A kinetic model for 2D grain boundary coarsening

A fundamental aspect of 2D cellular networks with isotropic line tension 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.

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 von Neumann rule, and a hydrodynamic limit theorem for population densities when $N \rightarrow \infty$. This model is compared with a set of models derived in the physics and materials science communities, as well as extensive numerical simulations by applied mathematicians.

This is joint work with Joe Klobusicky (Brown University and Geisinger Health Systems) and Bob Pego (Carnegie Mellon University).