Understanding Disordered Systems Through Numerical Simulation and Algorithm Development

<p>Disordered systems arise in many physical contexts. Not all matter is uni-</p> <p>form, and impurities or heterogeneities can be modeled by fixed random disor-</p> <p>der. Numerous complex networks also possess fixed disorder, leading to appli-</p> <p>cations in transportation systems [1], telecommunications [2], social networks</p> <p>[3, 4], and epidemic modeling [5], to name a few.</p> <p>Due to their random nature and power law critical behavior, disordered</p> <p>systems are difficult to study analytically. Numerical simulation can help</p> <p>overcome this hurdle by allowing for the rapid computation of system states.</p> <p>In order to get precise statistics and extrapolate to the thermodynamic limit,</p> <p>large systems must be studied over many realizations. Thus, innovative al-</p> <p>gorithm development is essential in order reduce memory or running time</p> <p>requirements of simulations.</p> <p>This thesis presents a review of disordered systems, as well as a thorough</p> <p>study of two particular systems through numerical simulation, algorithm de-</p> <p>velopment and optimization, and careful statistical analysis of scaling proper-</p> <p>ties.</p> <p>Chapter 1 provides a thorough overview of disordered systems, the his-</p> <p>tory of their study in the physics community, and the development of tech-</p> <p>niques used to study them. Topics of quenched disorder, phase transitions, the</p> <p>renormalization group, criticality, and scale invariance are discussed. Several</p> <p>prominent models of disordered systems are also explained. Lastly, analysis</p> <p>techniques used in studying disordered systems are covered.</p> <p>In Chapter 2, minimal spanning trees on critical percolation clusters are</p> <p>studied, motivated in part by an analytic perturbation expansion by Jackson</p> <p>and Read [6] that I check against numerical calculations. This system has a</p> <p>direct mapping to the ground state of the strongly disordered spin glass [7].</p> <p>We compute the path length fractal dimension of these trees in dimensions</p> <p>d = {2, 3, 4, 5} and find our results to be compatible with the analytic results</p> <p>suggested by Jackson and Read.</p> <p>In Chapter 3, the random bond Ising ferromagnet is studied, which is es-</p> <p>pecially useful since it serves as a prototype for more complicated disordered</p> <p>systems such as the random field Ising model and spin glasses. We investigate</p> <p>the effect that changing boundary spins has on the locations of domain walls</p> <p>in the interior of the random ferromagnet system. We provide an analytic</p> <p>proof that ground state domain walls in the two dimensional system are de-</p> <p>composable, and we map these domain walls to a shortest paths problem. By</p> <p>implementing a multiple-source shortest paths algorithm developed by Philip</p> <p>Klein [8], we are able to efficiently probe domain wall locations for all possible</p> <p>configurations of boundary spins. We consider lattices with uncorrelated dis-</p> <p>order, as well as disorder that is spatially correlated according to a power law.</p> <p>We present numerical results for the scaling exponent governing the probabil-</p> <p>ity that a domain wall can be induced that passes through a particular location</p> <p>in the system’s interior, and we compare these results to previous results on</p> <p>the directed polymer problem.</p>

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Physics

Year

2015

Author dc:creator

• Sweeney, Sean M.

Contributors dc:contributor

• A. Alan Middleton

• Combinatorics
• Ising Model
• Minimal Spanning Tree
• Percolation
• Scaling
• Shortest Path
• Physical Sciences and Mathematics

Repository record dc:identifier

https://surface.syr.edu/etd/407

OAI identifier oai:identifier

oai:surface.syr.edu:etd-1407