Understanding disordered systems through numerical simulation and algorithm development
Sean Michael Sweeney · Syracuse University Libraries (Syracuse University) · 2015
Disordered systems arise in many physical contexts. Not all matter is uni- form, and impurities or heterogeneities can be modeled by fixed random disor- der. Numerous complex networks also possess fixed disorder, leading to appli- cations in transportation systems [1], telecommunications [2], social networks [3, 4], and epidemic modeling [5], to name a few. Due to their random nature and power law critical behavior, disordered systems are difficult to study analytically. Numerical simulation can help overcome this hurdle by allowing for the rapid computation of system states. In order to get precise statistics and extrapolate to the thermodynamic limit, large systems must be studied over many realizations. Thus, innovative al- gorithm development is essential in order reduce memory or running time requirements of simulations. This thesis presents a review of disordered systems, as well as a thorough study of two particular systems through numerical simulation, algorithm de- velopment and optimization, and careful statistical analysis of scaling proper- ties. Chapter 1 provides a thorough overview of disordered systems, the his- tory of their study in the physics community, and the development of tech- niques used to study them. Topics of quenched disorder, phase transitions, the renormalization group, criticality, and scale invariance are discussed. Several prominent models of disordered systems are also explained. Lastly, analysis techniques used in studying disordered systems are covered. In Chapter 2, minimal spanning trees on critical percolation clusters are studied, motivated in part by an analytic perturbation expansion by Jackson and Read [6] that I check against numerical calculations. This system has a direct mapping to the ground state of the strongly disordered spin glass [7]. We compute the path length fractal dimension of these trees in dimensions d = {2, 3, 4, 5} and find our results to be compatible with the analytic results suggested by Jackson and Read. In Chapter 3, the random bond Ising ferromagnet is studied, which is es- pecially useful since it serves as a prototype for more complicated disordered systems such as the random field Ising model and spin glasses. We investigate the effect that changing boundary spins has on the locations of domain walls in the interior of the random ferromagnet system. We provide an analytic proof that ground state domain walls in the two dimensional system are de- composable, and we map these domain walls to a shortest paths problem. By implementing a multiple-source shortest paths algorithm developed by Philip Klein [8], we are able to efficiently probe domain wall locations for all possible configurations of boundary spins. We consider lattices with uncorrelated dis- order, as well as disorder that is spatially correlated according to a power law. We present numerical results for the scaling exponent governing the probabil- ity that a domain wall can be induced that passes through a particular location in the system’s interior, and we compare these results to previous results on the directed polymer problem.