Altieri Jamming and Glass Transitions

In Mean-Field Theories and Beyond
1. Auflage 2019
ISBN: 978-3-030-23600-7
Verlag: Springer International Publishing
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 229 Seiten

Reihe: Springer Theses

The work described in this book originates from a major effort to develop a fundamental theory of the glass and the jamming transitions. The first chapters guide the reader through the phenomenology of supercooled liquids and structural glasses and provide the tools to analyze the most frequently used models able to predict the complex behavior of such systems. A fundamental outcome is a detailed theoretical derivation of an effective thermodynamic potential, along with the study of anomalous vibrational properties of sphere systems. The interested reader can find in these pages a clear and deep analysis of mean-field models as well as the description of advanced beyond-mean-field perturbative expansions. To investigate important second-order phase transitions in lattice models, the last part of the book proposes an innovative theoretical approach, based on a multi-layer construction. The different methods developed in this thesis shed new light on important connections among constraint satisfaction problems, jamming and critical phenomena in complex systems, and lay part of the groundwork for a complete theory of amorphous solids.

Autoren/Hrsg.

Altieri, Ada

Inhaltsverzeichnis

1;Supervisors’ Foreword;8
2;Abstract;10
3;Publications Related to This Thesis;12
4;Acknowledgements;13
5;Contents;15
6;Acronyms;19
7;1 Introduction;20
7.1;References;24
8;Part I Glass and Jamming Transitions in Mean-Field Models;26
9;2 Supercooled Liquids and the Glass Transition;27
9.1;2.1 Arrhenius and Super-Arrhenius Behaviors;30
9.2;2.2 Mode Coupling Theory;33
9.2.1;2.2.1 Mori-Zwanzig Formalism;33
9.2.2;2.2.2 Application of the Mori-Zwanzig Formalism to the Physics of Supercooled Liquids;35
9.2.3;2.2.3 Dynamical Correlation Functions;37
9.3;2.3 A Paradigmatic Example of Disordered System: The p-spin Spherical Model;40
9.3.1;2.3.1 Connection with the Static Replica Computation;42
9.4;2.4 Fluctuation-Dissipation Theorem and the Dynamical Temperature;44
9.5;2.5 Dynamical Facilitation and Kinetically Constrained Models;47
9.6;2.6 A Static Approach: Random First Order Transition Theory;49
9.6.1;2.6.1 The Adam-Gibbs-Di Marzio Theory;50
9.6.2;2.6.2 The Mosaic Theory;52
9.7;2.7 Complexity in Mean-Field Systems;54
9.8;References;59
10;3 The Jamming Transition;62
10.1;3.1 Theoretical and Numerical Protocols in Jammed Systems;63
10.1.1;3.1.1 Edwards Conjecture: Equiprobability of Jammed Configurations;63
10.2;3.2 The Marginal Glass Phase;68
10.3;3.3 Anomalous Properties of Jammed Systems;70
10.3.1;3.3.1 Isostaticity;70
10.3.2;3.3.2 Coordination Number;71
10.3.3;3.3.3 Density of States;72
10.3.4;3.3.4 Diverging Correlation Lengths;75
10.3.5;3.3.5 Beyond the Spherical Symmetry;76
10.4;3.4 Force and Pair Distributions at Random Close Packing;77
10.5;References;79
11;4 An Exactly Solvable Model: The Perceptron;82
11.1;4.1 The Perceptron Model in Neural Networks;82
11.2;4.2 From Computer Science to Sphere-Packing Transitions;86
11.2.1;4.2.1 Free-Energy Behavior in the SAT Phase;91
11.2.2;4.2.2 Free-Energy Behavior in the UNSAT Phase;91
11.2.3;4.2.3 Jamming Regime;92
11.3;4.3 Computation of the Effective Potential in Fully-Connected Models;94
11.4;4.4 TAP Equations in the Negative Perceptron;96
11.4.1;4.4.1 Cavity Method;103
11.5;4.5 Logarithmic Interaction Near Random Close Packing Density;107
11.6;4.6 Third Order Corrections to the Effective Potential;110
11.7;4.7 Leading and Subleading Contributions to the Forces Near Jamming;112
11.7.1;4.7.1 Scalings and Crossover Regimes;115
11.8;4.8 Spectrum of Small Harmonic Fluctuations;116
11.8.1;4.8.1 Spectrum in the UNSAT Phase;118
11.8.2;4.8.2 Spectrum in the SAT Phase;120
11.8.3;4.8.3 Asymptotic Behavior of the Spectral Density in the Jamming Limit;124
11.9;4.9 Conclusions;127
11.10;References;128
12;5 Universality Classes: Perceptron Versus Sphere Models;131
12.1;5.1 TAP Formalism Generalized to Sphere Models;134
12.2;5.2 Optimal Packing of Polydisperse Hard Spheres in Finite and Infinite Dimensions;136
12.3;5.3 Condensation in High Dimensions?;139
12.3.1;5.3.1 Quenched Computation;140
12.3.2;5.3.2 Annealed Computation;143
12.4;5.4 Conclusions;146
12.5;References;146
13;6 The Jamming Paradigm in Ecology;148
13.1;6.1 Stability and Complexity in Ecosystems;148
13.2;6.2 MacArthur's Model;152
13.2.1;6.2.1 Beyond the MacArthur Model: The Role of High Dimensionality;153
13.3;6.3 Similarities with the Percetron Model;156
13.4;6.4 Spectral Density of Harmonic Fluctuations of the Lyapunov Function;158
13.5;6.5 Stability Analysis: The Replicon Mode;160
13.6;6.6 Conclusions;166
13.7;References;167
14;Part II Lattice Theories Beyond Mean-Field;168
15;7 The M-Layer Construction;169
15.1;7.1 The Curie-Weiss Model;171
15.2;7.2 M-Layer Expansion Around Fully-Connected Models;174
15.2.1;7.2.1 The Propagator in Momentum Space Near the Criticality;176
15.2.2;7.2.2 A Continuum Field Theory;177
15.3;7.3 The Bethe Approximation;180
15.3.1;7.3.1 A Simple Argument Behind the Bethe Expansion;183
15.4;7.4 Mathematical Formalism for a Hamiltonian System on a Bethe Lattice;184
15.5;7.5 The Bethe M-Layer;187
15.6;7.6 1/M Corrections by the Cavity Method;188
15.6.1;7.6.1 Critical Behavior;190
15.6.2;7.6.2 The Graph-Theoretical Expansion;190
15.6.3;7.6.3 The Graph-Theoretical Expansion on the M-Lattice;193
15.6.4;7.6.4 Critical Behavior;195
15.6.5;7.6.5 The Expression for Line-Connected Observables;198
15.7;7.7 Conclusions;202
15.8;References;202
16;Part III Conclusions;204
17;8 Conclusions and Perspectives;205
17.1;8.1 Summary of the Main Results;205
17.2;8.2 Perspectives and Future Developments;207
17.2.1;8.2.1 Perceptron Model;207
17.2.2;8.2.2 Hard-Sphere Models;208
17.2.3;8.2.3 Connections Between Jamming and Ecology;209
17.3;References;210
18;A O(?3) Corrections to the Effective Potential in the Perceptron Model;211
19;B Computation of the Replicon Mode in a High-Dimensional Model of Critical Ecosystems;216
20;C Diagrammatic Rules for the M-Layer Construction in the Bethe Approximation;223
21;C.0.1 Non-backtracking Walks;224
22;D The Symmetry Factor;228