Variational Problems in Analysis and Physics

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

Physical systems tend towards equilibrium. Mechanical systems achieve a state of lowest energy by giving off energy to the environment. Likewise, local disturbances in a thermodynamic system, such as temperature fluctuations in a room, tend to even out; i.e., the system relaxes towards a thermal equilibrium. However, there are instances where due to external influences, systems do not tend towards an equilibrium. A famous example is a piece of metal that is heated on one end and cooled at the other. In this case, the system does not relax into equilibrium but remains in a steady state as heat flows from the hot to the cold end. The objective of this proposal is to describe these states in terms of various mathematical models ranging from basic optimization problems to large systems of interacting agents. We will consider optimization problems that are important in mathematics or ubiquitous in descriptions of physical phenomena, such as superconductivity. Our goal is to answer questions such as: How fast does a large system equilibrate or tend towards a steady state? What are the properties of these equilibria? If a system has some underlying symmetry, do equilibria have the same symmetry? The fascinating aspect of this research area is that problems run the gamut from ones that can be solved using standard methods to those that need considerable ingenuity for their solution. These characteristics also make it an ideal training ground for graduate students. The analysis of systems out of equilibrium is still a wide open field; e.g., the interaction of a physical system with heat baths is not well understood. Such systems do not tend towards equilibrium but instead towards a non-equilibrium steady state (NESS). The PI proposes to examine such phenomena within the frame work of master equations. One of the research goals is to find Lyapunov functions that allow to control the approach towards an NESS. The investigation of such systems often leads to variational problems for the gap as well as for the entropy production, and their solution requires functional inequalities. It is important to prove these inequalities in their sharp form and to know all functions for which there is equality. One direction of research is to investigate such inequalities using flows which is the idea of Lyapunov functions in reverse. This method has been successful in some highly non-trivial cases and it is expected that it will deliver not only sharp inequalities but also additional correction terms. The Polaron model which describes an electron interacting with the vibrations of a crystal, fits into this framework as well. This is one of the simplest examples of a quantum field theory. The strong coupling limit of the energy is well understood and it is proposed to study the more subtle problem, namely the strong coupling limit of the electron density.