Chaos and statistical mechanics in the Hamiltonian mean field model

We study the dynamical and statistical behavior of the Hamiltonian mean fie ld (HMF) model in order to investigate the relation between microscopic cha os and phase transitions. HMF is a simple toy model of N fully coupled rota tors which shows a second-order phase transition. The solution in the canon ical ensemble is briefly recalled and its predictions are tested numericall y at finite N. The Vlasov stationary solution is shown to give the same con sistency equation of the canonical solution and its predictions for rotator angle and momenta distribution functions agree very well with numerical si mulations, A link is established between the behavior of the maximal Lyapun ov exponent and that of thermodynamical fluctuations, expressed by kinetic energy fluctuations or specific heat. The extensivity of chaos in the N-->i nfinity limit is tested through the scaling properties of Lyapunov spectra and of the Kolmogorov-Sinai entropy, Chaotic dynamics provides the mixing p roperty in phase space necessary for obtaining equilibration; however, the relaxation time to equilibrium grows with N, at least near the critical poi nt. Our results constitute an interesting bridge between Hamiltonian chaos in many degrees of freedom systems and equilibrium thermodynamics. (C) 1999 Elsevier Science B.V All rights reserved.