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We perform direct numerical simulations of the HMF model starting fromnon-magnetized initial conditions with a velocity distribution that is (i)gaussian, (ii) semi-elliptical, and (iii) waterbag. Below a critical energyE_c, depending on the initial condition, this distribution is Vlasovdynamically unstable. The system undergoes a process of violent relaxation andquickly reaches a quasi-stationary state (QSS). We find that the distributionfunction of this QSS can be conveniently fitted by a polytrope with index (i)n=2, (ii) n=1, and (iii) n=1/2. Using the values of these indices, we are ableto determine the physical caloric curve T_{kin}(E) and explain the negativekinetic specific heat region C_{kin}=dE/dT_{kin}<0 observed in the numericalsimulations. At low energies, we find that the system takes a "core-halo"structure. The core corresponds to the pure polytrope discussed above but it isnow surrounded by a halo of particles. We also consider unsteady initialconditions with magnetization M_0=1 and isotropic waterbag distribution andreport the complex dynamics of the system creating phase space holes and densefilaments. We show that the kinetic caloric curve is approximately constant,corresponding to a polytrope with index n_0= 3.56. Finally, we consider thecollisional evolution of an initially Vlasov stable distribution, and show thatthe time-evolving distribution function f(v,t) can be fitted by a sequence ofpolytropic distributions with a time-dependent index n(t) both in thenon-magnetized and magnetized regimes. These numerical results show thatpolytropic distributions (also called Tsallis distributions) provide in manycases a good fit of the QSSs. However, in order to moderate our message, wealso report a case where the Lynden-Bell theory provides an excellentprediction of an inhomogeneous QSS.

Caloric curves fitted by polytropic distributions in the HMF model