Abundance of regular orbits and out-of-equilibrium phase transitions in the thermodynamic limit for long-range systems (original) (raw)
Related papers
2006
We introduce a Hamiltonian dynamics for the description of long-range interacting systems in contact with a thermal bath (i.e., in the canonical ensemble). The dynamics confirms statistical mechanics equilibrium predictions for the Hamiltonian Mean Field model and the equilibrium ensemble equivalence. We find that long-lasting quasi-stationary states persist in presence of the interaction with the environment. Our results indicate that quasi-stationary states are indeed reproducible in real physical experiments. PACS numbers: 05.20.Gg, 05.10.-a, 05.70.Ln
Out–Of–Equilibrium Phase Transitions in Mean-Field Hamiltonian Dynamics
Chaos, Complexity and Transport - Theory and Applications - Proceedings of the CCT '07, 2008
Systems with long-range interactions display a short-time relaxation towards Quasi-Stationary States (QSSs), whose lifetime increases with system size. With reference to the Hamiltonian Mean Field (HMF) model, we here review Lynden-Bell's theory of "violent relaxation". The latter results in a maximum entropy scheme for a water-bag initial profile which predicts the presence of out-of-equilibrium phase transitions separating homogeneous (zero magnetization) from inhomogeneous (non-zero magnetization) QSSs. Two different parametric representations of the initial condition are analyzed and the features of the phase diagram are discussed. In both representations we find a second order and a first order line of phase transitions that merge at a tricritical point. Particular attention is payed to the condition of existence and stability of the homogenous phase.
Physical Review E, 2008
We study dynamical phase transitions in systems with long-range interactions, using the Hamiltonian Mean Field (HMF) model as a simple example. These systems generically undergo a violent relaxation to a quasi-stationary state (QSS) before relaxing towards Boltzmann equilibrium. In the collisional regime, the out-of-equilibrium one-particle distribution function (DF) is a quasi-stationary solution of the Vlasov equation, slowly evolving in time due to finite NNN effects. For subcritical energies 7/12<U<3/47/12<U<3/47/12<U<3/4, we exhibit cases where the DF is well-fitted by a Tsallis qqq-distribution with an index q(t)q(t)q(t) slowly decreasing in time from qsimeq3q\simeq 3qsimeq3 (semi-ellipse) to q=1q=1q=1 (Boltzmann). When the index q(t)q(t)q(t) reaches a critical value qcrit(U)q_{crit}(U)qcrit(U), the non-magnetized (homogeneous) phase becomes Vlasov unstable and a dynamical phase transition is triggered, leading to a magnetized (inhomogeneous) state. While Tsallis distributions play an important role in our study, we explain this dynamical phase transition by using only conventional statistical mechanics. For supercritical energies, we report for the first time the existence of a magnetized QSS with a very long lifetime.
Statistical mechanics of systems with long range interactions
2006
Many physical systems are governed by long range interactions, the main example being self-gravitating stars. Long range interaction implies a lack of additivity for the energy. As a consequence, the usual thermodynamic limit is not appropriate. However, by contrast with many claims, the statistical mechanics of such systems is a well understood subject. In this proceeding, we explain briefly the classical way to equilibrium and non equilibrium statistical mechanics, starting from first principles and emphasizing some new results. At equilibrium, we explain how the Boltzmann-Gibbs entropy can be proved to be the appropriate one, using large deviations tools. We explain the thermodynamics consequences of the lack of additivity, like the generic occurence of statistical ensemble inequivalence and negative specific heat. This well known behavior is not the only way inequivalence may occur, as emphasized by a recent new classification of phase transitions and ensemble inequivalence in systems with long range interaction. We note a number of generic situations that have not yet been observed in any physical systems. Out of equilibrium, we show that algebraic temporal correlations or anomalous diffusion may occur in these systems, and can be explained using usual statistical mechanics and kinetic theory.
Physical Review Letters, 2006
We introduce a Hamiltonian dynamics for the description of long-range interacting systems in contact with a thermal bath (i.e., in the canonical ensemble). The dynamics confirms statistical mechanics equilibrium predictions for the Hamiltonian Mean Field model and the equilibrium ensemble equivalence. We find that long-lasting quasi-stationary states persist in presence of the interaction with the environment. Our results indicate that quasi-stationary states are indeed reproducible in real physical experiments. PACS numbers: 05.20.Gg, 05.10.-a, 05.70.Ln
Action diffusion and lifetimes of quasistationary states in the Hamiltonian mean-field model
Physical Review E, 2013
Out-of-equilibrium quasistationary states (QSSs) are one of the signatures of a broken ergodicity in long-range interacting systems. For the widely studied Hamiltonian Mean-Field model, the lifetime of some QSSs has been shown to diverge with the number N of degrees of freedom with a puzzling N 1.7 scaling law, contradicting the otherwise widespread N scaling law. It is shown here that this peculiar scaling arises from the locality properties of the dynamics captured through the computation of the diffusion coefficient in terms of the action variable. Estimating the QSS lifetime by a mean first passage time successfully explains the non-trivial scaling at stake here. PACS numbers: 05.20.-y,05.70.Ln,05.20.Dd