Large deviations in physics: the legacy of the law of large numbers
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| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2014
|
| Series: | Lecture notes in physics
885 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XIV, 314 S. graph. Darst. 235 mm x 155 mm |
| ISBN: | 3642542506 9783642542503 |
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CONTENTS
1 FROM THE LAW OF LARGE NUMBERS TO LARGE DEVIATION
THEORY IN STATISTICAL PHYSICS: AN INTRODUCTION 1
FABIO CECCONI, MASSIMO CENCINI, ANDREA PUGLISI, DAVIDE
VERGNI, AND ANGELO VULPIANI
1.1 INTRODUCTION 1
1.2 AN INFORMAL HISTORICAL NOTE 3
1.2.1 LAW OF LARGE NUMBERS AND ERGODICITY 4
1.2.2 CENTRAL LIMIT THEOREMS 9
1.2.3 LARGE DEVIATION THEORY 11
1.3 LDT FOR THE SUM AND PRODUCT OF RANDOM INDEPENDENT
VARIABLES 13
1.3.1 A COMBINATORIAL EXAMPLE 13
1.3.2 PRODUCT OF RANDOM VARIABLES 15
1.4 LARGE DEVIATION THEORY: EXAMPLES FROM PHYSICS 17
1.4.1 ENERGY FLUCTUATIONS IN THE CANONICAL ENSEMBLE 17
1.4.2 MULTIPLICATIVE CASCADE IN TURBULENCE 18
1.4.3 CHAOTIC SYSTEMS 19
1.4.4 DISORDERED SYSTEMS 21
1.4.5 ENTROPY PRODUCTION IN MARKOV PROCESSES 24
REFERENCES 26
2 ERGODICITY: HOW CAN IT BE BROKEN? 29
GIANCARLO BENETTIN, ROBERTO LIVI, AND GIORGIO PARISI
2.1 THE ERGODIC HYPOTHESIS 30
2.1.1 THE FUNDAMENTAL PHYSICAL IDEAS 30
2.1.2 A WELL-POSED MATHEMATICAL SETTING 33
2.2 ERGODICITY BREAKING 36
2.2.1 ERGODICITY BREAKING IN THE FERMI-PASTA-ULAM
MODEL AT LOW ENERGIES 37
2.2.2 ERGODICITY BREAKING INDUCED BY BREATHER
STATES IN THE DISCRETE NONLINEAR SCHRODINGER EQUATION. 47
VII
HTTP://D-NB.INFO/1046584111
VIII CONTENTS
2.3 ERGODICITY BREAKING AT EQUILIBRIUM 53
2.3.1 GENERAL CONCEPTS 53
2.3.2 THE GIBBS STATES 54
2.3.3 THE LOCAL DRL EQUILIBRIUM STATES 60
2.4 GLASSY SYSTEM 62
2.4.1 A HEURISTIC CONSTRUCTION: FINITE VOLUME STATES 62
2.4.2 THE CASE OF MANY STATES 65
REFERENCES 68
3 LARGE DEVIATIONS IN STATIONARY STATES, ESPECIALLY NONEQUILIBRIUM. 71
GIOVANNI JONA-LASINIO
3.1 INTRODUCTION 71
3.2 ASSUMPTIONS 73
3.3 THE FUNDAMENTAL FORMULA 74
3.4 TIME REVERSAL AND ITS CONSEQUENCES 78
3.5 LONG RANGE CORRELATIONS 80
3.6 FLUCTUATIONS OF THE CURRENT AND DYNAMICAL PHASE TRANSITIONS 81
3.7 UNIVERSALITY IN CURRENT FLUCTUATIONS AND OTHER RESULTS 83
3.8 NONEQUILIBRIUM PHASE TRANSITIONS 84
3.9 LARGE DEVIATIONS FOR REACTION-DIFFUSION SYSTEMS 85
3.10 THERMODYNAMIC INTERPRETATION OF THE LARGE DEVIATION
FUNCTIONAL 85
3.11 CONCLUDING REMARKS AND ADDITIONAL REFERENCES 89
REFERENCES 90
4 FLUCTUATION-DISSIPATION AND FLUCTUATION RELATIONS: FROM
EQUILIBRIUM TO NONEQUILIBRIUM AND BACK 93
PAOLO ADAMO, ROMAN BELOUSOV, AND LAMBERTO RONDONI
4.1 CONCISE HISTORY 93
4.2 THE BROWNIAN MOTION AND THE LANGEVIN EQUATION 95
4.3 THE FLUCTUATION-DISSIPATION RELATION 99
4.4 EVOLUTION OF PROBABILITY DISTRIBUTIONS 103
4.4.1 ERGODICITY AND MIXING 107
4.5 LINEAR RESPONSE 108
4.6 ONSAGER-MACHLUP: RESPONSE FROM SMALL DEVIATIONS 112
4.7 FLUCTUATION RELATIONS: RESPONSE FROM LARGE DEVIATIONS 118
4.7.1 THE GALLAVOTTI-COHEN APPROACH 119
4.7.2 FLUCTUATION RELATIONS FOR THE DISSIPATION FUNCTION 121
4.7.3 GREEN-KUBO RELATIONS 125
4.7.4 JARZYNSKI EQUALITY 126
4.8 T-MIXING AND GENERAL RESPONSE THEORY 127
4.9 CONCLUDING REMARKS 129
REFERENCES 131
CONTENTS IX
5 LARGE DEVIATIONS IN DISORDERED SPIN SYSTEMS 135
ANDREA CRISANTI AND LUCA LEUZZI
5.1 SOME GENERAL RESULTS ON LARGE DEVIATIONS 135
5.1.1 AN EXAMPLE: THE MEAN FIELD ISING MODEL 139
5.2 SAMPLE-TO-SAMPLE FREE ENERGY FLUCTUATIONS AND REPLICA TRICK. 142
5.2.1 RANDOM ISING CHAIN 143
5.2.2 REPLICA TRICK 145
5.2.3 REPLICAS IN THE RANDOM ISING CHAIN 147
5.2.4 FROM SMALL TO LARGE DEVIATIONS 148
5.2.5 RANDOM DIRECTED POLYMER 151
5.2.6 MEAN-FIELD SPIN-GLASS 151
5.2.7 POSITIVE LARGE DEVIATIONS 153
5.2.8 SPHERICAL SPIN-GLASS MODEL AND GAUSSIAN
RANDOM MATRICES 154
5.3 SAMPLE-TO-SAMPLE FLUCTUATION OF THE OVERLAP 155
5.3.1 BACK TO THE REPLICATED RANDOM ISING CHAIN 156
5.3.2 RANDOM FIELD ISING MODEL 158
5.4 A FINAL WORD 159
REFERENCES 159
6 LARGE DEVIATIONS IN MONTE CARLO METHODS 161
ANDREA PELISSETTO AND FEDERICO RICCI-TERSENGHI
6.1 INTRODUCTION 161
6.2 DATA REWEIGHTING 163
6.3 MULTIPLE HISTOGRAM METHOD 168
6.4 UMBRELLA SAMPLING AND SIMULATED TEMPERING 172
6.4.1 UMBRELLA SAMPLING 172
6.4.2 SIMULATED TEMPERING 173
6.4.3 EQUIVALENCE OF SIMULATED TEMPERING AND
UMBRELLA SAMPLING 174
6.5 GENERALIZING THE UMBRELLA METHOD: MULTICANONICAL SAMPLING 178
6.6 PARALLEL TEMPERING 184
6.6.1 GENERAL CONSIDERATIONS 184
6.6.2 SOME GENERAL RIGOROUS RESULTS 186
6.6.3 OPTIMAL CHOICE OF TEMPERATURES 187
6.6.4 IMPROVING PARALLEL TEMPERING 189
6.7 CONCLUSIONS 190
REFERENCES 190
7 LARGE DEVIATIONS TECHNIQUES FOR LONG-RANGE INTERACTIONS 193
AURELIO PATELLI AND STEFANO RUFFO
7.1 LONG-RANGE INTERACTIONS 193
7.2 SOME USEFUL RESULTS OF LARGE DEVIATIONS THEORY 198
7.3 THERMODYNAMIC FUNCTIONS FROM LARGE DEVIATIONS THEORY 199
X CONTENTS
7.4 APPLICATIONS 202
7.4.1 THREE-STATES POTTS MODEL 203
7.4.2 THE BLUME-CAPEL MODEL 205
7.4.3 A SYSTEM WITH CONTINUOUS VARIABLES: THE XY MODEL . 208
7.4.4 NEGATIVE SUSCEPTIBILITY: 0
4
MODEL 212
7.4.5 THE FREE ELECTRON LASER 215
7.5 THE MIN-MAX PROCEDURE AND A MODEL WITH SHORT
AND LONG-RANGE INTERACTIONS 216
7.6 CONCLUSIONS AND PERSPECTIVES 219
REFERENCES 219
8 LARGE DEVIATIONS OF BROWNIAN MOTORS 221
ALESSANDRO SARRACINO AND DARIO VILLAMAINA
8.1 INTRODUCTION 221
8.2 NONEQUILIBRIUM FLUCTUATIONS AND BROWNIAN MOTORS 224
8.2.1 KINETIC RATCHETS 225
8.2.2 MOLECULAR MOTORS 227
8.3 EXPERIMENTS IN GRANULAR SYSTEMS 231
8.3.1 VELOCITY FLUCTUATIONS OF A SELF-PROPELLED
POLAR PARTICLE 232
8.3.2 ASYMMETRIC ROTOR IN A GRANULAR GAS 234
8.4 CONCLUSIONS 238
REFERENCES 239
9 STOCHASTIC FLUCTUATIONS IN DETERMINISTIC SYSTEMS 243
ANTONIO POLITI
9.1 INTRODUCTION 243
9.2 KOLMOGOROV-SINAI ENTROPY 245
9.3 LYAPUNOV EXPONENTS 250
9.3.1 PESIN RELATION 253
9.4 NON HYPERBOLICITY 254
9.4.1 A 3D MAP 256
9.5 SPACE-TIME CHAOS 257
9.6 STABLE CHAOS 259
REFERENCES 261
10 ANOMALOUS DIFFUSION: DETERMINISTIC AND STOCHASTIC
PERSPECTIVES 263
ROBERTO ARTUSO AND RAFFAELLA BURIONI
10.1 INTRODUCTION 263
10.2 STOCHASTIC ANOMALOUS TRANSPORT 264
10.2.1 MOMENTS AND SCALING 264
10.2.2 A FEW OBSERVATIONS ABOUT LEVY STABLE LAWS 267
10.2.3 AN EXTENSION: CONTINUOUS TIME RANDOM WALKS 269
10.2.4 TOPOLOGICAL EFFECTS IN SUBDIFFUSION: WEAK
ANOMALOUS DIFFUSION AND RANDOM WALKS ON GRAPHS . 272
CONTENTS XI
10.2.5 TOPOLOGICAL EFFECTS IN SUPERDIFFUSION: STRONG
ANOMALOUS DIFFUSION AND QUENCHED LEVY WALKS 277
10.3 DETERMINISTIC ANOMALOUS TRANSPORT 281
10.3.1 A BRIEF TOUR OF INTERMITTENCY 281
10.4 CHAIN OF INTERMITTENT MAPS 284
10.4.1 SUBDIFFUSION 287
10.4.2 SUPERDIFFUSION 287
10.5 DETERMINISTIC VS STOCHASTIC APPROXIMATION 289
10.6 A FINAL WARNING 291
REFERENCES 291
11 LARGE DEVIATIONS IN TURBULENCE 295
GUIDO BOFFETTA AND ANDREA MAZZINO
11.1 INTRODUCTION 295
11.2 GLOBAL SCALE INVARIANCE AND KOLMOGOROV THEORY 296
11.3 ACCOUNTING FOR THE FLUCTUATIONS: THE MULTIFRACTAL MODEL 299
11.3.1 THE STATISTICS OF VELOCITY GRADIENT 302
11.3.2 THE STATISTICS OF ACCELERATION 303
11.3.3 MULTIPLICATIVE PROCESSES FOR THE MULTIFRACTAL MODEL 305
11.4 FLUCTUATIONS OF THE ENERGY DISSIPATION RATE 307
11.5 CONCLUSIONS 309
REFERENCES 309
INDEX 311 |
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| spelling | Large deviations in physics the legacy of the law of large numbers Angelo Vulpiani ... eds. Berlin [u.a.] Springer 2014 XIV, 314 S. graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Lecture notes in physics 885 Große Abweichung (DE-588)4330658-5 gnd rswk-swf Gesetz der großen Zahlen (DE-588)4157077-7 gnd rswk-swf Statistische Physik (DE-588)4057000-9 gnd rswk-swf Gesetz der großen Zahlen (DE-588)4157077-7 s Große Abweichung (DE-588)4330658-5 s Statistische Physik (DE-588)4057000-9 s DE-604 Vulpiani, Angelo 1958- (DE-588)1054572976 edt Erscheint auch als Online-Ausgabe 978-3-642-54251-0 Lecture notes in physics 885 (DE-604)BV000003166 885 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4570867&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027362508&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Large deviations in physics the legacy of the law of large numbers Lecture notes in physics Große Abweichung (DE-588)4330658-5 gnd Gesetz der großen Zahlen (DE-588)4157077-7 gnd Statistische Physik (DE-588)4057000-9 gnd |
| subject_GND | (DE-588)4330658-5 (DE-588)4157077-7 (DE-588)4057000-9 |
| title | Large deviations in physics the legacy of the law of large numbers |
| title_auth | Large deviations in physics the legacy of the law of large numbers |
| title_exact_search | Large deviations in physics the legacy of the law of large numbers |
| title_full | Large deviations in physics the legacy of the law of large numbers Angelo Vulpiani ... eds. |
| title_fullStr | Large deviations in physics the legacy of the law of large numbers Angelo Vulpiani ... eds. |
| title_full_unstemmed | Large deviations in physics the legacy of the law of large numbers Angelo Vulpiani ... eds. |
| title_short | Large deviations in physics |
| title_sort | large deviations in physics the legacy of the law of large numbers |
| title_sub | the legacy of the law of large numbers |
| topic | Große Abweichung (DE-588)4330658-5 gnd Gesetz der großen Zahlen (DE-588)4157077-7 gnd Statistische Physik (DE-588)4057000-9 gnd |
| topic_facet | Große Abweichung Gesetz der großen Zahlen Statistische Physik |
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