  We study the condensate phase dynamics in a low-temperature equilibrium gas
of weakly interacting bosons, harmonically trapped and isolated from the
environment. We find that at long times, much longer than the collision time
between Bogoliubov quasiparticles, the variance of the phase accumulated by the
condensate grows with a ballistic term quadratic in time and a diffusive term
affine in time. We give the corresponding analytical expressions in the limit
of a large system, in the collisionless regime and in the ergodic approximation
for the quasiparticle motion. When properly rescaled, they are described by
universal functions of the temperature divided by the Thomas-Fermi chemical
potential. The same conclusion holds for the mode damping rates. Such
universality class differs from the previously studied one of the homogeneous
gas.
