The Mathematics of Interacting Particle Systems

One of the great successes of modern science has been the understanding that simple objects can combine to form complicated structures. For instance, most of the matter that surrounds us is made up of protons, neutrons, and electrons. These building blocks, taken individually, are rather simple, but, when enough of them interact with each other, they produce complex and varied structures, from the intricate patterns of tiny snowflakes to enormous spiraling galaxies. This is a very powerful idea: in principle, it suffices to understand the behavior of simple particles to derive everything. However, in practice, this is a very challenging task, as keeping track of large numbers of interacting particles is impossibly difficult. Instead, sophisticated techniques need to be developed to extract the relevant collective behaviors. This project consists in developing and investigating several such techniques. This project focuses on three types of particle systems, both classical and quantum, which exhibit different types of collective behavior. The first is a model of interacting quantum particles called Bosons. This is a toy model for helium atoms, which are known to form a superfluid phase at low temperature, in which the helium flows without viscosity. The principal investigator (PI) is studying the so-called "Simplified Approach", which has been shown to reproduce much of the complex behavior of the interacting Bose gas, while being much more tractable. The second is a classical model in which the PI is proving the existence of crystalline phases, in which infinite large-scale regular patterns spontaneously emerge. The third is a model of interacting quantum particles called Fermions. This is a toy model for the electrons in conductors, which are known to form a superconducting phase at low temperature, in which electricity flows without resistance. The PI is investigating "hierarchical models", for which exact solutions can be found, and complex behavior can be proved. This project includes a significant educational component at various levels. The PI is developing graduate, undergraduate, and master's level courses that incorporate the techniques developed in the project, thus introducing students to the tools and techniques of mathematical research. In addition, the PI is producing and distributing educational videos aimed at high school students, undergraduates, and the general public, which are informed by the PI's perspective as a researcher. In addition, the PI is involved in a project to design new mathematical reasoning courses at Rutgers, based on the formal proof assistant called "Lean". This project lies in the field of mathematical physics and aims to develop new tools and refine existing ones to analyze the effect of interactions in a systematic and mathematically rigorous way. Specifically, it consists of the analysis of three types of systems: interacting Bose gases, classical hard-core particle models at high density, and interacting lattice Fermi gases. To analyze the interacting Bose gas, the PI is investigating the "Simplified Approach", which is a nonlinear, nonlocal partial differential equation (PDE) in three dimensions. Its analysis has yielded very promising results: it reproduces all known and conjectured behavior of the Bose gas for all densities. The objectives of this part are to solve the more important problems that are still open about this PDE and study its relation to the original many-Boson problem. The PI has developed a framework to study a large class of hard-core particle models at high density and prove that these behave like crystals in that regime. The objectives of this part are to extend the family of hard-core particle models for which we can rigorously prove ordering phase transitions to include three dimensional models, liquid crystals, as well as continuum models. To study interacting lattice Fermi gases, the PI is using the Renormalization Group (RG), which is a powerful tool to study systems of interacting quantum particles, but it is notoriously difficult to implement. The PI has introduced a family of models for which the RG analysis can be carried out easily, rigorously, and exactly, and nontrivial properties can be proved. The main objective of this part is to define and study Fermionic hierarchical models that exhibit superconductivity. 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.