Mesoscale Systems
David Kinderlehrer
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.
David Kinderlehrer
Carnegie Mellon University
Department of Mathematical Sciences
Pittsburgh, PA 15213
email: davidk@andrew.cmu.edu
