Variational Problems in Analysis and Physics

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

The aim of mathematical physics is to provide a well reasoned connection between the world around us and the laws of physics. We understand why matter is extended on the basis of non-relativistic quantum mechanics. We are able to explain why certain materials change their quality when the temperature is lowered, as for example, when water turns to ice. There are, however, many phenomena that are not understood in a rigorous way. It is an everyday experience that physical systems tend towards equilibrium: Hot coffee cools by giving up energy to the environment until the temperatures are the same. Symmetry can be broken; for instance matter that appears to be homogeneous at high temperature tends to loose its homogeneity, that is, it forms clumps when cooled down. Physical systems try to achieve a state of lowest energy. Central questions are how to describe this state and how the system makes the transition. How does hot coffee approach an equilibrium with its environment by cooling down? How can matter lower its energy by transitioning to a less homogeneous shape? Another example, maybe less obvious, is a heat-conducting rod with one end held at a high temperature and the other at low temperature. This system is not in equilibrium but in a steady state; heat keeps flowing from hot to cold. Although a very old problem, there is no satisfactory mathematically-rigorous microscopic explanation for this observation. The aim of this proposal is to study these questions in specific mathematical and physical models. Some of these are large physical systems with many interacting agents. Others are, from a superficial perspective, quite simple, such as a single charged particle in a magnet. Finding answers to these questions requires new mathematical insights. An important feature of the project is to exploit the interaction of physical insight and mathematical techniques. This interdisciplinary quality makes it an ideal training ground for students at all levels. The project interweaves several strands of mathematical physics: non-equilibrium statistical mechanics through classical and quantum mechanical master equations, variational problems involving magnetic fields and more general vector fields as well as charged systems interacting with a classical radiation field. The investigation has variational inequalities as a common theme. One endeavor is to carry over recent robust advances concerning approach to equilibrium to other problems, both in the classical and quantum mechanical realm. Likewise, the gap for Kac type master equations for realistic models is now within reach. The Kac master equation is ideal for approximating thermostats by finite reservoirs. The PI will investigate the properties of the non-equilibrium steady state (NESS) for a system of two thermostats at different temperatures. The next step is to study the approximation of such systems by finite reservoirs, in particular determining the various time scales over which these approximations hold. A wide open area is the calculus of variations involving vector fields. This project will investigate a class of conformally invariant inequalities that includes the computation of the sharp constants. The goal is to use these insights to shed some light on systems with magnetic fields where the wave function is complex. A different but closely related problem is the analysis of the Maxwell-Pauli-Coulomb equations for supercritical charges. The focus of this inquiry will be on the question of whether there is blow up of solutions. 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.