Glassy phase in the Hamiltonian mean-field model

Nonextensive Footprints in Dissipative and Conservative Dynamical Systems

Despite its centennial successes in describing physical systems at thermal equilibrium, Boltzmann–Gibbs (BG) statistical mechanics have exhibited, in the last several decades, several flaws in addressing out-of-equilibrium dynamics of many nonlinear complex systems. In such circumstances, it has been shown that an appropriate generalization of the BG theory, known as nonextensive statistical mechanics and based on nonadditive entropies, is able to satisfactorily handle wide classes of anomalous emerging features and violations of standard equilibrium prescriptions, such as ergodicity, mixing, breakdown of the symmetry of homogeneous occupancy of phase space, and related features. In the present study, we review various important results of nonextensive statistical mechanics for dissipative and conservative dynamical systems. In particular, we discuss applications to both discrete-time systems with a few degrees of freedom and continuous-time ones with many degrees of freedom, as well as to asymptotically scale-free networks and systems with diverse dimensionalities and ranges of interactions, of either classical or quantum nature.

Finite-size scaling of quasi-stationary-state temperature

We numerically study, from first principles, the temperature T_{QSS} and duration t_{QSS} of the longstanding initial quasi-stationary state of the isolated d-dimensional classical inertial α-XY ferromagnet with two-body interactions decaying as 1/r_{ij}^{α} (α≥0). It is shown that this temperature T_{QSS} (defined proportional to the kinetic energy per particle) depends, for the long-range regime 0≤α/d≤1, on (α,d,U,N) with numerically negligible changes for dimensions d=1,2,3, with U the energy per particle and N the number of particles. We verify the finite-size scaling T_{QSS}-T_{∞}∝1/N^{β} where T_{∞}≡2U-1 for U≲U_{c}, and β appears to depend sensibly only on α/d. Our numerical results indicate that neither the scaling with N of T_{QSS} nor that of t_{QSS} depend on U.

Criticality in the duration of quasistationary state

The duration of the quasistationary states (QSSs) emerging in the d-dimensional classical inertial α-XY model, i.e., N planar rotators whose interactions decay with the distance r_{ij} as 1/r_{ij}^{α} (α≥0), is studied through first-principles molecular dynamics. These QSSs appear along the whole long-range interaction regime (0≤α/d≤1), for an average energy per rotator U<U_{c} (U_{c}=3/4), and they do not exist for U>U_{c}. They are characterized by a kinetic temperature T_{QSS}, before a crossover to a second plateau occurring at the Boltzmann-Gibbs temperature T_{BG}>T_{QSS}. We investigate here the behavior of their duration t_{QSS} when U approaches U_{c} from below, for large values of N. Contrary to the usual belief that the QSS merely disappears as U→U_{c}, we show that its duration goes through a critical phenomenon, namely t_{QSS}∝(U_{c}-U)^{-ξ}. Universality is found for the critical exponent ξ≃5/3 throughout the whole long-range interaction regime.

Quasi-stationary-state duration in the classical d-dimensional long-range inertial XY ferromagnet

A classical α-XY inertial model, consisting of N two-component rotators and characterized by interactions decaying with the distance r_{ij} as 1/r_{ij}^{α} (α≥0) is studied through first-principle molecular-dynamics simulations on d-dimensional lattices of linear size L (N≡L^{d} and d=1,2,3). The limits α=0 and α→∞ correspond to infinite-range and nearest-neighbor interactions, respectively, whereas the ratio α/d>1 (0≤α/d≤1) is associated with short-range (long-range) interactions. By analyzing the time evolution of the kinetic temperature T(t) in the long-range-interaction regime, one finds a quasi-stationary state (QSS) characterized by a temperature T_{QSS}; for fixed N and after a sufficiently long time, a crossover to a second plateau occurs, corresponding to the Boltzmann-Gibbs temperature T_{BG} (as predicted within the BG theory), with T_{BG}>T_{QSS}. It is shown that the QSS duration (t_{QSS}) depends on N, α, and d, although the dependence on α appears only through the ratio α/d; in fact, t_{QSS} decreases with α/d and increases with both N and d. Considering a fixed energy value, a scaling for t_{QSS} is proposed, namely, t_{QSS}∝N^{A(α/d)}e^{-B(N)(α/d)^{2}}, analogous to a recent analysis carried out for the classical α-Heisenberg inertial model. It is shown that the exponent A(α/d) and the coefficient B(N) present universal behavior (within error bars), comparing the XY and Heisenberg cases. The present results should be useful for other long-range systems, very common in nature, like those characterized by gravitational and Coulomb forces.

d-Dimensional Classical Heisenberg Model with Arbitrarily-Ranged Interactions: Lyapunov Exponents and Distributions of Momenta and Energies

We numerically study the first-principle dynamics and thermostatistics of a d-dimensional classical inertial Heisenberg ferromagnetic model (d=1,2,3) with interactions decaying with the distance rij as 1/rijα (α≥0), where the limit α=0 (α→∞) corresponds to infinite-range (nearest-neighbour) interactions, and the ratio α/d>1 (0≤α/d≤1) characterizes the short-ranged (long-ranged) regime. By means of first-principle molecular dynamics we study: (i) The scaling with the system size N of the maximum Lyapunov exponent λ in the form λ∼N−κ, where κ(α/d) depends only on the ratio α/d; (ii) The time-averaged single-particle angular momenta probability distributions for a typical case in the long-range regime 0≤α/d≤1 (which turns out to be well fitted by q-Gaussians), and (iii) The time-averaged single-particle energies probability distributions for a typical case in the long-range regime 0≤α/d≤1 (which turns out to be well fitted by q-exponentials). Through the Lyapunov exponents we observe an intriguing, and possibly size-dependent, persistence of the non-Boltzmannian behavior even in the α/d>1 regime. The universality that we observe for the probability distributions with regard to the ratio α/d makes this model similar to the α-XY and α-Fermi-Pasta-Ulam Hamiltonian models as well as to asymptotically scale-invariant growing networks.

Synchronous harmony in an ensemble of Hamiltonian mean-field oscillators and inertial Kuramoto oscillators

We study a dynamic interplay between Hamiltonian mean-field oscillators and inertial Kuramoto oscillators. We present several sufficient frameworks leading to asymptotic complete synchronization for the mixed ensemble. For a two-oscillator system with the same natural frequencies, we prove that the mixed ensemble exhibits asymptotic complete synchronization for any initial data, whereas we also show that the two-oscillator system tends to asymptotic complete synchronization under an a priori assumption on the uniform boundedness on the phase diameter. For the many-body system, we show that asymptotic complete frequency synchronization occurs for Kuramoto oscillators with inertia if the oscillators have the same natural frequencies. Moreover, we show that overall phase concentration can be controlled by increasing the coupling strengths. We also provide several numerical experiments and compare them with analytical results.

Dynamics, thermodynamics, and phase transitions of classical spins interacting through the magnetic field

We introduce and study a one dimensional model of classical planar spins interacting self-consistently through magnetic field. The spins and the magnetic field evolve in time according to the Hamiltonian dynamics which mimics that of a free electron laser. We show that by rescaling the energy due to magnetic field inhomogeneity, in equilibrium, this system can be mapped onto a model very similar to the paradigmatic globally coupled Hamiltonian mean-field (HMF) model. The system exhibits a continuous equilibrium phase transition from paramagnetic to ferromagnetic phase, however unlike HMF, we do not see any magnetized quasistationary states.

The inherent dynamics of isotropic- and nematic-phase liquid crystals

The geodesic (shortest) pathways through the potential energy landscape of a liquid can be thought of as defining what its dynamics would be if thermal noise were removed, revealing what we have called the "inherent dynamics" of the liquid. We show how these inherent paths can be located for a model liquid crystal former, showing, in the process, how the molecular mechanisms of translation and reorientation compare in the isotropic and nematic phases of these systems. These mechanisms turn out to favor the preservation of local orientational order even under macroscopically isotropic conditions (a finding consistent with the experimental observation of pseudonematic domains in these cases), but disfavor the maintenance of macroscopic orientational order, even in the nematic phase. While the most efficient nematic pathways that maintain nematic order are indeed shorter than those that do not, it is apparently difficult for the system to locate these paths, suggesting that molecular motion in liquid-crystal formers is dynamically frustrated, and reinforcing the sense that there are strong analogies between liquid crystals and supercooled liquids.

Statistical theory of quasistationary states beyond the single water-bag case study

An analytical solution for the out-of-equilibrium quasistationary states of the paradigmatic Hamiltonian mean field (HMF) model can be obtained from a maximum entropy principle. The theory has been so far tested with reference to a specific class of initial condition, the so called (single-level) water-bag type. In this paper a step forward is taken by considering an arbitrary number of overlapping water bags. The theory is benchmarked to direct microcanonical simulations performed for the case of a two-level water-bag. The comparison is shown to return an excellent agreement.

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.

Diffusive anomalies in a long-range Hamiltonian system

We scrutinize the anomalies in diffusion observed in an extended long-range system of classical rotors, the HMF model. Under suitable preparation, the system falls into long-lived quasi-stationary states for which superdiffusion of rotor phases has been reported. In the present work, we investigate diffusive motion by monitoring the evolution of full distributions of unfolded phases. After a transient, numerical histograms can be fitted by the q -Gaussian form P(x) proportional to {1+(q-1)[x/beta]2}(1/(1-q)) , with parameter q increasing with time before reaching a steady value q approximately 32 (squared Lorentzian). From the analysis of the second moment of numerical distributions, we also discuss the relaxation to equilibrium and show that diffusive motion in quasistationary trajectories depends strongly on system size.

Quasistationary trajectories of the mean-field XY Hamiltonian model: a topological perspective

We employ a topological approach to investigate the nature of quasistationary states of the mean-field XY Hamiltonian model. We focus on the quasistationary states reached when the system is initially prepared in a fully magnetized configuration. By means of numerical simulations and analytical considerations, we show that, along the quasistationary trajectories, the system evolves in a manifold of critical points of the potential energy function. Although these critical points are maxima, the large number of directions with marginal stability may be responsible for the slow relaxation dynamics and the trapping of the system in such trajectories.