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.

Compromise and Synchronization in Opinion Dynamics

We discuss two models of opinion dynamics. First wepresent a brief review of the Hegselmann and Krause (HK) compromise model in two dimensions, showing that it is possible to simulate the dynamics in the limit of an infinite number of agents by solving numerically a rate equation for a continuum distribution of opinions. Then, we discuss the Opinion Changing Rate (OCR) model, which allows to study under which conditions a group of agents with a different natural tendency (rate) to change opinion can find the agreement. In the context of the this model, consensus is viewed as a synchronization process.Comment: Talk presented at the international conference Next05 Sigma Phi, 13-18 august 2005, Kolymbari, Crete. EPJ B (2006) in press. Typos corrected, refs adde

Lyapunov instability and finite size effects in a system with long-range forces

We study the largest Lyapunov exponent $\lambda$ and the finite size effects of a system of N fully-coupled classical particles, which shows a second order phase transition. Slightly below the critical energy density $U_c$, $\lambda$ shows a peak which persists for very large N-values (N=20000). We show, both numerically and analytically, that chaoticity is strongly related to kinetic energy fluctuations. In the limit of small energy, $\lambda$ goes to zero with a N-independent power law: $\lambda \sim \sqrt{U}$. In the continuum limit the system is integrable in the whole high temperature phase. More precisely, the behavior $\lambda \sim N^{-1/3}$ is found numerically for $U > U_c$ and justified on the basis of a random matrix approximation.Comment: 5 pages, Revtex, 3 figures included. Both text and figures have been changed. New Version accepted for publication in Physical Review Letter

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