{"@context":"http://iiif.io/api/presentation/2/context.json","@id":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/manifest.json","@type":"sc:Manifest","label":"Potts Model and Generalizations: Exact Results and Statistical Physics","metadata":[{"label":"dc.description.sponsorship","value":"This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree."},{"label":"dc.format","value":"Monograph"},{"label":"dc.format.medium","value":"Electronic Resource"},{"label":"dc.identifier.uri","value":"http://hdl.handle.net/1951/59921"},{"label":"dc.language.iso","value":"en_US"},{"label":"dc.publisher","value":"The Graduate School, Stony Brook University: Stony Brook, NY."},{"label":"dcterms.abstract","value":"The q-state Potts model is a spin model that has been of longstanding interest as a many body system in statistical mechanics. Via a cluster expansion, the Potts model partition function Z(G,q,v), defined on a graph G=(V,E), where V is the set of vertices (sites) and E is the set of edges (bonds), is expressed as a polynomial in terms of q and a temperature-dependent Boltzmann variable v. An important special case (v=-1) is the zero-temperature Potts antiferromagnet, for which Z(G,q,-1)=P(G,q), where P(G,q) is the chromatic polynomial, counting the number of ways of assigning q colors to the vertices of graph G such that no two adjacent vertices have the same color. A natural generalization is to consider this model in a generalized external field that favors or disfavors spin values in a subset I<sub>s</sub>={1, . . . ,s} of the total set of q-state spin values. In this dissertation, we calculate the exact partition functions of the generalized Potts model Z(G,q,s,v,w), where w is a field-dependent Boltzmann variable, for certain families of graphs. We also investigate its special case, viz. Z(G,q,s,-1,w)=Ph(G,q,s,w), which describes a weighted-set graph coloring problem. Nonzero ground-state entropy (per lattice site), S<sub>0</sub>>0, is an important subject in statistical physics, as an exception to the third law of thermodynamics and a phenomenon involving large disorder even at zero temperature. The q-state Potts antiferromagnet is a model exhibiting ground-state entropy for sufficiently large q on a given lattice graph. Another part of the dissertation is devoted to the study of ground-state entropy, for which lower bounds on slabs of the simple cubic lattice and exact results on homeomorphic expansions of kagom&eacute lattice strips are presented. Next, we focus on the structure of chromatic polynomials for a particular class of graphs, viz. planar triangulations, and discuss implications for chromatic zeros and some asymptotic limiting quantities."},{"label":"dcterms.available","value":"2013-05-22T17:35:50Z"},{"label":"dcterms.contributor","value":"Abanov, Alexander"},{"label":"dcterms.creator","value":"Xu, Yan"},{"label":"dcterms.dateAccepted","value":"2013-05-22T17:35:50Z"},{"label":"dcterms.dateSubmitted","value":"2013-05-22T17:35:50Z"},{"label":"dcterms.description","value":"Department of Physics"},{"label":"dcterms.extent","value":"135 pg."},{"label":"dcterms.format","value":"Monograph"},{"label":"dcterms.identifier","value":"Xu_grad.sunysb_0771E_10867"},{"label":"dcterms.issued","value":"2012-05-01"},{"label":"dcterms.language","value":"en_US"},{"label":"dcterms.provenance","value":"Made available in DSpace on 2015-04-24T14:47:38Z (GMT). No. of bitstreams: 3\nXu_grad.sunysb_0771E_10867.pdf.jpg: 1894 bytes, checksum: a6009c46e6ec8251b348085684cba80d (MD5)\nXu_grad.sunysb_0771E_10867.pdf.txt: 223961 bytes, checksum: 3bc22cb0b5f6b6aad8bbf298dd8332e7 (MD5)\nXu_grad.sunysb_0771E_10867.pdf: 1546960 bytes, checksum: 582784d319ebd15ee6d674c12f23ea8d (MD5)\n  Previous issue date:    1"},{"label":"dcterms.publisher","value":"The Graduate School, Stony Brook University: Stony Brook, NY."},{"label":"dcterms.subject","value":"Chromatic polynomial, Ground state entropy, Planar triangulation, Potts model in an external field, Statistical physics, Weighted-set graph colorings"},{"label":"dcterms.title","value":"Potts Model and Generalizations: Exact Results and Statistical Physics"},{"label":"dcterms.type","value":"Dissertation"},{"label":"dc.type","value":"Dissertation"}],"description":"This manifest was generated dynamically","viewingDirection":"left-to-right","sequences":[{"@type":"sc:Sequence","canvases":[{"@id":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/canvas/page-1.json","@type":"sc:Canvas","label":"Page 1","height":1650,"width":1275,"images":[{"@type":"oa:Annotation","motivation":"sc:painting","resource":{"@id":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/98%2F76%2F03%2F9876035591909272935723669012020741638/full/full/0/default.jpg","@type":"dctypes:Image","format":"image/jpeg","height":1650,"width":1275,"service":{"@context":"http://iiif.io/api/image/2/context.json","@id":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/98%2F76%2F03%2F9876035591909272935723669012020741638","profile":"http://iiif.io/api/image/2/level2.json"}},"on":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/canvas/page-1.json"}]}]}]}