CAREER: Statistical mechanics of superconductors and other macroscopic phenomena

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

The PI will work on problems related to the statistical mechanics of quantum systems and spin models for ferromagnets and superconductors. A fundamental problem in mathematical physics is to explain macroscopic phenomena from the first principles of interacting microscopic particles. This project will build on previous results of the PI and collaborators on the statistical mechanics of Bose-Einstein condensation and the Gross-Pitaevskii equation, Gibbs measures for quantum systems, spin models of ferromagnetism, and probabilistic algorithms for family pedigree relationships. The proposed problems include developing a new Stein's method for quantum many-body systems; extending recent methods to other questions such as large deviations for quantum systems; studying metastability in XY and related models; formulating new spin models of superconductivity and analyzing their properties; and finding connections to the phenomenological theories of Ginzburg-Landau and Gross-Pitaevskii. The PI will integrate her research expertise into probability courses and train students in analytical and computational techniques valuable for a variety of careers. The PI will also continue her interest in encouraging women and other underrepresented groups in math. There are fundamental challenges in the mathematical physics of condensed matter, the cool and unusual phases of matter near absolute zero that have important applications in physics and engineering. One is to understand how the unique properties of condensed matter emerge from the large-scale interaction of many microscopic particles. Superconductors, for instance, allow current to flow freely with no loss and can expel magnetic fields; superconducting magnets are used in particle accelerators and MRI machines. Mathematical physicists would like to explain these phenomena (and the associated equations) starting from microscopic quantum many-body systems, but to do so, will need innovative mathematical tools developed at the interface between mathematics and physics. The main goals of this project are to develop such tools, to formulate new models with interesting phase transitions, and to find connections to phenomenological theories. Other goals are to strengthen ties with physics and engineering for mutual benefit, integrating research expertise into probability courses, and training students in analytical and computational techniques valuable for a variety of careers. The PI will also continue her interest in encouraging women and other underrepresented groups in math.