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

统计力学及其在偏微分方程、凝聚态物质和生物学中的应用

• 批准号: 1106770
• 负责人: Kay Kirkpatrick
• 金额: $ 15.88万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1106770&HistoricalAwards=false
• 关键词: Statistical; mechanics; applications; PDEs; condensed

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.

这项研究旨在探索物理和生物系统中出现的非线性薛定谔方程(NLS)和其他偏微分方程组和晶格模型的统计力学。PI等人最近的工作建立了玻色-爱因斯坦凝聚(BEC)的量子多体物理与作为其宏观模型的立方NLS之间的严格联系。PI和一位合作者还表明，聚焦立方离散NLS的热力学在三维及更高的维度上是渐近精确可解的，并过渡到新的、物理上集中的BEC相。她的目标是继续研究这个模型和相关模型，包括噪声量子系统，经典的铁磁性海森堡模型，理论和计算量子多体系统，以及电子在DNA上移动时具有概率长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所产生的现象的有力方法。由于统计力学位于数学和理论物理的几个领域之间，它具有提高我们对各种物理和生物现象的认识及其在科学和工程中的应用的潜力。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，一团量子粒子聚集在一起，行为就像一个巨大的量子粒子，在干涉测量、可能的流氓波和量子计算中有着重要的应用。生物现象推动了这项研究的其他方面，包括遗传动力学和对基因突变的应用。另一个目标是与物理学和计算生物学团体建立联系，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。