Special Seminar in Applied Mathematics

The ultra-cold non-interacting Boson gases form so-called Bose-Einstein condensates (BECs). It is desirable to have a rigorous mathematical theory for such macroscopic quantum phenomenon. In this talk, I will report our numerical and analytical results on the ground state patterns and their phase transitions of spin-1 BECs w/o a uniform magnetic field, based on a mean field model--a generalized Gross-Pitaevskii equation. For numerical studies, I will present a complete investigation of the ground state patterns and phase diagrams of the spin-1 BECs. Two types of bifurcations are found. The first type is a transition from a two-component(2C) state to a three-component(3C) state. The second type is a symmetry breaking in the 3C state, followed by a phase separation of the spin components. In the semi-classical regime, it is found that these two bifurcation curves are gradually merged as the Planck parameter tends to zero. For analytic studies, we provide a rigorous proof for the bifurcation from 2C to 3C for antiferromagnetic systems. The key is a mass-redistribution lemma, that is, a redistribution of the mass densities between different components can be used to decrease bulk energy, but does not increase the kinetic energy. In the semi-classical regime, the Thomas-Fermi approximation and the Gamma-convergence theory are adopted. The bifurcation curves in the Thomas-Fermi regime are given explicitly. The ground state patterns are determined by the zero mean curvature interface with contact angle determined by Young's relation, a generalization of classical wetting theory to the quantum cases.