Statistics of knots and entangled random walks:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Singapore [u.a.]
World Scientific
1996
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 190 S. graph. Darst. |
| ISBN: | 9810225199 |
Internformat
MARC
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| 100 | 1 | |a Nečaev, Sergej K. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Statistics of knots and entangled random walks |c S. K. Nechaev |
| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 1996 | |
| 300 | |a XIV, 190 S. |b graph. Darst. | ||
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Datensatz im Suchindex
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|---|---|
| adam_text | CONTENTS
Preface v
CHAPTER 1. KNOT DIAGRAMS AS DISORDERED
SPIN SYSTEMS
1.1. Introduction: Statistical Problems in Topology 1
1.2. Review of Abelian Problems in Statistics of Entangled
Random Walks and Incompleteness of Gauss Invariant . . 6
1.3. Nonabelian Algebraic Knot Invariants 12
1.3.1. Disordered Potts Model and Generalized Dichromatic
Polynomials 13
1.3.2. Reidemeister Moves and State Model for Construction of
Algebraic Invariants 15
1.4. Lattice Knot Diagrams as Disordered Potts Model . . 21
1.4.1. Algebraic Invariants of Regular Isotopy 22
1.4.2. Algebraic Invariants of Ambient Isotopy 27
1.5. Annealed and Quenched Realizations of Topological
Disorder 28
1.5.1. Averaged Kauffman Invariant for Annealed Topological
Disorder 29
1.5.2. Entropy of Knots. Replica Methods 33
1.6. Remarks and Conclusions 38
References 40
xi
xii Statistics of Knots and Entangled Random Walks
CHAPTER 2. RANDOM WALKS ON LOCAL
NONCOMMUTATIVE GROUPS
2.1. Introduction 42
2.2. Brownian Bridges on Simplest Noncommutative
Groups and Knot Statistics 43
2.2.1. Basic Definitions and Statistical Model 43
2.2.2. Statistics of Random Walks and Joint Distributions of
Brownian Bridges of Free Group 51
2.2.3. Random walks on PSL(2,Z), B3 and Distribution of
Powers of Alexander Invariants 56
2.3. Random Walks on Locally Free Groups 67
2.4. Brownian Bridges on Lobachevskii Plane and Products
of Noncommutative Random Matrices 74
2.5. Remarks and Conclusions 79
References 81
CHAPTER 3. CONFORMAL METHODS IN STATISTICS
OF ENTANGLED RANDOM WALKS
3.1. Introduction: Random Walk with Topological
Constraints 83
3.2. Construction of Nonabelian Connections for T2 and
PSL(2, Z) from Conformal Methods 84
3.3. Random Walk on Double Punctured Plane and
Conformal Field Theory 90
3.4. Statistics of Random Walks with Topological Constraints
in 2D Lattice of Obstacles 94
3.4.1. Rare Lattice of Obstacles (a L c).
Correspondence to Edwards Approach 98
3.4.2. Dense Lattice of Obstacles (c2 La).
Random Walk on Lobachevskii Plane 100
Contents xiii
3.5. Remarks and Conclusions 103
References 105
CHAPTER 4. PHYSICAL APPLICATIONS
4.1. Introduction: Polymer Language in Statistics of
Entangled Chain Like Objects 106
4.2. Polymer Chain in 3D Array of Obstacles: Critical
Exponents for Gyration Radius 108
4.3. High Elasticity of Polymer Networks 114
4.3.1. Free Energy of Polymer Network 116
4.3.2. Calculation of Generating Function 118
4.3.3. Statistical Weight of Coarsened Trajectory and Averaging
over Primitive Paths 122
4.4. Collapsed Phase of Unknotted Polymer 125
4.4.1. Crumpled Globule Concept in Statistics of Strongly
Collapsed Unknotted Polymer Loops 125
4.4.2. Knot Formation Probability 130
4.4.3. Quasi Knot Concept in Collapsed Phase of Unknotted
Polymers 133
4.5. Ordering Phase Transition in Entangled Directed
Polymers 134
4.5.1. Model 135
4.5.2. Mean Field Theory of Phase Transition in System of
Entangled Directed Polymers 139
4.6. Remarks and Conclusions 143
References 144
xiv Statistics of Knots and Entangled Random Walks
APPENDIX A. LIMIT THEOREM FOR CONDITIONAL
DSISTRIBUTION OF PRODUCTS OF INDEPENDENT
UNIMODULAR 2x2 MATRICES 146
References 157
APPENDIX B. POLYMER CHAIN IN RANDOM
ARRAY OF TOPOLOGICAL OBSTACLES
1.1. Phase Transition in Poplymer Loops Induced by
Topological Disorder 158
1.1.1. Description of the System 159
1.1.2. Field Representation of the Model 161
1.1.3. Method of Effective Potential 166
1.1.4. Role of Fluctuations 169
1.1.5. Remarks and Conclusions 171
1.2. Classification and Statistics of Complex Loops 173
1.2.1. Cactus Like Representation of Complex Loop . . . .174
1.2.2. Effective Hamiltonian and Mean Field Free Energy of
Complex Loops 177
1.2.3. Variational Principle 180
1.2.4. Remarks and Conclusions 184
References 185
Subject Index 187
|
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| id | DE-604.BV011162741 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:05:01Z |
| institution | BVB |
| isbn | 9810225199 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007482729 |
| oclc_num | 247573847 |
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| owner_facet | DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-83 DE-11 |
| physical | XIV, 190 S. graph. Darst. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Nečaev, Sergej K. Verfasser aut Statistics of knots and entangled random walks S. K. Nechaev Singapore [u.a.] World Scientific 1996 XIV, 190 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Knoten Mathematik (DE-588)4164314-8 gnd rswk-swf Irrfahrtsproblem (DE-588)4162442-7 gnd rswk-swf Irrfahrtsproblem (DE-588)4162442-7 s Knoten Mathematik (DE-588)4164314-8 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007482729&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nečaev, Sergej K. Statistics of knots and entangled random walks Knoten Mathematik (DE-588)4164314-8 gnd Irrfahrtsproblem (DE-588)4162442-7 gnd |
| subject_GND | (DE-588)4164314-8 (DE-588)4162442-7 |
| title | Statistics of knots and entangled random walks |
| title_auth | Statistics of knots and entangled random walks |
| title_exact_search | Statistics of knots and entangled random walks |
| title_full | Statistics of knots and entangled random walks S. K. Nechaev |
| title_fullStr | Statistics of knots and entangled random walks S. K. Nechaev |
| title_full_unstemmed | Statistics of knots and entangled random walks S. K. Nechaev |
| title_short | Statistics of knots and entangled random walks |
| title_sort | statistics of knots and entangled random walks |
| topic | Knoten Mathematik (DE-588)4164314-8 gnd Irrfahrtsproblem (DE-588)4162442-7 gnd |
| topic_facet | Knoten Mathematik Irrfahrtsproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007482729&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT necaevsergejk statisticsofknotsandentangledrandomwalks |