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

本研究旨在探讨非线性薛定谔方程（NLS）和其他物理和生物系统中出现的PDE和晶格模型的统计力学。PI和其他人最近的工作已经建立了玻色-爱因斯坦凝聚（BEC）的量子多体物理和作为其宏观模型的立方NLS之间的严格联系。PI还与合作者一起证明了聚焦三次离散NLS的热力学在三维及更高的维度上是渐进精确可解的，并过渡到BEC的一个新的物理集中相。她的目标是继续研究这个和相关模型，包括噪声量子系统，经典的海森堡铁磁性模型，理论和计算量子多体系统，以及电子在DNA上移动时具有概率远程相互作用的分数阶非线性薛定谔方程模型。统计力学是理解许多微观粒子大规模相互作用产生的现象的有力方法。由于统计力学是数学和理论物理几个领域的交叉点，它有可能提高我们对各种物理和生物现象的认识，以及它们在科学和工程中的应用。一个很酷的物理现象是玻色-爱因斯坦凝聚，这是一种接近绝对零度的物质状态，在这种状态下，量子粒子气体聚集在一起，表现得像一个巨大的量子粒子，在干涉测量、可能的异常波和量子计算中有着重要的应用。生物现象激发了这项研究的其他方面，包括遗传动力学和基因突变的应用。另一个目标是与物理学和计算生物学社区建立联系，以便在实验室中研究这些现象，并为未来的数学研究汲取灵感。

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.