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.
