Behavior of the time-dependent Heisenberg spin system Stein's method and physically interesting processes

This dissertation explores the mean field Heisenberg spin system and its evolution in time. We first study the system in equilibrium, where we explore the tool known as Stein's method, used for determining convergence rates to thermodynamic limits, both in an example proof for a mean field Ising system and in tightening a previous result for the equilibrium mean field Heisenberg system. We then model the evolution of the mean field Heisenberg model using Glauber dynamics and use this method to test the equilibrium results of two previous papers, uncovering a typographical error in one. Agreement in other aspects between theory and our simulations validates our approach in the equilibrium case. Next, we compare the evolution of the Heisenberg system under Glauber dynamics to a number of forms of Brownian motion and determine that Brownian motion is a poor match in most situations. Turning back to Stein's method, we consider what sort of proof regarding the behavior of the mean field Heisenberg model over time is obtainable and look at several possible routes to that path. We finish up by offering a Stein's method approach to understanding the evolution of the mean field Heisenberg model and offer some insight into its convergence in time to a thermodynamic limit. This demonstrates the potential usefulness of Stein's method in understanding the finite time behavior of evolving systems. In our efforts, we encounter several holes in current mathematical and physical knowledge. In particular, we suggest the development of tools for Markov chains currently unavailable and the development of a more physically based algorithm for the evolution of Heisenberg systems. These projects lie beyond the scope of this dissertation but it is our hope that these ideas may be useful to future research.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Physics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2020

Author dc:creator

• Ross, Leslie

Contributors dc:contributor

• Kirkpatrick, Kay
• Mason, Nadya
• Stone, Michael
• Katz, Sheldon

• Stein's Method
• Glauber Dynamics
• Heisenberg Spin Systems
• Spin Systems
• Brownian Motion

Statement dc:rights

• Copyright 2019 Leslie Ross

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/106198

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/106198