Diffusion In Fuzzy Lattice Systems: Exploring the Anomalous Regime, Connecting the Steady-State, and Fat-Tailed Distributions

Abstract

Diffusion and random walks have been studied for more than 100 years. However, there are still details in the methodology that are overlooked, and more information can be extracted from the typical data that is studied. In this thesis, I simulate random walks on two dimensional lattices with immobile obstacles configured in a variety of ways: periodic, random, and "Fuzzy" (a cross intermediate state of disorder between periodic and random). The primary goal is to develop a deeper understanding of "Fuzzy" systems by designing different ways of generating tunable disorder. An example of this is the universal Fz parameter that we developed to unify the natural disorder parameters of the various disorder generation methods we developed. Often times the importance of analysing the transient/anomalous regime with more precision and consistency is overlooked. In our work, we expand on random walk dynamics by applying non-standard probabilities, and justify our choice analytically and through a comparison of results. Furthermore we discuss how the transient regime should be analyzed so that there is consistency in the field. Other than discussing semantics of algorithms and analysis, we study the connection between the transient regime and the steady-state. We introduce two measures of the width of the transient/anomalous regime, and compare them to the crossover time. Using the width of the transient/anomalous regime we are able to provide an estimate of the steady-state diffusion coefficient without access to the steady-state simulation data.