  The theoretical description of non-equilibrium Bose--Einstein condensate
(BEC) is one of the main challenges in modern statistical physics and kinetics.
The non-equilibrium nature of BEC makes it impossible to employ the
well-established formalism of statistical mechanics. We develop a framework for
the analytical description of a non-equilibrium phase transition to BEC that,
in contrast to previously developed approaches, takes into account the infinite
number of continuously distributed states. We consider the limit of fast
thermalization and obtain an analytical expression for the full density matrix
of a non-equilibrium ideal BEC which also covers the equilibrium case. For the
particular cases of 2D and 3D, we investigate the non-equilibrium formation of
BEC by finding the temperature dependence of the ground state occupation and
second-order coherence function. We show that for a given pumping rate, the
macroscopic occupation of the ground state and buildup of coherence may occur
at different temperatures. Moreover, the buildup of coherence strongly depends
on the pumping scheme. We also investigate the condensate linewidth and show
that the Schawlow--Townes law holds for BEC in 3D and does not hold for BEC in
2D.
