{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/102487"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/102487","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Spatial statistical-physical systems","abstract":"Statistical physics, as a branch of modern physics, uses methods of probability theory and statistics to solve physical problems with large populations and approximations. In this thesis, we use numerical simulations to study two statistical physical models — Ising model under topological constraints and particle systems with anomalous behavior. Ising model is a mathematical model of ferromagnetism, which describes how magnetic spins, with values -1 or 1, change their states under nearest neighbor interactions and the external magnetic field. We study a topologically constrained Ising model, where several pre-selected anchored sites are fixed to be value 1, and the topology of the active domain (the union of all value 1 sites) remains invariant under the evolution of the system. When the sites change their values with less preference of 1, the system tends to an equilibrium that approximates the Steiner tree structure. For two- to four-anchor cases, we calculate the theoretical equilibrium configurations, and in particular for three and four anchors, the positions of the Steiner points. For one-anchor case, we consider a reversed model that a single active site grows to a coral-shape active domain. In all analysis, we provide simulation results for verification. The second part of the thesis is devoted to study particle system with anomalous behavior. Anomalous behavior originates from the Braess Paradox, which states that adding an extra path to a network could in some cases impede the overall performance. We study and reproduce a spring-string model by Cohen and Horowitz in mechanical network exhibiting such paradoxical behavior. We simulate their model in two different ways and in both ways the anomalous behavior is observed. We also identify the conditions of the system parameters for the anomalous behavior and verify our theoretical results via simulations.","abstract_html":"Statistical physics, as a branch of modern physics, uses methods of probability theory and statistics to solve physical problems with large populations and approximations. In this thesis, we use numerical simulations to study two statistical physical models — Ising model under topological constraints and particle systems with anomalous behavior. Ising model is a mathematical model of ferromagnetism, which describes how magnetic spins, with values -1 or 1, change their states under nearest neighbor interactions and the external magnetic field. We study a topologically constrained Ising model, where several pre-selected anchored sites are fixed to be value 1, and the topology of the active domain (the union of all value 1 sites) remains invariant under the evolution of the system. When the sites change their values with less preference of 1, the system tends to an equilibrium that approximates the Steiner tree structure. For two- to four-anchor cases, we calculate the theoretical equilibrium configurations, and in particular for three and four anchors, the positions of the Steiner points. For one-anchor case, we consider a reversed model that a single active site grows to a coral-shape active domain. In all analysis, we provide simulation results for verification. The second part of the thesis is devoted to study particle system with anomalous behavior. Anomalous behavior originates from the Braess Paradox, which states that adding an extra path to a network could in some cases impede the overall performance. We study and reproduce a spring-string model by Cohen and Horowitz in mechanical network exhibiting such paradoxical behavior. We simulate their model in two different ways and in both ways the anomalous behavior is observed. We also identify the conditions of the system parameters for the anomalous behavior and verify our theoretical results via simulations.","abstract_has_math":false,"creators":["Wang, Xiao"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Baryshnikov, Yuliy","Song, Renming","DeVille, Lee","Kirkpatrick, Kay"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-02-06T19:36:37Z","date_published":"2019-02-06T19:36:37Z","updated_at":"2026-07-22T22:24:42Z","subjects":["Particle systems","Anomalous behavior","Braess paradox","Ising model","Simulations."],"languages":["en"],"rights":["2018 by Xiao Wang. All rights reserved."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/102487","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Baryshnikov, Yuliy","Song, Renming","DeVille, Lee","Kirkpatrick, Kay"]},{"key":"dc:creator","label":"Author","values":["Wang, Xiao"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2019-02-06T19:36:37Z","2018-12-06","2018-12"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Particle systems","Anomalous behavior","Braess paradox","Ising model","Simulations."]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["2018 by Xiao Wang. All rights reserved."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/102487"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Statistical physics, as a branch of modern physics, uses methods of probability theory and statistics to solve physical problems with large populations and approximations. In this thesis, we use numerical simulations to study two statistical physical models — Ising model under topological constraints and particle systems with anomalous behavior. Ising model is a mathematical model of ferromagnetism, which describes how magnetic spins, with values -1 or 1, change their states under nearest neighbor interactions and the external magnetic field. We study a topologically constrained Ising model, where several pre-selected anchored sites are fixed to be value 1, and the topology of the active domain (the union of all value 1 sites) remains invariant under the evolution of the system. When the sites change their values with less preference of 1, the system tends to an equilibrium that approximates the Steiner tree structure. For two- to four-anchor cases, we calculate the theoretical equilibrium configurations, and in particular for three and four anchors, the positions of the Steiner points. For one-anchor case, we consider a reversed model that a single active site grows to a coral-shape active domain. In all analysis, we provide simulation results for verification. The second part of the thesis is devoted to study particle system with anomalous behavior. Anomalous behavior originates from the Braess Paradox, which states that adding an extra path to a network could in some cases impede the overall performance. We study and reproduce a spring-string model by Cohen and Horowitz in mechanical network exhibiting such paradoxical behavior. We simulate their model in two different ways and in both ways the anomalous behavior is observed. We also identify the conditions of the system parameters for the anomalous behavior and verify our theoretical results via simulations.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2019-02-05 without embargo terms","The student, Xiao Wang, accepted the attached license on 2018-12-05 at 23:56.","The student, Xiao Wang, submitted this Dissertation for approval on 2018-12-06 at 00:08.","This Dissertation was approved for publication on 2018-12-06 at 11:50.","DSpace SAF Submission Ingestion Package generated from Vireo submission #13228 on 2019-02-05 at 11:15:18","Made available in DSpace on 2019-02-06T19:36:37Z (GMT). No. of bitstreams: 2 WANG-DISSERTATION-2018.pdf: 4185552 bytes, checksum: e3d26ff249f7c25a777ed57966bf29f9 (MD5) LICENSE.txt: 4206 bytes, checksum: 25d8ca591f1a2e11dd17cf0f08d31c23 (MD5) Previous issue date: 2018-12-06"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Spatial statistical-physical systems"]}]}],"canonical_facts":{"dc:contributor":["Baryshnikov, Yuliy","Song, Renming","DeVille, Lee","Kirkpatrick, Kay"],"dc:creator":["Wang, Xiao"],"dc:date":["2019-02-06T19:36:37Z","2018-12-06","2018-12"],"dc:description":["Statistical physics, as a branch of modern physics, uses methods of probability theory and statistics to solve physical problems with large populations and approximations. In this thesis, we use numerical simulations to study two statistical physical models — Ising model under topological constraints and particle systems with anomalous behavior. Ising model is a mathematical model of ferromagnetism, which describes how magnetic spins, with values -1 or 1, change their states under nearest neighbor interactions and the external magnetic field. We study a topologically constrained Ising model, where several pre-selected anchored sites are fixed to be value 1, and the topology of the active domain (the union of all value 1 sites) remains invariant under the evolution of the system. When the sites change their values with less preference of 1, the system tends to an equilibrium that approximates the Steiner tree structure. For two- to four-anchor cases, we calculate the theoretical equilibrium configurations, and in particular for three and four anchors, the positions of the Steiner points. For one-anchor case, we consider a reversed model that a single active site grows to a coral-shape active domain. In all analysis, we provide simulation results for verification. The second part of the thesis is devoted to study particle system with anomalous behavior. Anomalous behavior originates from the Braess Paradox, which states that adding an extra path to a network could in some cases impede the overall performance. We study and reproduce a spring-string model by Cohen and Horowitz in mechanical network exhibiting such paradoxical behavior. We simulate their model in two different ways and in both ways the anomalous behavior is observed. We also identify the conditions of the system parameters for the anomalous behavior and verify our theoretical results via simulations.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2019-02-05 without embargo terms","The student, Xiao Wang, accepted the attached license on 2018-12-05 at 23:56.","The student, Xiao Wang, submitted this Dissertation for approval on 2018-12-06 at 00:08.","This Dissertation was approved for publication on 2018-12-06 at 11:50.","DSpace SAF Submission Ingestion Package generated from Vireo submission #13228 on 2019-02-05 at 11:15:18","Made available in DSpace on 2019-02-06T19:36:37Z (GMT). No. of bitstreams: 2 WANG-DISSERTATION-2018.pdf: 4185552 bytes, checksum: e3d26ff249f7c25a777ed57966bf29f9 (MD5) LICENSE.txt: 4206 bytes, checksum: 25d8ca591f1a2e11dd17cf0f08d31c23 (MD5) Previous issue date: 2018-12-06"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/102487"],"dc:language":["en"],"dc:rights":["2018 by Xiao Wang. All rights reserved."],"dc:subject":["Particle systems","Anomalous behavior","Braess paradox","Ising model","Simulations."],"dc:title":["Spatial statistical-physical systems"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:42Z"}