average of F ( t )  are zero.It is well known that this stochastic process can be recast in terms of the probabilitydensity W(X, t )  that satisfies the Fokker-Planck equation m(x, t )  = - dzJ(x ,  t ) ,  withperiodic boundary conditions in space. Here(2)is the current density, and D ( ’ ) ( x ,  t )  = cos(kx) + (1/2) cos(2kx) + A sin(&) is the driftvelocity. -We are mainly interested in asymptotic averages. Thus we define the averagecurrent J asJ ( x ,  t )  = [D“’(x, t )  - Da,] W(X,t )T + t  Lwith temporal period T = 2 x / w  and spatial period L = 2 x / k .  We note that the current in (3) isindependent of the initial preparation, because W(z, t + CQ ) assumes a unique limitingperiodic distribution in time and space [7,8]. The result of a unique current holds true even inthe limit of zero noise intensity D, because the drift is periodic and different deterministictrajectories do not cross.To cover an extensive range of parameters, we solved eq. (3) using a matrix continued-fraction method [7,8], supplemented by analytic results in limiting cases, such as theadiabatic limit.Figures la) and b)  contain our main findings for generic cases: the current is plotted forseveral driving frequencies o as a function of the noise strength D and the driving ampli-tude A. The adiabatic limit w = 0, drawn as solid lines in fig. l a )  and b), is readily evaluated0.0150.01‘50.00500 0.05 0.1 0.15 0.2 0.25 0.3-0 1 1 1 2  3 4 5 6 7I ,D Ai A, AFig. 1. - a) The mean current us. thermal-noise strength D for fxed driving amplitude A = 0.5 canchange sign. The current for the driving frequencies w = 0.01, 1, 4, 5.5, and 7 is depicted from top tobottom. The analytic rssult obtained in the adiabatic limit, eq. (4), coincides with the solid line w = 0.01.b) The mean current J vs. A for f i e d  D = 0.1 becomes negative for sufficiently high frequencies. Thecurrent for the frequencies w = 0.01, 1,4,  7, and 10 is depicted from top to bottom. The adiabatic limitfalls onto the solid line w = 0.01.                                                          461from the first-order equation (1) yieldingTDJd = - Idt{[l  - exp[@(L, t ) ] ] - ' .T Owhere @(x, t )  = [V(x) - xA sin(wt)l/D.First, we observe that the current vanishes asymptotically, both for large A, as well as forlarge D. This can be understood upon noting that in these limits the influence of the ratchetpotential becomes negligible. At A = 0 the current vanishes, because here the open systemreduces to a system in thermal equilibrium [9]. In the deterministic limit, i.e. D = 0, thecurrent vanishes identically for all driving amplitudes below a critical threshold Al (seebelow), while for A > Al the current can assume zero or finite values for D = 0, depending onw. Note,_however, that for a periodic potential V(x) with reflection symmetry, the averagecurrent J vanishes identically for any A and D (with w z 0).More surprising than the facts mentioned up to this point is the result of current reversalin fig. la), b). This characteristic feature seemingly occurs only for driving frequencies whichexceed the intrawell relaxation frequency w o .  In addition one finds several extrema in thecurrent-amplitude characteristics.To understand these novel features, we first address the deterministic behaviour. Inparticular it turns out that the intriguing result of a sign change is not exhibited by anydeterministic motion! Hence only the mutual interplay between noise and finite-frequency42 ,0 0.5 1 1.5 2 2.5 3 0.1 -0.1tT-H1.510-0.5HtT-Fig. 2. - a) Asymptotic trajectory x, ( t )  'us. tjme t for futed w = 1. Small driving amplitudes A lead to aperiodic motion with zero average current J :  &he dash-dotted line shows A = 1. For A = 1.5 (dashedline), J is non-zero; but for A = 3 (solid line), J is smaller than that of A = 1.5. As a guide for the eye,the static potential is included on the right-hand side of the diagram. Note that the system has beenstarted at  large negative times to to be in its asymptotic regime a t  the times t 3 0 shown in the figure.b) Asymptotic trajectories x , ( t )  us. t for f i e d  w = 7 and A = 3. Note that for this special values of Aand U, there are two distinct stable trajectories (solid lines) that carry no current, and two separatrices(dashed lines). The upper separatrix in the figure is equivalent to the lower one. The mows within theplot of the potential on the right-hand side show that trajectory 1 is centred around the minimum of thepotential, trajectory 2 around the maximum.462                    driving (w f 0, i.e. non-adiabatic) is able to produce this effect. So far we have not been ableto find a satisfying (analytical) explanation. We want to point out that the mechanism of thiscurrent reversal differs distinctly from that put forward by Doering et al. [5], where the signchange of j occurs due to a differing statistics of non-equilibrium, but stochastic drivingforces F(t).In fig. 2a) and b)  we depict some deterministic, asymptotic trajectories z, ( t ) .  A glance a tthese trajectories reveals that the deterministic dynamics is indeed a non-trivial problem.First of all, a larger A does not necessarily imply a larger current J (fig. 2a)). Secondly, for asuitable choice of parameters (here A = 3, w = 7) one can find two distinct stable trajectories(fig. 2b)). Hence different initial positions can lie in different basins of attraction for thesolution x(t) .  By looking at  the potential on the right side of fig. 2b),  it can be seen that thepath of trajectory 1 is centred around the minimum of the static potential V(x), whereastrajectory 2 explores the region around the potential maximum.We next go on to consider the average deterministic current. Figures 3a) and b)  show thestep-like structure of j as a function of the driving amplitude. An important fwst differencecompared to the noise-driven case can be seen immediately: the deterministic current neverchanges sign. Furthermore, the onset of current occurs a t  a frequency-dependent thresholdvalue Ami, (U). Naively one would expect the onset to occur a t  the amplitude where the tilt ofthe potential is big enough that the local minima of the ratchet disappear. For the tilt to theright side this happens if A b AI = 0.75; for the tilt to the left if A b A, = 1.5. Theoreticallyand numerically one obtains Amin (U)  2 A ] ,  with Ami, ( w  + 0) + Al. In the regime where Jincreases monotonically, the particle can move freely to the right side, but gets trapped in avalley for a tilt to the left side (this mechanism can be termed <<mechanical diode.). ForIA sin(&) I > A,, the particle does no longer encounter minima; it, therefore, can move inboth directions. This makes a maximum current around A = A, plausible. Note that, for theadiabatic limit, the current j achieves its maximum exactly at A = A,! One obtains thesolution of the adiabatic limit for zero noise by integrating eq. (1). The time t a particle needsFig. 3. - a) The mean current j vs. driving amplitude A for zero noise strength D = 0 demonstrates howthe adiabatic limit is approached. The dashed line is for the frequency w = 0.01 and falls on top of theadiabatic limit, eq. (6). The solid line depicts the current for w = 0.25. The current is positive for alldriving amplitudes. The adiabatic curve looks smooth, because the height of the steps is proportional tow (see text). In the inset, the width of the n-th step for A > 7.8 is plotted us. n (with error bars). b) Themean current j vs. A for fEed w = 1 shows the smearing of the deterministic step-like behaviour withincreasing noise strength D = 0, 0.01, 0.05, from top to bottom. The arrows indicate the <<distance.between steps as defined in the text.                                                           463to move by a distance of length L equalswhere D")( 9, e) = cos(k9) + (1/2) cos(2ky) 3- A sin 8. Here, e is the phase of the slowlyvarying external force (hence the adiabatic limit). Equation (5) leads-immediately to thecurrent for a fixed tilt A sine. A final integration over 8 then yields J :Adding noise makes the connection to the diagrams 1 above. Finite noise € ( t )  smears thepronounced steps and plateaus of the current curve into a wave-like structure; cf. fig. 3b) andfig. lb); we see that the local extrema of fig. l b )  are deterministic features-not related toany noise-driving co-operation. Obviously, if the noise strength is increased further, theextrema will eventually disappear, as shown in fig.3b).Another striking property of the deterministic current is the quantisation of the stepheight in units of L w / ~ x ,  cf. fig. 3. Clearly, in the asymptotic limit, where d, ( t )  is a periodicfunction o f t  with periodicity T, the position of the particle at time t differs from the positionat time t+ T by an integer multiple of L. Put differently, the periodic part of x,( t ) ,  namelyx , ( t )  - Ja t ,  locks onto the phase of the periodic driving [8]. This immediately implies thequantisation. Finally we mention a most peculiar property of the <<distance>> and width of thesteps. Defining the .distance. as being measured from the onset of one step to the onset ofthe next step (see the two arrows in fig.3b)), our simulations seemingly yield a constantdistance hetween successive steps. Their widths decrease monotonically within constantcurrent J ,  as shown in the inset of fig.3~). An exponential function fits the curvesatisfactorily, but due to the numerical uncertainty, a definite conclusion cannot bedrawn.To summarize, the overdamped motion of a Brownian particle in an asymmetric potential,driven by a periodic force, exhibits a high degree of complexity, even in the deterministicregime. Mostly by means of numerical calculations we have discovered a number ofinteresting and unexpected features of this ratchet system. The deterministic ratchetproblem is a good starting point for analysing the system in the presence of thermal noise;many features of the latter are an immediate consequence of the deterministic one. But evenwith the simplest thermal noise, namely Gaussian white noise, a reversal of the currentoccurs. This feature is neither present in the adiabatic limit (with or without noise), nor in thecase of zero noise but finite driving frequency. We therefore conclude that it is caused by aco-operative interplay between noise and deterministic, finite-frequency driving.It should be noted that our arocking ratchet. device may also have a variety of importantapplications in nano-technology. For example, it can be utilized to efficiently separateparticles with nearly equal diffusion coefficient due t_o the Arrhenius-like sensitivity of thecurrent vs. diffusion strength (cf. the behaviour of J in fig. la) at small 0).There are a number of further studies suggested by the results presented here: the novelphenomena we have observed for pedodically driven ratchets are expected to be robustunder incoherent forcing, such as a stochastic, non-equilibrium harmonic-oscillator processF( t )  (harmonic noise [lo]). Additionally, the intriguing deterministic features of eq. (1) areexpected to stimulate future theoretical work on non-autonomous differential equations [U]that are periodic both in space and time.464                    R E F E R E N C E S[ll MADDOX J., Nature, 365 (1993) 203; 368 (1994) 287; 369 (1994) 181; LEIBLER S., Nature, 370 (1994)412.121 AJDARI A. and PROST J., C. R. Acad Sci. Paris, 315 (1992) 1635; PROST J., CHAUWIN J.-F., PELITIL. and AJDARI A., Phys. Rev. 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Periodically Rocked Thermal Ratchets

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