Chaos and Statistical Mechanics in the Hamiltonian Mean Field Model

Abstract

We study the dynamical and statistical behavior of the Hamiltonian Mean Field (HMF) model in order to investigate the relation between microscopic chaos and phase transitions. HMF is a simple toy model of $N$ fully-coupled rotators which shows a second order phase transition. The canonical thermodynamical solution is briefly recalled and its predictions are tested numerically at finite $N$. The Vlasov stationary solution is shown to give the same consistency equation of the canonical solution and its predictions for rotator angle and momenta distribution functions agree very well with numerical simulations. A link is established between the behavior of the maximal Lyapunov exponent and that of thermodynamical fluctuations, expressed by kinetic energy fluctuations or specific heat. The extensivity of chaos in the $N\to \infty$ limit is tested through the scaling properties of Lyapunov spectra and of the Kolmogorov-Sinai entropy. Chaotic dynamics provides the mixing property in phase space necessary for obtaining equilibration; however, the relaxation time to equilibrium grows with $N$, at least near the critical point. Our results constitute an interesting bridge between Hamiltonian chaos in many degrees of freedom systems and equilibrium thermodynamics.

19 pages, 10 postscript figures included, Latex, Elsevier macros included. Invited talk at the conference ``Classical Chaos and its quantum manifestations'' in honour of Boris Chirikov, Sputnik conference of STATPHYS 20 - Toulouse, France - July 16-18, 1998. Revised version (added refs, changed part of the text and some figures) accepted for publication in Physica D

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Cited by in corpus (46)

• Non-Gaussian equilibrium in a long-range Hamiltonian system
• Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model
• Geometric approach to Hamiltonian dynamics and statistical mechanics
• What should a statistical mechanics satisfy to reflect nature?
• The rate of entropy increase at the edge of chaos
• On the diffusive anomalies in a long-range Hamiltonian system
• Classical Infinite-Range-Interaction Heisenberg Ferromagnetic Model: Metastability and Sensitivity to Initial Conditions
• Long-range interacting classical systems: universality in mixing weakening
• Weak and strong chaos in FPU models and beyond
• Metastable states in a class of long-range Hamiltonian systems
• Metastability, negative specific heat and weak mixing in classical long-range many-rotator system
• Thermodynamics of Self-Gravitating Systems with Softened Potentials
• Extensive and Sub-Extensive Chaos in Globally-Coupled Dynamical Systems
• Glassy phase in the Hamiltonian Mean Field model
• Chaotic dynamics and superdiffusion in a Hamiltonian system with many degrees of freedom
• Lyapunov exponents as a dynamical indicator of a phase transition
• Critical fluctuations and slowing down of chaos
• Chaos in the Hamiltonian mean field model
• Chaos suppression in the large size limit for long-range systems
• Scaling with system size of the Lyapunov exponents for the Hamiltonian Mean Field model
• Dynamics and nonequilibrium states in the Hamiltonian mean-field model: A closer look
• Hamiltonian dynamics and geometry of phase transitions in classical XY models
• Phase Transitions and Chaos in Long-Range Models of Coupled Oscillators
• Linear response theory for long-range interacting systems in quasistationary states
• Controlling the Range of Interactions in the Classical Inertial Ferromagnetic Heisenberg Model: Analysis of Metastable States
• Thermodynamics is more powerful than the role to it reserved by Boltzmann-Gibbs statistical mechanics
• Theoretical estimates for the largest Lyapunov exponent of many-particle systems
• Clustering in a model with repulsive long-range interactions
• Statistical-mechanical formulation of Lyapunov exponents
• Quasi-stationary trajectories of the HMF model: a topological perspective
• Thermodynamics of a one-dimensional self-gravitating gas with periodic boundary conditions
• Inhomogeneous Quasi-stationary States in a Mean-field Model with Repulsive Cosine Interactions
• Effective spin-glass Hamiltonian for the anomalous dynamics of the HMF model
• Coarsening of two dimensional XY model with Hamiltonian dynamics: Logarithmically divergent vortex mobility
• Onset of Synchronization in the Disordered Hamiltonian Mean Field Model
• Lyapunov exponent of many-particle systems: testing the stochastic approach
• Remarks on Tsallis' Satistics
• Nontrapping arrest of Langmuir wave damping near the threshold amplitude
• Importance-sampling computation of statistical properties of coupled oscillators
• Lyapunov stability of Vlasov Equilibria using Fourier-Hermite modes
• Quantum fluctuations inhibit symmetry breaking in the HMF model
• Perfectly Translating Lattices on a Cylinder
• Renewal stochastic processes with correlated events. Phase transitions along time evolution
• Dynamics of Aligning Active Matter: Mapping to a Schrödinger Equation and Exact Diagonalization
• Dynamics and Thermodynamics of the mean field d-HMF model out-of-equilibrium
• Universal behavior in the static and dynamic properties of the $\alpha$-XY model