Long Time Convergence of the Bose–Einstein Condensation

Abstract

We study long time behavior of the Bose–Einstein condensation of measure-valued solutions $F_t$ of the space homogeneous and velocity isotropic Boltzmann equation for Bose–Einstein particles at low temperature. We prove that if $F_0\ge 0$ is a non-singular Borel measure on ${{\mathbb R}}_{\ge 0}$ satisfying a very low temperature condition and that the ratio $F_0([0,\varepsilon ])/\varepsilon ^{\alpha }$ is sufficiently large for all $\varepsilon \in (0, R]$ for some constants $0<\alpha <1, R>0$, then there exists a solution $F_t$ of the equation on $[0,+\infty )$ with the initial datum $F_0$ such that $F_t(\{0\})$ converges to the expected Bose–Einstein condensation as $t\rightarrow +\infty$. We also show that such initial data $F_0$ exist extensively.