  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.
