Abstract:
In the context of the energy landscape description of supercooled liquids, we propose an explanation for the different behaviour of fragile and strong liquids. Above the Goldstein crossover temperature Tx, diffusion is interpreted as a motion in the phase space among unstable stationary points of the potential energy, that is among saddles. In this way two mechanisms of diffusion arise: mechanism A takes place when the system crosses potential energy barriers along stable uphill directions, while mechanism B consists in finding unstable downhill directions out of a saddle. Depending on the mutual value of the efficiency temperatures of A and B, we obtain two very different behaviours of the viscosity, reproducing the usual classification of liquids in fragile and strong. Moreover, this scenario very naturally predicts the possibility of a fragile-to-strong crossover when lowering the temperature.

Abstract:
By means of extensive numerical simulations we show that all the distinctive features of the minority game introduced by Challet and Zhang (1997), are completely independent from the memory of the agents. The only crucial requirement is that all the individuals must posses the same information, irrespective of the fact that this information is true or false.

Abstract:
When we lower the temperature of a liquid, at some point we meet a first order phase transition to the crystal. Yet, under certain conditions it is possible to keep the system in its metastable phase and to avoid crystallization. In this way the liquid enters in the supercooled phase. Supercooled liquids have a very rich phenomenology, which is still far from being completely understood. To begin with, there is the problem of how to prevent crystallization and how deeply the liquid can be supercooled before a metastability limit is hit. But by far the most interesting feature of supercooled liquids is the dynamic glass transition: when the temperature is decreased below a certain point, the relaxation time increases so much that a dramatic dynamical arrest intervenes and we are unable to equilibrate the system within reasonable experimental times. The glass transition is a phenomenon whose physical origin has stirred an enormous interest in the last hundred years. Why does it occur? Is it just a conventional reference point, or does it have a more profound physical meaning? Is it a purely dynamical event, or the manifestation of a true thermodynamic transition? What is the correlation length associated to the sharp increase of the relaxation time? Can we define a new kind of amorphous order? A shared theory of supercooled liquids and the glass transition does not yet exist and these questions are still largely open. Here, I will illustrate in the most elementary fashion the main phenomenological traits of supercooled liquids and discuss in a very partial way a few theoretical ideas on the subject.

Abstract:
We introduce a novel method for calculating the size of the critical nucleus and the value of the surface tension in systems with first order phase transition. The method is based on classical nucleation theory, and it consists in studying the thermodynamics of a sphere of given radius embedded in a frozen metastable surrounding. The frozen configuration creates a pinning field on the surface of the free sphere. The pinning field forces the sphere to stay in the metastable phase as long as its size is smaller than the critical nucleus. We test our method in two first-order systems, both on a two-dimensional lattice: a system where the parameter tuning the transition is the magnetic field, and a second system where the tuning parameter is the temperature. In both cases the results are satisfying. Unlike previous techniques, our method does not require an infinite volume limit to compute the surface tension, and it therefore gives reliable estimates even by using relatively small systems. However, our method cannot be used at, or close to, the critical point, i.e. at coexistence, where the critical nucleus becomes infinitely large.

Abstract:
In these notes the main theoretical concepts and techniques in the field of mean-field spin-glasses are reviewed in a compact and pedagogical way, for the benefit of the graduate and undergraduate student. One particular spin-glass model is analyzed (the p-spin spherical model) by using three different approaches. Thermodynamics, covering pure states, overlaps, overlap distribution, replica symmetry breaking, and the static transition. Dynamics, covering the generating functional method, generalized Langevin equation, equations for the correlation and the response, the Mode Coupling approximation, and the dynamical transition. And finally complexity, covering the mean-field (TAP) free energy, metastable states, entropy crisis, threshold energy, and saddles. Particular attention has been paid on the mutual consistency of the results obtained from the different methods.

Abstract:
We consider the ordering kinetics of a nonconserved scalar field advected by a uniform shear flow. Using the Ohta-Jasnow-Kawasaki approximation, modified to allow for shear-induced anisotropy, we calculate the asymptotic time dependence of the characteristic length scales, L_parallel and L_perp, that describe the growth of order parallel and perpendicular to the mean domain orientation. In space dimension d=3 we find, up to constants, L_parallel = gamma t^{3/2}, L_perp = t^{1/2}, where gamma is the shear rate, while for d = 2 we find L_parallel = gamma^{1/2} t (ln t)^{1/4}, L_perp = gamma^{-1/2}(ln t)^{-1/4} . Our predictions for d=2 can be tested by experiments on twisted nematic liquid crystals.

Abstract:
We consider the p-spin spherical spin-glass model in the presence of an external magnetic field as a general example of a mean-field system where a one step replica symmetry breaking (1-RSB) occurs. In this context we compute the complexity of the Thouless-Anderson-Palmer states, performing a quenched computation. We find what is the general connection between this method and the standard static 1-RSB one, formulating a clear mapping between the parameters used in the two different calculations. We also perform a dynamical analysis of the model, by which we confirm the validity of our results.

Abstract:
We introduce a three replica potential useful to examine the structure of metastables states above the static transition temperature, in the spherical p-spin model. Studying the minima of the potential we are able to find which is the distance between the nearest equilibrium and local equilibrium states, obtaining in this way information on the dynamics of the system. Furthermore, the analysis of the potential at the dynamical transition temperature suggests that equilibrium states are not randomly distributed in the phase space.

Abstract:
In the context of the p-spin spherical model for generalized spin glasses, we give an estimate of the free energy barriers separating an equilibrium state from the metastable states close to it.

Abstract:
We introduce a lattice spin model where frustration is due to multibody interactions rather than quenched disorder in the Hamiltonian. The system has a crystalline ground state and below the melting temperature displays a dynamic behaviour typical of fragile glasses. However, the supercooled phase loses stability at an effective spinodal temperature, and thanks to this the Kauzmann paradox is resolved. Below the spinodal the system enters an off-equilibrium regime corresponding to fast crystal nucleation followed by slow activated crystal growth. In this phase and in a time region which is longer the lower the temperature we observe a violation of the fluctuation-dissipation theorem analogous to structural glasses. Moreover, we show that in this system there is no qualitative difference between a locally stable glassy configuration and a highly disordered polycrystal.