Jamming and Glass Transitions (eBook)

Supervisors’ Foreword 8
Abstract 10
Publications Related to This Thesis 12
Acknowledgements 13
Contents 15
Acronyms 19
1 Introduction 20
References 24
Part I Glass and Jamming Transitions in Mean-Field Models 26
2 Supercooled Liquids and the Glass Transition 27
2.1 Arrhenius and Super-Arrhenius Behaviors 30
2.2 Mode Coupling Theory 33
2.2.1 Mori-Zwanzig Formalism 33
2.2.2 Application of the Mori-Zwanzig Formalism to the Physics of Supercooled Liquids 35
2.2.3 Dynamical Correlation Functions 37
2.3 A Paradigmatic Example of Disordered System: The p-spin Spherical Model 40
2.3.1 Connection with the Static Replica Computation 42
2.4 Fluctuation-Dissipation Theorem and the Dynamical Temperature 44
2.5 Dynamical Facilitation and Kinetically Constrained Models 47
2.6 A Static Approach: Random First Order Transition Theory 49
2.6.1 The Adam-Gibbs-Di Marzio Theory 50
2.6.2 The Mosaic Theory 52
2.7 Complexity in Mean-Field Systems 54
References 59
3 The Jamming Transition 62
3.1 Theoretical and Numerical Protocols in Jammed Systems 63
3.1.1 Edwards Conjecture: Equiprobability of Jammed Configurations 63
3.2 The Marginal Glass Phase 68
3.3 Anomalous Properties of Jammed Systems 70
3.3.1 Isostaticity 70
3.3.2 Coordination Number 71
3.3.3 Density of States 72
3.3.4 Diverging Correlation Lengths 75
3.3.5 Beyond the Spherical Symmetry 76
3.4 Force and Pair Distributions at Random Close Packing 77
References 79
4 An Exactly Solvable Model: The Perceptron 82
4.1 The Perceptron Model in Neural Networks 82
4.2 From Computer Science to Sphere-Packing Transitions 86
4.2.1 Free-Energy Behavior in the SAT Phase 91
4.2.2 Free-Energy Behavior in the UNSAT Phase 91
4.2.3 Jamming Regime 92
4.3 Computation of the Effective Potential in Fully-Connected Models 94
4.4 TAP Equations in the Negative Perceptron 96
4.4.1 Cavity Method 103
4.5 Logarithmic Interaction Near Random Close Packing Density 107
4.6 Third Order Corrections to the Effective Potential 110
4.7 Leading and Subleading Contributions to the Forces Near Jamming 112
4.7.1 Scalings and Crossover Regimes 115
4.8 Spectrum of Small Harmonic Fluctuations 116
4.8.1 Spectrum in the UNSAT Phase 118
4.8.2 Spectrum in the SAT Phase 120
4.8.3 Asymptotic Behavior of the Spectral Density in the Jamming Limit 124
4.9 Conclusions 127
References 128
5 Universality Classes: Perceptron Versus Sphere Models 131
5.1 TAP Formalism Generalized to Sphere Models 134
5.2 Optimal Packing of Polydisperse Hard Spheres in Finite and Infinite Dimensions 136
5.3 Condensation in High Dimensions? 139
5.3.1 Quenched Computation 140
5.3.2 Annealed Computation 143
5.4 Conclusions 146
References 146
6 The Jamming Paradigm in Ecology 148
6.1 Stability and Complexity in Ecosystems 148
6.2 MacArthur's Model 152
6.2.1 Beyond the MacArthur Model: The Role of High Dimensionality 153
6.3 Similarities with the Percetron Model 156
6.4 Spectral Density of Harmonic Fluctuations of the Lyapunov Function 158
6.5 Stability Analysis: The Replicon Mode 160
6.6 Conclusions 166
References 167
Part II Lattice Theories Beyond Mean-Field 168
7 The M-Layer Construction 169
7.1 The Curie-Weiss Model 171
7.2 M-Layer Expansion Around Fully-Connected Models 174
7.2.1 The Propagator in Momentum Space Near the Criticality 176
7.2.2 A Continuum Field Theory 177
7.3 The Bethe Approximation 180
7.3.1 A Simple Argument Behind the Bethe Expansion 183
7.4 Mathematical Formalism for a Hamiltonian System on a Bethe Lattice 184
7.5 The Bethe M-Layer 187
7.6 1/M Corrections by the Cavity Method 188
7.6.1 Critical Behavior 190
7.6.2 The Graph-Theoretical Expansion 190
7.6.3 The Graph-Theoretical Expansion on the M-Lattice 193
7.6.4 Critical Behavior 195
7.6.5 The Expression for Line-Connected Observables 198
7.7 Conclusions 202
References 202
Part III Conclusions 204
8 Conclusions and Perspectives 205
8.1 Summary of the Main Results 205
8.2 Perspectives and Future Developments 207
8.2.1 Perceptron Model 207
8.2.2 Hard-Sphere Models 208
8.2.3 Connections Between Jamming and Ecology 209
References 210
A O(?3) Corrections to the Effective Potential in the Perceptron Model 211
B Computation of the Replicon Mode in a High-Dimensional Model of Critical Ecosystems 216
C Diagrammatic Rules for the M-Layer Construction in the Bethe Approximation 223
C.0.1 Non-backtracking Walks 224
D The Symmetry Factor 228