Emergence of a collective crystal in a classical system with long-range interactions

¹ INAF Osservatorio Astrofisico di Arcetri - Largo Enrico Fermi 5, I-50125 Firenze, Italy
² Dipartimento di Fisica e Astronomia, Università di Firenze and INFN and CSDC via Sansone, I-50019 Sesto Fiorentino, Firenze, Italy
³ Aix Marseille Université, Université de Toulon, CNRS, CPT UMR 7332 - F-13288 Marseille, France

Received: 5 May 2015
Accepted: 29 July 2015

Abstract

A one-dimensional long-range model of classical rotators with an extended degree of complexity, as compared to paradigmatic long-range systems, is introduced and studied. Working at constant density, in the thermodynamic limit one can prove the statistical equivalence with the Hamiltonian mean-field (HMF) model and α-HMF: a second-order phase transition is indeed observed at the critical energy threshold . Conversely, when the thermodynamic limit is performed at infinite density (while keeping the length of the hosting interval L constant), the critical energy is modulated as a function of L. At low energy, a self-organized collective crystal phase is reported to emerge, which converges to a perfect crystal in the limit . To analyze the phenomenon, the equilibrium one-particle density function is analytically computed by maximizing the entropy. The transition and the associated critical energy between the gaseous and the crystal phase is computed. Molecular dynamics show that the crystal phase is apparently split into two distinct regimes, depending on the energy per particle ε. For small ε, particles are exactly located on the lattice sites; above an energy threshold , particles can travel from one site to another. However, does not signal a phase transition but reflects the finite time of observation: the perfect crystal observed for corresponds to a long-lasting dynamical transient, whose lifetime increases when the approaches zero.