Statistical mechanics and applications to PDEs, condensed matter, and biology

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This research aims to explore the statistical mechanics of the nonlinear Schrodinger equation (NLS) and other PDE and lattice models arising in physical and biological systems. Recent work of the PI and others has established the rigorous connection between the quantum many-body physics of Bose-Einstein condensation (BEC) and the cubic NLS as its macroscopic model. The PI has also, together with a collaborator, shown that the thermodynamics of the focusing cubic discrete NLS are asymptotically exactly solvable in dimensions three and higher, with a transition to a new, physically concentrated phase of BEC. She aims to continue research on this and related models, including noisy quantum systems, the classical Heisenberg model of ferromagnetism, theoretical and computational quantum many-body systems, and the fractional nonlinear Schrodinger equation model of electrons with probabilistic long-range interactions as they move on DNA.Statistical mechanics is a powerful approach for understanding phenomena whose behavior emerges from the large-scale interaction of many microscopic particles. Since statistical mechanics lies in the interface between several fields of mathematics and theoretical physics, it has the potential for advancing our knowledge of various physical and biological phenomena and their applications in science and engineering. One cool physical phenomenon is Bose-Einstein condensation, a state of matter close to absolute zero in which a gas of quantum particles coalesces and behaves like a giant quantum particle, with important applications to interferometry and possibly rogue waves and quantum computing. Biological phenomena motivate other aspects of this research, including genetic dynamics and applications to genetic mutations. Another goal is to establish links with the physics and computational biology communities in order to study these phenomena in laboratories and draw inspiration for future mathematical research.