  We study the d-HMF model proposed by Atenas and Curilef, a mean field model
with long-range interactions inspired by the dipole-dipole interaction. Among
the challenges of this thesis is: the resolution of the d-HMF model in the
canonical and microcanonical ensembles and the analytical and numerical
description of the distribution function of the system. This model has been
studied both in equilibrium and out of equilibrium. In the equilibrium,
analytical solutions have been found for internal energy per particle and
temperature using standard procedures through statistical mechanics such as the
calculation of the partition function in the canonical ensemble and the
calculation of the number of microstates accessible in the microcanonical
ensemble. The results indicate that there is an equivalence of ensembles.
Additionally, we found the Boltzmann-Gibbs (BG) equilibrium distribution, which
coincides with the analytical results of the canonical and microcanonical
ensembles and molecular dynamics simulations. Then we focused on describing the
Quasi-Stationary-States (QSS) present in this system, by means of molecular
dynamics simulations. Two types of QSS states are found out of equilibrium. For
its description, a combination of two techniques is used; the methods of
molecular dynamics and stationary solutions of the Vlasov equation associated
with the equations of motion of the system. From molecular we describe the
marginal distributions in moments and orientations, which were describe
anaytically by means of stationary solutions of Vlasov equation. The stationary
solutions tested in this thesis correspond to distribution functions of the
q-exponential type. In particular, it is found that one of these solutions
describes precisely the marginal distributions of the moments and the
orientations of one of the QSS states found with the molecular dynamics.
