Existence of Quasi-stationary states at the Long Range threshold
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Existence of Quasi-stationary states at theLong Range threshold
Alessio Turchi ∗ Duccio Fanelli ∗∗ Xavier Leoncini ∗∗∗∗
Dipartimento di Energetica "Sergio Stecco", Universita’ di Firenze,via s. Marta 3, 50139 Firenze, Italia and Centre de PhysiqueThéorique, Aix-Marseille Université, CNRS, Luminy, Case 907,F-13288 Marseille cedex 9, France. ∗∗ Dipartimento di Energetica "Sergio Stecco", Universita’ di Firenze,via s. Marta 3, 50139 Firenze, Italia and Centro interdipartimentaleper lo Studio delle Dinamiche Complesse (CSDC) and INFN ∗∗∗
Centre de Physique Théorique, Aix-Marseille Université, CNRS,Luminy, Case 907, F-13288 Marseille cedex 9, France (e-mail:[email protected])
Abstract:
In this paper the lifetime of quasi-stationary states (QSS) in the α − HMF modelare investigated at the long range threshold ( α = 1 ). It is found that QSS exist and have adiverging lifetime τ ( N ) with system size which scales as τ (N) ∼ log N , which contrast to theexhibited power law for α < and the observed finite lifetime for α > . Another feature ofthe long range nature of the system beyond the threshold ( α > ) namely a phase transition isdisplayed for α = 1 . . The definition of a long range system is as well discussed.Keywords: Long range systems, Fractional dynamics, Hamiltonian chaos.1. INTRODUCTIONStudying the dynamics of Hamiltonian systems with alarge number of degrees of freedom and its connection toequilibrium statistical mechanics has been a long standingproblem. The relaxation to statistical equilibrium has beenunder scrutiny ever since the pioneering work of Fermi andthe FPU problem(1). Moreover, since the advent of pow-erful computers and for specific systems within a class ofinitial conditions, integrating numerically Hamiltonian dy-namics has proven to be competitive in regards to Monte-Carlo schemes for the study of statistical properties (seefor instance (2; 3) and references therein). The assumptionmade is that since the system admits only a few conservedquantities for generic initial conditions, once the dimen-sions of phase space are large enough, microscopic Hamil-tonian chaos should be at play and be sufficiently strong toprovide the foundation for the statistical approach withinthe micro-canonical ensemble. However recent studies haveshown that there is an increase of regularity with systemsize in the microscopic dynamics when considering sys-tems with long range interactions (4; 5; 6; 7). Indeed, thestatistical and dynamical properties of these systems arestill under debate. For instance extensivity is not alwaysprovided and discrepancies between canonical and micro-canonical ensembles can be found such as negative specificheats for the latter (8; 9). Moreover, phase transitions forsystems embedded in one dimension can be found.In particular, long range systems often display a slow re-laxation to equilibrium. Starting from an initial conditionthey are in fact trapped in long-lasting out of equilibriumregimes, termed in the literature Quasi Stationary States (QSS) which have distinct macroscopic characteristics,when compared to the equilibrium configuration.A now paradigmatic model of long range interactionsHamiltonian systems is the Hamiltonian Mean Field(HMF) model (10), which corresponds to a mean field XY -model with a kinetic energy term (rotators). In the limit ofinfinite system size the HMF model can be described usinga Vlasov equation (11; 9). More recently, stationary stateshave been constructed using invariant measures of systemscomposed of uncoupled pendula (6), more specifically itwas emphasized that the microscopic dynamics in the mag-netized stationary state is regular and explicitly known.This observation lead to explain the abundance of regularorbits as revealed in (5). These results first obtained for theHMF have been extended for the case when the couplingconstant depends on the distance between sites, namelyfor α − HMF model in its long range version ( α < )(7).This model was introduced for instance in (12) and anddisplays identical equilibrium features as the HMF(13; 9).In fact it was shown that all stationary states of theHMF model are as well stationary states of the α − HMFmodel, that microscopic dynamics is as well regular, at theprice of microscopic spatial complexity, which is locallyscale invariant(7). Before going on we write the governingHamiltonian of the model: H = N X i =1 p i N N X j = i − cos ( q i − q j ) k i − j k α , (1)where q i stands for some spin angle located on the latticesite i, and p i is its canonically conjugate momentum. Thedistance k i − j k is actually the shortest distance on theircle of perimeter N − , so that the systems can beisolated and still translational invariant along the lattice.The mean field model is recovered for α = 0 , and for N even, we write ˜ N = (cid:18) N (cid:19) α + 2 N/ − X i =1 i α , (2)to insure extensivity. The equations of motions of element i are derived from the Hamiltonian (1): ˙ p i = − sin( q i ) C i + cos( q i ) S i = M i sin( q i − ϕ i ) , (3) ˙ q i = p i , (4)where C i = 1˜ N X j = i cos q j k i − j k α (5) S i = 1˜ N X j = i sin q j k i − j k α . (6) C i and S i identify the two components of a magnetizationper site, with modulus M i = p C i + S i , and phase ϕ i = arctan( S i /C i ) . For large N , and assuming < α < ,we have ˜ N ≈ − α ( N/ − α . (7)We then can use (7) in Eq.(5) and, make the N → ∞ limit while introducing the continuous variables x = i/N and y = j/N to arrive at C ( x ) = 1 − α α / Z − / cos ( q ( y )) k x − y k α dy , (8)where k x − y k represents the minimal distance on a circleof perimeter one. We can recognize the fractional integral I − α and consequently write C ( x ) = 1 − α α Γ(1 − α ) I − α (cos q ( x )) . (9)In this large size limit, the α − HMF dynamics impliesstudying the evolution of the scalar fields q ( x, t ) and p ( x, t ) which are ruled by the fractional (non-local) partialdifferential equations ∂q∂t = p ( x, t ) ∂p∂t = µ α Γ( µ ) ( − sin( q ) I µ (cos q ) + cos( q ) I µ (sin q )) . where µ = 1 − α . It has then been shown in (7) thatstationary states are solutions of D α cos q = d α cos qdx α = 0 . (10)where the operator D α stands for the fractional derivative,and that actually this property was shared with nonstationary QSS’s. All these results were obtained for themodel in its long range version, meaning when α is smallerthan one. M ε N = 2 N = 2 N = 2 Fig. 1. Magnetization vs energy, for α = 1 . . A phasetransition is displayed with ǫ c ≈ . . The transitionpoint seem to be different from the long-range onewhich is ǫ c = 0 . , but the qualitative behavior ofcurve is the same.2. WHAT HAPPENS FOR > α ≥ We recall that systems are considered long range when thetwo body interacting potential V ( r ) decays at the most as /r α with α < d , where d stands for the dimension ofthe embedding space. Having only one degree of freedom d = 1 for the α − HMF model. In these regards, consideringsituations where < α < is actually studying shortrange models, In fact ˜ N is finite so there is no need ofsystem size renormalization of the couling constant for α > . However something is peculiar about this latticemodel. Indeed when considering the dynamics of a longrange system, we would expect the force to be ruled by a /r β decay with β = α + 1 , which is not the case for thelattice model for which the decay exponent is unchanged( β = α ). Moreover given the particular importance of themicroscopic dynamics and possible ergodicity breaking,one could naturally raise the question if the long rangenature of a system is not ruled by the dynamics, whichwould then imply a system to be long ranged if β < d + 1 ,which for the α − HMF model would imply α < . Most ofthe previous analysis of the model has been performed for α < , and can not simply be extended to < α < . Afirst numerical analysis is therefore necessary. One peculiarity of one dimensional systems, is that thereshould not bare any phase transition if the interaction isshort ranged. A first numerical study of the magnetizationversus density of energy is performed for α = 1 . in Fig. 1.The numerical integration of the microscopic dynamicsis performed using a simplectic scheme, (optimal fifthorder see(14)), a typical time step used is δt = 0 . , andthe initial conditions are Gaussian distributed. The fastFourier transforms are done with the fftw libraries. Ascan be seen in Fig. 1 a phase transition is displayed ata transition point ǫ c ≈ . which is then different fromthe universal value obtained for α < which is ǫ c = 0 . .Preliminary results show actually that the critical pointdepends on the value of α and results seem to show thatit approaches ǫ = 0 for α = 2 .This existence of a phase transition beyond the “classical”long range threshold in one dimension had already been
10 100200 1.000 10.00000,10,20.250,30.3650,40,50,60,70,80,91 t M
100 150 2000.250.30.3650.40.450.5 t M t t
11 12 13 14 15 16 17 18 19 2055,35,65.755,96,26,5 log (N) t / t Fig. 2. Top: Magnetization curve vs time for α = 1 . , N =2 and ǫ = 1 . . During the QSS the magnetizationis oscillating for a time τ , then it relaxes down to it’sequilibrium value in a time τ .Bottom: τ τ for different values of N , α = 1 and ǫ =0 . . The two times are approximately proportional,so the knowledge of τ gives a good approximation for τ . This proportionality is respected for all α values inthis paper, even if the proportionality constant mayvary.noticed for the Ising model by Dyson in the sixties, but thisfeature seems to favor the dynamical definition of what along range system ought be. However an important featureto assert this new definition would be to find as well quasi-stationary states in this region of α ’s. In long range interacting systems, generally, the limit N →∞ and t → ∞ doesn’t commute, so the thermodynamiclimit is not unequivocally defined and one may end up indifferent equilibrium states depending on the order of theprevious limits. Physically it’s more feasible to computethe continuous limit before the time limit, so if the lifetime τ of the QSS diverges with N it becomes the effectivereal equilibrium of the system, which in general is notobeying Boltzmann’s statistics (4). We studied how theQSS lifetime scales with the exponent α in the decayparameter of the potential.First we studied the behavior of the lifetime τ around thecrucial value α = 1 to better understand the transitionbetween a long range system and a supposedly shortrange one, but as mentioned in the case of α -HMF there −3 −2 −1 0 1 2 3−3−2−10123456 q p −3 −2 −1 0 1 2 3−3−2−10123456 q p Fig. 3. Poincaré sections for α = 1 , ǫ = 1 . and N = 2 .Top refers to the QSS, while bottom represent therelaxation state between τ and τ . It can be easilyseen that the second small isle disappear during therelaxation and the phase space becomes symmetric in q , thus ending the oscillations of the magnetization.are convincing arguments that the requirement for theemergence of long range behavior could be relaxed, andwe can expect some long range feature to survive above α = 1 .We considered the initial condition already used in (7)which was giving rise to a QSS, namely a long livedmagnetized state above the critical energy. The initialcondition used is all q i = 0 , wile the p i ’s are Gaussian.To characterize the QSS lifetime we monitor the behaviorof the global macroscopic parameter magnetization whichcan be M = | P j e iq j | . In Fig. 2 we show the behaviorfor α = 1 , which is qualitatively representative for the allstudied values of α in this paper. Here a first transitionat t = τ can be identified until which the systemoscillates around an almost constant magnetization value,and beyond which the system starts to relax towards theequilibrium M = 0 value. A second transition at t = τ is as well identified, it corresponds to the time at whichthe system finally reaches it’s equilibrium state. As can beseen in Fig. 2, we find that these two values are almostlinearly proportional for each α -value that we took into (N) t t (N), a =0.9 t =547*log(N)−3674 t (N), alpha=1.0
11 12 13 14 15 16 17 18 19 200200040006000800010000 log (N) t t (N) t =A*log (N/C) t (N) t =B*log (N/C)
13 14 15 16 17 18 19 20200300400500600700 log (N) t t (N), a =1.1 t =62*log(N)−218 Fig. 4. Scaling of τ with N for different α values and ǫ = 1 . . Top shows the scaling for α = 0 . ; middlerefers to α = 1 . and shows both τ and τ ; whilebottom refers to α = 1 . . All curves seems to growat least logarithmically.consideration, so we will refer to the first lifetime τ as thelifetime of the QSS, since is an order of magnitude fasterto compute and we are interested only in the qualitativeform of the scaling law for the lifetimes.The difference of the two dynamical regimes defined bythe above thresholds are better understood when lookingat the Poincaré section captured in each of this regimesdisplayed in Fig. 3. At first the system is forms twodistinct islands in the phase space, which start movingaround and create the oscillations in the magnetization M M(N,t), N=2 M(N,t), N=2 M(N,t), N=2 M(N,t), N=2 M(N,t), N=2 Fig. 5. Magnetization curve vs time for α = 1 . , ǫ = 1 . and increasing N values. The initial oscillationlifetime τ is independent of N and the systemrelaxes to equilibrium in a time which should beconstant in the continuous limit.that characterize the QSS, and then one of the islesdisappear during the relaxation period, while the phasespace becomes symmetric in q thus ending the oscillations.Now we analyze the lifetime of the QSS versus the sizeof the system around the classical long range threshold α = 1 , if the system is long range it should diverge with N . The cases α = 0 . , α = 1 and α = 1 . are displayedin Fig. 4. We can see that around α = 1 the scaling of τ with N approaches a logarithmic curve, meaning that theQSS survive at least until α = 1 and maybe beyond as itappears as well true for α = 1 . . However when looking atthe data for larger value of α , namely α = 1 . the scalingof τ appears to saturate as displayed in Fig. 5, where themagnetization curves obtained for α = 1 . appear to allbe identical no matter the size of the system. At this valueof α there is still a short initial oscillation in M typical ofthe QSS, but now it’s lifetime seems to be independent of N and finite so the system will reach the equilibrium statein a large enough time. Conversely we may expect that wemay actually observe the same feature for α = 1 . , butthat we have not seen yet the saturation in the lifetime aswe were not able to simulate systems that would be largeenough. In other words, we have as now not enough data toidentify if there is a transition value between . < α < . where the system becomes suddenly short-range droppingthe logarithmic law of τ , or it still saturates at somelarger value of N > even for α = 1 . . Preliminaryresults shows that this saturation becomes quite fast for α ∼ . , . where even for low values of N it may still bepossible to observe a scaling which is sub-logarithmic, buteven if it did this would be only relevant for systems withan astronomical scale of constituants.However even if the QSS lifetime seems to saturate at somepoint, so that the system dynamics change into a shortrange one, the phase transition from a magnetized to anhomogeneous state, typical of the long range regime, is stillpresent (figure 1). Hence is appears that the dynamicaldefinition may be more relevant for macroscopic features,uch as the presence of a QSS (even with a finite lifetime)or a phase transition, while the more classical statisticaldefinition of a long range system corresponds actuallyto different dynamical behavior of the system and theexistence of QSS with diverging lifetimes.REFERENCES[1] E. Fermi, J. Pasta, and S. Ulam. Los Alamos Reports ,(LA-1940), 1955.[2] Xavier Leoncini, Alberto Verga, and Stefano Ruffo.Hamiltonian dynamics and the phase transition of thexy model.
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