Statistical mechanics of disordered systems a mathematical perspective

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1. Verfasser: Bovier, Anton 1957- (VerfasserIn)
Format: Buch
Sprache:English
Veröffentlicht: Cambridge Cambridge University Press 2006
Ausgabe:First published
Schriftenreihe:Cambridge series on statistical and probabilistic mathematics [18]
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adam_text STATISTICAL MECHANICS OF DISORDERED SYSTEMS A MATHEMATICAL PERSPECTIVE ANTON BOVIER WEIERSTRAFI-INSTITUT FIIR ANGEWANDTE ANALYSIS UND STOCHASTIK, BERLIN AND INSTITUT FIIR MATHEMATIK, TECHNISCHE UNIVERSITDT BERLIN CAMBRIDGE UNIVERSITY PRESS CONTENTS PREFACE PAGE IX NOMENCLATURE XIII PART I STATISTICAL MECHANICS 1 1 INTRODUCTION 3 1.1 THERMODYNAMICS 4 2 PRINCIPLES OF STATISTICAL MECHANICS 9 2.1 THE IDEAL GAS IN ONE DIMENSION 9 2.2 THE MICRO-CANONICAL ENSEMBLE 13 2.3 THE CANONICAL ENSEMBLE AND THE GIBBS MEASURE 19 2.4 NON-IDEAL GASES IN THE CANONICAL ENSEMBLE 22 2.5 EXISTENCE OF THE THERMODYNAMIC LIMIT 24 2.6 THE LIQUID-VAPOUR TRANSITION AND THE VAN DER WAALS GAS 28 2.7 THE GRAND CANONICAL ENSEMBLE 31 3 LATTICE GASES AND SPIN SYSTEMS 33 3.1 LATTICE GASES 33 3.2 SPIN SYSTEMS 34 3.3 SUBADDITIVITY AND THE EXISTENCE OF THE FREE ENERGY 36 3.4 THE ONE-DIMENSIONAL ISING MODEL 37 3.5 THE CURIE-WEISS MODEL 39 4 GIBBSIAN FORMALISM FOR LATTICE SPIN SYSTEMS 49 4.1 SPIN SYSTEMS AND GIBBS MEASURES 49 4.2 REGULAR INTERACTIONS 52 4.3 STRUCTURE OF GIBBS MEASURES: PHASE TRANSITIONS 59 5 CLUSTER EXPANSIONS 73 5.1 HIGH-TEMPERATURE EXPANSIONS 73 5.2 POLYMER MODELS: THE DOBRUSHIN-KOTECKY-PREISS CRITERION 76 5.3 CONVERGENCE OF THE HIGH-TEMPERATURE EXPANSION 82 5.4 LOW-TEMPERATURE EXPANSIONS 88 VIII CONTENTS PART II DISORDERED SYSTEMS: LATTICE MODELS 95 6 GIBBSIAN FORMALISM AND METASTATES 97 6.1 INTRODUCTION 97 6.2 RANDOM GIBBS MEASURES AND METASTATES 99 6.3 REMARKS ON UNIQUENESS CONDITIONS 106 6.4 PHASE TRANSITIONS 107 6.5 THE EDWARDS-ANDERSON MODEL 109 7 THE RANDOM-FIELD ISING MODEL 111 7.1 THE IMRY-MA ARGUMENT 111 7.2 ABSENCE OF PHASE TRANSITIONS: THE AIZENMAN-WEHR METHOD 118 7.3 THE BRICMONT-KUPIAINEN RENORMALIZATION GROUP 125 PART III DISORDERED SYSTEMS: MEAN-FIELD MODELS 159 8 DISORDERED MEAN-FIELD MODELS 161 9 THE RANDOM ENERGY MODEL 165 9.1 GROUND-STATE ENERGY AND FREE ENERGY 165 9.2 FLUCTUATIONS AND LIMIT THEOREMS 169 9.3 THE GIBBS MEASURE 175 9.4 THE REPLICA OVERLAP 180 9.5 MULTI-OVERLAPS AND GHIRLANDA-GUERRA RELATIONS 182 10 DERRIDA S GENERALIZED RANDOM ENERGY MODELS 186 10.1 THE STANDARD GREM AND POISSON CASCADES 186 10.2 MODELS WITH CONTINUOUS HIERARCHIES: THE CREM 195 10.3 CONTINUOUS STATE BRANCHING AND COALESCENT PROCESSES 207 11 THE SK MODELS AND THE PARISI SOLUTION 218 11.1 THE EXISTENCE OF THE FREE ENERGY 218 11.2 2ND MOMENT METHODS IN THE SK MODEL . 220 11.3 THE PARISI SOLUTION AND GUERRA S BOUNDS 227 11.4 THE GHIRLANDA-GUERRA RELATIONS IN THE SK MODELS 238 11.5 APPLICATIONS IN THE /?-SPIN SK MODEL 240 12 HOPFIELD MODELS 247 12.1 ORIGINS OF THE MODEL 247 12.2 BASIC IDEAS: FINITE M 250 12.3 GROWING M 257 12.4 THE REPLICA SYMMETRIC SOLUTION 265 13 THE NUMBER PARTITIONING PROBLEM 285 13.1 NUMBER PARTITIONING AS A SPIN-GLASS PROBLEM 285 13.2 AN EXTREME VALUE THEOREM 287 13.3 APPLICATION TO NUMBER PARTITIONING 289 REFERENCES 297 INDEX 309
adam_txt STATISTICAL MECHANICS OF DISORDERED SYSTEMS A MATHEMATICAL PERSPECTIVE ANTON BOVIER WEIERSTRAFI-INSTITUT FIIR ANGEWANDTE ANALYSIS UND STOCHASTIK, BERLIN AND INSTITUT FIIR MATHEMATIK, TECHNISCHE UNIVERSITDT BERLIN CAMBRIDGE UNIVERSITY PRESS CONTENTS PREFACE PAGE IX NOMENCLATURE XIII PART I STATISTICAL MECHANICS 1 1 INTRODUCTION 3 1.1 THERMODYNAMICS 4 2 PRINCIPLES OF STATISTICAL MECHANICS 9 2.1 THE IDEAL GAS IN ONE DIMENSION 9 2.2 THE MICRO-CANONICAL ENSEMBLE 13 2.3 THE CANONICAL ENSEMBLE AND THE GIBBS MEASURE 19 2.4 NON-IDEAL GASES IN THE CANONICAL ENSEMBLE 22 2.5 EXISTENCE OF THE THERMODYNAMIC LIMIT 24 2.6 THE LIQUID-VAPOUR TRANSITION AND THE VAN DER WAALS GAS 28 2.7 THE GRAND CANONICAL ENSEMBLE 31 3 LATTICE GASES AND SPIN SYSTEMS 33 3.1 LATTICE GASES 33 3.2 SPIN SYSTEMS 34 3.3 SUBADDITIVITY AND THE EXISTENCE OF THE FREE ENERGY 36 3.4 THE ONE-DIMENSIONAL ISING MODEL 37 3.5 THE CURIE-WEISS MODEL 39 4 GIBBSIAN FORMALISM FOR LATTICE SPIN SYSTEMS 49 4.1 SPIN SYSTEMS AND GIBBS MEASURES 49 4.2 REGULAR INTERACTIONS 52 4.3 STRUCTURE OF GIBBS MEASURES: PHASE TRANSITIONS 59 5 CLUSTER EXPANSIONS 73 5.1 HIGH-TEMPERATURE EXPANSIONS 73 5.2 POLYMER MODELS: THE DOBRUSHIN-KOTECKY-PREISS CRITERION 76 5.3 CONVERGENCE OF THE HIGH-TEMPERATURE EXPANSION 82 5.4 LOW-TEMPERATURE EXPANSIONS 88 VIII CONTENTS PART II DISORDERED SYSTEMS: LATTICE MODELS 95 6 GIBBSIAN FORMALISM AND METASTATES 97 6.1 INTRODUCTION 97 6.2 RANDOM GIBBS MEASURES AND METASTATES 99 6.3 REMARKS ON UNIQUENESS CONDITIONS 106 6.4 PHASE TRANSITIONS 107 6.5 THE EDWARDS-ANDERSON MODEL 109 7 THE RANDOM-FIELD ISING MODEL 111 7.1 THE IMRY-MA ARGUMENT 111 7.2 ABSENCE OF PHASE TRANSITIONS: THE AIZENMAN-WEHR METHOD 118 7.3 THE BRICMONT-KUPIAINEN RENORMALIZATION GROUP 125 PART III DISORDERED SYSTEMS: MEAN-FIELD MODELS 159 8 DISORDERED MEAN-FIELD MODELS 161 9 THE RANDOM ENERGY MODEL 165 9.1 GROUND-STATE ENERGY AND FREE ENERGY 165 9.2 FLUCTUATIONS AND LIMIT THEOREMS 169 9.3 THE GIBBS MEASURE 175 9.4 THE REPLICA OVERLAP 180 9.5 MULTI-OVERLAPS AND GHIRLANDA-GUERRA RELATIONS 182 10 DERRIDA'S GENERALIZED RANDOM ENERGY MODELS 186 10.1 THE STANDARD GREM AND POISSON CASCADES 186 10.2 MODELS WITH CONTINUOUS HIERARCHIES: THE CREM 195 10.3 CONTINUOUS STATE BRANCHING AND COALESCENT PROCESSES 207 11 THE SK MODELS AND THE PARISI SOLUTION 218 11.1 THE EXISTENCE OF THE FREE ENERGY 218 11.2 2ND MOMENT METHODS IN THE SK MODEL . 220 11.3 THE PARISI SOLUTION AND GUERRA'S BOUNDS 227 11.4 THE GHIRLANDA-GUERRA RELATIONS IN THE SK MODELS 238 11.5 APPLICATIONS IN THE /?-SPIN SK MODEL 240 12 HOPFIELD MODELS 247 12.1 ORIGINS OF THE MODEL 247 12.2 BASIC IDEAS: FINITE M 250 12.3 GROWING M 257 12.4 THE REPLICA SYMMETRIC SOLUTION 265 13 THE NUMBER PARTITIONING PROBLEM 285 13.1 NUMBER PARTITIONING AS A SPIN-GLASS PROBLEM 285 13.2 AN EXTREME VALUE THEOREM 287 13.3 APPLICATION TO NUMBER PARTITIONING 289 REFERENCES 297 INDEX 309
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spellingShingle Bovier, Anton 1957-
Statistical mechanics of disordered systems a mathematical perspective
Cambridge series on statistical and probabilistic mathematics
Mécanique statistique
Mécanique statistique ram
Ordre et désordre (physique) ram
Probabilités
Statistique mathématique
Statistique mathématique ram
Théorie des systèmes
Mathematical statistics
Probabilities
Statistical mechanics
System theory
Ungeordnetes System (DE-588)4124353-5 gnd
Statistische Mechanik (DE-588)4056999-8 gnd
Mathematische Physik (DE-588)4037952-8 gnd
subject_GND (DE-588)4124353-5
(DE-588)4056999-8
(DE-588)4037952-8
title Statistical mechanics of disordered systems a mathematical perspective
title_auth Statistical mechanics of disordered systems a mathematical perspective
title_exact_search Statistical mechanics of disordered systems a mathematical perspective
title_exact_search_txtP Statistical mechanics of disordered systems a mathematical perspective
title_full Statistical mechanics of disordered systems a mathematical perspective Anton Bovier: Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin and Institut für Mathematik, Technsiche Universität Berlin
title_fullStr Statistical mechanics of disordered systems a mathematical perspective Anton Bovier: Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin and Institut für Mathematik, Technsiche Universität Berlin
title_full_unstemmed Statistical mechanics of disordered systems a mathematical perspective Anton Bovier: Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin and Institut für Mathematik, Technsiche Universität Berlin
title_short Statistical mechanics of disordered systems
title_sort statistical mechanics of disordered systems a mathematical perspective
title_sub a mathematical perspective
topic Mécanique statistique
Mécanique statistique ram
Ordre et désordre (physique) ram
Probabilités
Statistique mathématique
Statistique mathématique ram
Théorie des systèmes
Mathematical statistics
Probabilities
Statistical mechanics
System theory
Ungeordnetes System (DE-588)4124353-5 gnd
Statistische Mechanik (DE-588)4056999-8 gnd
Mathematische Physik (DE-588)4037952-8 gnd
topic_facet Mécanique statistique
Ordre et désordre (physique)
Probabilités
Statistique mathématique
Théorie des systèmes
Mathematical statistics
Probabilities
Statistical mechanics
System theory
Ungeordnetes System
Statistische Mechanik
Mathematische Physik
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