International audienceAbstract A two-dimensional class of mean-field models that may serve as a minimal model to study the properties of long-range systems in two space dimensions is considered. The statistical equilibrium mechanics is derived in the microcanonical ensemble using Monte Carlo simulations for different combinations of the coupling constants in the potential leading to fully repulsive, fully attractive and mixed attractive?repulsive potential along the Cartesian axis and diagonals. Then, having in mind potential realizations of long-range systems using cold atoms, the linear theory of this two-dimensional mean-field Hamiltonian models is derived in the low temperature limit

In this study we present a Microcanonical Monte Carlo investigation of one dimensional selfgravitating toy models. We study the effect of hard-core potentials and compare to those results obtained with softening parameters and also the effect of the geometry of the models. In order to study the effect of the geometry and the borders in the system we introduce a model with the symmetry of motion in a line instead of a circle, which we denominate as 1/r model. The hard-core particle potential introduces the effect of the size of particles and, consequently, the effect of the density of the system that is redefined in terms of the packing fraction of the system. The latter plays a role similar to the softening parameter in the softened particles' case. In the case of low packing fractions both models with hard-core particles show a behavior that keeps the intrinsic properties of the three dimensional gravitational systems such as negative heat capacity. For higher values of the packing fraction the ring the system behaves as the Hamiltonian Mean Field model and while for the 1/r it is similar to the one-dimensional systems. * marco@fis.unb.br 1 arXiv:1512.01391v1 [cond-mat.stat-mech] 4 Dec 2015