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A characteristic feature of long-range interacting systems is that theybecome trapped in a non-equilibrium and long-lived quasi-stationary state (QSS)during the early stages of their development. We present a comprehensive reviewof recent studies of the core-halo structure of QSSs, in the Hamiltonianmean-field model, which is a mean-field model of mutually coupled ferromagneticXY spins located at a point, obtained by starting from various unsteadyrectangular water-bag type initial phase-space distributions. The main resultexposed in this review is that the core-halo structure can be described by thesuperposition of two independent Lynden-Bell distributions. We discuss thecompleteness of collisionless relaxation of this double Lynden-Belldistribution by using both of Lynden-Bell entropy and double Lynden-Bellentropy for the systems at low energies per particle.

Core-Halo Quasi-Stationary States in the Hamiltonian Mean-Field Model