Computation of the replicated entropy in the RS ansatz 

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Strengths and weaknesses

FLD theory has the great merit of actually posing deep questions about the nature of cooperative regions seen in glassy systems. RFOT just postulates them as originated by an underlying disordered FEL, and DFT only focuses on how they move. FLD instead describes them explicitly, in a very practical and very grounded way, and proposes a coherent and elegant picture for their origin. FLD has also the merit of bringing back the attention of researchers on a fact so obvious it is often forgotten: glasses behave like solids, and the rigidity of solids is indeed due to structure, not dynamics, so perhaps it is too soon to rule out the existence of any structure in glassy materials: they may actually hide more order than we think (see for example [119]).
This grounded and real-space bound description of FLD theory is, however, also its weakness: as of now there have not been any direct observations of FLDs [5, 23]. We have focused before on how the static structure factor S(q) does not seem to capture anything unusual on crossing Tg, although the fragmentation in FLDs could be so severe that the residual order eludes a bulk tool like the S(q). In principle it could be possible to observe FLDs and their related LPSs through numerical simulations, but this is more complicated than it looks [120], and there is not even agreement on which is the locally preferred order one should look for. For example in [121], the growth of icosahedral order is found to be more pronounced in fragile liquids as expected from the FLD approach, but in [122] it is argued that everything can be understood in terms of bond-orientational order, rather than icosahedral, so that the situation looks very convoluted. Moreover, one would also appreciate to go beyond scaling arguments and coarsegrained models like the one defined in equation (1.55), and perform calculations on microscopic models of glass formers, but this does not look easy. It is indeed possible to implement numerically a Nelson-like treatment for a Lennard-Jones mixture [123] with very encouraging results, but at present there is no apparent way of translating this into a statistical-mechanical calculation. As a matter of fact, models like the one in equation (1.55), when treated with the replica method, show a Kauzmann transition like the one predicted in RFOT (see for example [124]), so it could very well be that a first-principles treatment of FLD will end up giving back RFOT results, which could be a very interesting turn of events.

The jamming transition

Let us now make a detour to talk about a phenomenon which at first glance is completely unrelated to glasses: the jamming transition [42–44]. The aim of this paragraph is to convince to reader that it is not at all a detour, at least within RFOT.
The jamming transition is probably one of the most ubiquitous phenomena one can conceive. The canonical example of jamming system is a fistful of sand: when it is not compressed, it responds to stresses by flowing, more or less like a liquid would. However, if we clench our fist, at a certain point we will not be able to squeeze the sand anymore and the response will be solid-like: the grains of sand are mechanically in contact, forming an amorphous, tight packing. The jamming transition is a transition between a loose, liquid-like system to a jammed, solid-like one.
This may look like a calorimetric glass transition, but it is not the same thing. In the glass transition the solid that originates from the glass former is due to caging and vibrations inside the cage, which render it capable of bearing loads and respond like a solid; however, the system is still compressible and pressure is finite. In the jamming case, there is no temperature and the solid-like behavior is due to the forming of a network of mechanical contacts between the grains; and if those
grains can be reasonably modeled as mechanically undeformable, hard particles, the resulting solid is incompressible: its pressure is infinite.
To study jammed packings, one usually constructs them using a certain protocol (the choice of words is not casual). In experiments, one can for example throw the grains in a shaking box one at a time until the packing jams [134]. In simulations, a very popular algorithm is the one by Lubachesky and Stillinger [135] (LS), wherein the packing is created by inflating the spheres at a fixed rate γ during a molecular dynamics run. Another possibility is to consider soft particles with a potential that vanishes outside the particles (tennis balls, essentially): one starts from a random configuration, compresses it, then minimizes the potential energy, and then compresses it again, until a zero energy configuration cannot be found anymore [136, 137]. The jamming problem can then be formulated as: “Given a procedure to construct an amorphous packing, what are the properties of the packing so obtained? First of all, what is its jamming density ϕj 3? How does the contact network behave?
Which properties depend on the actual procedure, and which ones do not?”. A ponderous research effort has ensued, in the last years, to answer these questions. Luckily for us, this effort has been successful (see for example the reviews [138, 139]), at least for frictionless spherical particles, so we can reap and summarize the most relevant results:
1. The jamming density ϕj does depend on the protocol used. In the paradigmatic case of the LS algorithm for hard spheres, it can be seen that a lower rate γ corresponds to a lower ϕj [140, 141]. In 3d, an fcc crystal is produced for small rates (ϕj = ϕFCC ≈ 0.74), while for a fairly large range of intermediate rates an amorphous packing with ϕj = ϕGCP ≈ 0.68 is produced4. For even higher rates, ϕj goes down smoothly with γ.
2. All amorphous packings of frictionless particles at the jamming threshold ϕj are isostatic [142, 143], which means that the average number of contacts z in the packing is just the one needed to ensure mechanical stability, ziso = 2d5, in agreement with Maxwell’s criterion [144].
3. The probability distribution of the absolute P value of forces in a packing P(f) ≡ Nc i=1 δ(f −fi) (where Nc is the number of contacts) has a power-law behavior (a pseudogap) for small forces, P(f) ≃ fθ, where the exponent θ is apparently the same for every d ≥ 2 [145–147].
4. The pair distribution function g(h)[18].

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Theoretical approaches to aging

As we mentioned in the previous paragraph, a satisfactory “theory of aging” must be able, given a certain preparation protocol, to predict the properties of the glass so produced. Since a glass is out of equilibrium, using standard statistical mechanics would only give back trivial results relative to the equilibrated supercooled liquid, which means that one must in principle resort to off-equilibrium dynamical tools. In this case, to predict the properties of an aged glass, one must
1. Write the equations for the dynamical process that reproduces the protocol under consideration,
2. solve them and compute the values of observables from the solution. Specifying to our case, in the case of brutal quenching protocols [125] one needs to study a dynamical process starting from a random initial configuration, while in the case of the annealing protocols (like those employed for ultrastable glasses) [126] one must consider a dynamics starting from an initial configuration equilibrated at Tf [21].

Table of contents :

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


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