Hamiltonian Mean Field Model Research Articles

Out-of-equilibrium phase re-entrance(s) in long-range interacting systems

Systems with long-range interactions display a short-time relaxation toward quasistationary states (QSSs) whose lifetime increases with system size. The application of Lynden-Bell's theory of "violent relaxation" to the Hamiltonian Mean Field model leads to the prediction of out-of-equilibrium first- and second-order phase transitions between homogeneous (zero magnetization) and inhomogeneous (nonzero magnetization) QSSs, as well as an interesting phenomenon of phase re-entrances. We compare these theoretical predictions with direct N -body numerical simulations. We confirm the existence of phase re-entrance in the typical parameter range predicted from Lynden-Bell's theory, but also show that the picture is more complicated than initially thought. In particular, we exhibit the existence of secondary re-entrant phases: we find unmagnetized states in the theoretically magnetized region as well as persisting magnetized states in the theoretically unmagnetized region. We also report the existence of a region with negative specific heats for QSSs both in the numerical and analytical caloric curves.

Kinetic theory of stellar systems, two-dimensional vortices and HMF model

We discuss the kinetic theories of stellar systems, two-dimensional vortices and Hamiltonian mean field model, stressing their analogies and differences. We describe the evolution of the system as a whole and discuss the timescale of relaxation towards the Boltzmann distribution predicted by statistical mechanics. We also consider the relaxation of a “test” particle in a bath of “field” particles and analyze it with the aid of a Fokker–Planck equation involving a term of diffusion counterbalanced by a friction or a drift.

Dynamical stability of systems with long-range interactions: application of the Nyquist method to the HMF model

We apply the Nyquist method to the Hamiltonian mean field (HMF) model in order to settle the linear dynamical stability of a spatially homogeneous distribution function with respect to the Vlasov equation. We consider the case of Maxwell (isothermal) and Tsallis (polytropic) distributions and show that the system is stable above a critical kinetic temperature Tc and unstable below it. Then, we consider a symmetric double-humped distribution, made of the superposition of two decentered Maxwellians, and show the existence of a re-entrant phase in the stability diagram. When we consider an asymmetric double-humped distribution, the re-entrant phase disappears above a critical value of the asymmetry factor Δ > 1.09. We also consider the HMF model with a repulsive interaction. In that case, single-humped distributions are always stable. For asymmetric double-humped distributions, there is a re-entrant phase for 1 ≤ Δ < 25.6, a double re-entrant phase for 25.6 < Δ < 43.9 and no re-entrant phase for Δ > 43.9. Finally, we extend our results to arbitrary potentials of interaction and mention the connexion between the HMF model, Coulombian plasmas and gravitational systems. We discuss the relation between linear dynamical stability and formal nonlinear dynamical stability and show their equivalence for spatially homogeneous distributions. We also provide a criterion of dynamical stability for spatially inhomogeneous systems.

Small traveling clusters in attractive and repulsive Hamiltonian mean-field models

Long-lasting small traveling clusters are studied in the Hamiltonian mean-field model by comparing between attractive and repulsive interactions. Nonlinear Landau damping theory predicts that a Gaussian momentum distribution on a spatially homogeneous background permits the existence of traveling clusters in the repulsive case, as in plasma systems, but not in the attractive case. Nevertheless, extending the analysis to a two-parameter family of momentum distributions of Fermi-Dirac type, we theoretically predict the existence of traveling clusters in the attractive case; these findings are confirmed by direct N -body numerical simulations. The parameter region with the traveling clusters is much reduced in the attractive case with respect to the repulsive case.

Comment on “Ergodicity and central-limit theorem in systems with long-range interactions” by Figueiredo A. et al.

In the present paper we refute the criticism advanced in a recent preprint by Figueiredo et al [1] about the possible application of the $q$-generalized Central Limit Theorem (CLT) to a paradigmatic long-range-interacting many-body classical Hamiltonian system, the so-called Hamiltonian Mean Field (HMF) model. We exhibit that, contrary to what is claimed by these authors and in accordance with our previous results, $q$-Gaussian-like curves are possible and real attractors for a certain class of initial conditions, namely the one which produces nontrivial longstanding quasi-stationary states before the arrival, only for finite size, to the thermal equilibrium.

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

We investigate the dynamics of many-body long-range interacting systems, taking the Hamiltonian mean-field model as a case study. We show that regular trajectories, associated with invariant tori of the single-particle dynamics, prevail. The presence of such tori provides a dynamical interpretation of the emergence of long-lasting out-of-equilibrium regimes observed generically in long-range systems. This is alternative to a previous statistical mechanics approach to such phenomena which was based on a maximum entropy principle. Previously detected out-of-equilibrium phase transitions are also reinterpreted within this framework.

Systems with negative specific heat in thermal contact: Violation of the zeroth law

Using both numerical simulations and exact expressions for the free energy and microcanonical entropy for a modified Hamiltonian mean-field (HMF) model, we show that when two similar systems with the same intensive parameters but with negative specific heat are weakly coupled, they undergo a process in which the total entropy increases irreversibly. We find that the final equilibrium is such that two phases appear at a temperature (equal in both systems) that is generally different from the initial temperature. We corroborate our results using two different kinds of couplings between the HMF systems. We confirm that our results hold also for the Ising model with long- and short-range interactions, which also has a parameter region with negative specific heat in the microcanonical ensemble. Further, we show that we can couple each system having negative specific heat to a third system that can be used as a thermometer, as long as this thermometer is small enough not to drive the system out of the microcanonical ensemble. Therefore, we show an instance of violation of the zeroth law of thermodynamics.

Dynamical phase transitions in long-range Hamiltonian systems and Tsallis distributions with a time-dependent index

We study dynamical phase transitions in systems with long-range interactions, using the Hamiltonian mean field model as a simple example. These systems generically undergo a violent relaxation to a quasistationary state (QSS) before relaxing towards Boltzmann equilibrium. In the collisional regime, the out-of-equilibrium one-particle distribution function (DF) is a quasistationary solution of the Vlasov equation, slowly evolving in time due to finite- N effects. For subcritical energy densities, we exhibit cases where the DF is well fitted by a Tsallis q distribution with an index q(t) slowly decreasing in time from q approximately = 3 (semiellipse) to q=1 (Boltzmann). When the index q(t) reaches an energy-dependent critical value q_(crit) , the nonmagnetized (homogeneous) phase becomes Vlasov unstable and a dynamical phase transition is triggered, leading to a magnetized (inhomogeneous) state. While Tsallis distributions play an important role in our study, we explain this dynamical phase transition by using only conventional statistical mechanics. For supercritical energy densities, we report the existence of a magnetized QSS with a very long lifetime.

Some thermodynamic non-Fermi liquid properties of correlated electron systems. (...)Cu&lt;sub&gt;&lt;i&gt;m&lt;/i&gt;&lt;/sub&gt;O&lt;sub&gt;&lt;i&gt;n-x&lt;/i&gt;&lt;sub&gt;

A mean-field Hamiltonian model has been used to investigate some thermodynamic properties of the normal states of non-Fermi liquid (NFL) systems,(...)CumOn-x . This Hamiltonian is like that of the Bardeen-Cooper-Schrieffer model [Phys. Rev. 108 (1957) 1175] but differs from the latter in (i) being multiband, (ii) the gap in energy being a function of the hopping integral and (iii) band energies of electrons being dependent upon spin orientation. The Hamiltonian is, therefore, similar to the Paring t-model [Physica 258 (9166) 30] but differs from it in not incorporating hybridization term and hybrid pair superconductivity. The analysis of the model yields magnetic energy spectrum for Cu (3d) bands` and non-magnetic energy spectrum for the O (2p) bands. Inverse temperature dependences of electronic specific heat Cv , entropy function (S) and pair susceptibility (&#935&#8593&#8595) are computed and exhibited. The specific heat dependence upon inverse temperature shows a linear form at very high temperature. It displays inverse-square-law temperature dependence, approximately, for lower temperatures. In the very low temperature range, the actual curve of the theoretical specific heat with temperature is rather like that of the Cp versus T-2 curve obtained for Bi4Cu4O16+y and Ti2Ba2Cn-1CunO2n+4 down to millikelvin temperature. This is in contradistinction to the linear temperature dependence (Cv = &#947T) of Fermi liquid systems. The specific entropy dependence on temperature shows correct physical response of systems to order (disorder) with varying temperatures. The pair susceptibility is linear at very high temperature and constant (X=X0) at moderate/low temperatures. The latter is as in Fermi liquid systems, but the former is an NFL manifestation. Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 149-156

A closer look at the indications of q-generalized Central Limit Theorem behavior in quasi-stationary states of the HMF model

We give a closer look at the Central Limit Theorem (CLT) behavior in quasi-stationary states of the Hamiltonian Mean Field model, a paradigmatic one for long-range-interacting classical many-body systems. We present new calculations which show that, following their time evolution, we can observe and classify three kinds of long-standing quasi-stationary states (QSS) with different correlations. The frequency of occurrence of each class depends on the size of the system. The different microscopic nature of the QSS leads to different dynamical correlations and therefore to different results for the observed CLT behavior.

Relaxation times of unstable states in systems with long range interactions

We consider several models with long range interactions evolving via Hamiltoniandynamics. The microcanonical dynamics of the basic Hamiltonian mean field(HMF) model and perturbed HMF models with either global anisotropy or anon-site potential are studied both analytically and numerically. We find that, inthe magnetic phase, the initial zero magnetization state remains stable above acritical energy and is unstable below it. In the dynamically stable state, thesemodels exhibit relaxation timescales that increase algebraically with the numberN of particles, indicating the robustness of the quasistationary state seen in previous studies.In the unstable state, the corresponding timescale increases logarithmically inN.

Long-time behavior of quasistationary states of the Hamiltonian mean-field model

The Hamiltonian mean-field model has been investigated, since its introduction about a decade ago, to study the equilibrium and dynamical properties of long-range interacting systems. Here we study the long-time behavior of long-lived, out-of-equilibrium, quasistationary dynamical states, whose lifetime diverges in the thermodynamic limit. The nature of these states has been the object of a lively debate in the recent past. We introduce a numerical tool, based on the fluctuations of the phase of the instantaneous magnetization of the system. Using this tool, we study the quasistationary states that arise when the system is started from different classes of initial conditions, showing that the new observable can be exploited to compute the lifetime of these states. We also show that quasistationary states are present not only below, but also above the critical temperature of the second-order magnetic phase transition of the model. We find that at supercritical temperatures the lifetime is much larger than at subcritical temperatures.

Nonequilibrium Tricritical Point in a System with Long-Range Interactions

Systems with long-range interactions display a short-time relaxation towards quasistationary states whose lifetime increases with system size. With reference to the Hamiltonian mean field model, we here show that a maximum entropy principle, based on Lynden-Bell's pioneering idea of "violent relaxation," predicts the presence of out-of-equilibrium phase transitions separating the relaxation towards homogeneous (zero magnetization) or inhomogeneous (nonzero magnetization) quasistationary states. When varying the initial condition within a family of "water bags" with different initial magnetization and energy, first- and second-order phase transition lines are found that merge at an out-of-equilibrium tricritical point. Metastability is theoretically predicted and numerically checked around the first-order phase transition line.

On the non-Boltzmannian nature of quasi-stationary states in long-range interacting systems

We discuss the non-Boltzmannian nature of quasi-stationary states in the Hamiltonian mean field (HMF) model, a paradigmatic model for long-range interacting classical many-body systems. We present a theorem excluding the Boltzmann–Gibbs exponential weight in Gibbs Γ -space of microscopic configurations, and comment a paper recently published by Baldovin and Orlandini [Phys. Rev. Lett. 96 (2006) 240602]. On the basis of the points here discussed, the ongoing debate on the possible application, within appropriate limits, of the generalized q-statistics to long-range Hamiltonian systems remains open.

Dynamical properties of cluster in the Hamiltonian Mean Field model

We investigate some properties of the trapping/untrapping mechanism of a single particle into/outside the cluster in the Hamiltonian Mean Field model. Particle are clustered in the ordered low-energy phase in this model. However, when the number of particles is finite, some particles can acquire a high energy and leave the cluster. Hence, below the critical energy, the fully-clustered and excited states appear by turns. First, we show that the numerically computed time-averaged trapping ratio agrees with that obtained by a statistical average performed for the Boltzmann–Gibbs stationary solution of the Vlasov equation. Second, we found numerically that the probability distribution of the lifetime of the fully-clustered state is not exponential but follows instead a power law. This means that the excitation of a particle from the cluster is not a Poisson process and might be controlled by some type of collective motion with long memory. Therefore, although an average trapping ratio exists, there appear to be no typical trapping ratio in the probabilistic sense. Finally, we discuss the dynamical mechanism using a modified model.

Exploring the Thermodynamic Limit of Hamiltonian Models: Convergence to the Vlasov Equation

We here discuss the emergence of quasistationary states (QSS), a universal feature of systems with long-range interactions. With reference to the Hamiltonian mean-field model, numerical simulations are performed based on both the original N-body setting and the continuum Vlasov model which is supposed to hold in the thermodynamic limit. A detailed comparison unambiguously demonstrates that the Vlasov-wave system provides the correct framework to address the study of QSS. Further, analytical calculations based on Lynden-Bell's theory of violent relaxation are shown to result in accurate predictions. Finally, in specific regions of parameters space, Vlasov numerical solutions are shown to be affected by small scale fluctuations, a finding that points to the need for novel schemes able to account for particle correlations.

Relaxation dynamics and topology in the Hamiltonian Mean Field model

In this work we analyze, for the Hamiltonian Mean Field model, the relationship between the existence of quasi-stationary long-standing trajectories and the topology of the potential energy, following the ideas recently introduced in the literature [1]. In particular, we study the way topology alters the distribution of momenta along the trajectories as well asthe long-time behavior of the system.

Phase transition in d-dimensional long-range interacting systems

We present new iterative method for the derivation one-point distribution function in the equilibrium state. For derivation of the distribution function, we must solve Lane-Emden equation. In general case, we solve the equation with iterative method. However the traditional method is not ensured convergence of the algorithm, we cannot often obtain solutions. In order to obtain the stable stationary distribution function, we apply an iterative method, inspired by a previous one used in 2D turbulence. Our method ensures entropy increase and convergence of the algorithm. Furthermore, our method can obtain the distribution function quickly [1]. Here we present the phase transition in longrange interacting systems. The Hamiltonian Mean-Field (HMF) model describes the motion of globally coupled particles on a 1D circle. Nevertheless the interaction is described only one cosine function, both the dynamical and the thermodynamical properties of this mode are quite various and complicated. For the HMF model, the extension for 2D model had been proposed [3]. We have extended the HMF model for 3D and 4D models. Then we analyze the phase-transition for d-dimensional models. The Hamiltonian of the HMF models is written as

Maximum entropy principle explains quasistationary states in systems with long-range interactions: The example of the Hamiltonian mean-field model

A generic feature of systems with long-range interactions is the presence of quasistationary states with non-Gaussian single particle velocity distributions. For the case of the Hamiltonian mean-field model, we demonstrate that a maximum entropy principle applied to the associated Vlasov equation explains known features of such states for a wide range of initial conditions. We are able to reproduce velocity distribution functions with an analytic expression which is derived from the theory with no adjustable parameters. A normal diffusion of angles is detected, which is consistent with Gaussian tails of velocity distributions. A dynamical effect, two oscillating clusters surrounded by a halo, is also found and theoretically justified.

Maximum entropy principle explains quasi-stationary states in systems with long-range interactions: the example of the Hamiltonian Mean Field model

Maximum entropy principle explains quasi-stationary states in systems with long-range interactions: the example of the Hamiltonian Mean Field model