Thermodynamic formalism: the mathematical structures of equilibrium statistical mechanics
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge University Press
2004
|
| Edition: | 2. ed. |
| Series: | Cambridge mathematical library
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XX, 174 S. |
| ISBN: | 0521546494 |
Staff View
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| 245 | 1 | 0 | |a Thermodynamic formalism |b the mathematical structures of equilibrium statistical mechanics |c David Ruelle |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge University Press |c 2004 | |
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| 650 | 4 | |a Mathematische Physik - Statistische Mechanik - Thermodynamik | |
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Record in the Search Index
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| adam_text | Contents page xv xvii xix Foreword to the first edition Preface to the first edition Preface to the second edition 0.1 0.2 0.3 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 Introduction Generalities Description of the thermodynamic formalism Summary of contents Theory of Gibbs states Configuration space Interactions Gibbs ensembles and thermodynamic limit Proposition Gibbs states Thermodynamic limit of Gibbs ensembles Boundary terms Theorem Theorem Algebra at infinity Theorem (characterization of pure Gibbs states) The operators 91іл Theorem (characterization of unique Gibbs states) Remark Notes Exercises vii 1 1 3 9 11 11 12 13 14 14 15 16 18 18 19 20 20 21 22 23 23
viii Contents 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Gibbs states: complements Morphisms of lattice systems Example The interaction Е*Ф Lemma Proposition Remarks Systems of conditional probabilities Properties of Gibbs states Remark Notes Exercises 24 24 25 25 26 26 27 28 29 30 30 31 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 Translation invariance. Theory of equilibrium states Translation invariance The function Аф Partition functions Theorem Invariant states Proposition Theorem Entropy Infinite limit in the sense of van Hove Theorem Lemma Theorem Corollary Corollary Physical interpretation Theorem Corollary Approximation of invariant states by equilibrium states Lemma Theorem Coexistence of phases Notes Exercises 33 33 34 35 36 39 39 40 42 43 43 45 45 47 48 48 49 49 50 50 52 53 54 54
Contents 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 Connection between Gibbs states and Equilibrium states Generalities Theorem Physical interpretation Proposition Remark Strict convexity of the pressure Proposition Zl’-lattice systems and Zl-morphisms Proposition Corollary Remark Proposition Restriction of Ζυ to a subgroup G Proposition Undecidability and non-periodicity Notes Exercises One-dimensional systems Lemma Theorem Theorem Lemma Proof of theorems 5.2 and 5.3 Corollaries to theorems 5.2 and 5.3 Theorem Mixing Z-lattice systems Lemma Theorem The transfer matrix and the operator £ The function ψ Proposition The operator § Lemma Proposition Remark Exponentially decreasing interactions ix 57 57 58 59 59 60 61 61 62 62 63 63 64 64 65 65 66 66 69 70 70 71 72 73 75 76 78 78 79 80 81 81 82 82 82 83 83
x Contents 5.19 5.20 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.30 The space S0 and related spaces Proposition Theorem Remarks Lemma Proposition Remark Theorem Corollary Zeta functions Theorem Remark Notes Exercises 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22 Extension of the thermodynamic formalism Generalities Expansiveness Covers Entropy Proposition Pressure Other definitions of the pressure Properties of the pressure The action r“ Lemma Lemma Theorem (variational principle) Equilibrium states Theorem Remark Commuting continuous maps Extension to a Ζυ-action Results for Z -actions Remark Topological entropy Relative pressure Theorem 84 85 85 86 86 87 88 88 89 89 90 93 93 94 101 101 101 102 103 103 104 105 106 107 107 107 108 110 111 111 112 112 113 115 115 115 116
Contents 6.23 Corollary Notes Exercises 7 Statistical mechanics on Smale spaces 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19 7.20 7.21 7.22 7.23 7.24 7.25 7.26 7.27 7.28 7.29 7.30 7.31 Smale spaces Example Properties of Smale spaces Smale’s “spectral decomposition” Markov partitions and symbolic dynamics Theorem Holder continuous functions Pressure and equilibrium states Theorem Corollary Remark Corollary Corollary Equilibrium states for A not Holder continuous Conjugate points and conjugating homeomorphisms Proposition Theorem Gibbs states Periodic points Theorem Study of periodic points by symbolic dynamics Proposition Zeta functions Theorem Corollary Expanding maps Remarks Results for expanding maps Markov partitions Theorem Applications Notes Exercises xi 117 117 118 121 121 123 123 124 124 125 126 126 128 128 128 129 129 130 131 132 132 133 133 134 135 135 135 137 137 138 139 140 140 141 141 143 144
xii Contents Appendix A.l Miscellaneous definitions and results A. 1.1 Order A. 1.2 Residual sets A. 1.3 Upper semi-continuity A. 1.4 Subadditivity 146 146 146 147 147 Appendix A.2 Topological dynamics 148 Appendix A.3 Convexity A.3.1 Generalities A.3.2 Hahn-Banach theorem A.3.3 Separation theorems A.3.4 Convex compact sets A.3.5 Extremal points A.3.6 Tangent functionals to convex functions A.3.7 Multiplicity of tangent functionals 150 150 150 151 151 151 152 152 Appendix A.4 Measures and abstractdynamical systems A.4.1 Measures on compact sets A.4.2 Abstract measure theory A.4.3 Abstract dynamical systems A.4.4 Bernoulli shifts A.4.5 Partitions A.4.6 Isomorphism theorems 153 153 154 154 155 155 156 Appendix A. 5 Integral representations on convex compact sets A.5.1 Resultant of a measure A.5.2 Maximal measures A.5.3 Uniqueness problem A.5.4 Maximal measures and extremal points A.5.5 Simplexes of measures A.5.6 Zv-invariant measures 157 157 158 158 158 159 159 Appendix В Open problems B.l Systems of conditional probabilities (Chapter 2) B.2 Theory of phase transitions (Chapter 3) B.3 Abstract measure-theory viewpoint (Chapter 4) B.4 A theorem of Dobrushin (Chapter 5) B.5 Definition of the pressure (Chapter 6) 160 160 160 160 160 161
Contents В.6 B.7 В.8 B.9 B. 10 Shub’s entropy conjecture (Chapter 6) The condition (SS3) (Chapter 7) Gibbs states on Smale spaces (Chapter 7) Cohomological interpretation (Chapter 7) Smale flows (Chapter 7 and Appendix C) Appendix C С. 1 C.2 C.3 C.4 Flows Thermodynamic formalism on a metrizable compact set Special flows Special flow over a Smale space Problems Appendix D References Index Update of open problems xiii 161 161 161 161 161 162 162 163 163 164 165 167 172
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| discipline | Physik Mathematik |
| edition | 2. ed. |
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| physical | XX, 174 S. |
| publishDate | 2004 |
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| spelling | Ruelle, David 1935- Verfasser (DE-588)113082711 aut Thermodynamic formalism the mathematical structures of equilibrium statistical mechanics David Ruelle 2. ed. Cambridge [u.a.] Cambridge University Press 2004 XX, 174 S. txt rdacontent n rdamedia nc rdacarrier Cambridge mathematical library Mathematische Physik - Statistische Mechanik - Thermodynamik Lattice theory Statistical mechanics Thermodynamics Statistische Mechanik (DE-588)4056999-8 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Statistische Mechanik (DE-588)4056999-8 s Mathematische Physik (DE-588)4037952-8 s DE-604 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013179536&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ruelle, David 1935- Thermodynamic formalism the mathematical structures of equilibrium statistical mechanics Mathematische Physik - Statistische Mechanik - Thermodynamik Lattice theory Statistical mechanics Thermodynamics Statistische Mechanik (DE-588)4056999-8 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| subject_GND | (DE-588)4056999-8 (DE-588)4037952-8 |
| title | Thermodynamic formalism the mathematical structures of equilibrium statistical mechanics |
| title_auth | Thermodynamic formalism the mathematical structures of equilibrium statistical mechanics |
| title_exact_search | Thermodynamic formalism the mathematical structures of equilibrium statistical mechanics |
| title_full | Thermodynamic formalism the mathematical structures of equilibrium statistical mechanics David Ruelle |
| title_fullStr | Thermodynamic formalism the mathematical structures of equilibrium statistical mechanics David Ruelle |
| title_full_unstemmed | Thermodynamic formalism the mathematical structures of equilibrium statistical mechanics David Ruelle |
| title_short | Thermodynamic formalism |
| title_sort | thermodynamic formalism the mathematical structures of equilibrium statistical mechanics |
| title_sub | the mathematical structures of equilibrium statistical mechanics |
| topic | Mathematische Physik - Statistische Mechanik - Thermodynamik Lattice theory Statistical mechanics Thermodynamics Statistische Mechanik (DE-588)4056999-8 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| topic_facet | Mathematische Physik - Statistische Mechanik - Thermodynamik Lattice theory Statistical mechanics Thermodynamics Statistische Mechanik Mathematische Physik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013179536&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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