  The paper is a continuation of our previous work on the spatially homogeneous
Boltzmann equation for Bose-Einstein particles with quantum collision kernel
that includes the hard sphere model. Solutions $F_t$ under consideration that
conserve the mass, momentum, and energy and converge at least weakly to
equilibrium $F_{{\rm be}}$ as $t\to\infty$ have been proven to exist at least
for radially symmetric and non-singular initial data, and for the case of low
temperature, $F_t$ have to be positive Borel measures. The new progress is as
follows: we prove that the long time convergence of $F_t(\{0\})$ to the
Bose-Einstein condensation $F_{{\rm be}}(\{0\})$ for low temperature holds for
all radially symmetric and non-singular initial data $F_0$. This immediately
implies the long time strong convergence to equilibrium. We also obtain an
algebraic rate of the strong convergence for arbitrary temperature. Our proofs
are based on the entropy control, Villani's inequality for the entropy
dissipation, a suitable time-dependent convex combination between the solution
and a fixed positive function (in order to overcome the lack of positive lower
bound), the convex-positivity of the cubic collision integral, and an iteration
technique for obtaining a positive lower bound of condensation.
