  We study the N-dependence of the thermodynamical variables and the dynamical
behavior of the well-known Hamiltonian Mean Field model. Microcanonical
analysis revealed a thermodynamic limit which defers from the a priory
traditional assumption of the N-dependence of the coupling constant g as 1/N:
to tend N?infinity keeping constant E/N^3 and g/N, prescription which
guarantees the extensivity of the Boltzmann entropy. The analysis of dynamics
leads to approximate the time evolution of the magnetization density m by means
of a Langevin equation with multiplicative noise. This equation leads to a
Fokker-Planck's equation which is N-independent when the time variable is
scaled by the N-dependent time constant T_{mac}=sqrt(IN/g), which represents
the characteristic time scale for the dynamical evolution of the macroscopic
observables derived from the magnetization density. This results explains the
origin of the slow relaxation regimen observed in microcanonical numerical
computations of dynamics of this model system. Connection with the system
self-similarity is suggested.
