PHYSICAL REVIEW E 67, 031106 ~2003!

Aging in an infinite-range Hamiltonian system of coupled rotators

Marcelo A. Montemurro,1,* Francisco A. Tamarit,1,† and Celia Anteneodo2,‡

1Facultad de Matema´tica, Astronomı´a y Fısica, Universidad Nacional de Co´rdoba, Ciudad Universitaria, 5000 Co´rdoba, Argentina2Centro Brasileiro de Pesquisas Fı´sicas Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, Brazil~Received 16 May 2002; revised manuscript received 25 February 2003; published 28 March 2003!

We analyze numerically the out-of-equilibrium relaxation dynamics of a long-range Hamiltonian system ofN fully coupled rotators. For a particular family of initial conditions, this system is known to enter a particularregime in which the dynamic behavior does not agree with thermodynamic predictions. Moreover, there isevidence that in the thermodynamic limit, whenN→` is taken prior tot→`, the system will never attain trueequilibrium. By analyzing the scaling properties of the two-time autocorrelation function we find that, in thatregime, a very complex dynamics unfolds, in whichaging phenomena appear. The scaling law stronglysuggests that the system behaves in a complex way, relaxing towards equilibrium through intricate trajectories.The present results are obtained for conservative dynamics, where there is no thermal bath in contact with thesystem.

DOI: 10.1103/PhysRevE.67.031106 PACS number~s!: 05.20.2y, 64.60.My

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At the very foundations of statistical mechanics, therestill some hypotheses whose validity rests merely on thetrapolation of observational facts and that have to be justia posteriori. Among them, let us mention two assumptiothat are intimately related to the issue addressed in this paThe first one refers to the introduction of a probabilistic dscription of the evolution of a physical system. The seconrelated to the mechanical specifications that a system mfulfill so that the results of statistical mechanics can beplied @1#. These two points are closely related to the fundmental problem of establishing a connection between thenamical behavior of a system, described by the HamiltonH, and its thermodynamics. In that sense, statistical mechics requires the existence of adequate conditions allowinreplace the dynamical temporal predictions by a probabiliensemble calculation that yields the correct equilibriumean value of the relevant quantities.

A very fast relaxation and a high degree of chaos amixing are usually required in order to guarantee that a stem orbit will cover most of its phase space in a short timThese questions have been extensively investigated fordimensional systems, whereas, for extended systems,infinite degrees of freedom, the matter is still far from sett@2#.

Due to the analytical and numerical efforts of many athors, it is today a well established fact that even very simmodels, when analyzed in the thermodynamical limitN→`, can yield results that bring to surface central questiabout the foundations of statistical mechanics. In particuin this work, we will concentrate on an infinite-range~mean-field! Hamiltonian system that, despite its simplicity, exhiba very peculiar dynamical behavior: depending on the inipreparation of the system its evolution can get trappedtrajectories that will prevent the system from attaining eqlibrium in finite time when theN→` limit is taken before

*Email address: mmontemu@famaf.unc.edu.ar†Email address: tamarit@fis.uncor.edu‡Email address: celia@cbpf.br

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the t→` limit @3–8#. Unlike most models that display complex macroscopic behavior, this infinite-range modelcludes neither randomness nor frustration in its microscointeractions. Furthermore, on one hand it can be exasolved in the canonical ensemble, while on the other hancan be efficiently integrated in the microcanonical ensemIn that sense, it is an excellent starting point for analyzthe above mentioned basic questions.

The system consists ofN fully coupled rotators whosedynamics is described by the following Hamiltonian:

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2N (i , j

@12cos~u i2u j !#[K1V. ~1!

It is worth mentioning that this is a rescaled version ofnonextensive infinite-range Hamiltonian@9#. However, bothof them share the same dynamical behavior after approprescaling of the dynamic variables.

In Fig. 1, we display the plot,T vs U, where T52^K&/N is the temperature andU5(K1V)/N is the totalenergy per particle. The solid line corresponds to the can

FIG. 1. The full line corresponds to the canonical theoreticaloric curve and the symbols correspond to numerical simulatfor systems of sizeN51000 ~circles! and N55000 ~triangles!, att51000, averaged over 50 realizations. Initial conditions are ‘‘wter bag.’’

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MONTEMURRO, TAMARIT, AND ANTENEODO PHYSICAL REVIEW E67, 031106 ~2003!

cal calculations@3#, which predict a second-order transitioat Uc50.75. AboveUc , constant specific heat is found anbelow Uc the system orders in a clustered phase. The sbols correspond to the numerical results obtained by integing the equations of motion forN51000, 5000 rotators anduntil t51000. The system is initially prepared in a ‘‘watebag’’ configuration, that is, all the angles are set to zero whthe momenta are randomly chosen from an uniform distrition such that the system has total energyNU. By measuringthe nonequilibrium temperature~or equivalently the magnetization! of a system started in these out-of-equilibrium intial conditions, one observes that, for a range of energyues below the transition, the system enters inquasistationaryregime characterized by a mean kinetic eergy that varies very slowly. Moreover, the value of this noequilibrium temperature remains different from that pdicted by canonical calculations. Actually, standaequilibrium is attained only after a time which grows withe size of the system, hence an infinite system will nereach true equilibrium@7,8#. In the quasistationary regimpreceding equilibrium, trajectories are non-ergodic anddynamics is weakly chaotic with Lyapunov exponent vaniing in the thermodynamic limit@5#.

It is our objective here to show that the discrepancytween the results drawn from the dynamics and those derfrom the canonical ensemble is closely associated topresence of strong long-term memory effects and slow reation dynamics, a phenomenon usually namedaging. Agingis one of the most striking features in the off-equilibriudynamics of complex systems. It refers to the presencestrong memory effects spanning time lengths that in socases exceed any available observational time. Althoughing has been seen in a wide variety of contexts and syst@10#, some of them, actually very simple ones@2,11#, it isperhaps in the realm of spin glass dynamics where a sysatic study of these phenomena has been carried out~see Ref.@10#, and references therein!. Systems thatage can be clas-sified into dynamical universality classes according toscaling properties of their relaxation function. Moreovthese scaling properties contribute to a quantitative desction of complex phenomena, even in cases where a gentheory is lacking@10,12#.

Aging can be characterized by measuring the two-tiautocorrelation function along the system trajectories. Ifstate of the system in phase space can be completely chterized giving a state vectorxW , then the two-time autocorrelation function is defined as follows:

C~ t1tw ,tw!5^xW~ t1tw!•xW~ tw!&2^xW~ t1tw!&•^xW~ tw!&

s t1tws tw

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~2!

where s t8 are standard deviations and the symbol^•••&stands for average over several realizations of the dynamIn the case of a Hamiltonian system withN degrees of free-dom the state vector is decomposed in coordinates andconjugate momenta, therefore we establish the followingtation: xW[(uW ,LW ).

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For systems that have attained ‘‘true’’ thermodynamicequilibrium, only time differences make physical sense whcalculating relaxation quantities. In this case, it is expecthat on an average the system will show only very shmemory of past configurations. However, for systems exhiting aging, a complex time dependence is observed inbehavior ofC(t1tw ,tw), indicating long-term memory ef-fects. In such a case, even at macroscopic time scalestwo-time autocorrelation function shows an explicit depedence on both times (t and tw) together with a slow relax-ation regime.

In order to integrate the motion equations numerically,employed a fourth-order symplectic method@13# with a fixedtime step selected so as to keep a constant value of theergy within a relative errorDE/E of order 1024. All thesimulations were started from the water-bag initial contions explained above.

In Fig. 2, we present the results of the numerical calcution of the two-time autocorrelation function~2! for U50.69. This value of the energy, together with the water-binitial conditions set the system into a particular dynamiregime, in which ensemble discrepancy is more pronounwhen finite size results are extrapolated to the thermonamic limit. In the graph, features characteristic of agiphenomena can be distinguished. For a giventw the systemfirst enters a quasiequilibrium stage, in the sense that temral translational invariance holds, withC(t1tw,tw)'1, upto a time of ordertw . After that, the system enters a secorelaxation characterized by a slow power law decay anstrong dependence on both times. This phenomenologybe clearly seen in the curves for the largesttw’s.

In Fig. 3, we show the best data collapse for the long-tibehavior of the autocorrelation function, using the dataFig. 2 corresponding to the three largest waiting timestw52048, 8192, and 32 768!. The resulting scaling law clearlyindicates that for the whole range of values oft/tw consid-ered,

FIG. 2. Two-time autocorrelation functionC(t1tw ,tw) vs t forsystems of sizeN51000 and energy per particleU50.69. The datacorrespond to an average over 200 trajectories initialized in wabag configurations. The waiting times aretw58, 32, 128, 512,2048, 8192, and 32 768.

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AGING IN AN INFINITE-RANGE HAMILTONIAN . . . PHYSICAL REVIEW E 67, 031106 ~2003!

C~ t1tw ,tw!5 f S t

twb D , ~3!

where f (t/twb);(t/tw

b)2l. It is worth mentioning that thisscaling is the same as observed experimentally in spin gsystems@14#. The values obtained for the scaling parametare b'0.90 andl'0.74. In the inset, we exhibit an altenative representation of the data which yields a linear plocorresponds to lnq@C(t1tw ,tw)# vs t/tw

b , where the functionlnq(x), namedq logarithm, is defined as follows@15#:

lnq~x!5x12q21

12q. ~4!

In this expression,q5111/l, which for the data in Fig. 2,yields q'2.35. Therefore, we can obtain a complete funtional form of the autocorrelation function valid over thwhole range of the scaling variablet/tw

b just by identifyingthe functionf (x) in Eq. ~3! with the inverse of lnq(x), that is,the q exponential@15#:

f ~x![eq2x5@12~12q!x#

112q, ~5!

which naturally arises within the nonextensive statisticstroduced by Tsallis inspired by the probabilistic descriptiof multifractal geometries@16#. The same qualitative behavior has been observed for other systems sizes, nameN5500, 2000.

This aging scenario contrasts with the time invarianthavior observed within the high energy phase, where no qsistationary regime is detected. In fact, let us discuss Fiwhere we present the results of the calculation of the twtime autocorrelation function~2! for U55.0, well above thesecond-order phase transition~i.e., inside the homogeneouphase!, with water-bag initial conditions. What we observhere is essentially that the autocorrelation function depeon the two times only through their difference, that is,C(t

FIG. 3. Data collapse for the long-time behavior of the autocrelation functionC(t1tw ,tw). The data are the same shown in F2 for the three largesttw . The gray solid line corresponds teq(20.2t/tw

b). Inset: lnq-linear representation of the same dawith q.2.35.

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1tw ,tw)'C(t). Therefore, the presence of aging is relatedthe existence of quasistationary states. It is important tophasize that, although the dynamics presents temporal trlational invariance in the high energy regime, the relaxatof the system is very slow.

There is nowadays growing evidence that aging is a vcommon dynamical phenomenon, associated to a greatety of physical systems. So far, there are two scenawithin which aging can emerge. On one hand, the onseaging in many systems derives from the presence of coening processes that give place to critical slowing downthe dynamics. In this case the scaling law of the two-timautocorrelation function is ruled by the following expressi

C~ t1tw ,tw!; f „L~ t !/L~ tw!…, ~6!

whereL(t) is the mean linear size of the domains at timet@17#. On the other hand, aging also appears as a consequof weak ergodicity breaking@10# and it is related to the complex fractal structure of the region of phase space thatsystem explores in time. This is the case, for instance, inSherrington-Kirkpatrick~SK! model and other spin glasmodels in which the complexity of the energy landscapeassociated to a certain degree of randomness and/or frution in the Hamiltonian.

What is particularly remarkable in this work is the appeance of a complex aging behavior in amean-fieldmodellacking both randomness and frustration. Since the modeare analyzing in this paper is an infinite-range one, suchthe SK model, a coarsening scenario has to be ruled out fthe outset. Furthermore, all our results are obtained in a cservative system without any thermal bath in contact with

Moreover, it is worth noting that the scaling law found fothe two-time autocorrelation functions below the transitiowhere a quasistationary regime is detected, points to anario very similar to that observed in spin glasses@14#. Asoccurs in spin glasses, there is weak breakdown of ergoity, which is consistent with the observation of weakly ch

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FIG. 4. Two-time autocorrelation functionC(t1tw ,tw) vs t forsystems ofN51000 andU55.0. The data correspond to an aveage over 10 trajectories initialized in water-bag configurations. Twaiting times aretw58, 32, 128, 512, 2048, 8192, and 32 76Inset: semilog representation of the same data.

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MONTEMURRO, TAMARIT, AND ANTENEODO PHYSICAL REVIEW E67, 031106 ~2003!

otic orbits, i.e., with a vanishing Lyapunov exponent in tthermodynamic limit@5#. Drawing the analogy with spinglasses even further, our results seem to confirm thatsystem visits phase space confined inside very intricatejectories ~presumably nonergodic!. This conjecture is alsosupported by features observed inm space@6–8#. The factthat the relaxation of the two-time autocorrelation functican be well fitted by aq-exponential decay over the whorange oft/tw deserves further investigation. Although a posible connection with nonextensive statistics@16# is still notclear, we believe that it would be interesting to examine tpossibility.

In summary, in this paper we have characterized the srelaxation dynamics of a long-range Hamiltonian syst

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through its aging dynamics. Our observation of the existeof aging in this Hamiltonian system and its characterizatby scaling properties reminiscent of spin glass dynamicsresult that can contribute to establish a unified frame fordiscussion of the out-of-equilibrium dynamics of systemwith many degrees of freedom.

We thank Constantino Tsallis for very stimulating discusions. We also want to thank Andrea Rapisarda, Vito LatoAlessandro Pluchino, and Rau´l O. Vallejos for very usefulcomments. This work was partially supported by grants frCONICET ~Argentina!, Agencia Co´rdoba Ciencia~Argen-tina!, SEYCT/UNC~Cordoba, Argentina!, and FAPERJ~Riode Janeiro, Brazil!.

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