  The central problem of this chapter is temporal coherence of a
three-dimensional spatially homogeneous Bose-condensed gas, initially prepared
at finite temperature and then evolving as an isolated interacting system. A
first theoretical tool is a number-conserving Bogoliubov approach that allows
to describe the system as a weakly interacting gas of quasi-particles. This
approach naturally introduces the phase operator of the condensate: a central
actor since loss of temporal coherence is governed by the spreading of the
condensate phase-change. A second tool is the set of kinetic equations
describing the Beliaev-Landau processes for the quasi-particles. We find that
in general the variance of the condensate phase-change at long times $t$ is the
sum of a ballistic term $\propto t^2$ and a diffusive term $\propto t$ with
temperature and interaction dependent coefficients. In the thermodynamic limit,
the diffusion coefficient scales as the inverse of the system volume. The
coefficient of $t^2$ scales as the inverse volume squared times the variance of
the energy of the system in the initial state and can also be obtained by a
quantum ergodic theory (the so-called eigenstate thermalisation hypothesis).
