Large deviation techniques applied to systems with long-range interactions

Abstract

We discuss a method to solve models with long-range interactions in the microcanonical and canonical ensemble. The method closely follows the one introduced by Ellis, Physica D 133, 106 (1999), which uses large deviation techniques. We show how it can be adapted to obtain the solution of a large class of simple models, which can show ensemble inequivalence. The model Hamiltonian can have both discrete (Ising, Potts) and continuous (HMF, Free Electron Laser) state variables. This latter extension gives access to the comparison with dynamics and to the study of non-equilibri um effects. We treat both infinite range and slowly decreasing interactions and, in particular, we present the solution of the alpha-Ising model in one-dimension with $0\le \alpha <1$.

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Cited by in corpus (69)

• The large deviation approach to statistical mechanics
• Nonequilibrium Statistical Mechanics of Systems with Long-Range Interactions: Ubiquity of Core-Halo Distributions
• Continuous Limit of Discrete Systems with Long-Range Interaction
• Coupled oscillators with power-law interaction and their fractional dynamics analogues
• The Vlasov equation and the Hamiltonian Mean-Field model
• Long time behavior of quasi-stationary states of the Hamiltonian Mean-Field model
• Phase space gaps and ergodicity breaking in systems with long range interactions
• Slow relaxation in long-range interacting systems with stochastic dynamics
• Map of Discrete System into Continuous
• Conservation Laws and Hamilton's Equations for Systems with Long-Range Interaction and Memory
• Partial equivalence of statistical ensembles and kinetic energy
• Hamiltonian dynamics reveals the existence of quasi-stationary states for long-range systems in contact with a reservoir
• Singularities in large deviation functions
• Dynamics and thermodynamics of rotators interacting with both long and short range couplings
• Solvable phase diagrams and ensemble inequivalence for two-dimensional and geophysical turbulent flows
• The mean-field phi4-model: entropy, analyticity, and configuration space topology
• Negative magnetic susceptibility and nonequivalent ensembles for the mean-field ${\varphi }^4$ spin model
• Application of large deviation theory to the mean-field phi^4-model
• Ensemble equivalence for general many-body systems
• Relaxation times of unstable states in systems with long range interactions
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• Phase transitions in systems with non-additive long-range interactions
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• Anisotropic Long-Range Spin Systems
• The Non--Ergodicity Threshold: Time Scale for Magnetic Reversal
• Microcanonical solution of the mean-field ${\varphi }^4$ model: comparison with time averages at finite size
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• On the mean-field spherical model
• Inhomogeneous Tsallis distributions in the HMF model
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• Extended gaussian ensemble solution and tricritical points of a system with long-range interactions
• Large Lattice Fractional Fokker-Planck Equation
• Algebraic Correlation Function and Anomalous Diffusion in the HMF model
• Models with short and long-range interactions: phase diagram and reentrant phase
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• Buckling in a rotationally invariant spin-elastic model
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• Phase diagram of the one-dimensional three-state Potts model with an additional mean-field interaction
• Lattice Gas Models with Long Range Interactions
• Ensemble dependence of the critical behavior of a system with long-range interaction and quenched randomness
• On the Conditions for a Quantum Violent Relaxation