Two-dimensional grain boundary networks: stochastic particle models and kinetic limits

We study kinetic theories for isotropic, two-dimensional grain boundary networks which evolve by curvature flow. The number densities $f_s(x,t)$ for $s$-sided grains, $s =1,2,\ldots$, of area $x$ at time $t$, are modeled by kinetic equations of the form $\partial_t f_s + v_s \partial_x f_s =j_s$. The velocity $v_s$ is given by the Mullins-von Neumann rule and the flux $j_s$ is determined by the topological transitions caused by the vanishing of grains and their edges. The foundations of such kinetic models are examined through simpler particle models for the evolution of grain size, as well as purely topological models for the evolution of trivalent maps. These models are used to characterize the parameter space for the flux $j_s$. Several kinetic models in the literature, as well as a new kinetic model, are simulated and compared with direct numerical simulations of mean curvature flow on a network. Existence and uniqueness of mild solutions to the kinetic equations with continuous initial data is established.