Stochastic Dynamics on Large-Scale Networks

对微观概率模型的严格分析增强了我们对流体动力学流动、晶体生长、磁性和波传播机制的理解。它还为具有可定制属性的材料提供了更智能的设计。这个项目旨在利用单粒子扩散拉普拉斯和狄里克莱特能量形式的基本概念，在包括网络、树和分形图在内的各种状态空间上，发展对这些宏观现象的强有力的理解。这个项目的一个组成部分将包括通过PI与本科生的教学和研究来促进和培训概率和统计力学。本项目集中于分析状态空间上相互作用的粒子系统，这些状态空间不需要具有空间对称性，但在由电阻确定的度规中有界。PI将在这些阻力空间上进行严格的非线性(随机)偏微分方程导出，作为弱不对称排斥过程中经验观测的标度极限。重点将放在控制整体排斥和边界川崎动力学的参数所起的作用，从而导致宏观相变和非线性效应。PI将利用新的函数不等式、代数恒等式、粗粒化方法和对(S)偏微分方程组的分析来建立极限定理。总体目标是通过概率工具和非平稳全球分析的组合，获得对大规模网络上的非平衡统计力学的自我一致的理解。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Rigorous analysis of microscopic probabilistic models enhances our understanding of the mechanisms governing hydrodynamic flow, crystal growth, magnetism, and wave propagation. It also informs smarter design of materials with customizable properties. This project aims to develop a robust understanding of these macroscopic phenomena, using fundamental concepts from single-particle diffusion Laplacians and Dirichlet energy forms on a wide variety of state spaces, including networks, trees, and fractals. An integral part of this project will involve promotion of, and training in, probability and statistical mechanics through the PI's teaching and research with undergraduate students.This project focuses on the analysis of interacting particle systems on state spaces which need not possess spatial symmetries, but are bounded in the metric determined by electrical resistance. The PI will carry out, on these resistance spaces, rigorous derivations of nonlinear (Stochastic) Partial Differential Equations as scaling limits of empirical observables in the weakly asymmetric exclusion processes. Emphasis will be on the role played by the parameters governing the bulk exclusion as well as the boundary Kawasaki dynamics, leading to macroscopic phase transitions and nonlinear effects. The PI will utilize novel functional inequalities, algebraic identities, coarse-graining methods, and analysis of (S)PDEs to establish limit theorems. The overall goal is to obtain a self-consistent understanding of non-equilibrium statistical mechanics on large-scale networks, through a combination of probabilistic tools and non-smooth global analysis.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.