Out of equilibrium magnetised solutions of the $XY$-Hamiltonian Mean Field
($XY$-HMF) model are build using an ensemble of integrable uncoupled pendula.
Using these solutions we display an out-of equilibrium phase transition using a
specific reduced set of the magnetised solutions

Bonifacio R.

Chavanis P. H.

D. Fanelli

Elskens Y.

Lynden-Bell D.

T. L. Van Den Berg

Van den Berg T. L.

X. Leoncini

Zaslavsky G. M.

English

arXiv

International audienceOut of equilibrium magnetised solutions of the $XY$-Hamiltonian Mean Field ($XY$-HMF) model are build using an ensemble of integrable uncoupled pendula. Using these solutions we display an out-of equilibrium phase transition using a specific reduced set of the magnetised solutions

Leoncini, Xavier

van den Berg, Tineke, L.

Fanelli, Duccio

HAL AMU

Out of Equilibrium Solutions in the XY -HamiltonianMean Field modelXavier Leoncini, Tineke Van den Berg, Duccio FanelliTo cite this version:Xavier Leoncini, Tineke Van den Berg, Duccio Fanelli. Out of Equilibrium Solutions in theXY -Hamiltonian Mean Field model. EPL, European Physical Society/EDP Sciences/Societa`Italiana di Fisica/IOP Publishing, 2009, 86 (2), pp.20002. <10.1209/0295-5075/86/20002>.<hal-00323591v2>HAL Id: hal-00323591https://hal.archives-ouvertes.fr/hal-00323591v2Submitted on 30 Apr 2009HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.epl draftOut of Equilibrium Solutions in the XY -Hamiltonian Mean FieldmodelX. Leoncini1, T. L. Van Den Berg1 and D. Fanelli21 Centre de Physique The´orique(a), Aix-Marseille Universite´, CNRS, Luminy, Case 907, F-13288 Marseille cedex 9,France2 Dipartimento di Energetica Sergio Stecco, Universita di Firenze, via s. Marta 3, 50139 Firenze, Italia e Centrointerdipartimentale per lo Studio delle Dinamiche Complesse (CSDC)PACS 05.20.-y – Classical statistical mechanicsPACS 05.45.-a – Nonlinear dynamics and chaosAbstract. - Out of equilibrium magnetised solutions of the XY -Hamiltonian Mean Field (XY -HMF) model are build using an ensemble of integrable uncoupled pendula. Using these solutionswe display an out-of equilibrium phase transition using a specific reduced set of the magnetisedsolutions.Long-range interactions are such that the two-body po-tential decays at large distances with a power–law ex-ponent which is smaller than the space dimension. Alarge number of fundamental physical systems falls insuch a broad category, including for instance gravitationalforces and unscreened Coulomb interactions [1], vorticesin two dimensional fluid mechanics [2–4], wave-particlesystems relevant to plasma physics [5, 6], Free-ElectronLasers (FELs) [7–9] and even condensed matter physicsfor easy-axis anti-ferromagnetic spin chains, where dipolareffects become dominant. While for short–range systemsonly adjacent elements are effectively coupled, long-rangeforces result in a global network of inter-particles con-nections, each element soliciting every other constitutiveunit. Clearly, from such an enhanced degree of complexitystems the difficulties in addressing the fascinating realmof long-range systems, for which standard techniques inphysics, notably in the framework of equilibrium statis-tical mechanics, proves essentially inadequate. It is inparticular customarily accepted that long–range systemsdisplay universal out-of-equilibrium features: Long-livedintermediate states can in fact emerge, where the systemgets, virtually indefinitely, trapped (the time of escape di-verging with the number of particles), before relaxing to-wards its deputed thermodynamic equilibrium. These arethe so–called Quasi Stationary States (QSSs) which have(a)Unite´ Mixte de Recherche (UMR 6207) du CNRS, et des univer-site´s Aix-Marseille I, Aix-Marseille II et du Sud Toulon-Var. Labo-ratoire affilie´ la FRUMAM (FR 2291).been shown to arise in several different physical contexts,ranging from laser physics to cosmology, via plasma ap-plications. A surprising, though general, aspect relates tothe role of initial conditions of which QSS keeps memory.More intriguingly, such a dependence can eventually ma-terialise in a genuine out-of-equilibrium phase transition:By properly adjusting dedicated control parameters whichrefers to the initial state, one observes the convergence to-wards intimately distinct macroscopic regimes (e.g. homo-geneous/non homogeneous) [10, 11]. Recently, such phasetransitions for systems embedded in one spatial dimensionhave been re-interpreted as a topological change in the sin-gle particles orbits [12]. This conclusion is achieved by per-forming a stroboscopic analysis of individual trajectories,which are being sampled at a specific rate imposed by theemerging time evolution of a collective variable and con-sequently sensitive to the intrinsic degree of microscopicself-organisation.QSSs out of equilibrium regimes have been explainedby resorting to a maximum entropy principle inspired tothe Lynden-Bell’s seminal work on the so-called violentrelaxation theory, an analytical treatment based on theVlasov equation and originally developed for astrophysicalapplications [13].In this letter we shall provide a strategy to constructa whole family of out-of-equilibrium solutions with ref-erence to the paradigmatic XY -Hamiltonian Mean Field(XY -HMF) model [14]. This procedure exploits the anal-ogy with an ensemble made of uncoupled pendula andp-1X. Leoncini et al.explicitly accommodate for self-consistency as a crucialingredient. Even more importantly, out-of equilibriumphase transitions are displayed using a reduced set of thenon-homogeneous (magnetised) solutions. The proposedapproach is inspired by the observation that in the con-tinuum limit (for an infinite number of particles) the dis-crete set of equations describing the physical system underscrutiny, converges towards the Vlasov equation, whichgoverns the evolution of the one-particle distribution func-tion.The N -body Hamiltonian for the XY -HMF model withferromagnetic interactions writesH =N∑i=1p2i2+12NN∑j=11− cos (qi − qj) , (1)where pi and qi are respectively the (canonically conju-gate) momentum and position of particle (rotor) i. Tomonitor the time evolution of the system, one can intro-duce the “magnetisation” asM =1N(∑cos qi,∑sin qi)=M (cosϕ, sinϕ) . (2)The equations of motion for the particles can be thereforecast in the form{p˙i = −M sin (qi − ϕ)q˙i = pi, (3)where the dot denotes the time derivative. Notice thatit is tempting to imagine equations (3) as resulting froma set of uncoupled, possibly driven, one dimensional pen-dula Hamiltonian. Inspired by this analogy, we here in-tend to shed light onto the out-of-equilibrium dynamicsof the original N -body model, by investigating the equi-librium properties of an associated pendula system. Morespecifically, let us imagine that the system of coupled ro-tators has reached some equilibrium state , such that inthe N → ∞ limit the magnetisation M of the XY -HMFmodel is constant and equal to m. Equations of motion(3) implies that the system formally reduces to an infiniteset of uncoupled pendula. Our strategy to construct sta-tionary solutions for (1) is to consider a finite ensemble ofN uncoupled pendula whose Hamiltonian readsH =N∑i=1p2i2+m(1− cos qi) , (4)and compute stationary solutions in the thermodynamiclimit of this m−pendula system [15].Given an initial condition each pendulum i is confinedon a specific torus of the pendulum phase portrait depictedin Fig 1.To build a stationary state we naturally consider theergodic measure on the torus which originate from thependulum motion and time averages. In order to proceedfurther in the analysis and due to integrability, we employ−3 −2 −1 0 1 2 3−3−2−10123qpFig. 1: Phase portrait of the pendulum for m = 0.5. Thegray region is here used as a tool to estimate numerically thefunction g(I) as introduced in eq. (9). It is termed in the litera-ture a water-bag and has no reference to the selected particles’initial condition.the canonical transformation to the action-angle variables(I, θ) of the system (see for instance [16]). We thus obtainHm = Hm(Ii), with θ˙i = ∂Hm/∂I = ω(Ii), where Iistands for the constant action, which is fixed by the initialstate of the uncoupled pendulum i. For any selected initialcondition, as time evolves, θ covers uniformly the circle[−pi, pi[, while the action I keeps its constant value. Theergodic measure reduces hence to ρi(I, θ) =12pi δ(I − Ii),which immediately translates inρE =N∏i=1ρi , (5)when considering an ensemble of N particles. Becauseof the above and since particles are identical and non in-teracting one can straightforwardly deduce the followingexpression for the one particle density function:f(I, θ) =g(I)2pi, (6)where g(·) is a discrete valued function, solely determinedby the selected initial conditions. In the limitN →∞, g(·)can change smoothly with the (continuous) action vari-able.In order to get a stationary equilibrium solution of thesystem of pendula we can then consider a given posi-tive and integrable function g, associate to it the func-tion f(I, θ) according to (6). Then we perform an “in-verse” transform to obtain an explicit expression for theone particle density function f˜(p, q) as defined in the origi-nal phase space Γpq. This latter represents an equilibrium(time invariant) solution, and enables us to estimate anymacroscopic observable, defined as a function of the seaof microscopic constituents. More concretely let us turnto consider the global magnetisation M as specified by eq.(2)). In the infinite N limit the time average of the mag-netisation, which coincides with the ergodic-spatial aver-p-2Titleage, reads:M¯ = 〈M〉 =(∫f˜(p, q) cos q dpdq , 0), (7)where the observation that f˜ is even in p and q has beenused. Expressing the above in term action-angle variables,one can writeM¯ = 〈M〉 =(12pi∫g(I) cos q(I, θ) dIdθ , 0). (8)The integration over the angle can be performed (see Ap-pendix), yielding to the final formM =∫ 8pi√m0g(I)(2E(κ)K(κ)− 1)dI+∫ ∞8pi√mg(I)(1 + 2κ2(E(κ−1)K(κ−1)− 1))dI, (9)with κ = κ(I) = (H(I) +m)/2m. Notice that if we con-sider an initial distribution given by f˜ , then the mag-netisation (9) stays constant as, by construction f˜ is sta-tionary. In order to reconcile the m-model of uncoupledpendula to the XY -HMF interacting rotors, we need toimpose the condition〈M〉 = m . (10)Should there exist an f˜ for which equation (10) had anm 6= 0 solution, then we would have obtained a stationarysolution of the system of pendula, which is in turn alsomagnetised stationary solution of the XY -HMF model inthe N →∞ limit. Indeed the equations of motion for thesystem of pendula write{p˙i = −m sin qiq˙i = pi, (11)and given the imposed condition (10) they are formallyidentical to (3) with a constant ϕ = 0. Note that thecondition of the phase ϕ = 0, can be modified at will bya simple shift in the m-pendula Hamiltonian.Such an out of equilibrium solution is displayed in Fig. 2,for a specific choice of the initial condition and outputmagnetisation amount. One typically recognises the un-derlying pendulum phase portrait, with each tori beingdifferently populated according to the function g(I). Thenon-uniformity of f˜(p, q) on each torus stems from thenonlinearity of the transformation q = q(I, θ), p = p(I, θ).As previously mentioned out of equilibrium, phase tran-sition have been previously reported separating macro-scopically distinct QSSs phases. These findings are hererevisited in the framework of the proposed approach,which holds promise to generalise the conclusion beyondthe water-bag regime so far inspected via the Lynden-Bell ansatz. The water-bag regimes correspond to ini-tially assign the particles to populate, randomly and uni-formly, a bound domain [−q0, q0] × [−p0, p0] depictedFig. 2: Stationary out of equilibrium distribution f˜(p, q) withm = 〈M〉 = 0.5 obtained for g(I) reconstructed from a water-bag with q0 = 2pi/5 and p0 = 1.37.in gray in Fig. 2. This latter initial condition is thenunivoquely specified by the magnetisation at time zero,namely M0 = sin(q0)/q0, and the energy per particleU = p20/6 + (1 −M20 )/2. Notice that the water bag con-cept will be here invoked as a mere numerical strategyto calculate the needed function g(I). We are hence notlimiting the present analysis to a specific class of initialconditions, as e.g. done in [10], but we rather present acompelling evidence on the existence of a phase transitionin a broader perspective.Equation (10) is implicit inm parametrised through theinitial conditions which enter the definition of the functiong. Such an equation admits m = 0 as a trivial solution.One can then look for more general solutions of Eq. (10)with m 6= 0. Even though f˜(p, q) water bag type is def-initely not of, the water-bag type, as clearly depicted inFig. 2, we have here decided to facilitate the forthcominganalysis, namely the calculation of the associated g(I), byfocusing on a finite portion of phase space as delimitedby a water-bag window. This choice allows us to obtain afamily of solutions monitored by the two same parameters,namely U and M0. To construct g(I) it is possible to ana-lytically compute the length of the intersection of each toriwith the water-bag (see Fig. 2), however we settled for asimple numerical procedure. Namely we consider a largeensemble of particles whose distribution approximates awaterbag, then for each particle and a given m we com-pute its corresponding action to construct an histogram ofI and use it as an approximate form of g(I). Finally weuse this expression and check whether m = 〈M〉 and lookfor possible solutions. The drawback of this choice resultsin a dependence on the number N of particles used to con-struct the approximate waterbag wich may be importantwhen we are close to the m = 〈M〉 = 0 transition line. Wehowever believe its accuracy is sufficient to present ourpoint.Results for differentM0 are depicted in Fig. 3. One canclearly appreciate the transition from a magnetised stateto a non magnetised one, as well as a first order type ofp-3X. Leoncini et al.0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.91U<M>Fig. 3: Magnetisation 〈M〉 as a function of energy density Ufor different values of M0. The symbols +, ×, +, ◦ correspondto non-zero solutions of the system of equations (9) and (10)computed respectively forM0 = 0.854, 0.637, 0.368, 0. One cannotice a dependence on M0, as well as a first order transitionfor M0 = 0. Computation of the function g(I) is made usingan approximate water-bag filled with 1.6 107 particles. Notealso that the that the transition point around U = 0.59 forM0 = 0 is very close to that found, numerically in [10] probablybecause the initial condition is “closer” to a water-bag onewhen M0 = 0.transition for M0 = 0.A few comments are mandatory at this point. First,we insist on the fact that the phase space trajectories cor-responding to derived solutions and its associated timeevolution in Γpq apply to two, intrinsically different, dy-namical systems, namely (1) and (4). Second, considerthe average energy per particle U = E/N . We point outthat the constants in Hamiltonians (1) and (4) are cho-sen in order to have 0 as a minimal value for the energy,which, in the thermodynamic limit, implies U = 0 for azero temperature. Now focus on UHMF for the XY -HMFmodel: We obtain UHMF = T/2−M2/2 + 1/2, while forthe pendula one gets Up = T/2−M2 +M . Here, in bothcases, T/2 is the average kinetic energy per particle. Dif-ferent energies are thus associated to the same trajectory,depending on the dynamical system that is being chosento generate it. In order to reconcile the two models onecan infer that the chemical potential of the particle is dif-ferent, yielding to δµ = δU = (M − 1)2/2 for respectivelythe integrable uncoupled model and the globally coupledone. Moreover, solutions with constant M = m = 0 docorrespond to a one dimensional perfect gas: the observedphase transition can hence be seen as a sort of sublimation.In conclusion, with reference to theXY -HMF model, wehave designed an analytical scheme which allows to iden-tify all possible stationary solutions with constant mag-netisation M using a set of integrable uncoupled pendula.This includes as a subset the celebrated QSSs, which aretherefore formally understood within a consistent mathe-matical framework. Following these lines, it can be in-ferred that the out of equilibrium states predicted bythe statistical mechanics scenario pioneered by Lynden-Bell [11], should belong to the class of solutions here dis-played. We are then providing de facto a testbed for ac-curacy of the controversial Lynden-Bell theory [13]. Notethough that the stability of the solutions has not been dis-cussed, it is currently under investigation and is likely toprovide further restrictions on the possible out of equilib-rium stationary states.∗ ∗ ∗We acknowledge usefull conversations with B. Fernan-dez, S. Ruffo andM. Vittot. X. Leoncini thanks the Centerfor Nonlinear Science at Georgia Tech for their hospiltal-ity and support during part of this work and acknowledgespartial support from the PICS program of the CNRS num-ber 4973.Appendix. – We shall here review the main mathe-matical tools which are employed in the above derivation.When it comes to the pendulum motion, trapped orbits(libration) are characterised by:q = 2 sin−1[κsn(2K(κ)θpi, κ)]I =8pi√m[E(κ)− κ′2K(κ)]〈cos q〉 = 2E(κ)K(κ)− 1while for the untrapped ones (rotation) the following re-lations hold:q = 2am(2K(1/κ)θpi, κ−1)I =8κpi√mE(1/κ)〈cos q〉 = 1 + 2κ2(E(κ−1)K(κ−1)− 1)with κ2 = (h+m)/2m. Here < · >= ∫ 2pi0·dθ/(2pi).REFERENCES[1] Dauxois T., Ruffo S., Arimondo E. and WilkensM., (Editors) Dynamics and Thermodynamics of Systemswith Long Range Interactions Vol. 602 of Lect. Not. Phys.(Springer) 2002.[2] Onsager L., Nuovo Cimento, Suppl. , 6 (1949) 279.[3] Edwards S. F. and Taylor J. B., Proc. R. Soc. Lond.A , 336 (1974) 257.[4] Weiss J. B. and McWilliams J. C., Phys. Fluids A , 3(1991) 835.[5] Elskens Y. and Escande D. F., Microscopic Dynamicsof Plasmas and Chaos (IoP Publishing, Bristol) 2002.[6] Be´nisti D. and Gremiller L., Phys. Plasmas, 14 (2007)042302.[7] Bonifacio R., Riv. Nuovo Cimento , 13 (1990) 1.p-4Title[8] Barre´ J., Dauxois T., De Ninno G., Fanelli D. andRuffo S., Phys. Rev. E , 69 (2004) 045501(R).[9] Bachelard R., Antoniazzi A., Chandre C., FanelliD., Leoncini X. and Vittot M., Eur. Phys. J. D , 42(2007) 125.[10] Antoniazzi A., Fanelli D., Ruffo S. and YamaguchiY., Phys. Rev. Lett. , 99 (2007) 040601.[11] Chavanis P. H., Ninno G. D., Fanelli D. and RuffoS., Out of equilibrium phase transitions in mean-fieldhamiltonian dynamics in proc. of Chaos, Complexity andTransport, edited by Chandre C., Leoncini X. and Za-slavsky G., (World Scientific, Singapore) 2008 pp. 3–26.[12] Bachelard R., Chandre C., Fanelli D., Leoncini X.and Ruffo S., Phys. Rev. Lett., 101 (2008) 260603.[13] Lynden-Bell D.,Mon. Not. R. Astron. Soc. , 136 (1967)101.[14] Antoni M. and Ruffo S., Phys. Rev. E , 52 (1995) 3261.[15] Van den Berg T. L., Syste`mes avec interactions a` longueporte´e Master’s thesis Universite´ Paris VII (2008).[16] Zaslavsky G. M., The Physics of Chaos in Hamilto-nian Systems (Imperial College Press, London) 2007 sec-ond Edition.p-5

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Out-of-equilibrium magnetised solutions of the XY-Hamiltonian Mean-Field (XY-HMF) model are built using an ensemble of integrable uncoupled pendula. Using these solutions, we display an out-of-equilibrium phase transition using a specific reduced set of the magnetised solutions

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Out of Equilibrium Solutions in the $XY$ -Hamiltonian Mean Field model