Many-particle Systems with Singular Interactions: Statistical Mechanics and Mean-field Dynamics

New York University, New York NY

Investigators

Abstract

Mathematical analysis can help understand and derive effective laws or effective theories emerging from the collective behavior of many particles. This project is particularly interested in such rigorous derivations in the case where the many particles are interacting with singular forces, such as the Coulomb force, which is the fundamental electric force of nature. Understanding the statistical behavior of such systems, as well as their dynamical laws, is directly related to fundamental questions in several areas of physics and applied science: the Coulomb gas in statistical physics, models of plasmas in astrophysics and plasma physics, quantum mechanics models, analysis of random matrices (itself initially motivated by the analysis of the spectrum of large atoms), phase transitions in condensed matter physics (superconductors and superfluids), but also collective behavior in biology, social sciences, and neural networks. Recent progress has been made bringing forward new tools from analysis and probability to analyze such questions, with or without randomness, but much remains to be done. The project focuses in particular on two directions. The first is obtaining convergence results for dynamics that are valid for all time and with an explicit error rate, thus useful in practice. The second is in understanding the famous Kosterlitz-Thouless phase transition in the so-called "two component plasma". This is a two-dimensional gas made of positively and negatively charged particles with electrostatic interaction. Positive particles and negative particles attract and, depending on the temperature, they pair into collapsed dipoles (at low temperature) or behave as free charges (at high temperature). What was an initial surprise is that, according to the Nobel-prize winning prediction of Berezinsky, Kosterlitz, and Thouless, a third, intermediate and new state of matter exists, with quite unusual behavior that is explained by the formation of vortices. Much remains to be rigorously analyzed about this phase transition, and the project hopes to advance this theoretical understanding. The broader impacts of the project stem from its mentoring and training component, expository work, communication and outreach to broader audiences, as well as involvement with the community in various roles. The effective or mean-field behavior of systems with singular interactions, in particular Coulombic ones, has been understood for several situations of dynamics and statistical mechanics. In the case of equilibrium statistical mechanics, this consists in examining the behavior of the particle density under the canonical Gibbs measure, and this has been understood via large deviations techniques and potential theory. In the purely repulsive Coulomb case, much more has been understood, including the fluctuations around the mean-field limit and the microscopic behavior of the points. The project further extends this understanding by proving the connection to the Gaussian Multiplicative Chaos in the 2D Coulomb case, and by analyzing non-Coulomb Riesz repulsive interactions, which present further challenges. Much less has been understood about the case of a neutral plasma of oppositely charged particles (which then attract), which makes sense as a two-dimensional system. In particular, the project will turn to understanding the fine behavior of such a two-component Coulomb gas, in which a very particular phase transition, the Berezinski-Kosterlitz-Thouless phase transition, is predicted to happen. Bringing in an electrostatic and large deviations-based approach to this topic will provide a new approach to such problems, alternate to the renormalization methods of quantum field theory, and allow to understand the model below and above the critical temperature, with characterizations of the formation of dipoles and multipoles which explain the phase transition, and analysis of the fluctuations. The last main part of the project turns to gradient flow, conservative dynamics and Newtonian dynamics of systems with Coulomb or Riesz repulsive or attractive interactions. Thanks in particular to the recent modulated energy and modulated free energy methods, the mean-field limit can be derived, but much less has been understood beyond this than in the statistical mechanics setting. The project will allow us to understand whether and when global-in-time convergence holds, questions of instability in the case with attraction, and fluctuations and large deviations away from the mean-field behavior, thus providing much more precise information on such dynamics. 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.