Analytical results on the magnetization of the Hamiltonian Mean-Field model

2013

Abstract. 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 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. In case (iii), we recover the "uniform" core-halo structure previously found by Pakter & Levin [Phys. Rev. Lett. 106, 200603 (2011)]. We also consider unsteady initial conditions with magnetization M0 = 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 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 (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.

Section: Introductionmentioning

confidence: 75%

Caloric curves fitted by polytropic distributions in the HMF model

Campa

2013

“…where the evolution of P N (θ 1 , ..., θ N , t) is governed by Eq. (27). The N -body Smoluchowski equation satisfies an Htheorem for the free energy…”

Section: The N -Body Smoluchowski Equationmentioning

confidence: 99%

“…Substituting this factorization in Eq. (27) and integrating over N − 1 angular variables we find that the evolution of the density ρ(θ, t) is governed by the mean field Smoluchowski equation [45]:…”

Section: The Mean Field Smoluchowski Equationmentioning

confidence: 99%

See 1 more Smart Citation

The Brownian mean field model

2014

Abstract. We discuss the dynamics and thermodynamics of the Brownian Mean Field (BMF) model which is a system of N Brownian particles moving on a circle and interacting via a cosine potential. It can be viewed as the canonical version of the Hamiltonian Mean Field (HMF) model. The BMF model displays a second order phase transition between a homogeneous phase and an inhomogeneous phase below a critical temperature Tc = 1/2. We first complete the description of this model in the mean field approximation valid for N → +∞. In the strong friction limit, the evolution of the density towards the mean field Boltzmann distribution is governed by the mean field Smoluchowski equation. For T < Tc, this equation describes a process of self-organization from a non-magnetized (homogeneous) phase to a magnetized (inhomogeneous) phase. We obtain an analytical expression for the temporal evolution of the magnetization close to Tc. Then, we take fluctuations (finite N effects) into account. The evolution of the density is governed by the stochastic Smoluchowski equation. From this equation, we derive a stochastic equation for the magnetization and study its properties both in the homogenous and inhomogeneous phases. We show that the fluctuations diverge close to the critical point so that the mean field approximation ceases to be valid. Actually, the limits N → +∞ and T → Tc do not commute. The validity of the mean field approximation requires N (T − Tc) → +∞ so that N must be larger and larger as T approaches Tc. We show that the direction of the magnetization changes rapidly close to Tc but its amplitude takes a long time to relax. We also indicate that, for systems with long-range interactions, the lifetime of metastable states scales as e N except close to a critical point. The BMF model shares many analogies with other systems of Brownian particles with long range interactions such as self-gravitating Brownian particles, the Keller-Segel model describing the chemotaxis of bacterial populations, and the Kuramoto model describing the collective synchronization of coupled oscillators.

“…In particular, the magnetic susceptibility diverges at the critical point (ǫ → ǫ + c = 1 + ). For ǫ < ǫ c = 1 and h → 0, the magnetic susceptibility χ(ǫ) is given by equations (98), (32) and (33) with h = 0. These equations can be written…”

Section: The Limit H →mentioning

confidence: 99%

Thermodynamics of the HMF model with a magnetic field

2011

Abstract. We study the thermodynamics of the Hamiltonian Mean Field (HMF) model with an external potential playing the role of a "magnetic field". If we consider only fully stable states, this system does not present any phase transition. However, if we take into account metastable states (for a restricted class of perturbations), we find a very rich phenomenology. In particular, the system displays a region of negative specific heats in the microcanonical ensemble in which the temperature decreases as the energy increases. This leads to ensembles inequivalence and to zeroth order phase transitions similar to the "gravothermal catastrophe" and to the "isothermal collapse" of self-gravitating systems. In the present case, they correspond to the reorganization of the system from an "anti-aligned" phase (magnetization pointing in the direction opposite to the magnetic field) to an "aligned" phase (magnetization pointing in the same direction as the magnetic field). We also find that the magnetic susceptibility can be negative in the microcanonical ensemble so that the magnetization decreases as the magnetic field increases. The magnetic curves can take various shapes depending on the values of energy or temperature. We describe hysteretic cycles involving positive or negative susceptibilities. We also show that this model exhibits gaps in the magnetization at fixed energy, resulting in ergodicity breaking.