Convergence and non-convergence to Bose-Einstein condensation

The paper is a continuation of our previous work on the strong convergence to equilibrium for the spatially homogeneous Boltzmann equation for Bose-Einstein particles for isotropic solutions at low temperature. Here we study the influence of the particle interaction potentials on the convergence to Bose-Einstein condensation (BEC). Consider two cases of certain potentials that are such that the corresponding scattering cross sections are bounded and 1) have a lower bound ${\rm const.}\min\{1, |{\bf v-v}_*|^{2\eta}\}$ with ${\rm const}.>0, 0\le \eta<1$, and 2) have an upper bound ${\rm const.}\min\{1, |{\bf v-v}_*|^{2\eta}\}$ with $\eta\ge 1$. For the first case, the long time convergence to BEC i.e. $\lim\limits_{t\to\infty}F_t(\{0\})=F_{\rm be}(\{0\})$ is proved for a class of initial data having very low temperature and thus it holds the strong convergence to equilibrium. For the second case we show that if initially $F_0(\{0\})=0$, then $F_t(\{0\})=0$ for all $t\ge 0$ and thus there is no convergence to BEC hence no strong convergence to equilibrium.