Entropy, large deviations, and statistical mechanics:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin, Heidelberg
Springer
2006
|
| Edition: | Reprint of the 1985 ed. |
| Series: | Classics in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 338 - 351. - Originally published as Vol. 271 of the Grundlehren der mathematischen Wissenschaften |
| Physical Description: | XIV, 364 S. graph. Darst. |
| ISBN: | 3540290591 9783540290599 |
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Record in the Search Index
| _version_ | 1804135303975272448 |
|---|---|
| adam_text | CONTENTS
PREFACE VII
COMMENTS ON THE USE OF THIS BOOK X
PART I: LARGE DEVIATIONS AND STATISTICAL MECHANICS
CHAPTER I. INTRODUCTION TO LARGE DEVIATIONS 3
1.1. OVERVIEW 3
1.2. LARGE DEVIATIONS FOR I.I.D. RANDOM VARIABLES WITH A FINITE STATE
SPACE 9
1.3. LEVELS-1 AND 2 FOR COIN TOSSING 11
1.4. LEVELS-1 AND 2 FOR I.I.D. RANDOM VARIABLES WITH A FINITE STATE
SPACE 13
1.5. LEVEL-3: EMPIRICAL PAIR MEASURE 17
1.6. LEVEL-3: EMPIRICAL PROCESS 21
1.7. NOTES 26
1.8. PROBLEMS 27
CHAPTER II. LARGE DEVIATION PROPERTY AND ASYMPTOTICS OF INTEGRALS 30
ILL INTRODUCTION 30
11.2. LEVELS-1, 2, AND 3 LARGE DEVIATIONS FOR I.I.D. RANDOM VECTORS 30
11.3. THE DEFINITION OF LARGE DEVIATION PROPERTY 33
11.4. STATEMENT OF LARGE DEVIATION PROPERTIES FOR LEVELS-1, 2, AND 3 37
11.5. CONTRACTION PRINCIPLES 42
11.6. LARGE DEVIATION PROPERTY FOR RANDOM VECTORS AND EXPONENTIAL
CONVERGENCE 46
11.7. VARADHAN S THEOREM ON THE ASYMPTOTICS OF INTEGRALS 50
11.8. NOTES 55
11.9. PROBLEMS 56
CHAPTER III. LARGE DEVIATIONS AND THE DISCRETE IDEAL GAS 59
111.1. INTRODUCTION 59
111.2. PHYSICS PRELUDE: THERMODYNAMICS 60
111.3. THE DISCRETE IDEAL GAS AND THE MICROCANONICAL ENSEMBLE 64
XLL CONTENTS
111.4. THERMODYNAMIC LIMIT, EXPONENTIAL CONVERGENCE, AND
EQUILIBRIUM VALUES 66
111.5. THE MAXWELL-BOLTZMANN DISTRIBUTION AND TEMPERATURE 74
111.6. THE CANONICAL ENSEMBLE AND ITS EQUIVALENCE WITH THE
MICROCANONICAL ENSEMBLE 75
111.7. A DERIVATION OF A THERMODYNAMIC EQUATION 78
111.8. THE GIBBS VARIATIONAL FORMULA AND PRINCIPLE 79
111.9. NOTES 84
III. 10. PROBLEMS 86
CHAPTER IV. FERROMAGNETIC MODELS ON Z 88
IV. 1. INTRODUCTION 88
IV.2. AN OVERVIEW OF FERROMAGNETIC MODELS 88
IV.3. FINITE-VOLUME GIBBS STATES ON Z 94
IV.4. SPONTANEOUS MAGNETIZATION FOR THE CURIE-WEISS MODEL 98
IV.5. SPONTANEOUS MAGNETIZATION FOR GENERAL FERROMAGNETS ON Z 103
IV.6. INFINITE-VOLUME GIBBS STATES AND PHASE TRANSITIONS 109
IV.7. THE GIBBS VARIATIONAL FORMULA AND PRINCIPLE 125
IV.8. NOTES 131
IV.9. PROBLEMS 134
CHAPTER V. MAGNETIC MODELS ON Z
D
AND ON THE CIRCLE 138
V.I. INTRODUCTION 138
V.2. FINITE-VOLUME GIBBS STATES ON Z
D
, D 1 140
V.3. MOMENT INEQUALITIES 142
V.4. PROPERTIES OF THE MAGNETIZATION AND THE GIBBS FREE ENERGY 146
V.5. SPONTANEOUS MAGNETIZATION ON 1, D 2, VIA THE PEIERLS ARGUMENT
153
V.6. INFINITE-VOLUME GIBBS STATES AND PHASE TRANSITIONS 157
V.7. INFINITE-VOLUME GIBBS STATES AND THE CENTRAL LIMIT THEOREM 162
V.8. CRITICAL PHENOMENA AND THE BREAKDOWN OF THE CENTRAL LIMIT
THEOREM 170
V.9. THREE FACES OF THE CURIE-WEISS MODEL 179
V.10. THE CIRCLE MODEL AND RANDOM WAVES 190
V.LL. A POSTSCRIPT ON MAGNETIC MODELS 198
V.12. NOTES 199
V.I 3. PROBLEMS 203
PART II: CONVEXITY AND PROOFS OF LARGE DEVIATION
THEOREMS
CHAPTER VI. CONVEX FUNCTIONS AND THE LEGENDRE-FENCHEL TRANSFORM 211
VI.L. INTRODUCTION 211
VI.2. BASIC DEFINITIONS 211
VI.3. PROPERTIES OF CONVEX FUNCTIONS 213
CONTENTS XLLL
VI.4. A ONE-DIMENSIONAL EXAMPLE OF THE LEGENDRE-FENCHEL TRANSFORM 218
VI.5. THE LEGENDRE-FENCHEL TRANSFORM FOR CONVEX FUNCTIONS ON R
D
220
VI.6. NOTES 225
VI.7. PROBLEMS 226
CHAPTER VII. LARGE DEVIATIONS FOR RANDOM VECTORS 229
VII. 1. STATEMENT OF RESULTS 229
VII.2. PROPERTIES OF V 231
VII.3. PROOF OF THE LARGE DEVIATION BOUNDS FOR D = 1 232
VII.4. PROOF OF THE LARGE DEVIATION BOUNDS FOR D 1 235
VII.5. LEVEL-1 LARGE DEVIATIONS FOR I.I.D. RANDOM VECTORS 238
VII.6. EXPONENTIAL CONVERGENCE AND PROOF OF THEOREM II.6.3 242
VII.7. NOTES 243
VII.8. PROBLEMS 245
CHAPTER VIII. LEVEL-2 LARGE DEVIATIONS FOR I.I.D. RANDOM VECTORS 250
VIII. 1. INTRODUCTION 250
VIII.2. THE LEVEL-2 LARGE DEVIATION THEOREM 250
VIII.3. THE CONTRACTION PRINCIPLE RELATING LEVELS-1 AND 2 (D = 1) 253
VIII.4. THE CONTRACTION PRINCIPLE RELATING LEVELS-1 AND 2 (D 2) 258
VIII.5. NOTES 264
VIII.6. PROBLEMS 265
CHAPTER IX. LEVEL-3 LARGE DEVIATIONS FOR I.I.D. RANDOM VECTORS 269
IX. 1. STATEMENT OF RESULTS 269
IX.2. PROPERTIES OF THE LEVEL-3 ENTROPY FUNCTION 270
IX.3. CONTRACTION PRINCIPLES 276
IX.4. PROOF OF THE LEVEL-3 LARGE DEVIATION BOUNDS 279
IX.5. NOTES 287
IX.6. PROBLEMS 288
APPENDICES
APPENDIX A: PROBABILITY 295
A.I. INTRODUCTION 295
A.2. MEASURABILITY 295
A.3. PRODUCT SPACES 296
A.4. PROBABILITY MEASURES AND EXPECTATION 297
A.5. CONVERGENCE OF RANDOM VECTORS 299
A.6. CONDITIONAL EXPECTATION, CONDITIONAL PROBABILITY, AND REGULAR
CONDITIONAL DISTRIBUTION 299
A.7. THE KOLMOGOROV EXISTENCE THEOREM 301
XIV CONTENTS
A.8. WEAK CONVERGENCE OF PROBABILITY MEASURES ON A METRIC SPACE 303
A.9. THE SPACE JT
S
((M
D
)
Z
) AND THE ERGODIC THEOREM 306
A. 10. ^-DEPENDENT MARKOV CHAINS 311
A. 11. PROBABILITY MEASURES ON THE SPACE {1, * 1 }
Z
312
APPENDIX B: PROOFS OF TWO THEOREMS IN SECTION II.7 319
B. 1. PROOF OF THEOREM II.7.1 319
B.2. PROOF OF THEOREM II.7.2 321
APPENDIX C: EQUIVALENT NOTIONS OF INFINITE-VOLUME MEASURES FOR
SPIN SYSTEMS 323
C.I. INTRODUCTION 323
C.2. TWO-BODY INTERACTIONS AND INFINITE-VOLUME GIBBS STATES 323
C.3. MANY-BODY INTERACTIONS AND INFINITE-VOLUME GIBBS STATES 325
C.4. DLR STATES 326
C.5. THE GIBBS VARIATIONAL FORMULA AND PRINCIPLE 327
C.6. SOLUTION OF THE GIBBS VARIATIONAL FORMULA FOR FINITE-RANGE
INTERACTIONS ON Z 330
APPENDIX D: EXISTENCE OF THE SPECIFIC GIBBS FREE ENERGY 332
D.I. EXISTENCE ALONG HYPERCUBES 332
D.2. AN EXTENSION 334
LIST OF FREQUENTLY USED SYMBOLS 335
REFERENCES 338
AUTHOR INDEX 353
SUBJECT INDEX 359
|
| adam_txt |
CONTENTS
PREFACE VII
COMMENTS ON THE USE OF THIS BOOK X
PART I: LARGE DEVIATIONS AND STATISTICAL MECHANICS
CHAPTER I. INTRODUCTION TO LARGE DEVIATIONS 3
1.1. OVERVIEW 3
1.2. LARGE DEVIATIONS FOR I.I.D. RANDOM VARIABLES WITH A FINITE STATE
SPACE 9
1.3. LEVELS-1 AND 2 FOR COIN TOSSING 11
1.4. LEVELS-1 AND 2 FOR I.I.D. RANDOM VARIABLES WITH A FINITE STATE
SPACE 13
1.5. LEVEL-3: EMPIRICAL PAIR MEASURE 17
1.6. LEVEL-3: EMPIRICAL PROCESS 21
1.7. NOTES 26
1.8. PROBLEMS 27
CHAPTER II. LARGE DEVIATION PROPERTY AND ASYMPTOTICS OF INTEGRALS 30
ILL INTRODUCTION 30
11.2. LEVELS-1, 2, AND 3 LARGE DEVIATIONS FOR I.I.D. RANDOM VECTORS 30
11.3. THE DEFINITION OF LARGE DEVIATION PROPERTY 33
11.4. STATEMENT OF LARGE DEVIATION PROPERTIES FOR LEVELS-1, 2, AND 3 37
11.5. CONTRACTION PRINCIPLES 42
11.6. LARGE DEVIATION PROPERTY FOR RANDOM VECTORS AND EXPONENTIAL
CONVERGENCE 46
11.7. VARADHAN'S THEOREM ON THE ASYMPTOTICS OF INTEGRALS 50
11.8. NOTES 55
11.9. PROBLEMS 56
CHAPTER III. LARGE DEVIATIONS AND THE DISCRETE IDEAL GAS 59
111.1. INTRODUCTION 59
111.2. PHYSICS PRELUDE: THERMODYNAMICS 60
111.3. THE DISCRETE IDEAL GAS AND THE MICROCANONICAL ENSEMBLE 64
XLL CONTENTS
111.4. THERMODYNAMIC LIMIT, EXPONENTIAL CONVERGENCE, AND
EQUILIBRIUM VALUES 66
111.5. THE MAXWELL-BOLTZMANN DISTRIBUTION AND TEMPERATURE 74
111.6. THE CANONICAL ENSEMBLE AND ITS EQUIVALENCE WITH THE
MICROCANONICAL ENSEMBLE 75
111.7. A DERIVATION OF A THERMODYNAMIC EQUATION 78
111.8. THE GIBBS VARIATIONAL FORMULA AND PRINCIPLE 79
111.9. NOTES 84
III. 10. PROBLEMS 86
CHAPTER IV. FERROMAGNETIC MODELS ON Z 88
IV. 1. INTRODUCTION 88
IV.2. AN OVERVIEW OF FERROMAGNETIC MODELS 88
IV.3. FINITE-VOLUME GIBBS STATES ON Z 94
IV.4. SPONTANEOUS MAGNETIZATION FOR THE CURIE-WEISS MODEL 98
IV.5. SPONTANEOUS MAGNETIZATION FOR GENERAL FERROMAGNETS ON Z 103
IV.6. INFINITE-VOLUME GIBBS STATES AND PHASE TRANSITIONS 109
IV.7. THE GIBBS VARIATIONAL FORMULA AND PRINCIPLE 125
IV.8. NOTES 131
IV.9. PROBLEMS 134
CHAPTER V. MAGNETIC MODELS ON Z
D
AND ON THE CIRCLE 138
V.I. INTRODUCTION 138
V.2. FINITE-VOLUME GIBBS STATES ON Z
D
, D 1 140
V.3. MOMENT INEQUALITIES 142
V.4. PROPERTIES OF THE MAGNETIZATION AND THE GIBBS FREE ENERGY 146
V.5. SPONTANEOUS MAGNETIZATION ON 1, D 2, VIA THE PEIERLS ARGUMENT
153
V.6. INFINITE-VOLUME GIBBS STATES AND PHASE TRANSITIONS 157
V.7. INFINITE-VOLUME GIBBS STATES AND THE CENTRAL LIMIT THEOREM 162
V.8. CRITICAL PHENOMENA AND THE BREAKDOWN OF THE CENTRAL LIMIT
THEOREM 170
V.9. THREE FACES OF THE CURIE-WEISS MODEL 179
V.10. THE CIRCLE MODEL AND RANDOM WAVES 190
V.LL. A POSTSCRIPT ON MAGNETIC MODELS 198
V.12. NOTES 199
V.I 3. PROBLEMS 203
PART II: CONVEXITY AND PROOFS OF LARGE DEVIATION
THEOREMS
CHAPTER VI. CONVEX FUNCTIONS AND THE LEGENDRE-FENCHEL TRANSFORM 211
VI.L. INTRODUCTION 211
VI.2. BASIC DEFINITIONS 211
VI.3. PROPERTIES OF CONVEX FUNCTIONS 213
CONTENTS XLLL
VI.4. A ONE-DIMENSIONAL EXAMPLE OF THE LEGENDRE-FENCHEL TRANSFORM 218
VI.5. THE LEGENDRE-FENCHEL TRANSFORM FOR CONVEX FUNCTIONS ON R
D
220
VI.6. NOTES 225
VI.7. PROBLEMS 226
CHAPTER VII. LARGE DEVIATIONS FOR RANDOM VECTORS 229
VII. 1. STATEMENT OF RESULTS 229
VII.2. PROPERTIES OF V 231
VII.3. PROOF OF THE LARGE DEVIATION BOUNDS FOR D = 1 232
VII.4. PROOF OF THE LARGE DEVIATION BOUNDS FOR D 1 235
VII.5. LEVEL-1 LARGE DEVIATIONS FOR I.I.D. RANDOM VECTORS 238
VII.6. EXPONENTIAL CONVERGENCE AND PROOF OF THEOREM II.6.3 242
VII.7. NOTES 243
VII.8. PROBLEMS 245
CHAPTER VIII. LEVEL-2 LARGE DEVIATIONS FOR I.I.D. RANDOM VECTORS 250
VIII. 1. INTRODUCTION 250
VIII.2. THE LEVEL-2 LARGE DEVIATION THEOREM 250
VIII.3. THE CONTRACTION PRINCIPLE RELATING LEVELS-1 AND 2 (D = 1) 253
VIII.4. THE CONTRACTION PRINCIPLE RELATING LEVELS-1 AND 2 (D 2) 258
VIII.5. NOTES 264
VIII.6. PROBLEMS 265
CHAPTER IX. LEVEL-3 LARGE DEVIATIONS FOR I.I.D. RANDOM VECTORS 269
IX. 1. STATEMENT OF RESULTS 269
IX.2. PROPERTIES OF THE LEVEL-3 ENTROPY FUNCTION 270
IX.3. CONTRACTION PRINCIPLES 276
IX.4. PROOF OF THE LEVEL-3 LARGE DEVIATION BOUNDS 279
IX.5. NOTES 287
IX.6. PROBLEMS 288
APPENDICES
APPENDIX A: PROBABILITY 295
A.I. INTRODUCTION 295
A.2. MEASURABILITY 295
A.3. PRODUCT SPACES 296
A.4. PROBABILITY MEASURES AND EXPECTATION 297
A.5. CONVERGENCE OF RANDOM VECTORS 299
A.6. CONDITIONAL EXPECTATION, CONDITIONAL PROBABILITY, AND REGULAR
CONDITIONAL DISTRIBUTION 299
A.7. THE KOLMOGOROV EXISTENCE THEOREM 301
XIV CONTENTS
A.8. WEAK CONVERGENCE OF PROBABILITY MEASURES ON A METRIC SPACE 303
A.9. THE SPACE JT
S
((M
D
)
Z
) AND THE ERGODIC THEOREM 306
A. 10. ^-DEPENDENT MARKOV CHAINS 311
A. 11. PROBABILITY MEASURES ON THE SPACE {1, * 1 }
Z
312
APPENDIX B: PROOFS OF TWO THEOREMS IN SECTION II.7 319
B. 1. PROOF OF THEOREM II.7.1 319
B.2. PROOF OF THEOREM II.7.2 321
APPENDIX C: EQUIVALENT NOTIONS OF INFINITE-VOLUME MEASURES FOR
SPIN SYSTEMS 323
C.I. INTRODUCTION 323
C.2. TWO-BODY INTERACTIONS AND INFINITE-VOLUME GIBBS STATES 323
C.3. MANY-BODY INTERACTIONS AND INFINITE-VOLUME GIBBS STATES 325
C.4. DLR STATES 326
C.5. THE GIBBS VARIATIONAL FORMULA AND PRINCIPLE 327
C.6. SOLUTION OF THE GIBBS VARIATIONAL FORMULA FOR FINITE-RANGE
INTERACTIONS ON Z 330
APPENDIX D: EXISTENCE OF THE SPECIFIC GIBBS FREE ENERGY 332
D.I. EXISTENCE ALONG HYPERCUBES 332
D.2. AN EXTENSION 334
LIST OF FREQUENTLY USED SYMBOLS 335
REFERENCES 338
AUTHOR INDEX 353
SUBJECT INDEX 359 |
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| edition | Reprint of the 1985 ed. |
| format | Book |
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| id | DE-604.BV021548801 |
| illustrated | Illustrated |
| index_date | 2024-07-02T14:30:50Z |
| indexdate | 2024-07-09T20:38:22Z |
| institution | BVB |
| isbn | 3540290591 9783540290599 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014764930 |
| oclc_num | 607067582 |
| open_access_boolean | |
| owner | DE-384 DE-19 DE-BY-UBM DE-11 DE-29T DE-739 DE-188 |
| owner_facet | DE-384 DE-19 DE-BY-UBM DE-11 DE-29T DE-739 DE-188 |
| physical | XIV, 364 S. graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Springer |
| record_format | marc |
| series2 | Classics in mathematics |
| spelling | Ellis, Richard S. 1947-2018 Verfasser (DE-588)129738395 aut Entropy, large deviations, and statistical mechanics Richard S. Ellis Reprint of the 1985 ed. Berlin, Heidelberg Springer 2006 XIV, 364 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Classics in mathematics Literaturverz. S. 338 - 351. - Originally published as Vol. 271 of the Grundlehren der mathematischen Wissenschaften Entropy Large deviations Statistical mechanics Streuung (DE-588)4058056-8 gnd rswk-swf Entropie (DE-588)4014894-4 gnd rswk-swf Statistische Thermodynamik (DE-588)4126251-7 gnd rswk-swf Statistische Mechanik (DE-588)4056999-8 gnd rswk-swf Große Abweichung (DE-588)4330658-5 gnd rswk-swf Wahrscheinlichkeitsmaß (DE-588)4137556-7 gnd rswk-swf Asymptotik (DE-588)4126634-1 gnd rswk-swf Statistische Mechanik (DE-588)4056999-8 s Entropie (DE-588)4014894-4 s Große Abweichung (DE-588)4330658-5 s 1\p DE-604 Wahrscheinlichkeitsmaß (DE-588)4137556-7 s Asymptotik (DE-588)4126634-1 s 2\p DE-604 Streuung (DE-588)4058056-8 s 3\p DE-604 Statistische Thermodynamik (DE-588)4126251-7 s 4\p DE-604 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014764930&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 4\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Ellis, Richard S. 1947-2018 Entropy, large deviations, and statistical mechanics Entropy Large deviations Statistical mechanics Streuung (DE-588)4058056-8 gnd Entropie (DE-588)4014894-4 gnd Statistische Thermodynamik (DE-588)4126251-7 gnd Statistische Mechanik (DE-588)4056999-8 gnd Große Abweichung (DE-588)4330658-5 gnd Wahrscheinlichkeitsmaß (DE-588)4137556-7 gnd Asymptotik (DE-588)4126634-1 gnd |
| subject_GND | (DE-588)4058056-8 (DE-588)4014894-4 (DE-588)4126251-7 (DE-588)4056999-8 (DE-588)4330658-5 (DE-588)4137556-7 (DE-588)4126634-1 |
| title | Entropy, large deviations, and statistical mechanics |
| title_auth | Entropy, large deviations, and statistical mechanics |
| title_exact_search | Entropy, large deviations, and statistical mechanics |
| title_exact_search_txtP | Entropy, large deviations, and statistical mechanics |
| title_full | Entropy, large deviations, and statistical mechanics Richard S. Ellis |
| title_fullStr | Entropy, large deviations, and statistical mechanics Richard S. Ellis |
| title_full_unstemmed | Entropy, large deviations, and statistical mechanics Richard S. Ellis |
| title_short | Entropy, large deviations, and statistical mechanics |
| title_sort | entropy large deviations and statistical mechanics |
| topic | Entropy Large deviations Statistical mechanics Streuung (DE-588)4058056-8 gnd Entropie (DE-588)4014894-4 gnd Statistische Thermodynamik (DE-588)4126251-7 gnd Statistische Mechanik (DE-588)4056999-8 gnd Große Abweichung (DE-588)4330658-5 gnd Wahrscheinlichkeitsmaß (DE-588)4137556-7 gnd Asymptotik (DE-588)4126634-1 gnd |
| topic_facet | Entropy Large deviations Statistical mechanics Streuung Entropie Statistische Thermodynamik Statistische Mechanik Große Abweichung Wahrscheinlichkeitsmaß Asymptotik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014764930&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT ellisrichards entropylargedeviationsandstatisticalmechanics |