Abstract:
A relation between vibrational entropy and particles mean square displacement is derived in super-cooled liquids, assuming that the main effect of temperature changes is to rescale the vibrational spectrum. Deviations from this relation, in particular due to the presence of a Boson Peak whose shape and frequency changes with temperature, are estimated. Using observations of the short-time dynamics in liquids of various fragility, it is argued that (i) if the crystal entropy is significantly smaller than the liquid entropy at $T_g$, the extrapolation of the vibrational entropy leads to the correlation $T_K\approx T_0$, where $T_K$ is the Kauzmann temperature and $T_0$ is the temperature extracted from the Vogel-Fulcher fit of the viscosity. (ii) The jump in specific heat associated with vibrational entropy is very small for strong liquids, and increases with fragility. The analysis suggests that these correlations stem from the stiffening of the Boson Peak under cooling, underlying the importance of this phenomenon on the dynamical arrest.

Abstract:
We poorly understand the properties of amorphous systems at small length scales, where a continuous elastic description breaks down. This is apparent when one considers their vibrational and transport properties, or the way forces propagate in these solids. Little is known about the microscopic cause of their rigidity. Recently it has been observed numerically that an assembly of elastic particles has a critical behavior near the jamming threshold where the pressure vanishes. At the transition such a system does not behave as a continuous medium at any length scales. When this system is compressed, scaling is observed for the elastic moduli, the coordination number, but also for the density of vibrational modes. In the present work we derive theoretically these results, and show that they apply to various systems such as granular matter and silica, but also to colloidal glasses. In particular we show that: (i) these systems present a large excess of vibrational modes at low frequency in comparison with normal solids, called the "boson peak" in the glass literature. The corresponding modes are very different from plane waves, and their frequency is related to the system coordination; (ii) rigidity is a non-local property of the packing geometry, characterized by a length scale which can be large. For elastic particles this length diverges near the jamming transition; (iii) for repulsive systems the shear modulus can be much smaller than the bulk modulus. We compute the corresponding scaling laws near the jamming threshold. Finally, we discuss the applications of these results to the glass transition, the transport, and the geometry of the random close packing.

Abstract:
A layer of sand of thickness h flows down a rough surface if the inclination is larger than some threshold value theta which decreases with h. A tentative microscopic model for the dependence of theta with h is proposed for rigid frictional grains, based on the following hypothesis: (i) a horizontal layer of sand has some coordination z larger than a critical value z_c where mechanical stability is lost (ii) as the tilt angle is increased, the configurations visited present a growing proportion $_s of sliding contacts. Instability with respect to flow occurs when z-z_s=z_c. This criterion leads to a prediction for theta(h) in good agreement with empirical observations.

Abstract:
Assemblies of purely repulsive and frictionless particles, such as emulsions or hard spheres, display very curious properties near their jamming transition, which occurs at the random close packing for mono-disperse spheres. Although such systems do not contain the long and cross-linked polymeric chains characterizing a rubber, they behave macroscopically in a similar way: the shear modulus $G$ can become infinitely smaller than the bulk modulus $B$. After reviewing recent theoretical results on the structure of such packing (in particular their coordination) I will propose an explanation for the observed scaling of the elastic moduli, and explain why the arguments both apply to soft and hard particles.

Abstract:
Glasses have a large excess of low-frequency vibrational modes in comparison with continuous elastic body, the so-called Boson Peak, which appears to correlate with several crucial properties of glasses, such as transport or fragility. I review recent results showing that the Boson Peak is a necessary consequence of the weak connectivity of the solid. I explain why in assemblies repulsive spheres the boson peak shifts up to zero frequency as the pressure is lowered toward the jamming threshold, and derive the corresponding exponent. I show how these ideas capture the main low-frequency features of the vibrational spectrum of amorphous silica. These results extend arguments of Phillips on the presence of floppy modes in under-constrained covalent networks to glasses where the covalent network is rigid, or when interactions are purely radial.

Abstract:
The effect of coordination on transport is investigated theoretically using random networks of springs as model systems. An effective medium approximation is made to compute the density of states of the vibrational modes, their energy diffusivity (a spectral measure of transport) and their spatial correlations as the network coordination $z$ is varied. Critical behaviors are obtained as $z\to z_c$ where these networks lose rigidity. A sharp cross-over from a regime where modes are plane-wave-like toward a regime of extended but strongly-scattered modes occurs at some frequency $\omega^*\sim z-z_c$, which does not correspond to the Ioffe-Regel criterion. Above $\omega^*$ both the density of states and the diffusivity are nearly constant. These results agree remarkably with recent numerical observations of repulsive particles near the jamming threshold \cite{ning}. The analysis further predicts that the length scale characterizing the correlation of displacements of the scattered modes decays as $1/\sqrt{\omega}$ with frequency, whereas for $\omega<<\omega^*$ Rayleigh scattering is found with a scattering length $l_s\sim (z-z_c)^3/\omega^4$. It is argued that this description applies to silica glass where it compares well with thermal conductivity data, and to transverse ultrasound propagation in granular matter.

Abstract:
The requirement that packings of hard particles, arguably the simplest structural glass, cannot be compressed by rearranging their network of contacts is shown to yield a new constraint on their microscopic structure. This constraint takes the form a bound between the distribution of contact forces P(f) and the pair distribution function g(r): if P(f) \sim f^{\theta} and g(r) \sim (r-{\sigma})^(-{\gamma}), where {\sigma} is the particle diameter, one finds that {\gamma} \geq 1/(2+{\theta}). This bound plays a role similar to those found in some glassy materials with long-range interactions, such as the Coulomb gap in Anderson insulators or the distribution of local fields in mean-field spin glasses. There is ground to believe that this bound is saturated, offering an explanation for the presence of avalanches of rearrangements with power-law statistics observed in packings.

Abstract:
We estimate numerically the normal modes of the free energy in a glass of hard discs. We observe that, near the glass transition or after a rapid quench deep in the glass phase, the density of states (i) is characteristic of a marginally stable structure, in particular it di splays a frequency scale $\omega^*\sim p^{1/2}$, where $p$ is the pressure and (ii) gives a faithful representation of the short-time dyn amics. This brings further evidences that the boson peak near the glass transition corresponds to the relaxation of marginal modes of a we akly-coordinated structure, and implies that the mean square displacement in the glass phase is anomalously large and goes as $< \delta R^2 > \sim p^{-3/2}$, a prediction that we check numerically.

Abstract:
We study theoretically and numerically the microscopic cause of the mechanical stability of hard sphere glasses near their maximum packing. We show that, after coarse-graining over time, the hard sphere interaction can be described by an effective potential which is exactly logarithmic at the random close packing $\phi_c$. This allows to define normal modes, and to apply recent results valid for elastic networks: mechanical stability is a non-local property of the packing geometry, and is characterized by some length scale $l^*$ which diverges at $\phi_c$ [1, 2]. We compute the scaling of the bulk and shear moduli near $\phi_c$, and speculate on the possible implications of these results for the glass transition.

Abstract:
Failure and flow of amorphous materials are central to various phenomena including earthquakes and landslides. There is accumulating evidence that the yielding transition between a flowing and an arrested phase is a critical phenomenon, but the associated exponents are not understood, even at a mean-field level where the validity of popular models is debated. Here we solve a mean-field model that captures the broad distribution of the mechanical noise generated by plasticity, whose behavior is related to biased L\'evy flights near an absorbing boundary. We compute the exponent $\theta$ characterizing the density of shear transformation $P(x)\sim x^{\theta}$, where $x$ is the stress increment beyond which they yield. We find that after an isotropic thermal quench, $\theta=1/2$. However, $\theta$ depends continuously on the applied shear stress, this dependence is not monotonic, and its value at the yield stress is not universal. The model rationalizes previously unexplained observations, and captures reasonably well the value of exponents in three dimensions. These results support that it is the true mean-field model that applies in large dimension, and raise fundamental questions on the nature of the yielding transition.