Dynamics and Thermodynamics of a model with long-range interactions

Abstract

The dynamics and the thermodynamics of particles/spins interacting via long-range forces display several unusual features with respect to systems with short-range interactions. The Hamiltonian Mean Field (HMF) model, a Hamiltonian system of N classical inertial spins with infinite-range interactions represents a paradigmatic example of this class of systems. The equilibrium properties of the model can be derived analytically in the canonical ensemble: in particular the model shows a second order phase transition from a ferromagnetic to a paramagnetic phase. Strong anomalies are observed in the process of relaxation towards equilibrium for a particular class of out-of-equilibrium initial conditions. In fact the numerical simulations show the presence of quasi-stationary state (QSS), i.e. metastable states which become stable if the thermodynamic limit is taken before the infinite time limit. The QSS differ strongly from Boltzmann-Gibbs equilibrium states: they exhibit negative specific heat, vanishing Lyapunov exponents and weak mixing, non-Gaussian velocity distributions and anomalous diffusion, slowly-decaying correlations and aging. Such a scenario provides strong hints for the possible application of Tsallis generalized thermostatistics. The QSS have been recently interpreted as a spin-glass phase of the model. This link indicates another promising line of research, which is not alternative to the previous one.

12 pages, 5 figures. Recent review paper for Continuum Mechanics and Thermodynamics

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Cited by in corpus (16)

• Nonextensivity and the power-law distributions for the systems with self-gravitating long-range interactions
• A closer look at the indications of q-generalized Central Limit Theorem behavior in quasi-stationary states of the HMF model
• Nonextensive Thermodynamics and Glassy behaviour in Hamiltonian systems
• Central Limit Behavior in the Kuramoto model at the 'Edge of Chaos'
• Color Superconductivity and Tsallis Statistics
• Ergodicity and Central Limit Theorem in Systems with Long-Range Interactions
• Hoffmann-Infeld Black Hole Solutions in Lovelock Gravity
• Metastability in the Hamiltonian Mean Field model and Kuramoto model
• Aging in coherent noise models and natural time
• Novel Dynamics and Thermodynamics in systems with long range interactions
• On the non-Boltzmannian nature of quasi-stationary states in long-range interacting systems
• Generalized Thermostatistics and Bose-Einstein Condensation
• Glassy dynamics and nonextensive effects in the HMF model: the importance of initial conditions
• A Monte Carlo Investigation of the Hamiltonian Mean Field Model
• Finding the ground states of symmetric infinite-dimensional Hamiltonians: explicit constrained optimizations of tensor networks
• A solvable problem in statistical mechanics: the dipole-type Hamiltonian mean field model