Mathematical Analysis of the Coupling Condensate - Thermal Cloud Systems

Abstract

Quantum kinetic theory has recently become an important topic of research in the mathematics community. While the mathematical analysis and rigorous derivations of the Boltzmann-Nordheim equation and the related 4-wave kinetic equation has quite a large body of literature and breakthroughs have been made (see [19, 20, 21, 22, 34, 35, 37, 39, 40, 43, 58, 59, 60, 61, 64, 65, 66, 68, 75, 105, 107, 108, 109, 110, 111, 112, 113, 116, 117, 122, 137, 152, 154] and references therein), this chapter is only devoted to the discussion of recent research on the coupling condensate-thermal cloud systems in both regimes: low temperatures and ultra-low temperatures, with a focus on the works of the authors, in which a full mathematical proof of a new result is given in detail. Since both systems in the two regimes are very complicated, recent mathematical works are only devoted to partial approximations of them.