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 impact 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 const.min{1, |v − v∗|2η} with const. > 0, 0 ≤ η < 1, and (2) have both a lower bound const.min{1,|v−v∗|2η1} and an upper bound min{1, |v − v∗|2η} with η1 ≥ η ≥ 1. For the first case, the long time convergence to BEC, i.e., limt→∞Ft({0})=Fbe({0}), is proved for a class of initial data F0 that satisfy a very low temperature condition and thus it holds the strong convergence to equilibrium. For the second case we show that Ft semi-strongly converges to equilibrium, but if initially F0({0}) = 0, then Ft({0}) ≡ 0 and thus there is no convergence to BEC hence no strong convergence to equilibrium.