Particles driven through a periodic potential by an external constant force
are known to exhibit a pronounced peak of the diffusion around a critical force
that defines the transition between locked and running states. It has recently
been shown both experimentally and numerically that this peak is greatly
enhanced if some amount of spatial disorder is superimposed on the periodic
potential. Here we show that beyond a simple enhancement lies a much more
interesting phenomenology. For some parameter regimes the system exhibits a
rich variety of behaviors from normal diffusion to superdiffusion, subdiffusion
and even subtransport.Comment: Substantial improvements in presentatio

Khoury, M.

Lacasta, A. M.

Lindenberg, Katja

Sancho, J. M.

Physical Review Letters

English

arXiv

M. Khoury

A. M. Lacasta

J. M. Sancho

Katja Lindenberg

Crossref

Weak Disorder: Anomalous Transport and Diffusion Are Normal Yet Again

We carry out a detailed study of the motion of particles driven by a constant external force over a

landscape consisting of a periodic potential corrugated by a small amount of spatial disorder. We observe

anomalous behavior in the form of subdiffusion and superdiffusion and even subtransport over very long

time scales. Recent studies of transport over slightly random landscapes have focused only on parameters

leading to normal behavior, and while enhanced diffusion has been identified when the external force

approaches the critical value associated with the transition from locked to running solutions, the regime of

anomalous behavior had not been recognized. We provide a qualitative explanation for the origin of these

anomalies, and make connections with a continuous time random walk approac

Khoury Arvelo, María José

Lacasta Palacio, Ana María

Sancho, Jose Maria

UPCommons

We carry out a detailed study of the motion of particles driven by a constant external force over a landscape consisting of a periodic potential corrugated by a small amount of spatial disorder. We observe anomalous behavior in the form of subdiffusion and superdiffusion and even subtransport over very long time scales. Recent studies of transport over slightly random landscapes have focused only on parameters leading to normal behavior, and while enhanced diffusion has been identified when the external force approaches the critical value associated with the transition from locked to running solutions, the regime of anomalous behavior had not been recognized. We provide a qualitative explanation for the origin of these anomalies, and make connections with a continuous time random walk approach

Sancho, José M.

Diposit Digital de la Universitat de Barcelona

We carry out a detailed study of the motion of particles driven by a constant external force over a
landscape consisting of a periodic potential corrugated by a small amount of spatial disorder. We observe
anomalous behavior in the form of subdiffusion and superdiffusion and even subtransport over very long
time scales. Recent studies of transport over slightly random landscapes have focused only on parameters
leading to normal behavior, and while enhanced diffusion has been identified when the external force
approaches the critical value associated with the transition from locked to running solutions, the regime of
anomalous behavior had not been recognized. We provide a qualitative explanation for the origin of these
anomalies, and make connections with a continuous time random walk approachPostprint (published version

UPCommons (Universitat Politècnica de Catalunya)

Weak Disorder: Anomalous Transport and Diffusion Are Normal Yet AgainM. Khoury,1 A.M. Lacasta,2 J.M. Sancho,1 and Katja Lindenberg31Departament d’Estructura i Constituents de la Matèria, Diagonal 647, E-08028 Barcelona, Spain2Departament de Fı́sica Aplicada, Universitat Politècnica de Catalunya, Avinguda Doctor Marañon 44, E-08028 Barcelona, Spain3Department of Chemistry and Biochemistry 0340 and BioCircuits Institute, University of California,San Diego, La Jolla, California 92093-0340, USA(Received 9 September 2010; published 2 March 2011)We carry out a detailed study of the motion of particles driven by a constant external force over alandscape consisting of a periodic potential corrugated by a small amount of spatial disorder. We observeanomalous behavior in the form of subdiffusion and superdiffusion and even subtransport over very longtime scales. Recent studies of transport over slightly random landscapes have focused only on parametersleading to normal behavior, and while enhanced diffusion has been identified when the external forceapproaches the critical value associated with the transition from locked to running solutions, the regime ofanomalous behavior had not been recognized. We provide a qualitative explanation for the origin of theseanomalies, and make connections with a continuous time random walk approach.DOI: 10.1103/PhysRevLett.106.090602 PACS numbers: 05.40.a, 02.50.Ey, 05.60.kSolid state surfaces frequently present periodic poten-tials marred by some disorder. Herein we show that anoverdamped particle moving over such a potential in onedimension (1D) may exhibit anomalous behavior in theform of superdiffusion, subdiffusion, and even subtran-sport. Although we cannot prove that these are steady stateregimes, our numerical simulation data show them to bepresent over time spans of several orders of magnitude.That diffusion of particles over both periodic and ran-dom surfaces leads to some forms of anomalous behavioris of course well known and continues to attract a great dealof attention [1–10]. In periodic potentials with low friction,extremely long (in time) dispersionless transport regimescan be observed when forces exceed a critical force [6].Moreover, in these same systems, in both overdamped andunderdamped regimes, the diffusion coefficient versus theapplied force presents a pronounced peak around the criti-cal force that allows the coexistence of locked and runningstates [1,4–7]. The enhancement is quantitatively largerthan the free particle diffusion coefficient. This behaviorhas been observed experimentally when tracking the mo-tion of colloidal spheres through a periodic potential cre-ated with optical vortex traps [9].The enhancement of the diffusion coefficient is evenmore pronounced when disorder is also present [9]. Thisphenomenon has been tested by numerical simulations on asurface in which a small amount of spatial disorder in theform of a random potential is added to the periodic poten-tial [10]. Dramatic diffusive enhancement occurs even forvery small amounts of disorder, e.g., when the amplitude ofthe random contribution of the potential relative to that ofthe periodic distribution is as small as 5%.Although dramatic, diffusive enhancement turns out tobe only a limited aspect of the story. Here we present arange of additional anomalous transport and diffusionphenomena arising from weak disorder that have notbeen previously noted. Our model is inspired by [9,10].We consider the overdamped motion of identical non-interacting Brownian particles moving in a 1D potentiallandscape UðxÞ following the Langevin equation  _xðtÞ ¼U0ðxÞ þ Fþ ðtÞ. Here x is the position of the particle, tdenotes time,  is the dissipation parameter, F is theapplied force, and ðtÞ is Gaussian thermal noise at tem-perature T. The correlation function of the noise obeysthe fluctuation-dissipation relation hðtÞðt0Þi ¼2kBTðt t0Þ. The potential UðxÞ consists of a periodicpart, VpðxÞ ¼ V0 cosð2x=pÞ, and a Gaussian spatiallyrandom contribution VrðxÞ with correlation functiongrðxÞ  hVrðxÞVrð0Þi ¼ V202exp 22x2l2r: (1)The relative contributions are determined by the parameter 2 ½0; 1 in the combination UðxÞ ¼ ð1 ÞVpðxÞ þVrðxÞ. Furthermore, we have chosen the potential corre-lations grðxÞ and gpðxÞ ¼ ðV20=2Þ cosð2x=pÞ to be equalat x ¼ 0, gpð0Þ ¼ grð0Þ. This ensures that the total poten-tial amplitude is of order V0 independently of . Also, withthe particular choice lr ¼ p the two correlation functionsare identical up to second order in a Taylor expansion.In terms of the spatial variable z ¼ 2x=p and thetemporal variable  ¼ ½ð2Þ2V0=2pt, the equation ofmotion can be reduced into dimensionless form,_z ¼ ð1 ÞfpðzÞ þ ð=Þfrðz=Þ þF þ ðÞ; (2)where fp and fr are the dimensionless forces arising fromVp and Vr respectively, and  is the dimensionless noise.The dimensionless parameters arePRL 106, 090602 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending4 MARCH 20110031-9007=11=106(9)=090602(4) 090602-1  2011 American Physical Society ¼ lrp; F ¼ pF2V0; T ¼ kBTV0: (3)Throughout this work we set T ¼ 0:01 and p ¼ 2.Variations in T do not lead to any additional phenome-nology. The specific choice of p ¼ 2 is only importantas a reference value. In these variables a decrease of  evenfor fixed  leads to an increase in the relative contributionof the random force.We have carried out numerical simulations of Eq. (2)over a large number (100) of particle trajectories and alarge number of realizations of the random potential con-tribution (typically 50–100) in order to calculate the ve-locity and diffusion coefficient using the prescriptionsvðÞ ¼ hzðÞi; DðÞ ¼ hz2ðÞi  hzðÞi22: (4)The brackets h  i denote averages over many trajectories.The numerical procedures are entirely standard. We showthat the usual assumption that stationary values are reachedat long times is at the very least questionable.Representative outcomes of this procedure are shown inthe middle and lower panels of Fig. 1. These outcomesqualitatively capture an unanticipated range of behaviors inthe right middle panel.The velocity reaches a well-defined stationary value forany value of the force in the figure (left panels), even in theregime of sharp increase around F ¼ 1. Also, for parame-ters similar to those used in Ref. [10], we see the enhance-ment of the (well-defined) diffusion coefficient around thecritical deterministic force F c ¼ 1  ¼ 0:95 (lowerright panel). However, entirely different behavior is nowseen in the middle right panel of the figure, which showshuge variations in the diffusion coefficient. Note that thesevariations extend over orders of magnitude. The questionthen is—what leads to these variations and why were theynot identified in Ref. [10]?The answer lies in the choice of the correlation length ofthe random potential. In [10] the correlation length is muchgreater than a single period of the periodic potential, that is,the randomness is very smooth. However, entirely differentbehavior is seen when the random potential is more corru-gated, as it is in our case. We see that the traditionaldiffusion coefficient is no longer well defined in this re-gime but instead presents very strong fluctuations aroundits peak value, making a precise estimation of D question-able. Note also the very large scale differences of themiddle and lower right panels of the figure. These aresignatures of anomalous behavior. While it is not clearwhether a well-defined value of D would be obtained atmuch longer times beyond our computational reach, wehave repeated these calculations for many more particlesand potential realizations and continue to find thisvariability.To understand the origin of these anomalies, in the upperpanel of Fig. 1 we draw the effective barriers h extractedfrom the total potential UðzÞ F z for a particular realiza-tion of the random potential. The heights and locations ofthe barriers are random, and most of them exceed thethermal energy (dashed line). Smaller values of  lead toa greater number and height of the barriers. As a particlemoves along such a landscape, it must overcome thesebarriers, some of which are extremely high. Thus, evenwith a small amount of disorder the particle motion isdominated by random waiting times due to the dispersionof the barrier heights, and a few long waiting times thatgreatly influence the outcome. We go on to show that thediffusion anomaly in Fig. 1 is a consequence of suchlandscapes and that it is qualitatively different from asimple large enhancement of the diffusion coefficient. Infact, strong fluctuations in the usual ensemble calculationof the diffusion coefficient indicate that our system isexhibiting behavior reminiscent of aging or of weak ergo-dicity breaking [11–13] over the time scales of our simu-lations. We approach the problem with this observation inmind.Our numerical results are collected as follows. Particlesare initially located at random positions uniformly distrib-uted over a region of about 1000 sites, and for each weobserve the times tp that it takes a single particle to coverthe underlying spatial period p over the course of itstrajectory over a long time. We collect these statistics formany particles and many realizations of the random po-tential. If the motion of the particles is ‘‘normal’’ thenfollowing the reasoning in [1,8] and also used in [10] leadsto the average velocity and diffusion coefficient expres-sions [14]z00.05h00.20.40510150.9 100.20.40.9 100.10.2v(τ)D(τ)FIG. 1 (color online). Upper panel: Barriers h of one realiza-tion of the total potential in a spatial domain of32 periods with ¼ 0:5 and F ¼ 1. The horizontal dashed line is the thermalenergy T . Middle and lower panels: Numerical data for 50different disordered potentials and N ¼ 100 particles in eachpotential. Left panels: vðÞ. Right panels: DðÞ. Middle panels: ¼ 1. Lower panels:  ¼ 2 (regime of Ref. [10]).  ¼ 0:05, ¼ 104. Each line corresponds to an average over 100 particlesfor a particular random potential.PRL 106, 090602 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending4 MARCH 2011090602-2v ¼ phtpi ; D ¼2p2h2tpihtpi3: (5)Here htpi is the average of the tp over all trajectories,particles, and potential realizations and h2tpi is theirdispersion about the average. These expressions are onlywell defined if the first and second moments of the distri-bution of the tp are finite. We present numerical evidencethat indicates that in the presence of a small amount ofdisorder in the potential the distribution PðtpÞ of the time tpto cover a single period can have a power-law tail for longstretches of time,PðtpÞ  t	p : (6)This behavior leads to the observed anomalies if 	 issufficiently small. For 	> 3, the first and second momentsare finite, so transport and diffusion are normal. In therange 2<	< 3, the first moment is finite but the secondmoment diverges. This leads to a finite average velocity v,but the diffusion coefficient as defined in Eq. (5) divergesas D t3	 (superdiffusion). In the interval 3=2<	< 2the first moment diverges and the average velocity thusvanishes as v t	2 (subtransport). The second momentagain diverges, leading to D t2	3 (superdiffusion). For1<	< 3=2 we again have subtransport (v t	2), thediffusion coefficient decays to zero (D t2	3, subdiffu-sion), and the particle remains extremely localized aroundits point of origin. We observe all of these behaviors.To obtain values of 	, we focus on the tails of thedistribution. If they follow a power law, exponents 	 areestimated, as depicted in Fig. 2. In the upper panel ofTable I, we present a set of typical results of simulations.In many cases, a value 	< 3 associated with anomalousbehavior is found. Table I also shows that a lower level ofdisorder ( 2%) leads to higher exponents. For comparisonwith Ref. [10], we also show the case  ¼ 2 for a 5%level of disorder. We observe what we expected fromEq. (2) when  increases, that is, the effect of the randompart of the potential is greatly muted. This implies thatmost of the exponents are in the range 	> 3 (normalbehavior) for the range of forces selected, which explainswhy the anomalies explored herein were not found inRef. [10]. Much smaller forces would need to be exploredto find robust anomalous behavior there. For our parameterchoices, using  as a control parameter (keeping F fixed)also leads to	 values covering all the possible regimes (seebottom panel of Table I). An important caveat that isapparent in the dispersion of data for large values of tp inFig. 2 is the sensitivity of the estimated 	 to the length ofthe simulations since the simulation time limits the valuesof tp that can be probed.As the value of 	 increases with increasing , the shapeof the distribution PðtpÞ, particularly its tail, changes froma power-law form to the more typical exponential associ-ated with normal behavior. This occurs because changing induces a change of the effective contribution of the ran-dom part of the potential. Earlier we pointed out thatincreasing  diminishes the effects of the disorder. Infact, case (d) of Fig. 2 is no longer a power law, as shownin the inset of the figure, but is instead better described asan exponential, characteristic of normal diffusive behavior.Finally, temporal evolutions of the velocity and thediffusion coefficient for the four different regimes of be-havior are shown in Fig. 3. The solid lines are trajectorysimulation results while the circles and dashed lines are thepredictions in the text following Eq. (6). Qualitatively, thesubtransport regimes of the two cases with the lowestvalues of  are clearly seen, as are the subdiffusive andsuperdiffusive behaviors of the diffusion coefficient.In conclusion, we have carried out a detailed numericalstudy of the motion of particles driven by a constantexternal force over a one-dimensional landscape consistingof a periodic potential modified by a small amount ofspatial disorder. We have identified a set of dramaticanomalous behaviors as diverse as subtransport, subdiffu-sion, and superdiffusion on the same surfaces as the driving101102103104105106tp10-810-610-410-2100P(tp)0 50 100 150 200tp10-410-2100P(tp)(a)(b)(c)(c)(d)(d)FIG. 2 (color online). Log-log plot of the time distribution fordifferent values of . (a)  ¼ 0:1 (	 ¼ 1:3), (b)  ¼ 0:4 (	 ¼1:7), (c)  ¼ 0:8 (	 ¼ 2:5), and (d)  ¼ 2 (	 ¼ 6:9). Theparameter values are F ¼ 1 and  ¼ 0:05. Inset: Log plot ofPðtpÞ for cases (c) and (d). Also shown is the exponentialdistribution for a purely periodic potential with the same force(green [light gray]).Time   104 as needed to obtain reliablehistograms.TABLE I. Upper panel: Exponents 	 for different values of theforce and two values of  and . Bottom panel: Exponents 	 fordifferent values of , with F ¼ 1 and  ¼ 0:05.nF 0.82 0.92 0.95 0.98 1.0 1.02 1.050.05 ( ¼ 1) 1.1 1.7 2.2 2.7 2.9 3.6 4.80.02 ( ¼ 1) 1.4 2.3 3.2 4.3 4.6 >5 >50.05 ( ¼ 2) 2.2 3.0 >5 >5 >5 >5 >5n 0.1 0.4 0.6 0.8 1.2 1.4 20.05 (F ¼ 1) 1.3 1.7 2.1 2.5 3.5 4.0 >5PRL 106, 090602 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending4 MARCH 2011090602-3force or the random corrugation length is varied. Thesebehaviors are observed over very long time scales. Theirasymptotic persistence behavior is not known. Earlierstudies have focused only on parameters of normal behav-ior [9,10], and while they have identified the occurrence ofenhanced diffusion when the external force approaches thecritical value associated with the transition from locked torunning solutions, they have not recognized the regime ofanomalous behavior.The anomalous behaviors that we observe are similar tothose associated with continuous time random walk(CTRW) models with a priori power-law waiting timedistributions [14]. Connections between ordinary randomwalks in random environments and CTRWs in orderedenvironments are abundant in the literature [12,15], butthey are typically confined to master equation models withrandom transition rates. Our case is different in that wedeal with Langevin dynamics. We have arrived at CTRW-like dynamics from an overdamped Langevin model that isparticularly useful and used for the discussion of diffusionof particles on surfaces [1–10]. We have found anomalousbehavior when a periodic potential is slightly corrugatedover short distances. Earlier studies had focused on re-gimes of very long smooth variation of the random con-tribution to the potential. Experiments with morecorrugated surfaces than have been used so far [9] areclearly desirable to see the effects that we have identified.It would of course also be desirable to extend thesestudies to higher dimensions and to gain further insightinto the asymptotic behavior. Such work has recently beenpresented for the case of a piecewise linear random poten-tial [16], but seems not yet to be available for the morerealistic potentials considered here.This work was supported by the MICINN (Spain) underthe Project No. FIS2009-13360 (A.M. L., M.K. andJ.M. S.), by Generalitat de Catalunya ProjectsNo. 2009SGR14 (J.M. S., M.K.) and No. 2009SGR878(A.M. L.), and the Grant No. FPU-AP2005-4765 (M.K.).K. L. gratefully acknowledges the NSF under GrantNo. PHY-0855471.[1] P. Reimann, C. Van den Broeck, H. Linke, P. Hänggi, J.M.Rubi, and A. Pérez-Madrid, Phys. Rev. Lett. 87, 010602(2001).[2] J.M. Sancho, A.M. Lacasta, Katja Lindenberg, I.M.Sokolov, and A.H. Romero, Phys. Rev. Lett. 92, 250601(2004).[3] A.M. Lacasta, J.M. Sancho, A. H. Romero, I.M. Sokolov,and Katja Lindenberg, Phys. Rev. E 70, 051104 (2004).[4] Katja Lindenberg, A.M. Lacasta, J.M. Sancho, and A.H.Romero, Proc. SPIE Int. Soc. Opt. Eng. 5845, 201 (2005).[5] Katja Lindenberg, A.M. Lacasta, J.M. Sancho, and A.H.Romero, New J. Phys. 7, 29 (2005).[6] Katja Lindenberg, J.M. Sancho, A.M. Lacasta, and I.M.Sokolov, Phys. Rev. Lett. 98, 020602 (2007).[7] M. Khoury, James P. Gleeson, J.M. Sancho, A.M.Lacasta, and Katja Lindenberg, Phys. Rev. E 80, 021123(2009).[8] B. Lindner, M. Kostur, and L. Schimansky-Geier, Fluct.Noise Lett. 1, R25 (2001).[9] Sang-Hyuk Lee and David G. Grier, Phys. Rev. Lett. 96,190601 (2006).[10] P. Reimann and R. Eichhorn, Phys. Rev. Lett. 101, 180601(2008).[11] Y. He, S. Burov, R. Metzler, and E. Barkai, Phys. Rev.Lett. 101, 058101 (2008); I.M. Sokolov, E. Heinsalu, P.Hänggi, and I. Goychuck, Europhys. Lett. 86, 30 009(2009).[12] J. P. Bouchaud, J. Phys. I (France) 2, 1705 (1992).[13] E. Barkai, in Anomalous Transport, edited by R. Klages,G. Radons, and I.M. Sokolov (Wiley-VCH, Weinheim,2008).[14] M. F. Shlesinger, J. Stat. Phys. 10, 421 (1974).[15] B. H. Hughes, Random Walks and Random Environments,Volume 1: Random Walks (Clarendon Press, Oxford,1995).[16] S. I. Denisov, E. S. Denisova, and H. Kantz, Eur. Phys. J. B76, 1 (2010).101 102 103 104 105 106τ10-310-210-1100101v(τ)101 102 103 104 105 106τ10-310-210-1100101D(τ)τ−0.3τ0.4τ−0.7 τ−0.4τ0.5FIG. 3 (color online). Log-log plot of the time evolution of thevelocity (left) and of the diffusion coefficient (right) for differentvalues of . Left panel, bottom to top:  ¼ 0:1, 0.4, 0.8, and 2.Right panel, bottom to top:  ¼ 0:1, 2, 0.4, and 0.8. The solidcurves are obtained from trajectory simulations. The circles anddashed curves are the predictions in the text following Eq. (6).Parameter values:  ¼ 0:05, F ¼ 1. Number of potential real-izations: 100, except for  ¼ 2 where 20 realizations aresufficient.PRL 106, 090602 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending4 MARCH 2011090602-4

UPCommons. Portal del coneixement obert de la UPC

Weak Disorder: Anomalous Transport and Diffusion Are Normal Yet AgainM. Khoury,1 A.M. Lacasta,2 J.M. Sancho,1 and Katja Lindenberg31Departament d’Estructura i Constituents de la Mate`ria, Diagonal 647, E-08028 Barcelona, Spain2Departament de Fı´sica Aplicada, Universitat Polite`cnica de Catalunya, Avinguda Doctor Maran˜on 44, E-08028 Barcelona, Spain3Department of Chemistry and Biochemistry 0340 and BioCircuits Institute, University of California,San Diego, La Jolla, California 92093-0340, USA(Received 9 September 2010; published 2 March 2011)We carry out a detailed study of the motion of particles driven by a constant external force over alandscape consisting of a periodic potential corrugated by a small amount of spatial disorder. We observeanomalous behavior in the form of subdiffusion and superdiffusion and even subtransport over very longtime scales. Recent studies of transport over slightly random landscapes have focused only on parametersleading to normal behavior, and while enhanced diffusion has been identified when the external forceapproaches the critical value associated with the transition from locked to running solutions, the regime ofanomalous behavior had not been recognized. We provide a qualitative explanation for the origin of theseanomalies, and make connections with a continuous time random walk approach.DOI: 10.1103/PhysRevLett.106.090602 PACS numbers: 05.40.a, 02.50.Ey, 05.60.kSolid state surfaces frequently present periodic poten-tials marred by some disorder. Herein we show that anoverdamped particle moving over such a potential in onedimension (1D) may exhibit anomalous behavior in theform of superdiffusion, subdiffusion, and even subtran-sport. Although we cannot prove that these are steady stateregimes, our numerical simulation data show them to bepresent over time spans of several orders of magnitude.That diffusion of particles over both periodic and ran-dom surfaces leads to some forms of anomalous behavioris of course well known and continues to attract a great dealof attention [1–10]. In periodic potentials with low friction,extremely long (in time) dispersionless transport regimescan be observed when forces exceed a critical force [6].Moreover, in these same systems, in both overdamped andunderdamped regimes, the diffusion coefficient versus theapplied force presents a pronounced peak around the criti-cal force that allows the coexistence of locked and runningstates [1,4–7]. The enhancement is quantitatively largerthan the free particle diffusion coefficient. This behaviorhas been observed experimentally when tracking the mo-tion of colloidal spheres through a periodic potential cre-ated with optical vortex traps [9].The enhancement of the diffusion coefficient is evenmore pronounced when disorder is also present [9]. Thisphenomenon has been tested by numerical simulations on asurface in which a small amount of spatial disorder in theform of a random potential is added to the periodic poten-tial [10]. Dramatic diffusive enhancement occurs even forvery small amounts of disorder, e.g., when the amplitude ofthe random contribution of the potential relative to that ofthe periodic distribution is as small as 5%.Although dramatic, diffusive enhancement turns out tobe only a limited aspect of the story. Here we present arange of additional anomalous transport and diffusionphenomena arising from weak disorder that have notbeen previously noted. Our model is inspired by [9,10].We consider the overdamped motion of identical non-interacting Brownian particles moving in a 1D potentiallandscape UðxÞ following the Langevin equation  _xðtÞ ¼U0ðxÞ þ Fþ ðtÞ. Here x is the position of the particle, tdenotes time,  is the dissipation parameter, F is theapplied force, and ðtÞ is Gaussian thermal noise at tem-perature T. The correlation function of the noise obeysthe fluctuation-dissipation relation hðtÞðt0Þi ¼2kBTðt t0Þ. The potential UðxÞ consists of a periodicpart, VpðxÞ ¼ V0 cosð2x=pÞ, and a Gaussian spatiallyrandom contribution VrðxÞ with correlation functiongrðxÞ  hVrðxÞVrð0Þi ¼ V202exp 22x2l2r: (1)The relative contributions are determined by the parameter 2 ½0; 1 in the combination UðxÞ ¼ ð1 ÞVpðxÞ þVrðxÞ. Furthermore, we have chosen the potential corre-lations grðxÞ and gpðxÞ ¼ ðV20=2Þ cosð2x=pÞ to be equalat x ¼ 0, gpð0Þ ¼ grð0Þ. This ensures that the total poten-tial amplitude is of order V0 independently of . Also, withthe particular choice lr ¼ p the two correlation functionsare identical up to second order in a Taylor expansion.In terms of the spatial variable z ¼ 2x=p and thetemporal variable  ¼ ½ð2Þ2V0=2pt, the equation ofmotion can be reduced into dimensionless form,_z ¼ ð1 ÞfpðzÞ þ ð=Þfrðz=Þ þF þ ðÞ; (2)where fp and fr are the dimensionless forces arising fromVp and Vr respectively, and  is the dimensionless noise.The dimensionless parameters arePRL 106, 090602 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending4 MARCH 20110031-9007=11=106(9)=090602(4) 090602-1  2011 American Physical Society ¼ lrp; F ¼ pF2V0; T ¼ kBTV0: (3)Throughout this work we set T ¼ 0:01 and p ¼ 2.Variations in T do not lead to any additional phenome-nology. The specific choice of p ¼ 2 is only importantas a reference value. In these variables a decrease of  evenfor fixed  leads to an increase in the relative contributionof the random force.We have carried out numerical simulations of Eq. (2)over a large number (100) of particle trajectories and alarge number of realizations of the random potential con-tribution (typically 50–100) in order to calculate the ve-locity and diffusion coefficient using the prescriptionsvðÞ ¼ hzðÞi; DðÞ ¼ hz2ðÞi  hzðÞi22: (4)The brackets h  i denote averages over many trajectories.The numerical procedures are entirely standard. We showthat the usual assumption that stationary values are reachedat long times is at the very least questionable.Representative outcomes of this procedure are shown inthe middle and lower panels of Fig. 1. These outcomesqualitatively capture an unanticipated range of behaviors inthe right middle panel.The velocity reaches a well-defined stationary value forany value of the force in the figure (left panels), even in theregime of sharp increase around F ¼ 1. Also, for parame-ters similar to those used in Ref. [10], we see the enhance-ment of the (well-defined) diffusion coefficient around thecritical deterministic force F c ¼ 1  ¼ 0:95 (lowerright panel). However, entirely different behavior is nowseen in the middle right panel of the figure, which showshuge variations in the diffusion coefficient. Note that thesevariations extend over orders of magnitude. The questionthen is—what leads to these variations and why were theynot identified in Ref. [10]?The answer lies in the choice of the correlation length ofthe random potential. In [10] the correlation length is muchgreater than a single period of the periodic potential, that is,the randomness is very smooth. However, entirely differentbehavior is seen when the random potential is more corru-gated, as it is in our case. We see that the traditionaldiffusion coefficient is no longer well defined in this re-gime but instead presents very strong fluctuations aroundits peak value, making a precise estimation of D question-able. Note also the very large scale differences of themiddle and lower right panels of the figure. These aresignatures of anomalous behavior. While it is not clearwhether a well-defined value of D would be obtained atmuch longer times beyond our computational reach, wehave repeated these calculations for many more particlesand potential realizations and continue to find thisvariability.To understand the origin of these anomalies, in the upperpanel of Fig. 1 we draw the effective barriers h extractedfrom the total potential UðzÞ F z for a particular realiza-tion of the random potential. The heights and locations ofthe barriers are random, and most of them exceed thethermal energy (dashed line). Smaller values of  lead toa greater number and height of the barriers. As a particlemoves along such a landscape, it must overcome thesebarriers, some of which are extremely high. Thus, evenwith a small amount of disorder the particle motion isdominated by random waiting times due to the dispersionof the barrier heights, and a few long waiting times thatgreatly influence the outcome. We go on to show that thediffusion anomaly in Fig. 1 is a consequence of suchlandscapes and that it is qualitatively different from asimple large enhancement of the diffusion coefficient. Infact, strong fluctuations in the usual ensemble calculationof the diffusion coefficient indicate that our system isexhibiting behavior reminiscent of aging or of weak ergo-dicity breaking [11–13] over the time scales of our simu-lations. We approach the problem with this observation inmind.Our numerical results are collected as follows. Particlesare initially located at random positions uniformly distrib-uted over a region of about 1000 sites, and for each weobserve the times tp that it takes a single particle to coverthe underlying spatial period p over the course of itstrajectory over a long time. We collect these statistics formany particles and many realizations of the random po-tential. If the motion of the particles is ‘‘normal’’ thenfollowing the reasoning in [1,8] and also used in [10] leadsto the average velocity and diffusion coefficient expres-sions [14]z00.05h00.20.40510150.9 100.20.40.9 100.10.2v(τ)D(τ)FIG. 1 (color online). Upper panel: Barriers h of one realiza-tion of the total potential in a spatial domain of32 periods with ¼ 0:5 and F ¼ 1. The horizontal dashed line is the thermalenergy T . Middle and lower panels: Numerical data for 50different disordered potentials and N ¼ 100 particles in eachpotential. Left panels: vðÞ. Right panels: DðÞ. Middle panels: ¼ 1. Lower panels:  ¼ 2 (regime of Ref. [10]).  ¼ 0:05, ¼ 104. Each line corresponds to an average over 100 particlesfor a particular random potential.PRL 106, 090602 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending4 MARCH 2011090602-2v ¼ phtpi ; D ¼2p2h2tpihtpi3: (5)Here htpi is the average of the tp over all trajectories,particles, and potential realizations and h2tpi is theirdispersion about the average. These expressions are onlywell defined if the first and second moments of the distri-bution of the tp are finite. We present numerical evidencethat indicates that in the presence of a small amount ofdisorder in the potential the distribution PðtpÞ of the time tpto cover a single period can have a power-law tail for longstretches of time,PðtpÞ  t	p : (6)This behavior leads to the observed anomalies if 	 issufficiently small. For 	> 3, the first and second momentsare finite, so transport and diffusion are normal. In therange 2<	< 3, the first moment is finite but the secondmoment diverges. This leads to a finite average velocity v,but the diffusion coefficient as defined in Eq. (5) divergesas D t3	 (superdiffusion). In the interval 3=2<	< 2the first moment diverges and the average velocity thusvanishes as v t	2 (subtransport). The second momentagain diverges, leading to D t2	3 (superdiffusion). For1<	< 3=2 we again have subtransport (v t	2), thediffusion coefficient decays to zero (D t2	3, subdiffu-sion), and the particle remains extremely localized aroundits point of origin. We observe all of these behaviors.To obtain values of 	, we focus on the tails of thedistribution. If they follow a power law, exponents 	 areestimated, as depicted in Fig. 2. In the upper panel ofTable I, we present a set of typical results of simulations.In many cases, a value 	< 3 associated with anomalousbehavior is found. Table I also shows that a lower level ofdisorder ( 2%) leads to higher exponents. For comparisonwith Ref. [10], we also show the case  ¼ 2 for a 5%level of disorder. We observe what we expected fromEq. (2) when  increases, that is, the effect of the randompart of the potential is greatly muted. This implies thatmost of the exponents are in the range 	> 3 (normalbehavior) for the range of forces selected, which explainswhy the anomalies explored herein were not found inRef. [10]. Much smaller forces would need to be exploredto find robust anomalous behavior there. For our parameterchoices, using  as a control parameter (keeping F fixed)also leads to	 values covering all the possible regimes (seebottom panel of Table I). An important caveat that isapparent in the dispersion of data for large values of tp inFig. 2 is the sensitivity of the estimated 	 to the length ofthe simulations since the simulation time limits the valuesof tp that can be probed.As the value of 	 increases with increasing , the shapeof the distribution PðtpÞ, particularly its tail, changes froma power-law form to the more typical exponential associ-ated with normal behavior. This occurs because changing induces a change of the effective contribution of the ran-dom part of the potential. Earlier we pointed out thatincreasing  diminishes the effects of the disorder. Infact, case (d) of Fig. 2 is no longer a power law, as shownin the inset of the figure, but is instead better described asan exponential, characteristic of normal diffusive behavior.Finally, temporal evolutions of the velocity and thediffusion coefficient for the four different regimes of be-havior are shown in Fig. 3. The solid lines are trajectorysimulation results while the circles and dashed lines are thepredictions in the text following Eq. (6). Qualitatively, thesubtransport regimes of the two cases with the lowestvalues of  are clearly seen, as are the subdiffusive andsuperdiffusive behaviors of the diffusion coefficient.In conclusion, we have carried out a detailed numericalstudy of the motion of particles driven by a constantexternal force over a one-dimensional landscape consistingof a periodic potential modified by a small amount ofspatial disorder. We have identified a set of dramaticanomalous behaviors as diverse as subtransport, subdiffu-sion, and superdiffusion on the same surfaces as the driving101 102 103 104 105 106tp10-810-610-410-2100P(t p)0 50 100 150 200tp10-410-2100P(t p)(a)(b)(c)(c)(d)(d)FIG. 2 (color online). Log-log plot of the time distribution fordifferent values of . (a)  ¼ 0:1 (	 ¼ 1:3), (b)  ¼ 0:4 (	 ¼1:7), (c)  ¼ 0:8 (	 ¼ 2:5), and (d)  ¼ 2 (	 ¼ 6:9). Theparameter values are F ¼ 1 and  ¼ 0:05. Inset: Log plot ofPðtpÞ for cases (c) and (d). Also shown is the exponentialdistribution for a purely periodic potential with the same force(green [light gray]).Time   104 as needed to obtain reliablehistograms.TABLE I. Upper panel: Exponents 	 for different values of theforce and two values of  and . Bottom panel: Exponents 	 fordifferent values of , with F ¼ 1 and  ¼ 0:05.nF 0.82 0.92 0.95 0.98 1.0 1.02 1.050.05 ( ¼ 1) 1.1 1.7 2.2 2.7 2.9 3.6 4.80.02 ( ¼ 1) 1.4 2.3 3.2 4.3 4.6 >5 >50.05 ( ¼ 2) 2.2 3.0 >5 >5 >5 >5 >5n 0.1 0.4 0.6 0.8 1.2 1.4 20.05 (F ¼ 1) 1.3 1.7 2.1 2.5 3.5 4.0 >5PRL 106, 090602 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending4 MARCH 2011090602-3force or the random corrugation length is varied. Thesebehaviors are observed over very long time scales. Theirasymptotic persistence behavior is not known. Earlierstudies have focused only on parameters of normal behav-ior [9,10], and while they have identified the occurrence ofenhanced diffusion when the external force approaches thecritical value associated with the transition from locked torunning solutions, they have not recognized the regime ofanomalous behavior.The anomalous behaviors that we observe are similar tothose associated with continuous time random walk(CTRW) models with a priori power-law waiting timedistributions [14]. Connections between ordinary randomwalks in random environments and CTRWs in orderedenvironments are abundant in the literature [12,15], butthey are typically confined to master equation models withrandom transition rates. Our case is different in that wedeal with Langevin dynamics. We have arrived at CTRW-like dynamics from an overdamped Langevin model that isparticularly useful and used for the discussion of diffusionof particles on surfaces [1–10]. We have found anomalousbehavior when a periodic potential is slightly corrugatedover short distances. Earlier studies had focused on re-gimes of very long smooth variation of the random con-tribution to the potential. Experiments with morecorrugated surfaces than have been used so far [9] areclearly desirable to see the effects that we have identified.It would of course also be desirable to extend thesestudies to higher dimensions and to gain further insightinto the asymptotic behavior. Such work has recently beenpresented for the case of a piecewise linear random poten-tial [16], but seems not yet to be available for the morerealistic potentials considered here.This work was supported by the MICINN (Spain) underthe Project No. FIS2009-13360 (A.M. L., M.K. andJ.M. S.), by Generalitat de Catalunya ProjectsNo. 2009SGR14 (J.M. S., M.K.) and No. 2009SGR878(A.M. L.), and the Grant No. FPU-AP2005-4765 (M.K.).K. L. gratefully acknowledges the NSF under GrantNo. PHY-0855471.[1] P. Reimann, C. Van den Broeck, H. Linke, P. Ha¨nggi, J.M.Rubi, and A. Pe´rez-Madrid, Phys. Rev. Lett. 87, 010602(2001).[2] J.M. Sancho, A.M. Lacasta, Katja Lindenberg, I.M.Sokolov, and A.H. Romero, Phys. Rev. Lett. 92, 250601(2004).[3] A.M. Lacasta, J.M. Sancho, A. H. Romero, I.M. Sokolov,and Katja Lindenberg, Phys. Rev. E 70, 051104 (2004).[4] Katja Lindenberg, A.M. Lacasta, J.M. Sancho, and A.H.Romero, Proc. SPIE Int. Soc. Opt. Eng. 5845, 201 (2005).[5] Katja Lindenberg, A.M. Lacasta, J.M. Sancho, and A.H.Romero, New J. Phys. 7, 29 (2005).[6] Katja Lindenberg, J.M. Sancho, A.M. Lacasta, and I.M.Sokolov, Phys. Rev. Lett. 98, 020602 (2007).[7] M. Khoury, James P. Gleeson, J.M. Sancho, A.M.Lacasta, and Katja Lindenberg, Phys. Rev. E 80, 021123(2009).[8] B. Lindner, M. Kostur, and L. Schimansky-Geier, Fluct.Noise Lett. 1, R25 (2001).[9] Sang-Hyuk Lee and David G. Grier, Phys. Rev. Lett. 96,190601 (2006).[10] P. Reimann and R. Eichhorn, Phys. Rev. Lett. 101, 180601(2008).[11] Y. He, S. Burov, R. Metzler, and E. Barkai, Phys. Rev.Lett. 101, 058101 (2008); I.M. Sokolov, E. Heinsalu, P.Ha¨nggi, and I. Goychuck, Europhys. Lett. 86, 30 009(2009).[12] J. P. Bouchaud, J. Phys. I (France) 2, 1705 (1992).[13] E. Barkai, in Anomalous Transport, edited by R. Klages,G. Radons, and I.M. Sokolov (Wiley-VCH, Weinheim,2008).[14] M. F. Shlesinger, J. Stat. Phys. 10, 421 (1974).[15] B. H. Hughes, Random Walks and Random Environments,Volume 1: Random Walks (Clarendon Press, Oxford,1995).[16] S. I. Denisov, E. S. Denisova, and H. Kantz, Eur. Phys. J. B76, 1 (2010).101 102 103 104 105 106τ10-310-210-1100101v(τ)101 102 103 104 105 106τ10-310-210-1100101D(τ)τ−0.3τ0.4τ−0.7 τ−0.4τ0.5FIG. 3 (color online). Log-log plot of the time evolution of thevelocity (left) and of the diffusion coefficient (right) for differentvalues of . Left panel, bottom to top:  ¼ 0:1, 0.4, 0.8, and 2.Right panel, bottom to top:  ¼ 0:1, 2, 0.4, and 0.8. The solidcurves are obtained from trajectory simulations. The circles anddashed curves are the predictions in the text following Eq. (6).Parameter values:  ¼ 0:05, F ¼ 1. Number of potential real-izations: 100, except for  ¼ 2 where 20 realizations aresufficient.PRL 106, 090602 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending4 MARCH 2011090602-4

Weak disorder: anomalous transport and diffusion are normal yet again

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