Normal modes for [formula omitted] identical particles: A study of the evolution of collective behavior from few-body to many-body

Abstract

Normal mode dynamics are ubiquitous in nature underlying the motions of diverse interacting systems from the behavior of rotating stars to the vibration of crystal structures. These behaviors are composed of simple collective motions of the N interacting particles which move with the same frequency and phase, thus encapsulating many-body effects into simple dynamic motions. In this study, I investigate the evolution of collective motion as a function of N for five types of normal modes previously obtained from an L=0 group theoretic solution of a general Hamiltonian for confined, identical particles. I show using simple analytic forms for the N-body normal modes that the collective behavior of few-body systems, which have the well known motions of molecular equivalents such as ammonia and methane, evolves smoothly as N increases to the collective motions expected for large N ensembles (breathing, center of mass, particle–hole radial and angular excitations and phonon). Furthermore, the transition from few-body behavior (symmetric stretch, symmetric bend, asymmetric stretch, asymmetric bend and the opening and closing of alternative interparticle angles) to large N behavior occurs at quite low values of N. I extend this investigation to a Hamiltonian known to support collective behavior, the Hamiltonian for Fermi gases in the unitary regime. This analysis reveals two phenomena that could contribute to the viability of collective behavior. With the recent success using normal modes to describe the emergence of collective behavior in the form of superfluidity in ultracold Fermi gases in the unitary regime, understanding the character of these normal modes and the evolution of their behavior as a function of N has become of some interest due to the possibility of offering insight into the dynamics of regimes supporting collective behavior. In this study, I investigate both the macroscopic behavior associated with these N-body normal modes, as well as the microscopic motions underlying this behavior, and study the evolution of their collective behavior as a function of N.