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The paper is a continuation of our previous work on the spatially homogeneousBoltzmann equation for Bose-Einstein particles with quantum collision kernelthat includes the hard sphere model. Solutions $F_t$ under consideration thatconserve the mass, momentum, and energy and converge at least weakly toequilibrium $F_{{\rm be}}$ as $t\to\infty$ have been proven to exist at leastfor radially symmetric and non-singular initial data, and for the case of lowtemperature, $F_t$ have to be positive Borel measures. The new progress is asfollows: we prove that the long time convergence of $F_t(\{0\})$ to theBose-Einstein condensation $F_{{\rm be}}(\{0\})$ for low temperature holds forall radially symmetric and non-singular initial data $F_0$. This immediatelyimplies the long time strong convergence to equilibrium. We also obtain analgebraic rate of the strong convergence for arbitrary temperature. Our proofsare based on the entropy control, Villani's inequality for the entropydissipation, a suitable time-dependent convex combination between the solutionand a fixed positive function (in order to overcome the lack of positive lowerbound), the convex-positivity of the cubic collision integral, and an iterationtechnique for obtaining a positive lower bound of condensation.

The Spatially Homogeneous Boltzmann Equation for Bose-Einstein Particles: Rate of Strong Convergence to Equilibrium