Caloric curves fitted by polytropic distributions in the HMF model

We perform direct numerical simulations of the Hamiltonian mean field (HMF) model starting from non-magnetized initial conditions with a velocity distribution that is (i) Gaussian; (ii) semi-elliptical, and (iii) waterbag. Below a critical energy Ec, depending on the initial condition, this distribution is Vlasov dynamically unstable. The system undergoes a process of violent relaxation and quickly reaches a quasi-stationary state (QSS). We find that the distribution function 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 able to determine the physical caloric curve Tkin(E) and explain the negative kinetic specific heat region Ckin = dE/dTkin < 0 observed in the numerical simulations. At low energies, we find that the system has a “core-halo” structure. The core corresponds to the pure polytrope discussed above but it is now surrounded by a halo of particles. In case (iii), we recover the “uniform” core-halo structure previously found by Pakter and Levin [Phys. Rev. Lett. 106, 200603 (2011)]. We also consider unsteady initial conditions with magnetization M0 = 1 and isotropic waterbag velocity distribution and report the complex dynamics of the system creating phase space holes and dense filaments. We show that the kinetic caloric curve is approximately constant, corresponding to a polytrope with index n0 ≃ 3.56 (we also mention the presence of an unexpected hump). Finally, we consider the collisional evolution of an initially Vlasov stable distribution, and show that the time-evolving distribution function f(θ,v,t) can be fitted by a sequence of polytropic distributions with a time-dependent index n(t) both in the non-magnetized and magnetized regimes. These numerical results show that polytropic distributions (also called Tsallis distributions) provide in many cases a good fit of the QSSs. They may even be the rule rather than the exception. However, in order to moderate our message, we also report a case where the Lynden-Bell theory (which assumes ergodicity or efficient mixing) provides an excellent prediction of an inhomogeneous QSS. We therefore conclude that both Lynden-Bell and Tsallis distributions may be useful to describe QSSs depending on the efficiency of mixing