Metastability in the Hamiltonian Mean Field model and Kuramoto model

We briefly discuss the state of the art on the anomalous dynamics of the Hamiltonian Mean Field model. We stress the important role of the initial conditions for understanding the microscopic nature of the intriguing metastable quasi stationary states observed in the model and the connections to Tsallis statistics and glassy dynamics. We also present new results on the existence of metastable states in the Kuramoto model and discuss the similarities with those found in the HMF model. The existence of metastability seem to be quite a common phenomenon in fully coupled systems, whose origin could be also interpreted as a dynamical mechanism preventing or hindering sinchronization.Comment: 5 pages 3 figures. Talk presented at the international conference NEXT Sigma Phi 05, 13-18 August 2005 Kolymbari, Crete. To be published in the volume of the proceeding

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 microsocopic nature of the QSS leads to different dynamical correlations and therefore to different results for the observed CLT behavior.Comment: 11 pages, 8 figures. Text and figures added, Physica A in pres

Central Limit Behavior in the Kuramoto model at the 'Edge of Chaos'

We study the relationship between chaotic behavior and the Central Limit Theorem (CLT) in the Kuramoto model. We calculate sums of angles at equidistant times along deterministic trajectories of single oscillators and we show that, when chaos is sufficiently strong, the Pdfs of the sums tend to a Gaussian, consistently with the standard CLT. On the other hand, when the system is at the "edge of chaos" (i.e. in a regime with vanishing Lyapunov exponents), robust $q$-Gaussian-like attractors naturally emerge, consistently with recently proved generalizations of the CLT.Comment: 15 pages, 8 figure

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 $\Gamma$-space of microscopic configurations, and comment a paper recently published by Baldovin and Orlandini (2006). 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.Comment: 8 pages, 4 figures. New version accepted for publication in Physica

Glassy dynamics and nonextensive effects in the HMF model: the importance of initial conditions

We review the anomalies of the HMF model and discuss the robusteness of the glassy features vs the initial conditions. Connections to Tsallis statistics are also addressed.Comment: 11 pages, 5 figures. Talk presented at the International conference Complexity and Nonextensivity: New Trends in Statistical Mechanics. - Yukawa Institute for Theoretical Physics - (14-18 March 2005) Kyoto, Japan. New calculations on the glassy behaviour of the HMF model are discussed. Typos correctd. Please note that in the published version, the exponent of the power-law fit observed in fig.2 is erroneously reported as -1/6 instead of the correct value -1.

Dynamics and Thermodynamics of a model with long-range interactions

The dynamics and the thermodynamics of particles/spins interacting via long-range forces display several unusual features with respect to systems with short-range interactions. The Hamiltonian Mean Field (HMF) model, a Hamiltonian system of N classical inertial spins with infinite-range interactions represents a paradigmatic example of this class of systems. The equilibrium properties of the model can be derived analytically in the canonical ensemble: in particular the model shows a second order phase transition from a ferromagnetic to a paramagnetic phase. Strong anomalies are observed in the process of relaxation towards equilibrium for a particular class of out-of-equilibrium initial conditions. In fact the numerical simulations show the presence of quasi-stationary state (QSS), i.e. metastable states which become stable if the thermodynamic limit is taken before the infinite time limit. The QSS differ strongly from Boltzmann-Gibbs equilibrium states: they exhibit negative specific heat, vanishing Lyapunov exponents and weak mixing, non-Gaussian velocity distributions and anomalous diffusion, slowly-decaying correlations and aging. Such a scenario provides strong hints for the possible application of Tsallis generalized thermostatistics. The QSS have been recently interpreted as a spin-glass phase of the model. This link indicates another promising line of research, which is not alternative to the previous one.Comment: 12 pages, 5 figures. Recent review paper for Continuum Mechanics and Thermodynamic