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Normal mode dynamics are ubiquitous underlying the motions of diverse systemsfrom rotating stars to crystal structures. These behaviors are composed ofsimple collective motions of particles which move with the same frequency andphase, thus encapsulating many-body effects into simple dynamic motions. Inregimes such as the unitary regime for ultracold Fermi gases, a singlecollective mode can dominate, leading to simple behavior as seen insuperfluidity. I investigate the evolution of collective motion as a functionof N for five types of normal modes obtained from an L=0 group theoreticsolution of a general Hamiltonian for confined, identical particles. I showusing simple analytic forms that the collective behavior of few-body systems,with the well known motions of molecular equivalents such as ammonia andmethane, evolves smoothly to the collective motions expected for large Nensembles. The transition occurs at quite low values of N. I study aHamiltonian known to support collective behavior, the Hamiltonian for Fermigases in the unitary regime. I analyze the evolution of both frequencies andthe coefficients that mix the radial and angular coordinates which both dependon interparticle interactions. This analysis reveals two phenomena that couldcontribute to the viability of collective behavior. First the mixingcoefficients go to zero or unity, i.e. no mixing, as N becomes large resultingin solutions that do not depend on the details of the interparticle potentialas expected for this unitary regime, and that manifest the symmetry of anunderlying approximate Hamiltonian. Second, the five normal mode frequencieswhich are all close for low values of N, separate as N increases, creatinglarge gaps that can, in principle, offer stability to collective behavior ifmechanisms to prevent the transfer of energy to other modes exist (such as lowtemperature) or can be constructed.

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