  We show that the phase of a condensate in a finite temperature gas spreads
linearly in time at long times rather than in a diffusive way. This result is
supported by classical field simulations, and analytical calculations which are
generalized to the quantum case under the assumption of quantum ergodicity in
the system. This super-diffusive behavior is intimately related to conservation
of energy during the free evolution of the system and to fluctuations of energy
in the prepared initial state.
