Variational problems in physics

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The proposal is about disparate physical and mathematical applications that have the Calculus of Variations as a common theme. One area of inquiry is non-equilibrium statistical mechanics with the goal to understand simple aspects of non- equilibrium steady states in simple models, such as a Kac-type model describing the interaction of colliding particles with a thermostat. A second topic is related to symmetry breaking or the absence thereof in a class of non-linear variational problems. The plan is to find volution equations that let any function evolve into an optimizing function. This is a concept that has worked in a number of interesting cases already. The third topic concerns problems in quantum mechanics, in particular some problems about random Schroedinger operators with the goal to extend recent results on the Random Displacement Model to a larger class of geometries. The fourth topic that will be investigated, is the connection between the geometry of curves and certain special Lieb-Thirring-type inequalities. While these topics sound quite different, they all can be cast as optimization problems, and can lead to new and unexpected mathematical insights into the calculus of variations.Mathematics is important because of its universal nature, both as a language and as a toolbox for solving problems in science and engineering. The aim of this proposal is to understand a range of questions that arise from the physical sciences. A central problem is how to describe the evolution of large systems of interacting agents, be it colliding particles, flocking birds or individuals that exchange goods in an economy. Such systems are usually in an equilibrium but react to external disturbances. It is important to understand whether such a system returns to equilibrium and if so, how fast. One focus of this grant is on understanding these question in the context of large systems of colliding particles. A second area of inquiry concerns the effect of randomness on the motion of quantum mechanical particles. The goal is to understand in more detail than is presently known how randomly displaced obstacles influence the motion of an electron. An understanding of this question is important since it yields another way of modeling insulators. The mathematically fascinating aspect of this line of research is that most of these problems can be formulated as optimization problems, thus tapping into a large mathematical toolkit while at the same time adding to it. This research area also is an excellent training ground for students because they are forced to see beyond the purely mathematical aspect of a problem.