By applying a staggered driving force in a prototypical discrete model with a
quartic nonlinearity, we demonstrate the spontaneous formation and destruction
of discrete breathers with a selected frequency due to thermal fluctuations.
The phenomenon exhibits the striking features of stochastic resonance (SR): a
nonmonotonic behavior as noise is increased and breather generation under
subthreshold conditions. The corresponding peak is associated with a matching
between the external driving frequency and the breather frequency at the
average energy given by ambient temperature.Comment: Added references, figure 5 modified to include new dat

David Cubero

G. Parisi

Jesús Cuevas

P. G. Kevrekidis

R. Zwanzig

English

arXiv

This is the pre-published version harvested from arXiv. The published version is located at http://prl.aps.org/abstract/PRL/v102/i20/e205505By applying a staggered driving force in a prototypical discrete model with a quartic nonlinearity, we demonstrate the spontaneous formation and destruction of discrete breathers with a selected frequency due to thermal fluctuations. The phenomenon exhibits the striking features of stochastic resonance: a nonmonotonic behavior as noise is increased and breather generation under subthreshold conditions. The corresponding peak is associated with a matching between the external driving frequency and the breather frequency at the average energy given by ambient temperature.

Cubero, D

Cuevas, J

Kevrekidis, PG

ScholarWorks@UMass Amherst

Nucleation of Breathers via Stochastic Resonance in Nonlinear Lattices

By applying a staggered driving force in a prototypical discrete model with a quartic nonlinearity, we demonstrate the spontaneous formation and destruction of discrete breathers with a selected frequency due to thermal fluctuations. The phenomenon exhibits the striking features of stochastic resonance: a nonmonotonic behavior as noise is increased and breather generation under subthreshold conditions. The corresponding peak is associated with a matching between the external driving frequency and the breather frequency at the average energy given by ambient temperature

Crossref

By applying a staggered driving force in a prototypical discrete model with a quartic nonlinearity, we demonstrate the spontaneous formation and destruction of discrete breathers with a selected frequency due to thermal fluctuations. The phenomenon exhibits the striking features of stochastic resonance (SR): a nonmonotonic behavior as noise is increased and breather generation under subthreshold conditions. The corresponding peak is associated with a matching between the external driving frequency and the breather frequency at the average energy given by ambient temperatur

Cubero Gómez, David

Cuevas-Maraver, Jesús

Kevrekidis, Panayotis G.

idUS. Depósito de Investigación Universidad de Sevilla

Nucleation of Breathers via Stochastic Resonance in Nonlinear LatticesDavid Cubero,1 Jesu´s Cuevas,1 and P. G. Kevrekidis21Departamento de Fı´sica Aplicada I, EUP, Universidad de Sevilla, Calle Virgen de A´frica 7, 41011 Sevilla, Spain2Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA(Received 28 January 2009; published 21 May 2009)By applying a staggered driving force in a prototypical discrete model with a quartic nonlinearity, wedemonstrate the spontaneous formation and destruction of discrete breathers with a selected frequency dueto thermal fluctuations. The phenomenon exhibits the striking features of stochastic resonance: a non-monotonic behavior as noise is increased and breather generation under subthreshold conditions. Thecorresponding peak is associated with a matching between the external driving frequency and the breatherfrequency at the average energy given by ambient temperature.DOI: 10.1103/PhysRevLett.102.205505 PACS numbers: 63.20.Pw, 05.40.a, 05.45.Xt, 05.45.YvIntrinsic localized modes, often referred to as discretebreathers, have been the focus of a number of theoreticalstudies (for a recent review see, e.g., [1]). One of the mainreasons for the intense interest in these modes is the factthat they provide a natural setup for energy localization, aparadigm of interest to many areas of physics as well aschemistry and biology. Such discrete breathers have beenrigorously proven to exist as time-periodic localized ex-citations in nonlinear Hamiltonian lattices [2]. Ever since,they have been experimentally observed in a wide varietyof different media, ranging from optical waveguides andphotorefractive crystals to micromechanical cantilever ar-rays and Josephson junctions, as well as in Bose-Einsteincondensates and layered antiferromagnets [1,3].In this Letter, having in mind realistic physical systems,we will focus on the nucleation of such breathers in aprototypical model system in which friction and ambientnoise are also present. This will lead us to the considerationof stochastic resonance (SR), the counterintuitive phe-nomenon by which an appropriate dose of noise, insteadof degradation, produces an enhancement of sensitivity ofa nonlinear system to external forcing [4]. In spatiallyextended systems, SR has been shown for bistable [5] orexcitable media [6], usually associated with pattern for-mation [7]; see, e.g., the review of [4].Our prototypical nonlinear model will consist of aquartic potential (the so-called hard 4 lattice [8]) withthe following equation of motion for each oscillator xn(n ¼ 1; . . . ; N) with mass m:m €xn ¼ U0ðxnÞ þ kðxnþ1 þ xn1  2xnÞ  _xn þ n þ FnðtÞ; (1)where the overdot and the prime denote the time and spacederivative, respectively; k is the coupling parameter be-tween oscillators;  is the damping constant; FnðtÞ denotesexternal driving; and n is a Gaussian white noise with zeromean and autocorrelation hnðtÞmðt0Þi ¼ 2Dnmðt t0Þ.The noise strength parameter D obeys the fluctuation-dissipation relation kBT ¼ D=, where kB is theBoltzmann constant and T the ambient temperature.Thus, D is an indicator of the level of noise associatedwith thermal fluctuations.The on-site potential is given byUðxÞ ¼ ax2=2þ bx4=4with a, b > 0. Because of the monostable character of thispotential, the noiseless lattice has a single stable steadystate, allowing us to focus on the behavior of homoclinicorbits, rather than kinklike heretoclinic orbits, like in thebistable 4 case (a < 0). This is one way in which ourwork differs from the standard bistable models of SR, orthe interesting recent work of [9], which considers noise-induced transitions between steady states in very shortFermi-Pasta-Ulam type, boundary-driven chains.Moreover, this model possesses a phonon band whichcan be easily determined upon linearization of (1) as!ph 2 ðﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃU00ð0Þ=mp ; ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ½U00ð0Þ þ 4k=mp Þ [1]. Thus, by excit-ing breathers with a frequency larger than the maximumfrequency of the phonon spectrum, !max ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃðaþ 4kÞ=mp ,these localized modes are guaranteed not to interact withthe phonons, avoiding effects like radiation resonanceswhich slowly dissipate away the energy of discrete breath-ers [10].Indeed, the associated Hamiltonian system of this hardmodel [i.e., Eq. (1) without damping and noise] exhibitsexponentially localized in space and periodic in time high(i.e., higher than !max) frequency excitations. An exampleis the dynamically stable Sievers-Takeno (ST) [11] site-centered mode. As shown in Fig. 1, this mode is staggered;i.e., the neighboring sites oscillate out of phase. Hence,these breathers will necessitate that we impose staggeredperiodic driving in order to sustain them in the presence ofdissipation. For this reason, we shall consider the followingac driving force:FnðtÞ ¼ ð1ÞnF0 sinðtÞ: (2)Its effect on a deterministic Fermi-Pasta-Ulam lattice hasbeen studied in Ref. [12].Equation (1) is supplemented with (periodic) boundaryconditions x0 ¼ xN and xNþ1 ¼ x1. Generally, 4 modelshave been commonly used to describe a wide range ofPRL 102, 205505 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending22 MAY 20090031-9007=09=102(20)=205505(4) 205505-1  2009 The American Physical Societyphenomena from elementary particle collisions and cos-mological domain walls [13] to droplet nucleation andphase transitions [14], constituting a prototypical nonlinearmodel in statistical field theory [15].For the sake of a dimensionless description, we can scaleall lengths and times by the characteristic length  ¼ða=bÞ1=2 and time  ¼ ðm=aÞ1=2, i.e. ~xn ¼ xn=, ~t ¼ t=,which implies ~k ¼ k=a, ~ ¼ =ðamÞ1=2, ~D ¼bD=ða5mÞ1=2, and ~F0 ¼ F0ðb=a3Þ1=2. In the following wewill omit the tilde symbols for simplicity. The dimension-less form of the equations of motion (1) then reads€xn ¼ xn  x3n þ kðxnþ1 þ xn1  2xnÞ  _xn þ n þ ð1ÞnF0 sinðtÞ: (3)Let us discuss, for the moment, the deterministic behav-ior of (3) (i.e., without noise,D ¼ 0). As mentioned above,the staggering factor ð1Þn in (2) is important in order tosustain the breathers in the presence of dissipation. Whenthat factor is absent, the ST breather shown in Fig. 1 isalways observed to die out, regardless of the drivingstrength. However, in addition to this staggering factor,the driving amplitude F0 must be large enough so that theenergy supplied by the driving force is able to compensatefor the energy which has been dissipated by the frictionforce. Breathers in the dissipative deterministic lattice canbe calculated using a method similar to that developed in[16]. An analysis of those breathers shows that, for k ¼ 1and  ¼ 0:1 and a breather frequency of ! ¼  ¼ 3,there is a threshold at Fth ¼ 0:6092. When the drivingamplitude is below this threshold (subthreshold ampli-tude), the dissipation gradually drains away the system’senergy, and breather modes cannot be supported. In thefollowing, unless explicitly stated, we will assume k ¼ 1, ¼ 0:1, and  ¼ 3, which implies !max ¼ﬃﬃﬃ5p<.Having summarized the results of the deterministic case,let us now turn to the analysis of the effect of thermalfluctuations and how theymodify this picture. Starting withthe suprathreshold case (F0 > Fth), we would expect thatthe addition of noise provokes the spontaneous formationof breathers with the frequency dictated by the externaldriving, even if the system starts from a uniform configu-ration. This is indeed what is observed, as shown in Fig. 2,where a typical realization of the oscillators’ trajectorieshas been plotted. The frequencies and amplitudes of theobserved noise-induced breathers are very similar to thedeterministic ST breather depicted in Fig. 1. The STbreather has the smallest size and the lowest energy of allpossible collective modes at the frequency of the driver,thus being the most probable breather to be induced bynoise. However, unlike the deterministic case, the breathersare not permanent, and the same random force which hasgenerated them is able to destroy them after a while.In the subthreshold case the driving force is not able byitself to sustain the breathers. Remarkably, however, thesimulations show that these breathers are still produced inthe system by the concerted action of noise and the drivingforce. This is shown in Fig. 3. The bottom panel shows acomparison of one of these noise-induced breathers withthe deterministic stable breather obtained with a supra-threshold driving force. This effect resembles one of themain features of the earliest manifestations of SR: thenoise-induced hopping events of a Brownian particle in abistable potential subject to a externally applied forcewhich taken alone is not sufficient to produce the hoppingevents [4]. Noise appears then as beneficial, enabling sys-tem transitions between the uniform background state andthe coherent, localized breather patterns. On the otherFIG. 2 (color online). Time evolution of the oscillator posi-tions unðtÞ :¼ xnðtÞ þ 10n for a system subject to a suprathres-hold driving with F0 ¼ 1 and  ¼ 3, and noise strengthD ¼ 0:1 starting from a uniform initial condition xnð0Þ ¼ 0.Breathers are observed to form and vanish spontaneously. Thebottom panel shows a magnification of the time evolution of anoscillator which becomes the center of a high-frequencybreather.0 5 10 15 20n0123xnFIG. 1 (color online). Spatial profile of a staggered high-frequency (! ¼ 3) Hamiltonian breather generated using meth-ods based on the anticontinuous limit [21]. The line is a guide tothe eye.PRL 102, 205505 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending22 MAY 2009205505-2hand, we would expect that if the thermal fluctuations arevery large, they will destroy any degree of order in thesystem, producing the nonmonotonic behavior character-istic of the SR phenomenon. This scenario should offer ageneric mechanism for the formation of breatherlike ex-citations in realistic nonlinear lattices where fluctuationsare present.In order to quantify these effects, we have computednumerically the Fourier coefficient of an arbitrary oscilla-tor at the frequency of the external driver,q ¼ 2TZ 2=0dteithx1ðtÞi1; (4)where the notation h. . .i indicates an average over noiserealizations and the subindex 1 indicates the long timelimit of the noise average, i.e., its value after waiting a longenough time for the transients to die out. Every time theoscillator x1 is part of a breather, or a collective attempt toform a breather mode at the frequency of the driver, willresult in a larger coefficient q. The choice of the oscillatorin (4) is irrelevant because of the periodic boundary con-ditions. The simulations were performed with N ¼ 40 andN ¼ 60 oscillators, the larger value carried out in order tocheck that the results did not depend on the system size.We present in Fig. 4 the values of the amplitude jqj2 as afunction of the noise strength D for a suprathreshold andsubthreshold driving. The nonmonotonic behavior withnoise, a recognized signature of the SR phenomenon, isvery clear in both the suprathreshold and subthresholdsituations. Nevertheless, let us note that a nonmonotonicbehavior is also obtained when we inhibit breather forma-tion by canceling the interaction between the oscillators.The dotted line in Fig. 4 shows this feature for a systemwith k ¼ 0. This is a well-known phenomenon for anysingle-oscillator system subject to a nonlinear monostablepotential, both in the underdamped [17] or overdamped[18] regime. In the former, the peak is produced by amatching between the driving frequency and the intrinsicfrequency associated with the most probable excitationenergy, which is a monotonic function of the noise strength[17]. In fact, a simple classical mechanics analysis of theundriven Hamiltonian system (, D, F0 ¼ 0) with k ¼ 0shows that a periodic solution with frequency ! ¼ 3 cor-responds to an energy per particle value of E=N ¼ 36.Under thermal equilibrium (F0 ¼ 0), we need to raise thetemperature to a value such that the noise strength is D ¼1:67 in order to obtain an average energy per particle of thesame value. It can be seen in Fig. 4 that this value ofD is ingood agreement with the position of the maximum. Asimilar argument also holds when we switch on the inter-action between the oscillators, though now collective ex-citations associated with lower energies are allowed,resulting in larger amplitudes with a peak at a lower valueof D. The total energy associated with the breather pre-sented in Fig. 1 is E ¼ 30. If we consider that roughly onlyl ¼ 5 oscillators are needed to form the ST breather, anaverage energy per particle of E=N ¼ 6 would be neces-sary in order to optimize breather formation. This value isobtained at thermal equilibrium when D ¼ 0:7, which is areasonable estimation of the peak position observed inFig. 4. A better estimation, namely, a peak at D ¼ 0:47,is obtained if we choose l ¼ 7 oscillators. The appropriatevalue of l is determined by the correlation length, asdiscussed below.For an arbitrarily small driving amplitude F0, linearresponse theory [19,20] predictshx1ðtÞi1 ¼Z 10dsðsÞfðt sÞ; (5)0 0.5 1 1.5 2 2.5D00.10.20.30.4|q|2F0=1.5F0=0.6F0=0.6  k=0FIG. 4 (color online). The Fourier amplitude jqj2, defined inEq. (4), as a function of the noise strength D for a suprathresholddriving F0 ¼ 1:5 (solid line), a subthreshold driving F0 ¼ 0:6(dashed line), and a system with F0 ¼ 0:6 but without interac-tion between the oscillators, i.e. k ¼ 0 (dotted line).FIG. 3 (color online). Same as in Fig. 2 but for a subthresholdamplitude F0 ¼ 0:6 and noise strength D ¼ 0:3. The bottompanel shows a magnification of part of the trajectory xnðtÞ of oneoscillator (boxed in the upper panel), as well as the timeevolution of the center oscillator of a deterministic (D ¼ 0)and dissipative breather ( ¼ 0:1) with suprathreshold F0 ¼ 1and same driving frequency (dashed red line).PRL 102, 205505 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending22 MAY 2009205505-3where fðtÞ ¼ F0 sinðtÞ and ðtÞ is the response function.This function can be expressed in terms of equilibriumtime correlation functions,ðtÞ ¼  Dddthx1ðtÞMð0Þieq; (6)where h. . .ieq denotes equilibrium averages in the absenceof driving and MðtÞ ¼ PNn¼1ð1ÞnxnðtÞ is part of the per-turbing Hamiltonian MfðtÞ associated with the externaldriving. The calculation of the Fourier coefficient of (5),together with the definitions r ¼R10 ðtÞ cosðtÞ andi ¼R10 ðtÞ sinðtÞ, yields the following spectral ampli-fication 	 ¼ jqj2=F20:	 ¼ ½ð0Þ  i2 þ ðrÞ2: (7)Figure 5 shows the linear response results obtained fromsimulations at thermal equilibrium. It is seen that a drivingamplitude of F0 ¼ 0:05 is small enough to be within linearresponse theory. Because of the presence of noise anddissipation, the correlation between two separated oscilla-tors decreases rapidly as their separation is increased. Infact, Fig. 5 shows that a good approximation is obtained ifthe time correlation functions hx1ðtÞxnð0Þieq correspondingto oscillators separated by more than 2 sites (jn 1j> 2)are neglected in Eq. (6). This is consistent with the aboveenergy matching argument with l ¼ 5, and the dynamicalrole of ST breathers, leading to the peak estimation atD ¼0:7. Larger driving amplitudes are associated with largercorrelation lengths, producing a peak at slightly lowervalues of D.Finally, the calculated correlation functions allow us toreadily compute the system response when the drivingforce acting on each oscillator is not staggered but spatiallyuniform, i.e., MðtÞ ¼ PNn¼1 xnðtÞ. The dotted line in Fig. 5shows that the spectral amplification is very similar to thatwhen breather formation was strictly forbidden (the casek ¼ 0 shown in Fig. 4), lacking the stronger peak at lowernoise levels. As expected from the deterministic analysis,this uniform driving force is very ineffective in boostingbreather formation.In conclusion, we have demonstrated the spontaneousformation of breathers in a hard 4 lattice subject to astaggered driving force. The formation of noise-inducedbreathers with the selected driving frequency is optimizedfor a given level of fluctuations such that the averageenergy per particle matches the intrinsic energy of thebreather mode. We expect these results to be useful forthe generation and manipulation of breathers, and to beindependent of dimension and of the specific nature of thenonlinearity, provided that the model is monostable.This research was supported by the Ministerio deCiencia e Innovacio´n of Spain (FIS2008-02873 andFIS2008-04848), as well as by NSF-CAREER, NSF-DMS-0806762. P. G.K. is grateful to D. Cai for originallysuggesting this problem and numerous discussions.[1] S. Flach and A. Gorbach, Phys. Rep. 467, 1 (2008).[2] R. MacKay and S. Aubry, Nonlinearity 7, 1623 (1994).[3] D. K. Campbell, S. Flach, and Yu. S. Kivshar, Phys. Today57, No. 1, 43 (2004).[4] L. Gammaitoni et al., Rev. Mod. Phys. 70, 223 (1998).[5] F. Marchesoni, L. Gammaitoni, and A. R. Bulsara, Phys.Rev. Lett. 76, 2609 (1996).[6] P. Jung and G. Mayer-Kress, Phys. Rev. Lett. 74, 2130(1995).[7] J. Vilar and J.M. Rubı´, Phys. Rev. Lett. 78, 2886 (1997).[8] D. Chen, S. Aubry, and G. P. Tsironis, Phys. Rev. Lett. 77,4776 (1996).[9] G. Miloshevich et al., Phys. Rev. Lett. 102, 020602(2009).[10] M. Johansson and S. Aubry, Phys. Rev. E 61, 5864 (2000).[11] A. J. Sievers and S. Takeno, Phys. Rev. Lett. 61, 970(1988).[12] R. Khomeriki, S. Lepri, and S. Ruffo, Phys. Rev. E 64,056606 (2001); Physica (Amsterdam) 168–169D, 152(2002).[13] P. Anninos et al., Phys. Rev. D 44, 1147 (1991).[14] J. S. Langer, Ann. Phys. (Leipzig) 41, 108 (1967).[15] G. Parisi, Statistical Field Theory (Perseus Books,New York, 1998).[16] J. L. Marı´n et al., Phys. Rev. E 63, 066603 (2001).[17] N. G. Stocks et al., J. Phys. A 26, L385 (1993).[18] M. Evstigneev et al., Europhys. Lett. 65, 7 (2004).[19] P. Ha¨nggi and H. Thomas, Phys. Rep. 88, 207 (1982).[20] R. Zwanzig, Nonequilibrium Statistical Mechanics(Oxford University Press, New York, 2001).[21] J. L. Marı´n and S. Aubry, Nonlinearity 9, 1501 (1996).0 0.5 1 1.5 2D00.20.40.6ηF0=0.05F0=0.6uniform drivingFIG. 5 (color online). Spectral amplification 	 ¼ jqj2=F20 as afunction of noise. Filled circles depict the linear response results,computed numerically using Eq. (7), and the solid line thesimulation results for F0 ¼ 0:05. The dashed line correspondsto F0 ¼ 0:6. Crosses and diamonds correspond to a linearresponse calculation neglecting correlations between oscillatorsseparated by more than 2 and 3 sites, respectively. Filled tri-angles depict the linear response prediction for a system with auniform external driving FnðtÞ ¼ F0 sinðtÞ (for all n) instead of(2). The dotted line is a guide to the eye.PRL 102, 205505 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending22 MAY 2009205505-4

Physical Review Letters

Nucleation of breathers via stochastic resonance in nonlinear lattices

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Nucleation of breathers via stochastic resonance in nonlinear lattices

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idUS. Depósito de Investigación Universidad de Sevilla