Evolution of finite temperature Bose-Einstein Condensates: Some rigorous studies on condensate growth

Abstract

In trapped Bose-Einstein condensates (BECs), \emph{condensate growth} refers to the process in which an increasing number of quasi-particles are immediately transferred from the non-condensate state (the thermal cloud) into the condensate state following the initial formation of the BEC. Despite its physical significance, this phenomenon has not yet been studied rigorously from a mathematical standpoint. In this work, we investigate a kinetic equation whose collision operator includes three types of wave interactions: one corresponding to a 3-wave process, and two classified as 4-wave processes. This wave kinetic equation models the evolution of the density function of the thermal cloud. We establish the immediate formation of condensation in solutions to this equation, thus providing a rigorous demonstration of the condensate growth phenomenon.