Gradient descent dynamics in complex energy landscapes, i.e. featuring multiple minima, finds application in many different problems, from soft matter to machine learning. Here, we analyze one of the simplest examples, namely that of soft repulsive particles in the limit of infinite spatial dimension $d$. The gradient descent dynamics then displays a jamming transition: at low density, it reaches zero-energy states in which particles' overlaps are fully eliminated, while at high density the energy remains finite and overlaps persist. At the transition, the dynamics becomes critical. In the $d\to \infty$ limit, a set of self-consistent dynamical equations can be derived via mean field theory. We analyze these equations and we present some partial progress towards their solution. We also study the Random Lorentz Gas in a range of $d=2\ldots 22$, and obtain a robust estimate for the jamming transition in $d\to\infty$. The jamming transition is analogous to the capacity transition in supervised learning, and in the appendix we discuss this analogy in the case of a simple one-layer fully-connected perceptron.
The jamming transition of non-spherical particles is fundamentally different from the spherical case. Non-spherical particles are hypostatic at their jamming points, while isostaticity is ensured in the case of the jamming of spherical particles. This structural difference implies that the presence of asphericity affects the critical exponents related to the contact number and the vibrational density of states. Moreover, while the force and gap distributions of isostatic jamming present power-law behaviors, even an infinitesimal asphericity is enough to smooth out these singularities. In a rec…
We introduce a new model to study the effect of surface roughness on the jamming transition. By performing numerical simulations, we show that for a smooth surface, the jamming transition density and the contact number at the transition point both increase upon increasing asphericity, as for ellipsoids and spherocylinders. Conversely, for a rough surface, both quantities decrease, in quantitative agreement with the behavior of frictional particles. Furthermore, in the limit corresponding to the Coulomb friction law, the model satisfies a generalized isostaticity criterion proposed in previous …
We investigate the glass and the jamming transitions of hard spheres in finite dimensions $d$, through a revised cell theory, that combines the free volume and the Random First Order Theory (RFOT). Recent results show that in infinite dimension the ideal glass transition and jamming transitions are distinct, while based on our theory we argue that they indeed coincide for finite $d$. As a consequence, jamming results into a percolation transition described by RFOT, with a static length diverging with exponent $ν=2/d$, which we verify through finite size scaling, and standard critical exponents…
We investigate the jamming transition in packings of emulsions and granular materials via molecular dynamics simulations. The emulsion model is composed of frictionless droplets interacting via nonlinear normal forces obtained using experimental data acquired by confocal microscopy of compressed emulsions systems. Granular materials are modeled by Hertz-Mindlin deformable spherical grains with Coulomb friction. In both cases, we find power-law scaling for the vanishing of pressure and excess number of contacts as the system approaches the jamming transition from high volume fractions. We find …
In this Letter, we analyze the quantum dynamics of the perceptron model: a particle is constrained on a $N$-dimensional sphere, with $N\to \infty$, and subjected to a set of randomly placed hard-wall potentials. This model has several applications, ranging from learning protocols to the effective description of the dynamics of an ensemble of infinite-dimensional hard spheres in Euclidean space. We find that the jamming transition with quantum dynamics shows critical exponents different from the classical case. We also find that the quantum jamming transition, unlike the typical quantum critica…
A new class of lattice gas models with trivial interactions but constrained dynamics are introduced. These are proven to exhibit a dynamical glass transition: above a critical density, rho_c, ergodicity is broken due to the appearance of an infinite spanning cluster of jammed particles. The fraction of jammed particles is discontinuous at the transition, while in the unjammed phase dynamical correlation lengths and timescales diverge as exp[C(rho_c-rho)^(-mu)]. Dynamic correlations display two-step relaxation similar to glass-formers and jamming systems.
Recent theoretical advances offer an exact, first-principle theory of jamming criticality in infinite dimension as well as universal scaling relations between critical exponents in all dimensions. For packings of frictionless spheres near the jamming transition, these advances predict that nontrivial power-law exponents characterize the critical distribution of (i) small inter-particle gaps and (ii) weak contact forces, both of which are crucial for mechanical stability. The scaling of the inter-particle gaps is known to be constant in all spatial dimensions $d$ -- including the physically rel…
We have looked into an experiment that has been termed the ``canonical example'' of jamming: granular material, clogging the outlet of a container as it is discharged by gravity. We present quantitative data of such an experiment. The experimental control parameter is the ratio between the radius of the orifice and the radius of the beads. As this parameter is increased, the jamming probability decreases. However, in the range of parameters explored, no evidence of criticality --in the sense of a jamming probability that becomes infinitely small for a finite radius-- has been found. We draw in…
We study the influence of particle size asymmetry on structural evolution of randomly jammed binary sphere mixtures with varying large-sphere/small-sphere composition. Simulations of jammed packings are used to assess the transition from large-sphere dominant to small-sphere dominant mixtures. For weakly asymmetric particle sizes, packing properties evolve smoothly, but not monotonically, with increasing small sphere composition, $f$. Our simulations reveal that at high values of ratio $α$ of large to small sphere radii, ($α\geq α_c \approx 5.75$) evolution of structural properties such as pac…
As a function of packing fraction at zero temperature and applied stress, an amorphous packing of spheres exhibits a jamming transition where the system is sensitive to boundary conditions even in the thermodynamic limit. Upon further compression, the system should become insensitive to boundary conditions provided it is sufficiently large. Here we explore the linear response to a large class of boundary perturbations in 2 and 3 dimensions. We consider each finite packing with periodic-boundary conditions as the basis of an infinite square or cubic lattice and study properties of vibrational m…
We theoretically calculate the average fraction of frozen particles in rectangular systems of arbitrary dimensions for the Kob-Andersen and Fredrickson-Andersen kinetically-constrained models. We find the aspect ratio of the rectangle's length to width, which distinguishes short, square-like rectangles from long, tunnel-like rectangles, and show how changing it can effect the jamming transition. We find how the critical vacancy density converges to zero in infinite systems for different aspect ratios: for long and wide channels it decreases algebraically $v_{c}\sim W^{-1/2}$ with the system's …
We numerically study the relaxation dynamics of the single layer perceptron with the spherical constraint. This is the simplest model of neural networks and serves a prototypical mean-field model of both convex and non-convex optimization problems. The relaxation time of the gradient descent algorithm rapidly increases near the SAT-UNSAT transition point. We numerically confirm that the first non-zero eigenvalue of the Hessian controls the relaxation time. This first eigenvalue vanishes much faster upon approaching the SAT-UNSAT transition point than the prediction of Marchenko-Pastur law in r…
We compute the shear modulus of amorphous hard and soft spheres, using the exact solution in infinite spatial dimensions that has been developed recently. We characterize the behavior of this observable in the whole phase diagram, and in particular around the glass and jamming transitions. Our results are consistent with other theoretical approaches, that are unified within this general picture, and they are also consistent with numerical and experimental results. Furthermore, we discuss some properties of the out-of-equilibrium dynamics after a deep quench close to the jamming transition, and…
In order to develop a toy model for car's traffic in cities, in this paper we analyze, by means of numerical simulations, the transition among fluid regimes and a congested jammed phase of the flow of "kinetically constrained" hard spheres in planar random networks similar to urban roads. In order to explore as timescales as possible, at a microscopic level we implement an event driven dynamics as the infinite time limit of a class of already existing model (e.g. "Follow the Leader") on an Erdos-Renyi two dimensional graph, the crossroads being accounted by standard Kirchoff density conservati…
Marginally stable solids have peculiar physical properties that were discovered and analyzed in the context of the jamming transition. We theoretically investigate the existence of marginal stability in a prototypical model for structural glass-formers, combining analytical calculations in infinite dimensions to computer simulations in three dimensions. While mean-field theory predicts the existence of a Gardner phase transition towards a marginally stable glass phase at low temperatures, simulations show no hint of diverging timescales or lengthscales, but reveal instead the presence of spars…
We introduce a new model of hard spheres under confinement for the study of the glass and jamming transitions. The model is an one-dimensional chain of the $d$-dimensional boxes each of which contains the same number of hard spheres, and the particles in the boxes of the ends of the chain are quenched at their equilibrium positions. We focus on the infinite dimensional limit ($d \to \infty$) of the model and analytically compute the glass transition densities using the replica liquid theory. From the chain length dependence of the transition densities, we extract the characteristic length scal…
