Dynamics and Thermodynamics of a model with long-range interactions

TL;DR

This paper explores the unique dynamical and thermodynamic behaviors of the Hamiltonian Mean Field model with long-range interactions, highlighting quasi-stationary states with properties distinct from classical equilibrium.

Contribution

It provides an analytical and numerical analysis of the HMF model, revealing the existence and characteristics of quasi-stationary states and their implications for non-equilibrium statistical mechanics.

Findings

Identification of quasi-stationary states with negative specific heat

Observation of non-Gaussian velocity distributions and anomalous diffusion

Linking QSS to spin-glass phases and Tsallis statistics

Abstract

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.