  The Hamiltonian Mean Field model describes a system of N fully-coupled
particles showing a second-order phase transition as a function of the energy.
The dynamics of the model presents interesting features in a small energy
region below the critical point. In particular, when the particles are prepared
in a ``water bag'' initial state, the relaxation to equilibrium is very slow.
In the transient time the system lives in a dynamical quasi-stationary state
and exhibits anomalous (enhanced) diffusion and L\'evy walks. In this paper we
study temperature and velocity distribution of the quasi-stationary state and
we show that the lifetime of such a state increases with N. In particular when
the $N\to \infty$ limit is taken before the $t \to \infty$ limit, the results
obtained are different from the expected canonical predictions. This scenario
seems to confirm a recent conjecture proposed by C.Tsallis.
