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1. Klobusicky, Joe J. Kinetic Limits of Piecewise Deterministic Markov Processes and Grain Boundary Coarsening.

Degree: PhD, Applied Mathematics, 2014, Brown University

URL: https://repository.library.brown.edu/studio/item/bdr:386230/

The subject of this thesis is the development of a
stochastic process that can be interpreted as a model of grain
coarsening, a central problem in material science. Namely, our
objective is the creation of piecewise-deterministic Markov
processes (PDMPs) on particles whose empirical densities converge
to solutions of kinetic equations. We begin with a study of a
simplified model, where particles drift toward the origin on
ℝ^{+}. When particles hit the origin, they are
redistributed to ℝ^{+} according to a fixed probability
density p(x). This ``dynamic shuffler" is shown to be an instance
of a PDMP. From the infinitesimal generator associated with the
dynamic shuffler, we then construct a martingale that we show is an
approximation to a weak solution to limiting kinetic equations of
the density of particles. Properties of these equations are then
investigated using Laplace transforms and renewal theory. We then
generalize the one-tier model by considering a k-species model,
where particles travel on several tiers, and are redistributed with
probability distributions that change with time. Obtaining the
``correct" martingale in this ``k-species model" involves an
augmented PDMP, which adds variables that keep track of certain
types of jumps. The main theorem of this thesis is the existence of
limiting kinetic equations for densities of each species. We end by
aligning the k-species model to a mean field model of grain
coarsening. Several basic properties of this model which are
inherent for grain systems are then shown. Finally, we run
simulations of the grain coarsening PDMP and raise several
conjectures on the universality of certain grain
statistics.
*Advisors/Committee Members: Menon, Govind (Director), Ramanan, Kavita (Reader), Pego, Robert (Reader).*

Subjects/Keywords: grain coarsening

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APA (6^{th} Edition):

Klobusicky, J. J. (2014). Kinetic Limits of Piecewise Deterministic Markov Processes and Grain Boundary Coarsening. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:386230/

Chicago Manual of Style (16^{th} Edition):

Klobusicky, Joe J. “Kinetic Limits of Piecewise Deterministic Markov Processes and Grain Boundary Coarsening.” 2014. Doctoral Dissertation, Brown University. Accessed November 27, 2020. https://repository.library.brown.edu/studio/item/bdr:386230/.

MLA Handbook (7^{th} Edition):

Klobusicky, Joe J. “Kinetic Limits of Piecewise Deterministic Markov Processes and Grain Boundary Coarsening.” 2014. Web. 27 Nov 2020.

Vancouver:

Klobusicky JJ. Kinetic Limits of Piecewise Deterministic Markov Processes and Grain Boundary Coarsening. [Internet] [Doctoral dissertation]. Brown University; 2014. [cited 2020 Nov 27]. Available from: https://repository.library.brown.edu/studio/item/bdr:386230/.

Council of Science Editors:

Klobusicky JJ. Kinetic Limits of Piecewise Deterministic Markov Processes and Grain Boundary Coarsening. [Doctoral Dissertation]. Brown University; 2014. Available from: https://repository.library.brown.edu/studio/item/bdr:386230/