Normal modes for N identical particles: A study of the evolution of collective behavior from few-body to many-body

Normal mode dynamics are ubiquitous underlying the motions of diverse systems from rotating stars to crystal structures. These behaviors are composed of simple collective motions of particles which move with the same frequency and phase, thus encapsulating many-body effects into simple dynamic motions. In regimes such as the unitary regime for ultracold Fermi gases, a single collective mode can dominate, leading to simple behavior as seen in superfluidity. I investigate the evolution of collective motion as a function of N for five types of normal modes obtained from an L=0 group theoretic solution of a general Hamiltonian for confined, identical particles. I show using simple analytic forms that the collective behavior of few-body systems, with the well known motions of molecular equivalents such as ammonia and methane, evolves smoothly to the collective motions expected for large N ensembles. The transition occurs at quite low values of N. I study a Hamiltonian known to support collective behavior, the Hamiltonian for Fermi gases in the unitary regime. I analyze the evolution of both frequencies and the coefficients that mix the radial and angular coordinates which both depend on interparticle interactions. This analysis reveals two phenomena that could contribute to the viability of collective behavior. First the mixing coefficients go to zero or unity, i.e. no mixing, as N becomes large resulting in solutions that do not depend on the details of the interparticle potential as expected for this unitary regime, and that manifest the symmetry of an underlying approximate Hamiltonian. Second, the five normal mode frequencies which are all close for low values of N, separate as N increases, creating large gaps that can, in principle, offer stability to collective behavior if mechanisms to prevent the transfer of energy to other modes exist (such as low temperature) or can be constructed.