Problems in Particle and Interface Models

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

This project studies stochastic processes that model complex, interacting behavior, such as interacting particle systems and interface models. Basic examples of particle systems studied are exclusion-type processes and Hammersley's process. Special situations studied in this project include bottlenecks, traffic jams, boundaries between different phases, priorities between individuals, and other irregularities that disturb the individual particles. An interface model describes the growth or decay of one phase relative to another. Examples of mathematical models in this category are last-passage models, models of increasing sequences, and marching soldiers models. The questions addressed are the speed, the eventual shape, and the roughness of the interface. The results of this project will concern the macroscopic behavior of these models and the fluctuations and deviations around the expected behavior. A basic feature of the world around us is that natural processes occur at several different scales. Large scale features are formed through the combined effect of a vast number of operators on a smaller scale. This can be observed in physical processes, biological processes, and man-made processes. For example, the disorganized motions and collisions of individual fluid particles make up the flow of a river; competition and cooperation at the level of individuals determines the evolution of a population; and the behavior of vehicles on a freeway determines the flow of traffic. Many different complex systems share common basic principles of organization and behavior. Such shared features can be abstracted in mathematical models, and analysis of these models then has implications for many particular applications. This project is about the mathematical study of such models, and the outcome of this project is a better understanding of the behavior of certain classes of models of complex systems.