Metastable States of the Classical Inertial Infinite-Range-Interaction Heisenberg Ferromagnet: Role of Initial Conditions

A system of $N$ classical Heisenberg-like rotators, characterized by infinite-range ferromagnetic interactions, is studied numerically within the microcanonical ensemble through a molecular-dynamics approach. Such a model, known as the classical inertial infinite-range-interaction Heisenberg ferromagnet, exhibits a second-order phase transition within the standard canonical-ensemble solution. The present numerical analysis, which is restricted to an energy density slightly below criticality, compares the effects of different initial conditions for the orientations of the classical rotators. By monitoring the time evolution of the kinetic temperature, we observe that the system may evolve into a metastable state (whose duration increases linearly with $N$), in both cases of maximal and zero initial magnetization, before attaining a second plateau at longer times. Since the kinetic temperatures associated with the second plateau, in the above-mentioned cases, do not coincide, the system may present a three-plateaux (or even more complicated) structure for finite $N$. To our knowledge, this has never before been observed on similar Hamiltonian models, such as the XY version of the present model. It is also shown that the system is sensitive to the way that one breaks the symmetry of the paramagnetic state: different nonzero values for the initial magnetization may lead to sensibly distinct evolutions for the kinetic temperature, whereas different situations with zero initial magnetization all lead to the same structure.Comment: Communicated at the International Workshop on {\it Trends and Perspectives in Extensive and Non-Extensive Statistical Mechanics}, held in November 19-21, 2003, in Angra dos Reis, Brazil. Submitted to a Physica A issue dedicated to the event, and edited by E.M.F. Curado, H.J. Herrmann and M. Barbosa. 10 pages including 4 figure

Metastable states in a class of long-range Hamiltonian systems

We numerically show that metastable states, similar to the Quasi Stationary States found in the so called Hamiltonian Mean Field Model, are also present in a generalized model in which $N$ classical spins (rotators) interact through ferromagnetic couplings decaying as $r^{-\alpha}$, where $r$ is their distance over a regular lattice. Scaling laws with $N$ are briefly discussed.Comment: Latex 2e, 11 pages, 3 eps figures, contributed paper to the conf. "NEXT 2001", 23-30 May 2001, Cagliari (Italy), submitted to Physica

Glassy dynamics in the HMF model

We discuss the glassy dynamics recently found in the meta-equilibrium quasi stationary states (QSS) of the HMF model. The relevance of the initial conditions and the connection with Tsallis nonextensive thermostatistics is also addressed.Comment: 10 pages, 4 figures, Proceedings of the Int. Conference Next2003 21-28 september 2003, Villasimius (CA) Italy, submitted to Physica

Dynamical anomalies and the role of initial conditions in the HMF model

We discuss the role of the initial conditions for the dynamical anomalies observed in the quasi-stationary states of the Hamiltonian Mean Field (HMF) model.Comment: 8 pages, 5 figures, submitted to Physica A for the proceedings of the conference Frontier Science 2003 Pavia, Italy, 8-12 September 200

Fingerprints of nonextensive thermodynamics in a long-range Hamiltonian system

We study the dynamics of a Hamiltonian system of N classical spins with infinite-range interaction. We present numerical results which confirm the existence of metaequilibrium Quasi Stationary States (QSS), characterized by non-Gaussian velocity distributions, anomalous diffusion, L\'evy walks and dynamical correlation in phase-space. We show that the Thermodynamic Limit (TL) and the Infinite-Time Limit (ITL) do not commute. Moreover, if the TL is taken before the ITL the system does not relax to the Boltzmann-Gibbs equilibrium, but remains in this new equilibrium state where nonextensive thermodynamics seems to apply.Comment: ReVteX, 10 pages, 5 ps figures, talk presented by V. Latora at NEXT 2001. Revised version with improved figs and updated refs. To be published in Physica

Stability of families of probability distributions under reduction of the number of degrees of freedom

We consider two classes of probability distributions for configurations of the ideal gas. They depend only on kinetic energy and they remain of the same form when degrees of freedom are integrated out. The relation with equilibrium distributions of Tsallis' thermostatistics is discussed.Comment: Latex, 8 pages, no figure

Chaotic dynamics and superdiffusion in a Hamiltonian system with many degrees of freedom

We discuss recent results obtained for the Hamiltonian Mean Field model. The model describes a system of N fully-coupled particles in one dimension and shows a second-order phase transition from a clustered phase to a homogeneous one when the energy is increased. Strong chaos is found in correspondence to the critical point on top of a weak chaotic regime which characterizes the motion at low energies. For a small region around the critical point, we find anomalous (enhanced) diffusion and L\'evy walks in a transient temporal regime before the system relaxes to equilibrium.Comment: 7 pages, Latex, 6 figures included, Contributed paper to the Int. Conf. on "Statistical Mechanics and Strongly Correlated System", 2nd Giovanni Paladin Memorial, Rome 27-29 September 1999, submitted to Physica

Microscopic dynamics of a phase transition: equilibrium vs out-of-equilibrium regime

We present for the first time to the nuclear physics community the Hamiltonian Mean Field (HMF) model. The model can be solved analytically in the canonical ensemble and shows a second-order phase transition in the thermodynamic limit. Numerical microcanonical simulations show interesting features in the out-of-equilibrium regime: in particular the model has a negative specific heat. The potential relevance for nuclear multifragmentation is discussed.Comment: 9 pages, Latex, 4 figures included, invited talk to the Int. Conf. CRIS2000 on "Phase transitions in strong interactions: status and perspectives", Acicastello (Italy) May 22-26 2000, submitted to Nucl Phys.

Effective spin-glass Hamiltonian for the anomalous dynamics of the HMF model

We discuss an effective spin-glass Hamiltonian which can be used to study the glassy-like dynamics observed in the metastable states of the Hamiltonian Mean Field (HMF) model. By means of the Replica formalism, we were able to find a self-consistent equation for the glassy order parameter which reproduces, in a restricted energy region below the phase transition, the microcanonical simulations for the polarization order parameter recently introducted in the HMF model.Comment: Revtex, 9 pages, 2 figures. New revised version revised according to the referee report

Dynamics and nonequilibrium states in the Hamiltonian mean-field model: A closer look

We critically revisit the evidence for the existence of quasistationary states in the globally coupled XY (or Hamiltonian mean-field) model. A slow-relaxation regime at long times is clearly revealed by numerical realizations of the model, but no traces of quasistationarity are found during the earlier stages of the evolution. We point out the nonergodic properties of this system in the short-time range, which makes a standard statistical description unsuitable. New aspects of the evolution during the nonergodic regime, and of the energy distribution function in the final approach to equilibrium, are disclosed