Gibbs measures and phase transitions:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; New York
de Gruyter
1988
|
| Series: | De Gruyter studies in mathematics
9 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 525 Seiten |
| ISBN: | 3110104555 0899254624 |
Staff View
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| 100 | 1 | |a Georgii, Hans-Otto |d 1944-2017 |0 (DE-588)10807546X |4 aut | |
| 245 | 1 | 0 | |a Gibbs measures and phase transitions |c Hans-Otto Georgii |
| 264 | 1 | |a Berlin ; New York |b de Gruyter |c 1988 | |
| 300 | |a XIV, 525 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a De Gruyter studies in mathematics |v 9 | |
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| 650 | 4 | |a Transformations de phase (Physique statistique) | |
| 650 | 7 | |a Waarschijnlijkheidstheorie |2 gtt | |
| 650 | 4 | |a Measure theory | |
| 650 | 4 | |a Phase transformations (Statistical physics) | |
| 650 | 4 | |a Probabilities | |
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Record in the Search Index
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Contents
Introduction 1
Part I. General theory and basic examples 9
Chapter 1 Specifications of random fields 11
1.1 Preliminaries 11
1.2 Prescribing conditional probabilities 15
1.3 ^ specifications 18
Chapter 2 Gibbsian specifications 25
2.1 Potentials 26
2.2 Quasilocality 30
2.3 Gibbs representation of pre modifications 35
2.4 Equivalence of potentials 39
Chapter 3 Finite state Markov chains as Gibbs measures 44
3.1 Markov specifications on the integers 44
3.2 The one dimensional Ising model 49
3.A Appendix. Positive matrices 54
Chapter 4 The existence problem 57
4.1 Local convergence of random fields 58
4.2 Existence of cluster points 60
4.3 Continuity results 66
4.4 Existence and topological properties of Gibbs measures 71
4.A Appendix. Standard Borel spaces 73
Chapter 5 Specifications with symmetries 81
5.1 Transformations of specifications 81
5.2 Gibbs measures with symmetries 85
Chapter 6 Three examples of symmetry breaking 94
6.1 Inhomogeneous Ising chains 95
XII Contents
6.2 The Ising ferromagnet in two dimensions 99
6.3 Shlosman's random staircases 106
Chapter 7 Extreme Gibbs measures 114
7.1 Tail triviality and approximation 115
7.2 Some applications 125
7.3 Extreme decomposition 129
7.4 Macroscopic equivalence of Gibbs simplices 136
Chapter 8 Uniqueness 140
8.1 Dobrushin's condition of weak dependence 140
8.2 Further consequences of Dobrushin's condition 153
8.3 Uniqueness in one dimension 164
Chapter 9 Absence of symmetry breaking. Non existence 168
9.1 Discrete symmetries in one dimension 169
9.2 Continuous symmetries in two dimensions 178
Part II. Markov chains and Gauss fields as Gibbs measures 189
Chapter 10 Markov fields on the integers I 190
10.1 Two sided and one sided Markov property 191
10.2 Markov fields which are Markov chains 196
10.3 Uniqueness of the shift invariant Markov field 204
Chapter 11 Markov fields on the integers II 209
11.1 Boundary laws, uniqueness, and non existence 210
11.2 The Spitzer Cox example of phase transition 221
11.3 Kalikow's example of phase transition 228
11.4 Spitzer's example of totally broken shift in variance 233
Chapter 12 Markov fields on trees 238
12.1 Markov chains and boundary laws 238
12.2 The Ising model on Cayley trees 247
Chapter 13 Gaussian fields 256
13.1 Gauss fields as Gibbs measures 257
13.2 Gibbs measures for Gaussian specifications 267
13.3 The homogeneous case 273
13.A Appendix. Some tools of Gaussian analysis 284
Contents XIII
Part III. Shift invariant Gibbs measures 289
Chapter 14 Ergodicity 290
14.1 Ergodic random fields 290
14.2 Ergodic Gibbs measures 296
14.A Appendix. The multidimensional ergodic theorem 302
Chapter 15 A variational characterization of Gibbs measures 308
15.1 Relative entropy 309
15.2 Specific entropy 312
15.3 Specific energy and free energy 319
15.4 The variational principle 323
Chapter 16 Convex geometry and the phase diagram 328
16.1 The pressure and its tangent functionals 328
16.2 A geometric view of Gibbs measures 333
16.3 Phase transitions with prescribed order parameters 338
16.4 Ubiquity of pure phases 349
Part IV. Phase transitions in reflection positive models 355
Chapter 17 Reflection positivity 357
17.1 The chessboard estimate 357
17.2 Gibbs distributions with periodic boundary condition 364
Chapter 18 Low energy oceans and discrete symmetry breaking 372
18.1 Percolation of spin patterns 373
18.2 Discrete symmetry breaking at low temperatures 384
18.3 Examples 388
Chapter 19 Phase transitions without symmetry breaking 398
19.1 Potentials with degenerated ground states,
and perturbations thereof 398
19.2 Exploiting Sperner's lemma 405
19.3 Models with an entropy energy conflict 410
19.A Appendix. Sperner's lemma 421
Chapter 20 Continuous symmetry breaking in N vector models 423
20.1 Some preliminaries 424
20.2 Spin wave analysis, and spontaneous magnetization 429
XIV Contents
Bibliographical Notes 443
References 481
List of Symbols 519
Index 521 |
| any_adam_object | 1 |
| author | Georgii, Hans-Otto 1944-2017 |
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| callnumber-subject | QC - Physics |
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| dewey-full | 519.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
| dewey-search | 519.2 |
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| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| format | Book |
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| isbn | 3110104555 0899254624 |
| language | English |
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| physical | XIV, 525 Seiten |
| publishDate | 1988 |
| publishDateSearch | 1988 |
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| publisher | de Gruyter |
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| series | De Gruyter studies in mathematics |
| series2 | De Gruyter studies in mathematics |
| spelling | Georgii, Hans-Otto 1944-2017 (DE-588)10807546X aut Gibbs measures and phase transitions Hans-Otto Georgii Berlin ; New York de Gruyter 1988 XIV, 525 Seiten txt rdacontent n rdamedia nc rdacarrier De Gruyter studies in mathematics 9 Fase-overgangen gtt Maattheorie gtt Mesure, Théorie de la Probabilités Transformations de phase (Physique statistique) Waarschijnlijkheidstheorie gtt Measure theory Phase transformations (Statistical physics) Probabilities Phasenumwandlung (DE-588)4132140-6 gnd rswk-swf Gibbs-Maß (DE-588)4157328-6 gnd rswk-swf Gibbs-Maß (DE-588)4157328-6 s Phasenumwandlung (DE-588)4132140-6 s DE-604 De Gruyter studies in mathematics 9 (DE-604)BV000005407 9 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000545873&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Georgii, Hans-Otto 1944-2017 Gibbs measures and phase transitions De Gruyter studies in mathematics Fase-overgangen gtt Maattheorie gtt Mesure, Théorie de la Probabilités Transformations de phase (Physique statistique) Waarschijnlijkheidstheorie gtt Measure theory Phase transformations (Statistical physics) Probabilities Phasenumwandlung (DE-588)4132140-6 gnd Gibbs-Maß (DE-588)4157328-6 gnd |
| subject_GND | (DE-588)4132140-6 (DE-588)4157328-6 |
| title | Gibbs measures and phase transitions |
| title_auth | Gibbs measures and phase transitions |
| title_exact_search | Gibbs measures and phase transitions |
| title_full | Gibbs measures and phase transitions Hans-Otto Georgii |
| title_fullStr | Gibbs measures and phase transitions Hans-Otto Georgii |
| title_full_unstemmed | Gibbs measures and phase transitions Hans-Otto Georgii |
| title_short | Gibbs measures and phase transitions |
| title_sort | gibbs measures and phase transitions |
| topic | Fase-overgangen gtt Maattheorie gtt Mesure, Théorie de la Probabilités Transformations de phase (Physique statistique) Waarschijnlijkheidstheorie gtt Measure theory Phase transformations (Statistical physics) Probabilities Phasenumwandlung (DE-588)4132140-6 gnd Gibbs-Maß (DE-588)4157328-6 gnd |
| topic_facet | Fase-overgangen Maattheorie Mesure, Théorie de la Probabilités Transformations de phase (Physique statistique) Waarschijnlijkheidstheorie Measure theory Phase transformations (Statistical physics) Probabilities Phasenumwandlung Gibbs-Maß |
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