Abundance of regular orbits and out-of-equilibrium phase transitions in the thermodynamic limit for long-range systems
Romain Bachelard, Cristel Chandre, Duccio Fanelli, Xavier Leoncini, Stefano Ruffo
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n Abundan e of regular orbits and out-of-equilibrium phase transitions in thethermodynami limit for long-range systemsR. Ba helard , C. Chandre , D. Fanelli , X. Leon ini , S. Ru(cid:27)o
1. Centre de Physique Théorique , CNRS - Aix-Marseille Université,Luminy, Case 907, F-13288 Marseille edex 9, Fran e2. Dipartimento di Energeti a "Sergio Ste o", Universitá di Firenze, via s. Marta 3, 50139 Firenze,Italia and Centro interdipartimentale per lo Studio delle Dinami he Complesse (CSDC) and INFNWe investigate the dynami s of many-body long-range intera ting systems, taking the HamiltonianMean Field model as a ase study. We show that regular traje tories, asso iated with invariant tori ofthe single-parti le dynami s, prevail. The presen e of su h tori provides a dynami al interpretation ofthe emergen e of long-lasting out-of-equilibrium regimes observed generi ally in long-range systems.This is alternative to a previous statisti al me hani s approa h to su h phenomena whi h was basedon a maximum entropy prin iple. Previously dete ted out-of-equilibrium phase transitions are alsoreinterpreted within this framework.The vast majority of phenomena observed in natureresults from omplex intera tions present in large assem-blies of elementary onstituents. A widespread observa-tion is the emergen e of regular traje tories despite the omplexity of the underlying network of ouplings. Asu essful approa h to des ribe the olle tive behaviourof large assemblies of parti les is traditionally providedby statisti al me hani s. The theoreti al foundation ofequilibrium statisti al me hani s relies upon the hypoth-esis of ergodi ity, i.e. the agreement of time with ensem-ble averages. Arguing for an e(cid:27)e tive degree of globalmixing of dynami al traje tories in phase spa e impliesergodi ity and thus the validity of statisti al me hani s[1℄. Thermodynami behaviour is obtained in the limitwhere the number of degrees of freedom goes to in(cid:28)nity,whi h o(cid:27)ers innumerable pathways to haos. Indeed, inthis limit regular regions (invariant tori) do not possessenough dimensions in phase spa e to prevent traje toriesfrom spreading, while the largest fra tion of the phasespa e is o upied by haoti motion, hen e sustainingmixing [2℄.However, the fa t that any weak nonlinearity wouldimply ergodi ity has been vigorously debated sin e thepioneering work of Fermi, Pasta and Ulam (FPU) [3℄on the dynami s of os illators intera ting via short-range ouplings. Contrary to expe tations, the elebrated FPU hain exhibits a re urrent behavior on very long times, vi-olating ergodi ity. Nowadays, there is a growing eviden ethat, for generi initial onditions, the relaxation time toequilibrium remains (cid:28)nite in the thermodynami limit [4℄.On the ontrary, the question of relaxation to equilibriumis still open when long-range for es ome into play [5, 6℄.Indeed, systems with long-range intera tions have beenshown to display an extremely slow relaxation to equi-librium. More spe i(cid:28) ally, out-of-equilibrium metastableregimes have been identi(cid:28)ed, where the system getstrapped before eventually attaining its asymptoti state[7, 8℄. The equilibration time in reases with system sizeand formally diverges in the thermodynami limit, lead-ing to a breaking of ergodi ity. For gravitational sys-tems the approa h to equilibrium has never been proven and seems problemati . Galaxies ould therefore rep-resent the most spe ta ular example of su h far-from-equilibrium pro esses [9℄, but analogous phenomena havealso been reported in fundamental problems of plasmaphysi s [10℄. These metastable states have been termedQuasi-Stationary States (QSS) in the literature and of-ten represent the solely a essible experimental regimes(e.g., in the Free Ele tron Laser [11℄). A relevant featureof long-range systems is the self- onsistent nature of theintera tion of a parti le with its "lo al (cid:28)eld", whi h itselfresults from the ombined a tion of all the other parti- les or of an "external (cid:28)eld" like for wave-parti le systems[10℄. It is exa tly this self- onsisten y whi h (cid:28)nally en-tails the widespread regularity of the motion. Besidesthat, one observes the presen e of lasses of initial statesthat show di(cid:27)erent time evolutions.The development of a systemati theoreti al treatmentof the QSS, whi h would enable us to unravel the puzzleof their ubiquity, is still an open problem. Both the self- onsistent nature of the intera tion and the strong de-penden e on the initial ondition suggest that QSS ouldbe related to the presen e of some type of regular mo-tion. The traditional approa h to larifying the emer-gen e of regular traje tories is based on the following re-sult : If the Hamiltonian system under s rutiny is lose tointegrable, Kolmogorov-Arnold-Moser [12℄ theory provesthat the phase spa e is (cid:28)lled with invariant tori on whi hthe motion is quasi-periodi . In this framework, however,in reasing the number of parti les enhan es the ontribu-tion of haoti traje tories [2℄, in stringent ontradi tionwith the observation that QSS prevail in the large N limit. Therefore the aforementioned s enario annot beinvoked to explain the presen e of regular motion in sys-tems with long range intera tions.A (cid:28)rst purpose of this Letter is to put forward a dif-ferent interpretative framework. We argue that tori anform in phase spa e also as a result of a self- onsistentintera tion in the thermodynami limit. As we shalldemonstrate, while for a small number N of degrees offreedom the single parti le motion of a paradigmati sys-tem with long-range intera tion is errati , the traje to-ries be ome more and more regular as N → ∞ . Thesetraje tories arise from a low-dimensional time dependente(cid:27)e tive Hamiltonian.For systems with long-range intera tions, the depen-den e on the initial ondition an materialize in the formof a true out-of-equilibrium phase transition: By varyingsome ru ial parameters of the initial state, one observesa onvergen e towards asymptoti states (in the limit N → ∞ ) with di(cid:27)erent ma ros opi properties (e.g., ho-mogeneous/inhomogeneous) [13℄. This phase transitionhas been interpreted by resorting to Lynden-Bell's the-ory of "violent relaxation" [14℄, whi h is brie(cid:29)y dis ussedin the following with referen e to a spe i(cid:28) model. As ase ond purpose of this Letter, we provide a mi ros opi ,dynami al, interpretation of this transition in terms of asharp modi(cid:28) ation of the properties of the single parti leorbits, orresponding to a hange of the e(cid:27)e tive Hamil-tonian. Our ultimate aim is to suggest a unifying pi turethat potentially applies to systems where olle tive, orga-nized, phenomena emerge from the globally oupled seaof individual omponents.The Hamiltonian Mean Field (HMF) model [15℄ iswidely referred to as the ben hmark for long-range sys-tems and analysed for pedagogi al reasons. The model,whi h des ribes the evolution of N parti les oupledthrough an equally strong, attra tive, osine intera tionis spe i(cid:28)ed by the following Hamiltonian: H = N X i =1 p i N N X j =1 (1 − cos ( θ i − θ j )) , (1)where θ i and p i label respe tively the position of parti le i on the unit ir le and its orresponding momentum. Notethat Hamiltonian (1) an also be seen as a simpli(cid:28)ed ver-sion of the gravitational [16℄ or plasma [17℄ sheets modelwhen onsidering only the (cid:28)rst harmoni in the Fourierexpansion of the potential. To hara terize the behaviourof the system, it is onvenient to introdu e the (cid:16)magne-tization(cid:17) M = N ( P cos θ i , P sin θ i ) = M (cos φ, sin φ ) ,whi h quanti(cid:28)es the degree of spatial bun hing of theparti les (homogeneity vs. inhomogeneity). We here on-sider water-bag initial onditions, onsisting of parti lesuniformly distributed in a re tangle [ − θ , θ ] × [ − p , p ] in the ( θ, p ) -plane. These states bear a magnetization M = M = sin( θ ) /θ , the asso iated energy per parti lebeing U = p / − M ) / . When performing nu-meri al simulations, starting from the water-bag initial ondition, the system gets usually frozen in a QSS of thetype dis ussed above [7, 8℄.The individual parti le i obeys the following equationsof motion ˙ p i = − M sin ( θ i − φ ) and ˙ θ i = p i , where M and φ are fun tions of all the positions of the parti les. Nu-meri al observations suggest that, for large enough valuesof N , both the magnetization M and the phase φ developa spe i(cid:28) os illatory time dependen e. Hen e, the singleparti le motion turns out to be governed by a time de-pendent one degree of freedom (often referred to as a oneand a half degree of freedom) e(cid:27)e tive Hamiltonian. This −2 0 2−2−1012 θ − φ p θ − φ p −1.54−1.04−0.54−0.050.450.95 Figure 1: Poin aré se tions of a few sele ted parti les ofone traje tory of Hamiltonian (1) for N = 2 × in theQSS regime for two di(cid:27)erent water-bag initial onditions : ( M , U ) = (0 . , . (upper panel) and ( M , U ) = (0 . , . (lower panel). The former returns a single luster, whi h givesa non-zero magnetization QSS ( M QSS ≈ . ), while the lat-ter shows two symmetri lusters, whi h produ e a QSS witha small magnetization. In the bi luster regime (lower panel)the presen e of a large set of rotational tori implies a sub-stantially lower magnetization level, whereas the librationaltori around the two lusters are responsible for the residualmagnetization. The olor ode orresponds to the values ofthe a tion variable asso iated with individual parti les.justi(cid:28)es the investigation of the phase spa e properties ofthe QSS using a te hnique inspired by that of Poin arése tions. In parti ular, we onsider the time average ¯ M (after a transient) and re ord the positions and momentaof a few sele ted parti les ( θ i , p i ) when M ( t ) = ¯ M and dM/dt > (sin e M typi ally shows an os illatory be-havior). The resulting strobos opi se tions are displayedin Fig. 1. Two di(cid:27)erent phase spa e stru tures are founddepending on the hoi e of the initial pair ( M , U ) , onewith mono luster and the other with a bi luster. Themono luster QSS displays a nonzero magnetization (in-homogeneous phase), while the bi luster QSS has a smallresidual magnetization (homogeneous phase).The mono luster QSS an be ideally mapped onto a olle tion of weakly intera ting pendula. As revealedby our strobos opi analysis, parti les evolve on reg-ular tra ks, whi h are approximately one-dimensional,though they do manifest a degree of lo al di(cid:27)usion (thi k- −2 0 2−2−1012 θ − φ p N = 2 (a) −2 0 2−2−1012 θ − φ p N = 2 (b)−2 0 2−2−1012 θ − φ p N = 2 (c) 10 −4 −3 −2 −5 N ∆ J (d) Figure 2: Poin aré se tions of a few sele ted parti les of onetraje tory of Hamiltonian (1), when the system size is varied(for M = 0 . and U = 0 . ). The thi kness of the tori de- reases as N is in reased (see text). For large enough valuesof N , the magnetization M is found numeri ally to approxi-mately s ale as M ( t ) ≈ ¯ M + δM ( t ) cos ωt , with | δM | ≪ ¯ M and | ∂ t δM | ≪ ω | M | . Ignoring the time dependen e of δM and using a redu ed model of test parti les in the external(cid:28)eld M ( t ) , one obtains strobos opi se tions whi h are qual-itatively and quantitatively similar to the ones reported inthis (cid:28)gure, with the unique di(cid:27)eren e that the thi kness iszero [18℄. Considering a torus with a tion J ≈ . , we plotin panel d) its varian e ∆ J omputed over a time interval ∆ t = 300 as a fun tion of N . The s aling /N (dotted line)looks a urate over a wide range of N values.ness). For the bi luster QSS, the Poin aré se tion showsa phase portrait whi h losely resembles the one obtainedfor a parti le evolving in the potential of two ontra-propagating waves. These latter intera t very weakly, asthe asso iated propagation velo ities appear rather dif-ferent.In order to get a quantitative estimate of the thi knessof the tori as a fun tion of the total number of parti les,we fo us on the mono luster QSS. Panels a), b) and )in Fig. 2 display the single parti le phase spa e for in- reasing values of N . A lear trend towards integrabilityis observed as quanti(cid:28)ed in panel d), where the thi knessis plotted versus N .Summing up, we have assessed that the single parti- le motion of a typi al long-range intera ting system be- omes progressively more regular as the number of parti- les is in reased. This is at varian e with what happensfor systems with short-range intera tions and provides adi(cid:27)erent interpretation of the abundan e of regular mo-tion in long-range dynami s. Besides that, we have seenthat the features of the single parti le motion depend onthe hoi e of the initial ondition. A natural questionthen arises: what is the link between the ma ros opi properties of the di(cid:27)erent QSS with the hange observedin the single parti le dynami s? Anti ipating the answer,we will see that this is related to a bifur ation o urring U p M =0.6 Figure 3: Upper panel: Phase diagram in the ontrol param-eter plane ( M , U ) of the out(cid:21)of(cid:21)equilibrium phase transi-tion of the HMF model from a magnetized to a demagnetizedphase. The solid urve pinpoints the position of the bifur a-tion from the mono luster to the bi luster QSS. The dashedline stands for the theoreti al predi tion based on Lynden-Bell's violent relaxation theory (see text). The star refers tothe tri riti al point separating (cid:28)rst from se ond order phasetransitions. Lower panel: The bifur ation is monitored asfun tion of U , for M = 0 . . The grey zones highlight thewidth of the resonan es.in the e(cid:27)e tive Hamiltonian.In the thermodynami limit, the evolution of the sin-gle parti le distribution fun tion f ( θ, p, t ) is governed bythe Vlasov equation [19℄. This equation also des ribes themean-(cid:28)eld limit of wave-parti le intera ting systems [10℄.It an be reasonably hypothesized that QSS are sta-tionary stable solutions of the Vlasov equation [8℄. Fol-lowing these lines, a maximum entropy prin iple, previ-ously developed in the astrophysi al ontext by Lynden-Bell [14℄, allowed one to predi t [13, 20℄, for the HMFmodel, the o urren e of out-of-equilibrium phase tran-sitions, separating distin t ma ros opi regimes (magne-tized/demagnetized) by varying sele ted ontrol param-eters whi h represent the initial ondition.The entral idea of Lynden-Bell's approa h onsists in oarse-graining the mi ros opi one-parti le distributionfun tion f ( θ, p, t ) by introdu ing a lo al average in phasespa e. Starting from a water-bag initial pro(cid:28)le, with auniform distribution f , a fermioni -like entropy an beasso iated with the oarse grained pro(cid:28)le ¯ f , namely s [ ¯ f ] = − Z d p d θ (cid:20) ¯ ff ln ¯ ff + (cid:18) − ¯ ff (cid:19) ln (cid:18) − ¯ ff (cid:19)(cid:21) . (2)The orresponding statisti al equilibrium, whi h appliesto the relevant QSS regimes, is hen e determined by max-imizing su h an entropy, while imposing the onservationof the Vlasov dynami al invariants, namely energy, mo-mentum and norm of the distribution. The analysis re-veals the existen e of an out-of-equilibrium phase transi-tion from a magnetized to a demagnetized phase [13, 20℄.We here reinterpret the transition in a purely dynami- al framework, as a bifur ation from a mono luster QSSto a bi luster QSS. Aiming at shedding light on this issue,we pro eed as follows: For (cid:28)xed M and N , we graduallyin rease the energy U and ompute the Poin aré se tions,as dis ussed above. We then analyze the re orded se -tions by identifying the number of resonan es and mea-suring the asso iated width and position (both al ulatedin the p dire tion). Results for M = 0 . are displayed inthe lower panel of Fig. 3: the shaded region, boundedby the dashed lines, quanti(cid:28)es the width of the reso-nan es. As anti ipated, one an re ognize the typi alsignature of a bifur ation pattern. Repeating the aboveanalysis for di(cid:27)erent values of the initial magnetization M , allows us to draw a bifur ation line in the parameterspa e ( M , U ) . In the upper panel of Fig. 3 we reportboth this bifur ation (full) and the Lynden-Bell phasetransition (dashed) lines [13℄. The two pro(cid:28)les resem-ble ea h other qualitatively, and even quantitatively forsmall M0