  Previously we numerically showed that thermalization can occur in an
oscillating Bose-Einstein condensate (BEC) with a disordered harmonic trap when
healing length $\xi$ of the condensate is shorter than the correlation length
$\sigma_{D}$ of the disorder [see, for example, the experiment reported in
Phys. Rev. A \textbf{82}, 033603 (2010)]. In this work, we investigate the
weakly interacting or Anderson localization regime $\xi>\sigma_{D}$ and show
that the oscillating BEC can also exhibit a relaxation process from
nonequilibrium to equilibrium. In such an isolated quantum system, energy and
particle number are conserved and the irreversible evolution towards
thermodynamic equilibrium is induced by the disorder. The thermodynamic
equilibrium is evidenced by the maximized entropy $S\left[n_{k}\right]$ in
which the waveaction spectrum $n_{k}$ follows the Rayleigh-Jeans distribution.
Besides, unlike a monotonic irreversible process of thermalization to
equilibrium, the Fermi-Pasta-Ulam-Tsingou recurrence arises in this system,
manifested by the oscillation of the non-equilibrium entropy.
