The investigators will study a variety of physical problems ranging from a probabilistic investigation of the rates of equilibration in many particle systems, to the stability of matter in the presence of a quantized radiation field. More specifically, in the first problem mentioned, they will be studying the distance from equilibrium of a many particle system as measured by entropy or related Lyapunov functions. The goal will be inequalities that yield realistic rates of approach to equilibrium for initial conditions that are far away from the equilibrium state. In recent work on the Kac model of molecular collisions, the investigators developed a method for controlling the rate of dissipation as a function of the number of molecules in the system. New work will extend this to other measures of the distance from equilibrium that is better suited to initial data far from equilibrium. The second problem is concerned with understanding the stability of matter problem for relativistic models with pair creation taken into account. Moreover, various questions concerning the self-energy of systems interacting with radiation and the existence of ground states will be investigated. All these problems will be treated from a non-perturbative perspective.
It is a fundamental observation that many laws of nature can be formulated as maximization or minimization problems. It is therefore important to describe the configurations of the physical systems that yield these minima or maxima, as well as understanding how these equilibrium configurations are reached by the system. This proposal investigates such problems in specific but fundamental models. These models are physically diverse: both classical and quantum, deterministic and probabilistic. The common mathematical thread that binds these problems, and many others discussed in the proposal, is that they all lead to challenging problems in the calculus of variations. Thus, they are tied together in that their solutions require new a--priori estimates, in the form of inequalities. This can be an estimate on the ground state energy in field theory or an estimate on the rate of dissipation in kinetic theory. Ideally, these inequalities should be sharp enough to serve as the basis of exact calculations, and have applicability to problems other than the one that motivated them, as in the previous work on the Kac model. What makes it especially attractive to work in this area is that it cuts across the boundaries of mathematics and other scientific disciplines.
