Abstract
The analytical expressions for the frequency and damping of the radial quadrupole and scissors modes, as obtained from the method of moments, are limited to the harmonic potential. In addition, the analytical results may not be sufficiently accurate as an average relaxation time is used and the high-order moments are ignored. Here, we numerically solve the Boltzmann model equation in the hydrodynamic, transition, and collisionless regimes to study mode frequency and damping. When the gas is trapped by the harmonic potential, we find that the analytical expressions underestimate the damping in the transition regime. Furthermore, we demonstrate that the numerical simulations are able to provide reasonable predictions for the collective oscillations in the Gaussian potentials. The present method can also be used to study many other problems, e.g. formation of quantum shockwave, expansion of atom cloud, and effective heat conductivity in very elongated traps.
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