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Discontinuous percolation transitions in cluster merging processes : 클러스터 결합 과정에서의 불연속 여과 상전이
DC Field | Value | Language |
---|---|---|
dc.contributor.advisor | 강병남 | - |
dc.contributor.author | 조영설 | - |
dc.date.accessioned | 2017-07-19T06:07:02Z | - |
dc.date.available | 2017-07-19T06:07:02Z | - |
dc.date.issued | 2015-02 | - |
dc.identifier.other | 000000025294 | - |
dc.identifier.uri | http://dcollection.snu.ac.kr:80/jsp/common/DcLoOrgPer.jsp?sItemId=000000025294 | - |
dc.description | 학위논문(박사)--서울대학교 대학원 :자연과학대학 물리·천문학부,2015. 2. 강병남. | - |
dc.description.abstract | The notion of percolation transition is widely applied in a variety of disciplines;
it explains the formation of a spanning cluster connecting two opposite sides of a system in Euclidean space, such as occurs in metal-insulator or sol-gel transitions. Alternatively, percolation can also be interpreted as the formation of a macroscopic cluster in the system. One of the models of this category is the Erd˝os- R´enyi model. In this model, starting with N isolated nodes, an edge is connected between a randomly selected unconnected pair of nodes at each time step. Then, a macroscopic cluster is generated continuously at the percolation threshold. Recently, the Erd˝os-R´enyi model was modified by imposing additionally a so-called product rule, which suppresses the formation of a large cluster. Because of this suppressive bias, the percolation threshold is delayed; thus, when the giant cluster eventually emerges, it does so explosively. Hence, this kind of model is called the explosive percolation model. We studied the explosive percolation models in various aspects and found that many unusual behaviors revealed in this model. Initially, this explosive percolation model was regarded as a model showing a discontinuous transition; however, it was recently found that the transition is continuous in the thermodynamic limit. In these circumstances, it is needed to set up a theoretical basis to understand the discontinuous percolation transition. For this, we classify the discontinuous percolation transition into Type-I and Type-II and provide a necessary condition for each type. These necessary conditions successfully explain the origin of the discontinuous percolation transitions in unified framework. Finally, we suggest a diffusion-limited cluster aggregation model as one example for physical model showing discontinuous transition and calculate conductivity in a discontinuous percolation transition model. | - |
dc.description.tableofcontents | Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Discontinuous percolation transitions in non-cluster merging processes . . . . . . . . . . . . . . . . . . . 3 1.2.1 Bootstrap (k-core) percolation . . . . . . . . . . . . . . . 3 1.2.2 Percolation transition of the mutually connected clusters . 5 1.3 Explosive percolation model: First trial to find discontinuous percolation transition in cluster merging processes . . . . . . . . . . 5 2. Explosive percolation model . . . . . . . . . . . . . . . . . . . . . 9 2.1 Explosive percolation model on scale-free networks . . . . . . . . 9 2.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.2 Model description . . . . . . . . . . . . . . . . . . . . . 10 2.1.3 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Finite-size scaling for discontinuous percolation transition . . . . 18 iii 2.2.1 Finite-size scaling ansatz for discontinuous percolation transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.2 Physical meaning of the onset of the discontinuous percolation transition . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Continuity and discontinuity in the explosive percolation models . 24 2.4 Suppression effect on the explosive percolation models . . . . . . 24 2.4.1 Suppression principle failure . . . . . . . . . . . . . . . . 25 2.4.2 Modified detailed rules . . . . . . . . . . . . . . . . . . . 28 2.4.3 Application of the modified detailed rules to explosive percolation models . . . . . . . . . . . . . . . . . . . . . . . 29 3. Type-I discontinuous percolation transition . . . . . . . . . . . . . 35 3.1 Definition and necessary condition for type-I discontinuous percolation transition . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Cluster aggregation network model . . . . . . . . . . . . . . . . . 37 3.2.1 Description of the model . . . . . . . . . . . . . . . . . . 38 3.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.3 Testing the necessary condition for type-I discontinuous percolation transition . . . . . . . . . . . . . . . . . . . . 43 3.3 Avoiding a spanning cluster model . . . . . . . . . . . . . . . . . 44 3.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.2 Description of the model . . . . . . . . . . . . . . . . . . 46 3.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 Relation between compactness of clusters at the onset of the transtion and type-I discontinuous percolation transition . . . . . . . . . . . 52 3.5 Other models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 iv 3.5.1 Product rule with m > lnN . . . . . . . . . . . . . . . . . 59 3.5.2 Gaussian model . . . . . . . . . . . . . . . . . . . . . . . 60 4. Type-II discontinuous percolation transition . . . . . . . . . . . . 63 4.1 General properties of type-II discontinuous percolation transitions 64 4.1.1 Definition and necessary condition for type-II discontinuous percolation transition . . . . . . . . . . . . . . . . . . 64 4.1.2 Symmetry breaking dynamics . . . . . . . . . . . . . . . 65 4.2 Two-species cluster aggregation model . . . . . . . . . . . . . . . 67 4.2.1 Description of the model . . . . . . . . . . . . . . . . . . 67 4.2.2 Testing the necessary condition for type-II discontinuous percolation transition . . . . . . . . . . . . . . . . . . . . 70 4.2.3 Testing the symmetry breaking dynamics . . . . . . . . . 71 4.3 Other models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3.1 BFW model . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3.2 Half-restricted model . . . . . . . . . . . . . . . . . . . . 73 4.3.3 Ordinary percolation in a hierarchical network . . . . . . 74 5. Discontinuous percolation transitions in real physical systems . . . 79 5.1 Diffusion-limited cluster aggregation model . . . . . . . . . . . . 79 5.2 Diffusion-limited cluster aggregation model from the perspective of percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.3 Discontinuous percolation transitions in two, three and four dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.4 Analysis via an asymmetric Smoluchowski equation . . . . . . . . 84 5.5 Experiment for the diffusion-limited cluster aggregation model . . 88 v 6. Conductivity in a discontinuous percolation transition model . . . 89 6.1 Conductivity in an ordinary percolation . . . . . . . . . . . . . . 89 6.2 Effective medium theory . . . . . . . . . . . . . . . . . . . . . . 90 6.3 Theoretical formula for the conductivity in the avoiding a spanning cluster model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.4 Prediction of the behavior of conductivity as the system size increases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Appendix A. Erd˝os-R´enyi cluster aggregation with arbitrary initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 A.1 Derivation of G(t) . . . . . . . . . . . . . . . . . . . . . . . . . . 101 A.2 Power-law initial conditions . . . . . . . . . . . . . . . . . . . . 103 A.3 Analysis of the product rule . . . . . . . . . . . . . . . . . . . . . 104 Appendix B. Derivation of percolation transitions for the avoiding a spanning cluster model . . . . . . . . . . . . . . . . . . . . . . . . 109 B.1 Analytic solution for percolation threshold . . . . . . . . . . . . . 109 B.1.1 For spatial dimensions d < dc = 6 . . . . . . . . . . . . . 109 B.1.2 For spatial dimensions d ≥ dc = 6 . . . . . . . . . . . . . 111 B.2 Analytic solution for statistical fluctuation of percolation threshold 113 Appendix C. Analytic solution for the solvable two-species cluster aggregation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 vi Abstract in Korean. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 | - |
dc.format.extent | 128 | - |
dc.language.iso | eng | - |
dc.publisher | 서울대학교 대학원 | - |
dc.subject | Percolation transition, Spanning cluster, Explosive percolation model, Continuous percolation transition, Discontinuous percolation transition, Achlioptas process, Scale-free network, Finite size scaling theory, Cluster aggregation model, Suppression effect, Diffusion-limited cluster aggregation model, Electric resistivity and conductivity, Cluster merging process | - |
dc.subject.ddc | 523 | - |
dc.title | Discontinuous percolation transitions in cluster merging processes | - |
dc.title.alternative | 클러스터 결합 과정에서의 불연속 여과 상전이 | - |
dc.type | Thesis | - |
dc.type | Dissertation | - |
dc.contributor.AlternativeAuthor | Young Sul Cho | - |
dc.contributor.department | 자연과학대학 물리·천문학부 | - |
dc.description.degree | Doctor | - |
dc.date.awarded | 2015-02 | - |
dc.identifier.holdings | 000000000021▲000000000023▲000000025294▲ | - |
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