Core-halo quasi-stationary states in the Hamiltonian mean-field model

Abstract

A characteristic feature of long-range interacting systems is that they become trapped in a non-equilibrium and long-lived quasi-stationary state (QSS) during the early stages of their development. We present a comprehensive review of recent studies of the core-halo structure of QSSs, in the Hamiltonian mean-field model (HMF), which is a mean-field model of mutually coupled ferromagnetic XY spins located at a point, obtained by starting from various unsteady rectangular water-bag type initial phase-space distributions. The main result exposed in this review is that the core-halo structure can be described by the superposition of two independent Lynden–Bell distributions. We discuss the completeness of collisionless relaxation of this double Lynden–Bell distribution by using both of Lynden–Bell entropy and double Lynden–Bell entropy for the systems at low energies per particle.