Real-time Bose-Einstein condensation in a finite volume with a discrete spectrum

We show that Bose condensation in real time occurs in a finite system not only as an accumulation of the bosons in the ground state below a critical temperature, but also as a rapid enhancement of an arbitrary small symmetry breaking, followed by a very slow decay of the symmetry breaking order parameter from the almost ideal value to the vanishing equilibrium value. We show this analytically on an exactly soluble model and numerically on a model of noninteracting bosons in an oscillator potential.