  We present a kinetic description of Bose-Einstein condensation for particle
systems being out of thermal equilibrium, which may happen for gluons produced
in the early stage of ultra-relativistic heavy-ion collisions. The dynamics of
bosons towards equilibrium is described by a Boltzmann equation including Bose
factors. To solve the Boltzmann equation with the presence of a Bose-Einstein
condensate we make further developments of the kinetic transport model BAMPS
(Boltzmann Approach of MultiParton Scatterings). In this work we demonstrate
the correct numerical implementations by comparing the final numerical results
to the expected solutions at thermal equilibrium for systems with and without
the presence of Bose-Einstein condensate. In addition, the onset of the
condensation in an over-populated gluon system is studied in more details. We
find that both expected power-law scalings denoted by the particle and energy
cascade are observed in the calculated gluon distribution function at infrared
and intermediate momentum regions, respectively. Also, the time evolution of
the hard scale exhibits a power-law scaling in a time window, which indicates
that the distribution function is approximately self-similar during that time.
