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 nonequilibrium 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, 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.

Time Parametrizations in Long-Range Interacting Bose-Einstein Condensates

“…From the results of Refs. [35,36], we infer that, in the µ-space, the coarse-grained distribution of this system in the QSS, in the presence of collective effects, at zero temperature is the superposition of two energy step functions (see Fig. 1)…”

“…1: The inferred zero-temperature QSS distribution (6) of the system {C} from the results of Refs. [35,36]. This ground-state distribution is a two-step functional of the onecondensate energy function ε = ε(Q a , P a ).…”

“…for the one-particle (i.e., one-condensate) energy function ε = P 2 /2M + Φ MF (Q a with the mean-field potential energy Φ MF (Q a ) (i.e., the integral of the binary potential energy Φ(|Q a − Q a′ |) multiplied by the coarsegrained distribution, f (Q a′ , P a′ , t), of the system over the µ-space), the so-called Fermi energy ε F , the diluted phase-space density η, and the Heaviside step function θ. Here, each component f (i) (i = 1, 2) as a subsystem is the zero-temperature case of the Lynden-Bell distribution [37], which is the maximum entropy distribution in the coarse-grained µ-space subject to the total particle number, total energy conservation laws, and the Vlasov incompressibility of the diluted phase-space density η (i) [35,36], where the uniqueness of the non-zero fine-grained phase-space density η 0 = η (1) + η (2) in the system {C} is assumed, in the collisionless regime. [45] This Vlasov incompressibility plays the same role as the Pauli exclusion principle for fermions in the statistics.…”