We derive an equation of motion for the velocity fluctuations of a two-dimensional deformed trapped Bose gas just above the critical temperature in the hydrodynamical regime. From this equation, we calculate the eigenfrequencies and the corresponding density fluctuations for a few low-lying excitation modes. Using the method of averages, we derive a dispersion relation in a deformed trap at very high temperature that interpolates between the collisionless and hydrodynamic regimes. We make use of this dispersion relation to calculate the frequencies and the damping rates for monopole and quadrupole mode in both the regimes. We also discuss the time evolution of the wave-packet width of a Bose gas in a time-independent trap as well as a time-dependent trap.
Recently it was shown that the Landau-Khalatnikov two-fluid hydrodynamics describes the collision-dominated region of a trapped Bose condensate interacting with a thermal cloud. We use these equations to discuss the low frequency hydrodynamic collective modes in a trapped Bose gas at finite temperatures. We derive variational expressions based on these equations for both the frequency and damping of collective modes. A new feature is our use of frequency-dependent transport coefficients, which produce a natural cutoff by eliminating the collisionless low-density tail of the thermal cloud. Above the superfluid transition, our expression for the damping in trapped inhomogeneous gases is analogous to the result first obtained by Landau and Lifshitz for uniform classical fluids. We also use the moment method to discuss the crossover from the collisionless to the hydrodynamic region. Recent data for the monopole-quadrupole mode in the hydrodynamic region of a trapped gas of metastable
4He is discussed. We also present calculations for the damping of the analogous
m=0 monopole-quadrupole condensate mode in the superfluid phase.
The monopole and quadrupole oscillations of a Bose-Einstein condensate (BEC) gas trapped in a spherically symmetric potential are calculated using the time-dependent Hartree approximation which enables us to use an orthonormal single-particle basis for the description of both the condensate and noncondensate components. It is found that the monopole mode has a much smaller damping rate than the quadrupole mode. The obtained temperature dependence of the frequency shifts and the damping rates of the collective oscillations are found to be consistent with experimental data obtained for BEC gases in anisotropic trapping potentials. A comparison between the time-dependent Hartree approximation and the time-dependent Hartree-Fock approximation is also made and it is found that there is a rather small difference.