Uniform in time lower bound for solutions to a quantum Boltzmann equation of bosons

TL;DR

This paper proves that solutions to a quantum Boltzmann equation for bosons remain bounded below by a Gaussian over time, providing insights into the behavior of condensate growth with detailed analysis of collision operators.

Contribution

It establishes a uniform in time lower bound for solutions to the quantum Boltzmann equation considering the full Bogoliubov dispersion law, a novel analytical result.

Findings

Solutions are bounded below by a Gaussian uniformly in time

Detailed analysis of surface integrals on energy manifolds

Enhanced understanding of condensate growth dynamics

Abstract

We consider the quantum Boltzmann equation, which describes the growth of the condensate, or in other words, models the interaction between excited atoms and a condensate. In this work, the full form of Bogoliubov dispersion law is considered, which leads to a detailed study of surface integrals inside the collision operator on energy manifolds. We prove that positive radial solutions of the quantum Boltzmann equation are bounded from below by a Gaussian, uniformly in time.