The jamming transition

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Table of contents

Introduction
1 Supercooled liquids and RFOT
1.1 The glassy slowdown
1.1.1 The calorimetric glass transition
1.1.2 Fragility and the Vogel-Fulcher-Tammann law
1.1.3 Two-step relaxation
1.1.4 Real space: the cage
1.1.5 Real space: cooperativity
1.2 From the slowdown to RFOT
1.2.1 The foundations of RFOT
1.2.2 Dynamics: MCT and Goldstein’s picture
1.2.3 Complexity: Kauzmann’s paradox
1.2.4 Summary of RFOT: for TMCT to TK.
1.2.5 Beyond mean field: scaling and the mosaic
1.3 Other approaches
1.3.1 Dynamic facilitation theory
1.3.2 Frustration limited domains
2 Metastable glasses
2.1 Thermodynamics and aging
2.1.1 Protocol dependence
2.1.2 Ultrastable glasses
2.1.3 The jamming transition
2.1.4 Theoretical approaches to aging
2.2 Driven dynamics and rheology
2.2.1 Athermal startup shear protocols
2.2.2 The yielding transition
2.2.3 Theoretical approaches
3 The State Following construction
3.1 The real replica method
3.1.1 Quenching: the threshold
3.1.2 Annealing: isocomplexity
3.1.3 Summary
3.2 The two-replica Franz-Parisi potential
3.3 Beyond two replicas: the replica chain and pseudodynamics
4 The replica symmetric ansatz
4.1 Computation of the FP potential
4.1.1 Perturbations
4.1.2 The replicated entropy and the RS ansatz
4.1.3 Final result for the entropy of the glassy state
4.1.4 Saddle point equations
4.1.5 Physical observables
4.2 Stability of the RS ansatz
4.2.1 The unstable mode
4.3 Results
4.3.1 Special limits
4.3.2 Compression-decompression
4.3.3 Shear strain
4.4 Discussion
4.4.1 The Gardner transition
5 The full replica symmetry breaking ansatz
5.1 The potential
5.1.1 The fRSB parametrization
5.1.2 Expression of the potential and the observables
5.2 Results
5.2.1 Phase diagrams and MSDs
5.2.2 Critical slowing down
5.2.3 Fluctuations
5.2.4 Shear moduli
5.3 Discussion
5.3.1 Yielding within the fRSB ansatz
6 Numerics in the Mari-Kurchan model
6.1 Model
6.2 Results
6.2.1 Isocomplexity
6.2.2 State Following and the Gardner transition
7 Conclusions
7.1 Summary of main predictions
7.2 Strengths and weaknesses of our approach
7.2.1 The current status of RFOT
7.3 Proposals for further research
7.3.1 The Gardner transition in shear
7.3.2 State following in AQS protocols
7.3.3 Avalanches
7.3.4 Yielding
7.3.5 Non-linear rheology
A The infinite-d solution of hard spheres
B Computation of the replicated entropy in the RS ansatz
B.1 Entropic term
B.2 Interaction term
B.2.1 General expression of the replicated Mayer function
B.2.2 Computation of the interaction term for a RS displacement matrix
C Computation of the replicon mode
C.1 The structure of the unstable mode
C.2 Entropic term
C.3 Interaction term
D Computation of the replicated entropy in the fRSB ansatz
D.1 Entropic term
D.2 Interaction term
D.3 Simplifications for m = 1
E Variational equations in the fRSB ansatz
E.1 Lagrange multipliers
E.2 A different equation for G(x)
E.3 Scaling analysis near jamming
E.3.1 Scaling form of the equations
E.3.2 Asymptotes and scaling of b P and b f
Bibliography