Asympotic Problems in Random Transport

University Of Maryland, College Park, College Park MD

Investigators

Abstract

The main goal of this proposal is to investigate various aspects of transport phenomena. One type of problems concerns the motion of a particle in a random force field. We shall prove that the position, velocity, and energy of the particle, when scaled appropriately, converge, as time tends to infinity, to certain limiting processes. The limiting processes will be described explicitly in terms of the distribution of the force field. We shall also study measures carried by random velocity fields and describe their limiting behavior as time tends to infinity. Another set of problems concerns random perturbations of Hamiltonian flows. We shall strive to obtain a description of the effective behavior of randomly perturbed Hamiltonian flows on compact symplectic manifolds. A separate part of the proposal is devoted to an inverse problem for Gibbs fields. We shall prove that there always exist Gibbs fields with prescribed correlation functions, provided that the correlation functions satisfy certain consistency conditions. Most of the proposed problems naturally arise in the study of various phenomena in physics, oceanography, theory of turbulence, and statistical mechanics. Consider, for example, a particle moving in a force field, such as a charged particle in an electro-magnetic field. We intend to describe the behavior of the particle (such as its position, velocity and energy) based on the probabilistic properties of the field. The main assumption is that the time at which the particle is observed is much larger than the typical size of the force field. We shall also study the transport properties of random flows. One could think of a pollutant carried by atmospheric currents as a physical model for this type of problems. Based on the probabilistic properties of the flow, one wishes to describe the long time behavior of the substance carried by it. Many results in this direction have been obtained be several groups of researchers under various simplifying assumptions, such as the periodicity of the flow. We shall study the problem for a large class of physically relevant flows.