Speaker: Govind Menon, Brown University
Date: October 13, 2016
Title: A Kinetic Theory For 2D Grain Boundary Coursening
Abstract: An important recent development in applied mathematics is the growing interest in ‘stochastic topology’. Natural examples of stochastic topologies are provided by cellular networks such as a froth of soap bubbles, consisting of a large number of bubbles with different sizes and connectivity. This talk describes the interplay between kinetic theory, probability theory and topology in a basic example– isotropic 2D grain boundary networks. A fundamental aspect of these networks is the Mullins-von Neumann n − 6 rule: the rate of change of the area of a (topological) n-gon is proportional to n − 6. As a consequence, cells with fewer than 6 sides vanish in finite time, and the network coarsens. Numerical and physical experiments have revealed a form of statistical self-similarity in the long time dynamics that is not understood. We propose a kinetic description for the evolution of such networks. The ingredients in our model are an elementary N particle system that mimics essential features of the Mullins-von Neumann rule, and a hydrodynamic limit theorem for population densities when N → ∞. This theorem is broad enough to include all kinetic equations proposed by physicists in the 1980s and 1990s. In fact, many of these equations conflict with one another, and our model allows us to examine the foundations of each of the kinetic theories and to compare them with computational data. It also allows us to begin to attempt to connect kinetic theory with what are still preliminary attempts to understand the stochatic topology of cellular networks. This is joint work with Joe Klobusicky (Rensselaer Polytechnic Institute) and Bob Pego (Carnegie Mellon University).