(Georgia Institute of Technology, 2021-12-14)
Hauger, Lukas Alexander
We study the approach to equilibrium in relative entropy of systems of gas particles modeled via the Kac master equation in arbitrary dimensions. First, we study the Kac system coupled to a thermostat, and secondly connected to a heat reservoir. The use of the Fisher-information allows simple proofs with weak regularity assumptions. As a result, we obtain exponential decay rates for the entropy and information that are essentially independent of the size of the systems.
(Georgia Institute of Technology, 2021-04-13)
Ma, Yuanzhe
In this thesis, our main goal is to use numerical simulations to study some quantities related to the growing set B(t). Motivated by prior works, we mainly study quantities including the boundary size, the hole size, and the location of each hole for B(t). We discuss the theoretical background of this work, the algorithm we used to conduct simulations, and include an extensive discussion of our simulation results. Our results support some of the prior conjectures and further introduce several interesting open problems.
(Georgia Institute of Technology, 2020-05-05)
Li, Jiaheng
The main object of this thesis is to numerically estimate some conjectured arm exponents when there exist a number of open paths and closed dual paths that extend to the boundary of different sizes of boxes centering at the origin in bond invasion percolation on a plane square lattice by Monte-Carlo simulations. The results turn out to be supportive of the conjectured value in some case. The numerical estimate for the acceptance profile of invasion percolation at the critical point is also obtained, which suggests a neighborhood in which the liminf and limsup of the acceptance profile might fall. An efficient algorithm to simulate invasion percolation and to find disjoint paths on most regular 2-dimensional lattices are discussed.