Dynamics of Lattice and Mean-Field Spin Systems

晶格和平均场自旋系统的动力学

• 批准号: 2246780
• 负责人: Reza Gheissari
• 金额: $ 26万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246780&HistoricalAwards=false
• 关键词: Dynamics; Lattice; Mean; Field; Spin

Dynamical evolutions in complex high-dimensional energy landscapes are ubiquitous in modern science. Prototypical examples include protein folding, evolutionary dynamics, out-of-equilibrium thermodynamics, and training of neural networks. This project will investigate these dynamical evolutions by analyzing Markov chains arising from spin systems in statistical physics. These models are both central to modern probability theory and have implications for other fields, including theoretical computer science, high-dimensional statistics, and machine learning. The project also includes broadening participation efforts at Northwestern University and mathematical education outreach to local K-12 schools and correctional institutions.This project is focused on Markov chain evolutions in three important settings: (1) spin systems (e.g., the Ising and Potts models) on lattices, (2) random surfaces and interfaces between thermodynamically stable phases, and (3) disordered mean-field models (e.g., spin glasses) and loss functions in high dimensions. These diverse set of dynamical processes are expected to exhibit a wide range of overlapping mathematical and physical phenomena. We study them with an eye towards understanding the mechanisms dictating convergence times to equilibrium, especially the roles played by varying the boundary conditions, initializations, and restrictions to the state space.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

复杂的高维能量景观中的动态演化在现代科学中无处不在。典型的例子包括蛋白质折叠，进化动力学，平衡热力学和神经网络的训练。本计画将借由分析统计物理中自旋系统所产生的马尔可夫链来探讨这些动态演化。这些模型既是现代概率论的核心，也对其他领域有影响，包括理论计算机科学，高维统计和机器学习。 该项目还包括扩大西北大学的参与努力和当地K-12学校和惩教机构的数学教育推广。该项目的重点是三个重要环境中的马尔可夫链演化：（1）自旋系统（例如，晶格上的Ising和Potts模型），（2）无规表面和化学稳定相之间的界面，以及（3）无序平均场模型（例如，自旋玻璃）和高维损失函数。这些不同的动力学过程集预计将展示出广泛的重叠的数学和物理现象。我们研究他们的眼睛对了解的机制支配收敛时间的平衡，特别是通过改变边界条件，初始化和状态空间的限制所发挥的作用。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。