Cristina Toninelli

Dynamical susceptibility of glass formers: Contrasting the predictions of theoretical scenarios

We compute analytically and numerically the four-point correlation function that characterizes non-trivial cooperative dynamics in glassy systems within several models of glasses: elasto-plastic deformations, mode-coupling theory (MCT), collectively rearranging regions (CRR), diffusing defects and kinetically constrained models (KCM). Some features of the four-point susceptibility chi_4(t) are expected to be universal. at short times we expect an elastic regime characterized by a t or sqrt{t} growth. We find both in the beta, and the early alpha regime that chi_4 sim t^mu, where mu is directly related to the mechanism responsible for relaxation. This regime ends when a maximum of chi_4 is reached at a time t=t^* of the order of the relaxation time of the system. This maximum is followed by a fast decay to zero at large times. The height of the maximum also follows a power-law, chi_4(t^*) sim t^{*lambda}. The value of the exponents mu and lambda allows one to distinguish between different mechanisms. For example, freely diffusing defects in d=3 lead to mu=2 and lambda=1, whereas the CRR scenario rather predicts either mu=1 or a logarithmic behaviour depending on the nature of the nucleation events, and a logarithmic behaviour of chi_4(t^*). MCT leads to mu=b and lambda =1/gamma, where b and gamma are the standard MCT exponents. We compare our theoretical results with numerical simulations on a Lennard-Jones and a soft-sphere system. Within the limited time-scales accessible to numerical simulations, we find that the exponent mu is rather small, mu<1, with a value in reasonable agreement with the MCT predictions.

Group Testing With Random Pools: Optimal Two-Stage Algorithms

We study the group testing of a set of N items each of which is defective with probability p. We focus on the double limit of small defect probability, p ≪ 1, and large number of variables, N ≫ 1, taking either p → 0 after N → ∝ or p = 1/Nβ with β ∈ (0,1/2). In both settings the optimal number of tests which are required to identify with certainty the defectives via a two-stage procedure, T̅(N, p), is known to scale as Np |log p|. Here we determine the sharp asymptotic value of T̅(N,p)/(Np|log p|) and construct a class of two-stage algorithms over which this optimal value is attained. This is done by choosing a proper bipartite regular graph (of tests and variable nodes) for the first stage of the detection. Furthermore we prove that this optimal value is also attained on average over a random bipartite graph where all variables have the same degree and the tests connected to a given variable are randomly chosen with uniform distribution among all tests. Finally, we improve the existing upper and lower bounds for the optimal number of tests in the case p = 1/Nβ with β ∈ [1/2,1).

Kinetically Constrained Models

Kinetically constrained spin models (KCSM) are interacting particle systems which are intensively studied in physics literature as models for systems undergoing glass or jamming transitions. KCSM leave on discrete lattices and evolve via a Glauber-like dynamics which is reversible w.r.t. a simple product measure. The key feature is that the creation/destruction of a particle at a given site can occur only if the current configuration satisfies proper local constraints. Due to the fact that creation/destruction rates can be zero, the whole analysis of the long time behavior becomes quite delicate. From the mathematical point of view, the basic issues concerning positivity of the spectral gap inside the ergodicity region and its scaling with the particle density remained open for most KCSM (with the exception of the East model in d=1 Aldous and P. Diaconis, J. Stat. Phys. 107(5–6):945–975 2002). Here we review a novel multi-scale approach which we have developed in Cancrini et al. (Probab. Theory Relat. Fields 140:459–504, 2008; Lecture Notes in Mathematics, vol. 1970, pp. 307–340, Springer, 2009) trough which we: (i) prove positivity of the spectral gap in the whole ergodic region for a wide class of KCSM on ℤ d , (ii) establish (sometimes optimal) bounds on the behavior of the spectral gap near the boundary of the ergodicity region and (iii) prove pure exponential decay at equilibrium for the persistence function, i.e. the probability that the occupation variable at the origin does not change before time t. Our findings disprove certain conjectures which appeared in the physical literature on the basis of numerical simulations. In particular (i) above establishes exponential decay of auto-correlation functions disproving the stretched exponential decay which had been conjecture for some KCSM and (ii) disproves some of the scalings which had been extrapolated from numerical simulations for the relaxation times (inverse of the spectral gap).

Finite-size effects in the dynamics of glass-forming liquids

We present a comprehensive theoretical study of finite-size effects in the relaxation dynamics of glass-forming liquids. Our analysis is motivated by recent theoretical progress regarding the understanding of relevant correlation length scales in liquids approaching the glass transition. We obtain predictions both from general theoretical arguments and from a variety of specific perspectives: mode-coupling theory, kinetically constrained and defect models, and random first-order transition theory. In the last approach, we predict in particular a nonmonotonic evolution of finite-size effects across the mode-coupling crossover due to the competition between mode-coupling and activated relaxation. We study the role of competing relaxation mechanisms in giving rise to nonmonotonic finite-size effects by devising a kinetically constrained model where the proximity to the mode-coupling singularity can be continuously tuned by changing the lattice topology. We use our theoretical findings to interpret the results of extensive molecular dynamics studies of four model liquids with distinct structures and kinetic fragilities. While the less fragile model only displays modest finite-size effects, we find a more significant size dependence evolving with temperature for more fragile models, such as Lennard-Jones particles and soft spheres. Finally, for a binary mixture of harmonic spheres we observe the predicted nonmonotonic temperature evolution of finite-size effects near the fitted mode-coupling singularity, suggesting that the crossover from mode-coupling to activated dynamics is more pronounced for this model. Finally, we discuss the close connection between our results and the recent report of a nonmonotonic temperature evolution of a dynamic length scale near the mode-coupling crossover in harmonic spheres.

Cooperative Behavior of Kinetically Constrained Lattice Gas Models of Glassy Dynamics

Kinetically constrained lattice models of glasses introduced by Kob and Andersen (KA) are analyzed. It is proved that only two behaviors are possible on hypercubic lattices: either ergodicity at all densities or trivial non-ergodicity, depending on the constraint parameter and the dimensionality. But in the ergodic cases, the dynamics is shown to be intrinsically cooperative at high densities giving rise to glassy dynamics as observed in simulations. The cooperativity is characterized by two length scales whose behavior controls finite-size effects: these are essential for interpreting simulations. In contrast to hypercubic lattices, on Bethe lattices KA models undergo a dynamical (jamming) phase transition at a critical density: this is characterized by diverging time and length scales and a discontinuous jump in the long-time limit of the density autocorrelation function. By analyzing generalized Bethe lattices (with loops) that interpolate between hypercubic lattices and standard Bethe lattices, the crossover between the dynamical transition that exists on these lattices and its absence in the hypercubic lattice limit is explored. Contact with earlier results are made via analysis of the related Fredrickson--Andersen models, followed by brief discussions of universality, of other approaches to glass transitions, and of some issues relevant for experiments.

Hydrodynamic limit for a particle system with degenerate rates

We study the hydrodynamic limit for some conservative particle systems with degenerate rates, namely with nearest neighbor exchange rates which vanish for certain configurations. These models belong to the class of kinetically constrained lattice gases (KCLG) which have been introduced and intensively studied in physics literature as simple models for the liquid/glass transition. Due to the degeneracy of rates for KCLG there exists blocked configurations which do not evolve under the dynamics and in general the hyperplanes of configurations with a fixed number of particles can be decomposed into different irreducible sets. As a consequence, both the Entropy and Relative Entropy method cannot be straightforwardly applied to prove the hydrodynamic limit. In particular, some care should be put when proving the One and Two block Lemmas which guarantee local convergence to equilibrium. We show that, for initial profiles smooth enough and bounded away from zero and one, the macroscopic density profile for our KCLG evolves under the diffusive time scaling according to the porous medium equation. Then we prove the same result for more general profiles for a slightly perturbed dynamics obtained by adding jumps of the Symmetric Simple Exclusion. The role of the latter is to remove the degeneracy of rates and at the same time they are properly slowed down in order not to change the macroscopic behavior. The equilibrium fluctuations and the magnitude of the spectral gap for this perturbed model are also obtained.

Activity Phase Transition for Constrained Dynamics

We consider two cases of kinetically constrained models, namely East and FA-1f models. The object of interest of our work is the activity

Finite Size Scaling of the Dynamical Free-Energy in a Kinetically Constrained Model

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