Thermalization of dipole oscillations in confined systems by rare collisions

We study the relaxation of the center-of-mass, or dipole oscillations in the system of interacting fermions confined spatially. With the confinement frequency $\omega_{\perp}$ fixed the particles were considered to freely move along one (quasi-1D) or two (quasi-2D) spatial dimensions. We have focused on the regime of rare collisions, such that the inelastic collision rate, $1/\tau_{in} \ll \omega_{\perp}$. The dipole oscillations relaxation rate, $1/\tau_{\perp}$ is obtained at three different levels: by direct perturbation theory, solving the integral Bethe-Salpeter equation and applying the memory function formalism. As long as anharmonicity is weak, $1/\tau_{\perp} \ll 1/ \tau_{in}$ the three methods are shown to give identical results. In quasi-2D case $1/\tau_{\perp} \neq 0$ at zero temperature. In quasi-1D system $1/\tau_{\perp} \propto T^3$ if the Fermi energy, $E_F$ lies below the critical value, $E_F < 3 \omega_{\perp}/4$. Otherwise, unless the system is close to integrability, the rate $1/\tau_{\perp}$ has the temperature dependence similar to that in quasi-2D. In all cases the relaxation results from the excitation of particle-hole pairs propagating along unconfined directions resulting in the relationship $1/\tau_{\perp} \propto 1/\tau_{in}$, with the inelastic rate $1/\tau_{in} \neq 0$ as the phase-space opens up at finite energy of excitation, $\hbar \omega_{\perp}$. While $1/\tau_{\perp} \propto \tau_{in}$ in the hydrodynamic regime, $\omega_{\perp} \ll 1/\tau_{in}$, in the regime of rare collisions, $\omega_{\perp} \gg 1/\tau_{in}$, we obtain the opposite trend $1/\tau_{\perp} \propto 1/\tau_{in}$.