108/3/18(一) 謝天長 研究員(國立台灣大學數學系)
Asymptotic problems in the Calculus of Variations and its application to Spin-1 Bose-Einstein condensation
Dr. Tien-Tsan Shieh
The Department of Mathematics, National Taiwan University
In the first part of this talk, I will introduce the asymptotic problem in the Calculus of Variations and the Gamma-convergence approach. In the second part of the talk, I will apply the Gamma-convergence method to solve the ground states problem in the spin-1 Bose-Einstein condensation. We develop an analytic theory for the ground-state patterns and phase diagram of spin-1 Bose–Einstein condensates in a bounded domain in $\R^d$ in the presence of a uniform magnetic field.
Our main results include: (1) obtaining a complete phase diagram with explicit analytic formulae of Thomas–Fermi solutions on the parameter q–M plane (q: quadratic Zeeman energy, M: total magnetization) for both ferromagnetic and antiferromagnetic systems, and (2) an entire characterization of ground-state patterns in the Thomas–Fermi regime and the semi-classical regime. In particular, a class of interesting ground states are mixed states. A mixed state consists of two constant states separated by an interface.
This interface minimizes an effective energy, the sum of an interfacial energy and two boundary energies. As a result, the interface has constant mean curvature and the corresponding contact angle satisfies Young’s relation as that in the classical wetting process.
日 期：108年3月18日(星期一) 16:10~17:00
茶 會： 15:30~16:00數學館四樓409室舉行