Caloric curves fitted by polytropic distributions in the HMF model

Abstract

We perform direct numerical simulations of the 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 E_c, 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 T_{kin}(E) and explain the negative kinetic specific heat region C_{kin}=dE/dT_{kin}<0 observed in the numerical simulations. At low energies, we find that the system takes a "core-halo" structure. The core corresponds to the pure polytrope discussed above but it is now surrounded by a halo of particles. We also consider unsteady initial conditions with magnetization M_0=1 and isotropic waterbag 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 n_0= 3.56. Finally, we consider the collisional evolution of an initially Vlasov stable distribution, and show that the time-evolving distribution function f(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. However, in order to moderate our message, we also report a case where the Lynden-Bell theory provides an excellent prediction of an inhomogeneous QSS.

References in corpus (16)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations
• A maximum entropy principle explains quasi-stationary states in systems with long-range interactions: the example of the Hamiltonian Mean Field model
• Out-of-equilibrium tricritical point in a system with long-range interactions
• Abundance of regular orbits and out-of-equilibrium phase transitions in the thermodynamic limit for long-range systems
• Core-halo distribution in the Hamiltonian Mean-Field Model
• Lynden-Bell and Tsallis distributions for the HMF model
• Dynamical phase transitions in long-range Hamiltonian systems and Tsallis distributions with a time-dependent index
• Long time behavior of quasi-stationary states of the Hamiltonian Mean-Field model
• Statistical Mechanics of Unbound Two Dimensional Self-Gravitating Systems
• Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems
• Dynamical stability of collisionless stellar systems and barotropic stars: the nonlinear Antonov first law
• Hamiltonian and Brownian systems with long-range interactions: IV. General kinetic equations from the quasilinear theory
• Dynamics and thermodynamics of systems with long-range interactions: interpretation of the different functionals
• Action diffusion and lifetimes of quasistationary states in the Hamiltonian Mean Field model
• Analytical results on the magnetization of the Hamiltonian Mean Field model

Cited by in corpus (10)

• Nonequilibrium Statistical Mechanics of Systems with Long-Range Interactions: Ubiquity of Core-Halo Distributions
• Influence of the interaction range on the thermostatistics of a classical many-body system
• The Brownian Mean Field model
• Kinetic theory of collisionless relaxation for systems with long-range interactions
• Thermodynamics is more powerful than the role to it reserved by Boltzmann-Gibbs statistical mechanics
• Generalization of the possible algebraic basis of $q$-triplets
• Violent relaxation in the Hamiltonian mean field model: II. Non-equilibrium phase diagrams
• The quasilinear theory in the approach of long-range systems to quasi-stationary states
• On formal groups and geometric quantization
• Hard-core collisional dynamics in the hamiltonian mean-field model