Abundance of Regular Orbits and Nonequilibrium Phase Transitions in the Thermodynamic Limit for Long-Range Systems

Bachelard, Romain; Chandre, Cristel; Fanelli, Duccio; Leoncini, Xavier; Ruffo, Stefano

doi:10.1103/physrevlett.101.260603

Cited by 80 publications

(66 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…2. On the paramagnetic side of the phase transition, the systems becomes trapped in an out-of-equilibrium limit cycle associated with appearance of resonance islands in the phase space [13], characterized by strong oscillation of magnetization around M ¼ 0, see Fig. 3.…”

Section: Fig 1 (Color Online)mentioning

confidence: 99%

Core-Halo Distribution in the Hamiltonian Mean-Field Model

Pakter

Levin

2011

Phys. Rev. Lett.

View full text Add to dashboard Cite

We study a paradigmatic system with long-range interactions: the Hamiltonian mean-field (HMF) model. It is shown that in the thermodynamic limit this model does not relax to the usual equilibrium Maxwell-Boltzmann distribution. Instead, the final stationary state has a peculiar core-halo structure. In the thermodynamic limit, HMF is neither ergodic nor mixing. Nevertheless, we find that using dynamical properties of Hamiltonian systems it is possible to quantitatively predict both the spin distribution and the velocity distribution functions in the final stationary state, without any adjustable parameters. We also show that HMF undergoes a nonequilibrium first-order phase transition between paramagnetic and ferromagnetic states. DOI: 10.1103/PhysRevLett.106.200603 PACS numbers: 05.20.Ày, 05.45.Àa, 05.70.Ln Since the early work of Clausius, Boltzmann, and Gibbs it has been known that for particles interacting through short-range potentials, the final stationary state reached by a system corresponds to the thermodynamic equilibrium [1]. Although no exact proof exists, in practice it is found that nonintegrable systems with a fixed energy and number of particles (microcanonical ensemble) always relax to a unique stationary state which only depends on the global conserved quantities: energy, momentum, and angular momentum. The equilibrium state does not depend on the specifics of the initial particle distribution. The situation is very different for systems in which particles interact through long-ranged unscreened potentials. This is the case for gravitational systems and confined one component plasmas [2,3]. For these systems, in the thermodynamic limit, the collision duration time diverges, and the thermodynamic equilibrium is never reached [4]. Instead, as time t ! 1, these systems become trapped in a stationary state characterized by a broken ergodicity [5][6][7]. Unlike the thermodynamic equilibrium, the stationary state depends explicitly on the initial particle distribution. Over the last 50 years, there has been a great effort to predict the final stationary state without having to explicitly solve the N-body dynamics or the collisionless Boltzmann (Vlasov) equation. Qualitatively, it has been observed that for many different systems the nonequilibrium stationary state has a peculiar core-halo shape. Recently, an ansatz solution to the Vlasov equation has been proposed which allowed us to explicitly calculate the core-halo distribution function for confined plasmas and self-gravitating systems [2,3]. In this Letter we will show that an ansatz solution is also possible for the Hamiltonian mean-field (HMF) model. The theory proposed allows us also to locate the nonequilibrium para-to-ferromagnetic phase transition, which earlier theories incorrectly predicted to be of second order [8]. All of the results are compared with the molecular dynamics simulations performed using a symplectic integrator, and are found to be in excellent agreement.The HMF model consists of N, XY interacting spins, whose dynamics i...

show abstract

Section: Fig 1 (Color Online)mentioning

confidence: 99%

Core-Halo Distribution in the Hamiltonian Mean-Field Model

Pakter

Levin

2011

Phys. Rev. Lett.

View full text Add to dashboard Cite

show abstract

“…This represented a substantial leap forward in the understanding of the dynamical properties of the QSS in the HMF model, beyond the aforementioned statistical approach. Indeed, it was understood that the microscopic dynamics in the inhomogeneous stationary state is regular and explicitly known, an observation which contributed to shed light onto the puzzling abundance of emerging regular orbits as revealed in [7].These last results have been though obtained in the framework of mean field systems: The actual distance between particles does no explicitly appear in the HMF potential. In this letter we aim at bridging this gap, by focusing on the so called α−HMF model [8].…”

mentioning

confidence: 99%

mentioning

confidence: 99%

Stationary states and fractional dynamics in systems with long-range interactions

Berg

Fanelli²,

Leoncini³

2010

Europhys. Lett.

Self Cite

View full text Add to dashboard Cite

Abstract. -Dynamics of many-body Hamiltonian systems with long range interactions is studied, in the context of the so called α−HMF model. Building on the analogy with the related mean field model, we construct stationary states of the α−HMF model for which the spatial organization satisfies a fractional equation. At variance, the microscopic dynamics turns out to be regular and explicitly known. As a consequence, dynamical regularity is achieved at the price of strong spatial complexity, namely a microscopic inhomogeneity which locally displays scale invariance.The out-of-equilibrium dynamics of many body systems subject to long range couplings defines a fascinating field of investigations, which can potentially impact different domains of applications [1]. In a long range system every constituent is simultaneously solicited by the ensemble of microscopic actors, resulting in a complex dynamical picture. This latter scenario applies to a vast realm of fundamental problems, including gravity and plasma physics, and also extends to a rich variety of cross disciplinary studies [2]. In particular, and with specific emphasis on the peculiar non equilibrium features, long range systems often display a slow relaxation to equilibrium. They are in fact trapped in long-lasting out of equilibrium regimes, termed in the literature Quasi Stationary States (QSS) which bear distinct characteristics, as compared to the corresponding deputed equilibrium configuration. A paradigmatic representative of long range interactions, sharing the mean field viewpoint, is the so called Hamiltonian Mean Field model, which describes the coupled evolution of N rotators, populating the unitary circle and interacting via a cosines like potential. In the limit of infinite system size the discrete HMF model is described by the Vlasov equation for the evolution of the single particle distribution function [3]. This leads to a statistically based treatment, inspired by the seminal work of Lynden-Bell, which revealed the existence of two different classes of QSS, spatially homogeneous or inhomogeneous [4,5]. More recently, and still with reference to the HMF case study, stationary states have been constructed using a dynamical scheme which exploited the formal analogy with a set of uncoupled pendula [6]. This represented a substantial leap forward in the understanding of the dynamical properties of the QSS in the HMF model, beyond the aforementioned statistical approach. Indeed, it was understood that the microscopic dynamics in the inhomogeneous stationary state is regular and explicitly known, an observation which contributed to shed light onto the puzzling abundance of emerging regular orbits as revealed in [7].These last results have been though obtained in the framework of mean field systems: The actual distance between particles does no explicitly appear in the HMF potential. In this letter we aim at bridging this gap, by focusing on the so called α−HMF model [8]. We shall p-1

show abstract

“…Other realistic systems that contain long-range interaction include cold atomic clouds [10], natural light-harvesting complexes [11][12][13], helium Rydberg atoms [14], and cold Rydberg gases [15]. Long-range interacting systems display features that are not often observed in other systems, such as broken ergodicity [16][17][18][19] and long-lasting out-of-equilibrium regimes [20]. …”

mentioning

confidence: 99%

Cooperative Shielding in Many-Body Systems with Long-Range Interaction

2016

View full text Add to dashboard Cite

In recent experiments with ion traps, long-range interactions were associated with the exceptionally fast propagation of perturbation, while in some theoretical works they have also been related with the suppression of propagation. Here, we show that such apparently contradictory behavior is caused by a general property of longrange interacting systems, which we name Cooperative Shielding. It refers to shielded subspaces that emerge as the system size increases and inside of which the evolution is unaffected by long-range interactions for a long time. As a result, the dynamics strongly depends on the initial state: if it belongs to a shielded subspace, the spreading of perturbation satisfies the Lieb-Robinson bound and may even be suppressed, while for initial states with components in various subspaces, the propagation may be quasi-instantaneous. We establish an analogy between the shielding effect and the onset of quantum Zeno subspaces. The derived effective Zeno Hamiltonian successfully describes the short-ranged dynamics inside the subspaces up to a time scale that increases with system size. Cooperative Shielding can be tested in current experiments with trapped ions. Introduction.-A better understanding of the nonequilibrium dynamics of many-body quantum systems is central to a wide range of fields, from atomic, molecular, and condensed matter physics to quantum information and cosmology. New insights into the subject have been obtained thanks to the remarkable level of controllability and isolation of experiments with optical lattices [1][2][3][4][5][6][7] and trapped ions [8,9]. Recently there has been a surge of interest in the dynamics of systems with long-range interactions, triggered by experiments with ion traps [8,9], where the range of interactions in onedimensional (1D) spin models can be tuned with great accuracy. Other realistic systems that contain long-range interaction include cold atomic clouds [10], natural light-harvesting complexes [11][12][13], helium Rydberg atoms [14], and cold Rydberg gases [15]. Long-range interacting systems display features that are not often observed in other systems, such as broken ergodicity [16][17][18][19] and long-lasting out-of-equilibrium regimes [20].

show abstract

Abundance of Regular Orbits and Nonequilibrium Phase Transitions in the Thermodynamic Limit for Long-Range Systems

Cited by 80 publications

References 22 publications

Core-Halo Distribution in the Hamiltonian Mean-Field Model

Core-Halo Distribution in the Hamiltonian Mean-Field Model

Stationary states and fractional dynamics in systems with long-range interactions

Cooperative Shielding in Many-Body Systems with Long-Range Interaction

Contact Info

Product

Resources

About