Convergence and non-convergence to Bose-Einstein condensation

TL;DR

This paper investigates how different particle interaction potentials affect the convergence to Bose-Einstein condensation in the Boltzmann equation, showing conditions for both convergence and non-convergence scenarios.

Contribution

It provides a rigorous analysis of the influence of scattering cross section bounds on the convergence to Bose-Einstein condensation for isotropic solutions.

Findings

Convergence to BEC occurs for potentials with lower bounds and low-temperature initial data.

No convergence to BEC if initial condensate is zero and potentials have upper bounds.

Strong convergence to equilibrium is established under specific potential and initial data conditions.

Abstract

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.