| Many important problems in science and economics can be phrased in the language of probability. This thesis is concerned with a problem in Materials Science, namely Grain Growth. The study of how these grain boundaries evolve has important applications to the design of materials that possess a resistance to wear over time and lower temperatures, and also to the reliability of microelectronic devices that employ thin films as interconnects for integrated circuits. These are only two of many examples of how we encounter grains in our everyday life. An excellent reference for more examples and the physical description of the growth process is the thesis by Archibald [Arch].;In the following work, we study a system of nonlinear transport equations modeling grain growth first proposed by Barmak et al [Barmak]. This system of differential equations is derived empirically, and matches the evolution of the grain network with the evolution of its statistical properties. Although this view reduces the number of equations to solve from the order of tens of thousands to the order of tens, the coupled, nonlinear, and nonlocal system that result are difficult to analyze using standard methods from analysis alone. However, when we add tools from probability, such as Functional Integration, (first described by Mckean and later by Freidlin [F'reidlin],) and Martingales, then the problem takes on a life of it's own. The result we prove in this thesis is not only existence of a solution to the differential equations, but the existence of a stochastic process that is intertwined with the solution of the system.;One could begin reading this thesis from Chapter 3, where we simplify the grain network to allow only {5, 6, 7} - sided grains, and then prove existence via compactness methods. However, this method of attack quickly shows it's limitations when we extend our gaze to the complete network. The specific form of the switching rates proposed by Barmak et al. forces us to consider a stochastic interpretation. In the {5, 6, 7} - sided version, we construct the existence of integrals of these solutions via fixed-point iteration, and we take this as a clue that iterative schemes should continue to play a role in our analysis. By combining the martingale view of hyperbolic systems first proposed by Heath [Heath] in his thesis with the linearization scheme proposed by Freidlin [Freidlin] for nonlinear diffusion equations, we are able to construct a limiting pair of stochastic process and generalized solution to the system of partial differential equations. |
