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University of Notre Dame

1. Seng Kai Wong. A Cursory Study of the Thermodynamic and Mechanical Properties of Monte-Carlo Simulations of the Ising Model</h1>.

Degree: PhD, Physics, 2005, University of Notre Dame

URL: https://curate.nd.edu/show/9z902z12z0s

The Ising model has been very successful in
simulating ferromagnetic and antiferromagnetic materials. It is, in
fact, the pedagogical explanation for the behavior of magnetic
materials. Researchers have adapted the Ising model to simulate
materials as disparate as foams, cell aggregates and metallic
crystals. <pre><code>In this research, I used
different modifications of Metropolis algorithms to simulate
</code></pre> phase separation and Brownian
motion in the Ising model. One of the algorithms (Algorithm Three
in the text) is a choice popular with many researchers. My goal is
to illuminate the differences in the dynamical and equilibrium
properties of various algorithms and model parameters.
<pre><code>I found that the most popular choice is not
always the right choice. It produces a non-Boltzmann equilibrium
and its simulated </code></pre> droplets
diffuse much slower than other near-Boltzmann algorithms. In fact,
the non-Boltzmann algorithm does not have a critical point, while
the others do. <pre><code>In my phase
separation simulations, I investigated a number of thermodynamical
properties of the two-dimensional Ising model,
</code></pre> including the surface energy,
surface tension, partition function, free energy and entropy as a
function of model parameters and algorithms.
<pre><code>To obtain a functional form for analyzing
phase-separation, I developed a second-nearest neighbor
Solid-On-Solid (SOS) model. I found that the SOS model agrees well
with the Ising model up to about one-half the critical temperature.
I also used heuristic arguments to create a modified SOS model and
compared it to simulation results for up to fourth-nearest-neighbor
interactions. I discovered unexpected behavior when I used the
model to simulate Brownian motion. For all the algorithms, droplets
diffuse slower than </code></pre> predicted at
low temperatures, which I explain by assuming that the underlying
lattice is sticky. <pre><code>One could
devise more tests to further distinguish and delineate the
limitations of the algorithms, like simulating Stoke's flow. When
one </code></pre> modifies or add more terms
to the Ising Hamiltonian to simulate different phenomena, one needs
to modify the algorithmâ€™s acceptance probabilities accordingly in
order to preserve detailed balance and Boltzmann equilibrium. I
have presented a clear method to create algorithms that obey
detailed balance and produce near-Boltzmann
equilibria.
*Advisors/Committee Members: Kathie E. Newman, Committee Co-Chair, Jonathan R. Sapirstein, Committee Member, Jacek K. Furdyna, Committee Member, James A. Glazier, Committee Chair, Gerald L. Jones, Committee Member.*

Subjects/Keywords: Thermodynamic; Monte Carlo; Phase Separation; Brownian Motion; Ising Model; Solid-On-Solid Model

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APA (6^{th} Edition):

Wong, S. K. (2005). A Cursory Study of the Thermodynamic and Mechanical Properties of Monte-Carlo Simulations of the Ising Model</h1>. (Doctoral Dissertation). University of Notre Dame. Retrieved from https://curate.nd.edu/show/9z902z12z0s

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

Wong, Seng Kai. “A Cursory Study of the Thermodynamic and Mechanical Properties of Monte-Carlo Simulations of the Ising Model</h1>.” 2005. Doctoral Dissertation, University of Notre Dame. Accessed January 25, 2020. https://curate.nd.edu/show/9z902z12z0s.

MLA Handbook (7^{th} Edition):

Wong, Seng Kai. “A Cursory Study of the Thermodynamic and Mechanical Properties of Monte-Carlo Simulations of the Ising Model</h1>.” 2005. Web. 25 Jan 2020.

Vancouver:

Wong SK. A Cursory Study of the Thermodynamic and Mechanical Properties of Monte-Carlo Simulations of the Ising Model</h1>. [Internet] [Doctoral dissertation]. University of Notre Dame; 2005. [cited 2020 Jan 25]. Available from: https://curate.nd.edu/show/9z902z12z0s.

Council of Science Editors:

Wong SK. A Cursory Study of the Thermodynamic and Mechanical Properties of Monte-Carlo Simulations of the Ising Model</h1>. [Doctoral Dissertation]. University of Notre Dame; 2005. Available from: https://curate.nd.edu/show/9z902z12z0s