Doctoral Dissertations

Date of Award

8-2020

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Timothy Schulze

Committee Members

Carl Sundberg, Ohannes Karakashian, William Hix

Abstract

In the calculation of time evolution of an atomic system where a chemical reaction and/or diffusion occurs, off-lattice kinetic Monte Carlo methods can be used to overcome timescale and lattice based limitations from other methods such as Molecular Dynamics and kinetic Monte Carlo procedures. Off-lattice kinetic Monte Carlo methods rely on a harmonic approximation to Transition State Theory, in which the rate of the rare transitions from one energy minimum to a neighboring minimum require surmounting a minimum energy barrier on the Potential Energy Surface, which is found at an index-1 saddle point commonly referred to as a transition state. In modeling the evolution of an atomic system, it is desirable to find all the relevant transitions surrounding the current minimum to a neighboring minima. Due to the large number of minima on the potential energy surface, exhaustively searching the landscape for these saddle points is a challenging task. We introduce an Accumulation Plot, which examines the number of found index-1 saddle points as a function of successful searches. In most systems, the Accumulation Plot appears to grow with no bound. We investigate this behavior by examining the basins of attraction for index-1 saddle points on the energy landscape to help understand the difficulties with an exhaustive search. We will investigate the Accumulation Statistics with an eye toward understanding why the Accumulation Plot grows so slowly. Finally, we will discuss implementing Early Termination to a recently introduced Rejection scheme for off-lattice kinetic Monte Carlo in order to help further speed up the modeling of the evolution of atomic systems.

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