Long-range Interactions and Diluted Networks

Abstract

Long-range interactions appear in gravitational and Coulomb systems, two-dimensional hydrodynamics, plasmas, etc. These physical systems are studied by a variety of theoretical and numerical methods, but their description in terms of statistical mechanics and kinetic theory remains an open challenge. Recently, there has been a burst of activity in this field, since it has been realized that some simplified models can be solved exactly in different ensembles (microcanonical, canonical, grand-canonical, etc.). Besides that, numerical simulations and specific kinetic theory approaches have revealed the presence of out-of-equilibrium macrostates, called Quasi Stationary States (QSSs), whose lifetime increases with a power of the number of particles. This discovery opens the interesting and intriguing possibility that the states observed in experiments where long-range interactions are involved are not Boltzmann-Gibbs equilibrium states. In this chapter, after a brief review of recent results on systems with long-range interactions, we focus on the Hamiltonian Mean Field (HMF) model. We give a short presentation of its equilibrium properties and present the numerical evidence of the existence of QSSs. Then, we discuss an analytical approach to the characterization of QSSs, pioneered by Lynden-Bell, that uses a maximum entropy principle. This approach captures some macroscopic features of QSSs and predicts the existence of phase transitions from homogeneous to inhomogeneous QSSs, which are then verified successfully in numerical experiments on the HMF model. The HMF model is defined on a lattice where all sites are coupled with equal strength. We here generalize the model to one where only a fraction of pairs of N sites are coupled, in such a way that the number of links scales as N L ∼ N γ with 1 < γ < 2. We present numerical evidence that QSSs exist in all this range of values of γ and that their lifetime scales as N α(γ−1) with α = 1.5 for homogeneous QSSs and α = 1 for inhomogenous QSSs. We devote this paper to George W. Zaslavsky, who introduced long ago a model similar to the HMF in order to study structural transitions in crystals. George was also interested in two-dimensional hydrodynamics, and in particular in the point vortex model, which also shows QSSs, and has more recently developed a theoretical approach to lattices with long-range interactions, for which the kinetic equations turn out to possess fractional derivatives.KeywordsOrder Phase TransitionCanonical EnsemblePhysical Review LetterVlasov EquationTricritical PointThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.