  We study the dynamical and statistical behavior of the Hamiltonian Mean Field
(HMF) model in order to investigate the relation between microscopic chaos and
phase transitions. HMF is a simple toy model of $N$ fully-coupled rotators
which shows a second order phase transition. The canonical thermodynamical
solution is briefly recalled and its predictions are tested numerically at
finite $N$. The Vlasov stationary solution is shown to give the same
consistency equation of the canonical solution and its predictions for rotator
angle and momenta distribution functions agree very well with numerical
simulations. A link is established between the behavior of the maximal Lyapunov
exponent and that of thermodynamical fluctuations, expressed by kinetic energy
fluctuations or specific heat. The extensivity of chaos in the $N \to \infty$
limit is tested through the scaling properties of Lyapunov spectra and of the
Kolmogorov-Sinai entropy. Chaotic dynamics provides the mixing property in
phase space necessary for obtaining equilibration; however, the relaxation time
to equilibrium grows with $N$, at least near the critical point. Our results
constitute an interesting bridge between Hamiltonian chaos in many degrees of
freedom systems and equilibrium thermodynamics.
