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.
Statistical mechanics is a powerful approach for understanding phenomena whose behavior emerges from interactions between many microscopic particles. The outcomes of this project include mathematical results that advance our knowledge in statistical mechanics and that should have valuable effects in science and technology. These results have extended the state of the art in statistical mechanics, both through theory and technique. And since statistical mechanics lies in the interface between several fields of mathematics and theoretical physics, these results also have the potential for advancing our knowledge of various physical and biological phenomena, for instance, by better predictions in condensed matter physics and better data analysis in family genetics. There are several results on a physical phenomenon called Bose-Einstein condensation (BEC), a state of matter close to absolute zero in which many quantum particles unite and behave as if they are a single quantum particle. One of these results is a central limit theorem for quantum many-body systems, important for developing rigorous quantum control theory in addition to understanding the statistics of BEC, which should be helpful in applications such as interferometry and quantum computing. Another result is the discovery of a new, physically concentrated phase of BEC, by showing that the macroscopic quantum equation for BEC is exactly solvable and has a phase transition, with a clear description of each phase. This may lead to new experimental findings and better predictions in experiments on ultracold quantum systems. There are also several results motivated by computational biology, specifically understanding genetic dynamics and genetic data analysis. One result is a continuum limit for particles with long-range interactions, connecting the microscopic description of quantum particles in a nonlocal system to the macroscopic equation that models the dynamics of electrons on DNA. This should improve the accuracy of biophysics predictions, and possibly help us understand the mechanisms behind certain types of genetic mutations. Another result is on pedigree hidden Markov model and kinship algorithms are advancing our knowledge of certain models used in computational biology, which should improve medical analysis on large sets of genetic data. Other outcomes are training students and postdocs, including undergraduate students, graduate students, and postdoctoral scholars, especially women and other underrepresented groups. This has a big impact on diversity among future scientists and engineers.