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力发展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，过平衡的热力学以及神经网络的培训。该项目将通过分析由统计物理学中的自旋系统引起的马尔可夫链来研究这些动力发展。这些模型既是现代概率理论的核心，又对其他领域具有影响，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工作，并向当地的K-12学校和惩教机构进行数学教育宣传。该项目集中在三个重要环境中的马尔可夫链的进化上：（1）旋转系统（例如，Ising和Potts和Potts模型）在晶格上，（2）随机表面和际交往，（2）（2）型号的含义，（2）型号稳定的型号（3）以及（3（3），以及（3），以及（3），以及（3）（3）（3），以及（3），以及（3），以及（3），（3）模型（3）。眼镜）和损失功能在高维度中。这些动态过程有望表现出各种重叠的数学和物理现象。我们研究它们是为了了解将收敛时间与均衡的机制，尤其是改变边界条件，初始化和对国家空间的限制所扮演的角色。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力和更广泛影响的评估来通过评估来支持的，这是值得的。