arXiv subjects

Pierfrancesco Urbani

Publications and source records attributed to Pierfrancesco Urbani.

At least 55 records · Page 3Linked to original sources

arXiv2018

Out-of-equilibrium dynamical mean-field equations for the perceptron model

Perceptrons are the building blocks of many theoretical approaches to a wide range of complex systems, ranging from neural networks and deep learning machines, to constraint satisfaction problems, glasses and ecosystems. Despite their applicability and importance, a detailed study of their Langevin dynamics has never been performed yet. Here we derive the mean-field dynamical equations that describe the continuous random perceptron in the thermodynamic limit, in a very general setting with arbitrary noise and friction kernels, not necessarily related by equilibrium relations. We derive the equations in two ways: via a dynamical cavity method, and via a path-integral approach in its supersymmetric formulation. The end point of both approaches is the reduction of the dynamics of the system to an effective stochastic process for a representative dynamical variable. Because the perceptron is formally very close to a system of interacting particles in a high dimensional space, the methods we develop here can be transferred to the study of liquid and glasses in high dimensions. Potentially interesting applications are thus the study of the glass transition in active matter, the study of the dynamics around the jamming transition, and the calculation of rheological properties in driven systems.

cond-mat.dis-nn↗

arXiv2018

Universality of jamming of non-spherical particles

Amorphous packings of non-spherical particles such as ellipsoids and spherocylinders are known to be hypostatic: the number of mechanical contacts between particles is smaller than the number of degrees of freedom, thus violating Maxwell's mechanical stability criterion. In this work, we propose a general theory of hypostatic amorphous packings and the associated jamming transition. First, we show that many systems fall into a same universality class. As an example, we explicitly map ellipsoids into a system of `breathing' particles. We show by using a marginal stability argument that in both cases jammed packings are hypostatic, and that the critical exponents related to the contact number and the vibrational density of states are the same. Furthermore, we introduce a generalized perceptron model which can be solved analytically by the replica method. The analytical solution predicts critical exponents in the same hypostatic jamming universality class. Our analysis further reveals that the force and gap distributions of hypostatic jamming do not show power-law behavior, in marked contrast to the isostatic jamming of spherical particles. Finally, we confirm our theoretical predictions by numerical simulations.

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arXiv2018

A stability-reversibility map unifies elasticity, plasticity, yielding and jamming in hard sphere glasses

Amorphous solids, such as glasses, have complex responses to deformations, with significant consequences in material design and applications. In this respect two intertwined aspects are important: stability and reversibility. It is crucial to understand on the one hand how a glass may become unstable due to increased plasticity under shear deformations; on the other hand, to what extent the response is reversible, meaning how much a system is able to recover the original configuration once the perturbation is released. Here we focus on assemblies of hard spheres as the simplest model of amorphous solids such as colloidal glasses and granular matter. We prepare glass states quenched from equilibrium supercooled liquid states, which are obtained by using the swap Monte Carlo algorithm and correspond to a wide range of structural relaxation time scales. We exhaustively map out their stability and reversibility under volume and shear strains, using extensive numerical simulations. The region on the volume-shear strain phase diagram where the original glass state remains solid is bounded by the shear-yielding and the shear-jamming lines which meet at a yielding-jamming crossover point. This solid phase can be further divided into two sub-phases: the stable glass phase where the system deforms purely elastically and is totally reversible, and the marginal glass phase where it experiences stochastic plastic deformations at mesoscopic scales and is partially irreversible. The details of the stability-reversibility map depend strongly on the quality of annealing of the glass. This study provides a unified framework for understanding elasticity, plasticity, yielding and jamming in amorphous solids.

cond-mat.soft↗

arXiv2018

Microscopic theory of two-step yielding in attractive colloids

Attractive colloids display two distinct amorphous solid phases: the attractive glass, due to particle bonding, and the repulsive glass, due to the hard core repulsion. By means of a microscopic mean field approach, we analyze their response to a quasi-static shear strain. We find that the presence of two distinct interaction length scales may result in a sharp two-step yielding process, which can be associated with a hysteretic stress response, or with a reversible but non-monotonic stress-strain curve. We derive a generic phase diagram characterized by two distinct yielding lines, an inverse yielding, and a critical point separating the hysteretic and reversible regimes. Our results should be applicable to a large class of glassy materials characterized by two distinct interaction length scales.

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arXiv2018

Approximate Survey Propagation for Statistical Inference

Approximate message passing algorithm enjoyed considerable attention in the last decade. In this paper we introduce a variant of the AMP algorithm that takes into account glassy nature of the system under consideration. We coin this algorithm as the approximate survey propagation (ASP) and derive it for a class of low-rank matrix estimation problems. We derive the state evolution for the ASP algorithm and prove that it reproduces the one-step replica symmetry breaking (1RSB) fixed-point equations, well-known in physics of disordered systems. Our derivation thus gives a concrete algorithmic meaning to the 1RSB equations that is of independent interest. We characterize the performance of ASP in terms of convergence and mean-squared error as a function of the free Parisi parameter s. We conclude that when there is a model mismatch between the true generative model and the inference model, the performance of AMP rapidly degrades both in terms of MSE and of convergence, while ASP converges in a larger regime and can reach lower errors. Among other results, our analysis leads us to a striking hypothesis that whenever s (or other parameters) can be set in such a way that the Nishimori condition $M=Q>0$ is restored, then the corresponding algorithm is able to reach mean-squared error as low as the Bayes-optimal error obtained when the model and its parameters are known and exactly matched in the inference procedure.

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arXiv2018

Liu-Nagel phase diagrams in infinite dimension

We study Harmonic Soft Spheres as a model of thermal structural glasses in the limit of infinite dimensions. We show that cooling, compressing and shearing a glass lead to a Gardner transition and, hence, to a marginally stable amorphous solid as found for Hard Spheres systems. A general outcome of our results is that a reduced stability of the glass favors the appearance of the Gardner transition. Therefore using strong perturbations, e.g. shear and compression, on standard glasses or using weak perturbations on weakly stable glasses, e.g. the ones prepared close to the jamming point, are the generic ways to induce a Gardner transition. The formalism that we discuss allows to study general perturbations, including strain deformations that are important to study soft glassy rheology at the mean field level.

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arXiv2017

Universality of the SAT-UNSAT (jamming) threshold in non-convex continuous constraint satisfaction problems

Random constraint satisfaction problems (CSP) have been studied extensively using statistical physics techniques. They provide a benchmark to study average case scenarios instead of the worst case one. The interplay between statistical physics of disordered systems and computer science has brought new light into the realm of computational complexity theory, by introducing the notion of clustering of solutions, related to replica symmetry breaking. However, the class of problems in which clustering has been studied often involve discrete degrees of freedom: standard random CSPs are random K-SAT (aka disordered Ising models) or random coloring problems (aka disordered Potts models). In this work we consider instead problems that involve continuous degrees of freedom. The simplest prototype of these problems is the perceptron. Here we discuss in detail the full phase diagram of the model. In the regions of parameter space where the problem is non-convex, leading to multiple disconnected clusters of solutions, the solution is critical at the SAT/UNSAT threshold and lies in the same universality class of the jamming transition of soft spheres. We show how the critical behavior at the satisfiability threshold emerges, and we compute the critical exponents associated to the approach to the transition from both the SAT and UNSAT phase. We conjecture that there is a large universality class of non-convex continuous CSPs whose SAT-UNSAT threshold is described by the same scaling solution.

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arXiv2017

Glass and Jamming Transitions: From Exact Results to Finite-Dimensional Descriptions

Despite decades of work, gaining a first-principle understanding of amorphous materials remains an extremely challenging problem. However, recent theoretical breakthroughs have led to the formulation of an exact solution in the mean-field limit of infinite spatial dimension, and numerical simulations have remarkably confirmed the dimensional robustness of some of the predictions. This review describes these latest advances. More specifically, we consider the dynamical and thermodynamic descriptions of hard spheres around the dynamical, Gardner and jamming transitions. Comparing mean-field predictions with the finite-dimensional simulations, we identify robust aspects of the description and uncover its more sensitive features. We conclude with a brief overview of ongoing research.

cond-mat.dis-nn↗

arXiv2017

Shear yielding and shear jamming of dense hard sphere glasses

We investigate the response of dense hard sphere glasses to a shear strain, in a wide range of pressures ranging from the glass transition to the infinite-pressure jamming point. The phase diagram in the density-shear strain plane is calculated analytically using the mean field infinite dimensional solution. We find that just above the glass transition, the glass generically yields at a finite shear strain. The yielding transition, in the mean field picture, is a spinodal point in presence of disorder. At higher densities, instead, we find that the glass generically jams at a finite shear strain: the jamming transition prevents yielding. The shear yielding and shear jamming lines merge in a critical point, close to which the system yields at extremely large shear stress. Around this point, a highly non-trivial yielding dynamics characterized by system-spanning disordered fractures is expected.

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arXiv2016

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.

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arXiv2015

Following the evolution of glassy states under external perturbations: the full replica symmetry breaking solution

The state-following technique allows the study of metastable glassy states under external perturbations. Here we show how this construction can be used to study the behavior of glassy states of Hard Spheres in infinite dimensions under compression or shear strain. In a preceding work it has been shown that in both cases, when the external perturbation is sufficiently strong, glassy states undergo a second-order transition, called the Gardner transition, whereupon a hierarchical structure of marginal micro-states manifests within the original glass state. The purpose of this work is to study the solution of the state-following construction in this marginal phase. We show that upon compression, close to the jamming transition, the metastable states are described by a scaling solution characterized by a set of non-trivial critical exponents that agree with the results obtained in a preceding work, and we compute the value of the final jamming density for various glassy states. Moreover we show that under the action of the shear strain, beyond the Gardner point, the metastable states can be followed in the marginal phase and we detect an overshoot in the stress-strain curve in agreement with numerical and experimental observations. Finally we further characterize the Gardner transition point by computing both the critical 4-point susceptibility and the exponents that characterize the critical slowing down of the dynamics within a glassy state close to the transition.

cond-mat.soft↗

arXiv2015

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.

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arXiv2015

Quasi equilibrium construction for the long time limit of glassy dynamics

In this paper we review a recent proposal to understand the long time limit of glassy dynamics in terms of an appropriate Markov Chain. [1]. The advantages of the resulting construction are many. The first one is that it gives a quasi equilibrium description on how glassy systems explore the phase space in the slow relaxation part of their dynamics. The second one is that it gives an alternative way to obtain dynamical equations starting from a dynamical rule that is static in spirit. This provides a way to overcome the difficulties encountered in the short time part of the dynamics where current conservation must be enforced. We study this approach in detail in a prototypical mean field disordered spin system, namely the p-spin spherical model, showing how we can obtain the well known equations that describes its dynamics. Then we apply the same approach to structural glasses. We first derive a set of dynamical Ornstein-Zernike equations which are very general in nature. Finally we consider two possible closure schemes for them, namely the Hypernetted Chain approximation of liquid theory and a closure of the BBGKY hierarchy that has been recently introduced by G. Szamel. From both approaches we finally find a set of dynamical Mode-Coupling like equations that are supposed to describe the system in the long time/slow dynamics regime.

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arXiv2015

Nonperturbative fluctuations and metastability in a simple model: from observables to microscopic theory and back

Slow dynamics in glassy systems is often interpreted as due to thermally activated events between "metastable" states. This emphasizes the role of nonperturbative fluctuations, which is especially dramatic when these fluctuations destroy a putative phase transition predicted at the mean-field level. To gain insight into such hard problems, we consider the implementation of a generic back-and-forth process, between microscopic theory and observable behavior via effective theories, in a toy model that is simple enough to allow for a thorough investigation: the one-dimensional $φ^4$ theory at low temperature. We consider two ways of restricting the extent of the fluctuations, which both lead to a nonconvex effective potential (or free energy) : either through a finite-size system or by means of a running infrared cutoff within the nonperturbative Renormalization Group formalism. We discuss the physical insight one can get and the ways to treat strongly nonperturbative fluctuations in this context.

cond-mat.stat-mech↗

arXiv2015

Quasi-equilibrium in glassy dynamics: a liquid theory approach

We introduce a quasi-equilibrium formalism in the theory of liquids in order to obtain a set of coarse grained long time dynamical equations for the two point density correlation functions. Our scheme allows to use typical approximations devised for equilibrium to study long time glassy dynamics. We study the Hypernetted Chain (HNC) approximation and a recent closure scheme by Szamel. In both cases we get dynamical equations that have the structure of the Mode-Coupling (MCT) equations in the long time region. The HNC approach, that was so far used to get equilibrium quantities is thus generalized to a fully consistent scheme where long-time dynamic quantities can also be computed. In the context of this approximation we get an asymptotic description of both equilibrium glassy dynamics at high temperature and of aging dynamics at low temperature. The Szamel approximation on the other hand is shown to lead to the exact Mode Coupling equation of Götze for equilibrium dynamics. We clarify the way phase space is sampled according to MCT during dynamical relaxation.

cond-mat.stat-mech↗

arXiv2014

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.

cond-mat.soft↗

arXiv2014

The Gardner transition in finite dimensions

Recent works on hard spheres in the limit of infinite dimensions revealed that glass states, envisioned as meta-basins in configuration space, can break up in a multitude of separate basins at low enough temperature or high enough pressure, leading to the emergence of new kinds of soft-modes and unusual properties. In this paper we study by perturbative renormalisation group techniques the critical properties of this transition, which has been discovered in disordered mean-field models in the '80s. We find that the upper critical dimension $d_u$ above which mean-field results hold is strictly larger than six and apparently non-universal, i.e. system dependent. Below $d_u$, we do not find any perturbative attractive fixed point (except for a tiny region of the 1RSB breaking parameter), thus showing that the transition in three dimensions either is governed by a non-perturbative fixed point unrelated to the Gaussian mean-field one or becomes first order or does not exist. We also discuss possible relationships with the behavior of spin glasses in a field.

cond-mat.dis-nn↗

arXiv2014

Exact theory of dense amorphous hard spheres in high dimension. III. The full RSB solution

In the first part of this paper, we derive the general replica equations that describe infinite-dimensional hard spheres at any level of replica symmetry breaking (RSB) and in particular in the fullRSB scheme. We show that these equations are formally very similar to the ones that have been derived for spin glass models, thus showing that the analogy between spin glasses and structural glasses conjectured by Kirkpatrick, Thirumalai, and Wolynes is realized in a strong sense in the mean field limit. We also suggest how the computation could be generalized in an approximate way to finite dimensional hard spheres. In the second part of the paper, we discuss the solution of these equations and we derive from it a number of physical predictions. We show that, below the Gardner transition where the 1RSB solution becomes unstable, a fullRSB phase exists and we locate the boundary of the fullRSB phase. Most importantly, we show that the fullRSB solution predicts correctly that jammed packings are isostatic, and allows one to compute analytically the critical exponents associated with the jamming transition, which are missed by the 1RSB solution. We show that these predictions compare very well with numerical results.

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