Breakdown of Elasticity in Amorphous Solids

What characterises a solid is its way to respond to external stresses. Ordered solids, such crystals, display an elastic regime followed by a plastic one, both well understood microscopically in terms of lattice distortion and dislocations. For amorphous solids the situation is instead less clear, and the microscopic understanding of the response to deformation and stress is a very active research topic. Several studies have revealed that even in the elastic regime the response is very jerky at low temperature, resembling very much the one of disordered magnetic materials. Here we show that in a very large class of amorphous solids this behaviour emerges by decreasing the temperature as a phase transition where standard elastic behaviour breaks down. At the transition all non-linear elastic modulii diverge and standard elasticity theory does not hold anymore. Below the transition the response to deformation becomes history and time-dependent.Comment: 3 figure

Theory of fluctuations in disordered systems

The thesis is devoted to the study of various aspects of disordered and glassy systems. In the first part of the thesis, we have studied the problem of charachterizing the critical dynamical fluctuations of structural glasses at the dynamical transition point. A field theory approach has been developed combined with the replica method and it has been introduced an effective theory that is capable to describe the dynamical heterogeneities at the dynamical transition point. A Ginzburg criterion has been derived to understand the region of validity of the mean field approach. These results are valid for the critical behavior of the dynamics in the beta regime. To understand what happens in the alpha regime we have developed a Boltzmann Pseudodynamics approach to structural glasses that is able to cupture the quasi-equilibrium nature of the glassy dynamics in the long time regime. The third part of the thesis is devoted to the study of the glass and jamming physics of hard spheres in the infinite dimension limit. In this context we show that this model displays a Gardner transition that affects deeply the jamming part of the phase diagram. This means that to describe correctly the jamming properties of hard spheres we need to take into account the full replica symmetry breaking effects. The full replica symmetry breaking formalism for hard sphere systems is completely developed. The last part of the thesis is devoted to study mode coupling dynamics around a quasi-continuous transition

Statistical mechanics of the spherical hierarchical model with random fields

We study analytically the equilibrium properties of the spherical hierarchical model in the presence of random fields. The expression for the critical line separating a paramagnetic from a ferromagnetic phase is derived. The critical exponents characterising this phase transition are computed analytically and compared with those of the corresponding $D$-dimensional short-range model, leading to conclude that the usual mapping between one dimensional long-range models and $D$-dimensional short-range models holds exactly for this system, in contrast to models with Ising spins. Moreover, the critical exponents of the pure model and those of the random field model satisfy a relationship that mimics the dimensional reduction rule. The absence of a spin-glass phase is strongly supported by the local stability analysis of the replica symmetric saddle-point as well as by an independent computation of the free-energy using a renormalization-like approach. This latter result enlarges the class of random field models for which the spin-glass phase has been recently ruled out.Comment: 23 pages, 2 figure

Exact theory of dense amorphous hard spheres in high dimension. II. The high density regime and the Gardner transition

We consider the theory of the glass phase and jamming of hard spheres in the large space dimension limit. Building upon the exact expression for the free-energy functional obtained previously, we find that the Random First Order Transition (RFOT) scenario is realized here with two thermodynamic transitions: the usual Kauzmann point associated with entropy crisis, and a further transition at higher pressures in which a glassy structure of micro-states is developed within each amorphous state. This kind of glass-glass transition into a phase dominating the higher densities was described years ago by Elisabeth Gardner, and may well be a generic feature of RFOT. Micro states that are small excitations of an amorphous matrix -- separated by low entropic or energetic barriers -- thus emerge naturally, and modify the high pressure (or low temperature) limit of the thermodynamic functions.Comment: 26 pages, 7 figures -- to be published in a Special Issue of The Journal of Physical Chemistry B in honor of Peter G. Wolynes -- paper I is arXiv:1208.042

Universal Spectrum of Normal Modes in Low-Temperature Glasses: an Exact Solution

We report an analytical study of the vibrational spectrum of the simplest model of jamming, the soft perceptron. We identify two distinct classes of soft modes. The first kind of modes are related to isostaticity and appear only in the close vicinity of the jamming transition. The second kind of modes instead are present everywhere in the glass phase and are related to the hierarchical structure of the potential energy landscape. Our results highlight the universality of the spectrum of normal modes in disordered systems, and open the way towards a detailed analytical understanding of the vibrational spectrum of low-temperature glasses.Comment: 6 pages, 3 figures, submitted to PNA

Statistical physics of learning in high-dimensional chaotic systems

Recurrent neural network models are high-dimensional dynamical systems in which the degrees of freedom are coupled through synaptic connections and evolve according to a set of differential equations. When the synaptic connections are sufficiently strong and random, such models display a chaotic phase which can be used to perform a task if the network is carefully trained. It is fair to say that this setting applies to many other complex systems, from biological organisms (from cells, to individuals and populations) to financial markets. For all these out-of-equilibrium systems, elementary units live in a chaotic environment and need to adapt their strategies to survive by extracting information from the environment and controlling the feedback loop on it. In this work we consider a prototypical high-dimensional chaotic system as a simplified model for a recurrent neural network and more complex learning systems. We study the model under two particular training strategies: Hebbian training and FORCE training. In the first case we show that Hebbian training can be used to tune the level of chaos in the dynamics and this reproduces some results recently obtained in the study of standard models of RNN. In the latter case, we show that the dynamical system can be trained to reproduce a simple periodic function. We show that the FORCE algorithm drives the dynamics close to an asymptotic attractor the larger the training time. We also discuss possible extensions of our setup to other learning strategies.Comment: 21 pages, 8 figure

Following the evolution of glassy states under external perturbations: compression and shear-strain

We consider the adiabatic evolution of glassy states under external perturbations. Although the formalism we use is very general, we focus here on infinite-dimensional hard spheres where an exact analysis is possible. We consider perturbations of the boundary, i.e. compression or (volume preserving) shear-strain, and we compute the response of glassy states to such perturbations: pressure and shear-stress. We find that both quantities overshoot before the glass state becomes unstable at a spinodal point where it melts into a liquid (or yields). We also estimate the yield stress of the glass. Finally, we study the stability of the glass basins towards breaking into sub-basins, corresponding to a Gardner transition. We find that close to the dynamical transition, glasses undergo a Gardner transition after an infinitesimal perturbation.Comment: 4 pages (3 figures) + 24 pages (5 pages) of appendice