Mesoscale Systems

ABSTRACT: By their very nature, systems viewed at the mesoscale level interpolate among space and time scales. We would like to introduce some methods for investigating some of these systems and to decide if we can infer something about their behavior. Frequently we are confronted with metastable processes or at lease processes with no clear way to know that we are close ? whatever that may mean ? to equilibrium. To begin we discuss basics of mass transfer problems and some general coarse graining issues, including some work of Brenier and Gangbo. A simple example from magnetism, and an analogy with critical opalescence, will illustrate the need for introducing mechanisms for evolution or dynamics. We describe how this can be done in a practical setting. We shall then discuss some favorite mesocscale systems. These include: the evolution of microstructure; diffusion mediated transport and the Brownian rachet, implicated in molecular motor function; and some gambling. The general ideas give rise to new approaches to understanding grain growth in polycrystals. We explore this and discuss some fundamental questions that arise when modern experimental methods meet mathematics.