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 $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{.}\min \{1,\mathrm{\mid }{\mathbf{v}-\mathbf{v}}_{\ast }{\mathrm{\mid }}^{2\eta }\}$ with $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{.}>0,0\le \eta <1$, and 2) have an upper bound $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{.}\min \{1,\mathrm{\mid }{\mathbf{v}-\mathbf{v}}_{\ast }{\mathrm{\mid }}^{2\eta }\}$ with $\eta \ge 1$. For the first case, the long time convergence to BEC i.e. $\underset{t\to \mathrm{\infty }}{\lim }{F}_t(\{0\})=F_{\mathrm{b}\mathrm{e}}(\{0\})$ 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.