First-passage time for many-particle diffusion in space-time random environments.

The first-passage time for a single diffusing particle has been studied extensively, but the first-passage time of a system of many diffusing particles, as is often the case in physical systems, has received little attention until recently. We consider two models for many-particle diffusion-one treats each particle as independent simple random walkers while the other treats them as coupled to a common space-time random forcing field that biases particles nearby in space and time in similar ways. The first-passage time of a single diffusing particle under both models shows the same statistics and scaling behavior. However, for many-particle diffusions, the first-passage time among all particles (the extreme first-passage time) is very different between the two models, effected in the latter case by the randomness of the common forcing field. We develop an asymptotic (in the number of particles and location where first passage is being probed) theoretical framework to separate the impact of the random environment with that of the sampling trajectories within it. We identify a power law describing the impact of the environment on the variance of the extreme first-passage time. Through numerical simulations, we verify that the predictions from this asymptotic theory hold even for systems with widely varying numbers of particles, all the way down to 100 particles. This shows that measurements of the extreme first-passage time for many-particle diffusions provide an indirect measurement of the underlying environment in which the diffusion is occurring.

Anomalous fluctuations of extremes in many-particle diffusion.

In many-particle diffusions, particles that move the furthest and fastest can play an outsized role in physical phenomena. A theoretical understanding of the behavior of such extreme particles is nascent. A classical model, in the spirit of Einstein's treatment of single-particle diffusion, has each particle taking independent homogeneous random walks. This, however, neglects the fact that all particles diffuse in a common and often inhomogeneous environment that can affect their motion. A more sophisticated model treats this common environment as a space-time random biasing field which influences each particle's independent motion. While the bulk (or typical particle) behavior of these two models has been found to match to high degree, recent theoretical work of G. Barraquand and I. Corwin, Probab. Theory Relat. Fields 167, 1057 (2017)0178-805110.1007/s00440-016-0699-z and G. Barraquand and P. Le Doussal, J. Phys. A: Math. Theor. 53, 215002 (2020)1751-811310.1088/1751-8121/ab8b39 on a one-dimensional exactly solvable version of this random environment model suggests that the extreme behavior is quite different between the two models. We transform these asymptotic (in system size and time) results into physically applicable predictions. Using high-precision numerical simulations we reconcile different asymptotic phases in a manner that matches numerics down to realistic system sizes, amenable to experimental confirmation. We characterize the behavior of extreme diffusion in the random environment model by the presence of a new phase with anomalous fluctuations related to the Kardar-Parisi-Zhang universality class and equation.

Local stability of spheres via the convex hull and the radical Voronoi diagram.

Jamming is an emergent phenomenon wherein the local stability of individual particles percolates to form a globally rigid structure. However, the onset of rigidity does not imply that every particle becomes rigid, and indeed some remain locally unstable. These particles, if they become unmoored from their neighbors, are called rattlers, and their identification is critical to understanding the rigid backbone of a packing, as these particles cannot bear stress. The accurate identification of rattlers, however, can be a time-consuming process, and the currently accepted method lacks a simple geometric interpretation. In this manuscript, we propose two simpler classifications of rattlers in hard sphere systems based on the convex hull of contacting neighbors and the maximum inscribed sphere of the radical Voronoi cell, each of which provides geometric insight into the source of their instability. Furthermore, the convex hull formulation can be generalized to explore stability in hyperstatic soft sphere packings, spring networks, nonspherical packings, and mean-field non-central-force potentials.

Predicting Defects in Soft Sphere Packings near Jamming Using the Force Network Ensemble.

Amorphous systems of soft particles above jamming have more contacts than are needed to achieve mechanical equilibrium. The force network of a granular system with a fixed contact network is thus underdetermined and can be characterized as a random instantiation within the space of the force network ensemble. In this Letter, we show that defect contacts that are not necessary for stability of the system can be uniquely identified by examining the boundaries of this space of allowed force networks. We further show that, for simulations in the near jamming limit, this identification is nearly always correct and that defect contacts are broken under decompression of the system.

Emergence of zero modes in disordered solids under periodic tiling.

In computational models of particle packings with periodic boundary conditions, it is assumed that the packing is attached to exact copies of itself in all possible directions. The periodicity of the boundary then requires that all of the particles' images move together. An infinitely repeated structure, on the other hand, does not necessarily have this constraint. As a consequence, a jammed packing (or a rigid elastic network) under periodic boundary conditions may have a corresponding infinitely repeated lattice representation that is not rigid or indeed may not even be at a local energy minimum. In this manuscript, we prove this claim and discuss ways in which periodic boundary conditions succeed in capturing the physics of repeated structures and where they fall short.

Hyperuniform jammed sphere packings have anomalous material properties.

A spatial distribution is hyperuniform if it has local density fluctuations that vanish in the limit of long length scales. Hyperuniformity is a well known property of both crystals and quasicrystals. Of recent interest, however, is disordered hyperuniformity: the presence of hyperuniform scaling without long-range configurational order. Jammed granular packings have been proposed as an example of disordered hyperuniformity, but recent numerical investigation has revealed that many jammed systems instead exhibit a complex set of distinct behaviors at long, emergent length scales. We use the Voronoi tessellation as a tool to define a set of rescaling transformations that can impose hyperuniformity on an arbitrary weighted point process, and show that these transformations can be used in simulations to iteratively generate hyperuniform, mechanically stable packings of athermal soft spheres. These hyperuniform jammed packings display atypical mechanical properties, particularly in the low-frequency phononic excitations, which exhibit an isolated band of highly collective modes and a band gap around zero frequency.

Transient learning degrees of freedom for introducing function in materials.

SignificanceMany protocols used in material design and training have a common theme: they introduce new degrees of freedom, often by relaxing away existing constraints, and then evolve these degrees of freedom based on a rule that leads the material to a desired state at which point these new degrees of freedom are frozen out. By creating a unifying framework for these protocols, we can now understand that some protocols work better than others because the choice of new degrees of freedom matters. For instance, introducing particle sizes as degrees of freedom to the minimization of a jammed particle packing can lead to a highly stable state, whereas particle stiffnesses do not have nearly the same impact.

Marginal stability in memory training of jammed solids.

Memory encoding by cyclic shear is a reliable process to store information in jammed solids, yet its underlying mechanism and its connection to the amorphous structure are not fully understood. When a jammed sphere packing is repeatedly sheared with cycles of the same strain amplitude, it optimizes its mechanical response to the cyclic driving and stores a memory of it. We study memory by cyclic shear training as a function of the underlying stability of the amorphous structure in marginally stable and highly stable packings, the latter produced by minimizing the potential energy using both positional and radial degrees of freedom. We find that jammed solids need to be marginally stable in order to store a memory by cyclic shear. In particular, highly stable packings store memories only after overcoming brittle yielding and the cyclic shear training takes place in the shear band, a region which we show to be marginally stable.

Finite-size effects in the microscopic critical properties of jammed configurations: A comprehensive study of the effects of different types of disorder.

Jamming criticality defines a universality class that includes systems as diverse as glasses, colloids, foams, amorphous solids, constraint satisfaction problems, neural networks, etc. A particularly interesting feature of this class is that small interparticle forces (f) and gaps (h) are distributed according to nontrivial power laws. A recently developed mean-field (MF) theory predicts the characteristic exponents of these distributions in the limit of very high spatial dimension, d→∞ and, remarkably, their values seemingly agree with numerical estimates in physically relevant dimensions, d=2 and 3. These exponents are further connected through a pair of inequalities derived from stability conditions, and both theoretical predictions and previous numerical investigations suggest that these inequalities are saturated. Systems at the jamming point are thus only marginally stable. Despite the key physical role played by these exponents, their systematic evaluation has yet to be attempted. Here, we carefully test their value by analyzing the finite-size scaling of the distributions of f and h for various particle-based models for jamming. Both dimension and the direction of approach to the jamming point are also considered. We show that, in all models, finite-size effects are much more pronounced in the distribution of h than in that of f. We thus conclude that gaps are correlated over considerably longer scales than forces. Additionally, remarkable agreement with MF predictions is obtained in all but one model, namely near-crystalline packings. Our results thus help to better delineate the domain of the jamming universality class. We furthermore uncover a secondary linear regime in the distribution tails of both f and h. This surprisingly robust feature is understood to follow from the (near) isostaticity of our configurations.

Long-Range Anomalous Decay of the Correlation in Jammed Packings.

We numerically study the structure of the interactions occurring in three-dimensional systems of hard spheres at jamming, focusing on the large-scale behavior. Given the fundamental role in the configuration of jammed packings, we analyze the propagation through the system of the weak forces and of the variation of the coordination number with respect to the isostaticity condition, ΔZ. We show that these correlations can be successfully probed by introducing a correlation function weighted on the density-density fluctuations. The results of this analysis can be further improved by introducing a representation of the system based on the contact points between particles. In particular, we find evidence that the weak forces and the ΔZ fluctuations support the hypothesis of randomly jammed packings of spherical particles being hyperuniform by exhibiting an anomalous long-range decay. Moreover, we find that the large-scale structure of the density-density correlation exhibits a complex behavior due to the superimposition of two exponentially damped oscillating signals propagating with linearly depending frequencies.

A direct link between active matter and sheared granular systems.

The similarity in mechanical properties of dense active matter and sheared amorphous solids has been noted in recent years without a rigorous examination of the underlying mechanism. We develop a mean-field model that predicts that their critical behavior-as measured by their avalanche statistics-should be equivalent in infinite dimensions up to a rescaling factor that depends on the correlation length of the applied field. We test these predictions in two dimensions using a numerical protocol, termed "athermal quasistatic random displacement," and find that these mean-field predictions are surprisingly accurate in low dimensions. We identify a general class of perturbations that smoothly interpolates between the uncorrelated localized forces that occur in the high-persistence limit of dense active matter and system-spanning correlated displacements that occur under applied shear. These results suggest a universal framework for predicting flow, deformation, and failure in active and sheared disordered materials.

Interplay between percolation and glassiness in the random Lorentz gas.

The random Lorentz gas (RLG) is a minimal model of transport in heterogeneous media that exhibits a continuous localization transition controlled by void space percolation. The RLG also provides a toy model of particle caging, which is known to be relevant for describing the discontinuous dynamical transition of glasses. In order to clarify the interplay between the seemingly incompatible percolation and caging descriptions of the RLG, we consider its exact mean-field solution in the infinite-dimensional d→∞ limit and perform numerics in d=2...20. We find that for sufficiently high d the mean-field caging transition precedes and prevents the percolation transition, which only happens on timescales diverging with d. We further show that activated processes related to rare cage escapes destroy the glass transition in finite dimensions, leading to a rich interplay between glassiness and percolation physics. This advance suggests that the RLG can be used as a toy model to develop a first-principle description of particle hopping in structural glasses.

Mean-Field Predictions of Scaling Prefactors Match Low-Dimensional Jammed Packings.

No known analytic framework precisely explains all the phenomena observed in jamming. The replica theory for glasses and jamming is a mean-field theory which attempts to do so by working in the limit of infinite dimensions, such that correlations between neighbors are negligible. As such, results from this mean-field theory are not guaranteed to be observed in finite dimensions. However, many results in mean field for jamming have been shown to be exact or nearly exact in low dimensions. This suggests that the infinite dimensional limit is not necessary to obtain these results. In this Letter, we perform precision measurements of jamming scaling relationships between pressure, excess packing fraction, and number of excess contacts from dimensions 2-10 in order to extract the prefactors to these scalings. While these prefactors should be highly sensitive to finite dimensional corrections, we find the mean-field predictions for these prefactors to be exact in low dimensions. Thus the mean-field approximation is not necessary for deriving these prefactors. We present an exact, first-principles derivation for one, leaving the other as an open question. Our results suggest that mean-field theories of critical phenomena may compute more for d≥d_{u} than has been previously appreciated.

Vibrational Properties of Hard and Soft Spheres Are Unified at Jamming.

The unconventional thermal properties of jammed amorphous solids are directly related to their density of vibrational states. While the vibrational spectrum of jammed soft sphere solids has been fully described, the vibrational spectrum of hard spheres, a model glass former often related to physical colloidal glasses, is still unknown due to the difficulty of treating the nonanalytic interaction potential. We bypass this difficulty using the recently described effective interaction potential for the free energy of thermal hard spheres. By minimizing this effective free energy, we mimic the rapid compression of hard spheres and produce typical configurations of the thermal system. We measure the resulting vibrational spectrum and characterize its evolution toward the jamming point where configurations of hard and soft spheres are trivially unified. For densities approaching jamming from below, we observe low-frequency modes which agree with those found in numerical simulations of jammed soft spheres. Our measurements of the vibrational structure demonstrate that the jamming universality extends away from jamming: hard sphere thermal systems below jamming exhibit the same vibrational spectra as thermal and athermal soft sphere systems above the transition.

Direct measurement of force configurational entropy in jamming.

Thermal fluctuations are not large enough to lead to state changes in granular materials. However, in many cases, such materials do achieve reproducible bulk properties, suggesting that they are controlled by an underlying statistical mechanics analogous to thermodynamics. While themodynamic descriptions of granular materials have been explored, they have not yet been concretely connected to their underlying statistical mechanics. We make this connection concrete by providing a first-principles derivation of the multiplicity and thus the entropy of the force networks in granular packings. We directly measure the multiplicity of force networks using a protocol based on the phase space volume of allowed force configurations. Analogous to Planck's constant, we find a scale factor, h_{f}, that discretizes this phase space volume into a multiplicity. To determine this scale factor, we measure angoricity over a wide range of pressures using the method of overlapping histograms and find that in the absence of a fundamental quantum scale it is set solely by the system size and dimensionality. This concretely links thermodynamic approaches of angoricity with the microscopic multiplicity of the force network ensemble.

Experimental observation of the marginal glass phase in a colloidal glass.

The replica theory of glasses predicts that in the infinite dimensional mean field limit, there exist two distinct glassy phases of matter: stable glass and marginal glass. We have developed a technique to experimentally probe these phases of matter using a colloidal glass. We avoid the difficulties inherent in measuring the long time behavior of glasses by instead focusing on the very short time dynamics of the ballistic to caged transition. We track a single tracer particle within a slowly densifying glass and measure the resulting mean squared displacement (MSD). By analyzing the MSD, we find that upon densification, our colloidal system moves through several states of matter. At lowest densities, it is a subdiffusive liquid. Next, it behaves as a stable glass, marked by the appearance of a plateau in the MSD whose magnitude shrinks with increasing density. However, this shrinking plateau does not shrink to zero; instead, at higher densities, the system behaves as a marginal glass, marked by logarithmic growth in the MSD toward that previous plateau value. Finally, at the highest experimental densities, the system returns to the stable glass phase. This provides direct experimental evidence for the existence of a marginal glass in three dimensions.

Gardner physics in amorphous solids and beyond.

One of the most remarkable predictions to emerge out of the exact infinite-dimensional solution of the glass problem is the Gardner transition. Although this transition was first theoretically proposed a generation ago for certain mean-field spin glass models, its materials relevance was only realized when a systematic effort to relate glass formation and jamming was undertaken. A number of nontrivial physical signatures associated with the Gardner transition have since been considered in various areas, from models of structural glasses to constraint satisfaction problems. This perspective surveys these recent advances and discusses the novel research opportunities that arise from them.

Glassy, Gardner-like phenomenology in minimally polydisperse crystalline systems.

We report on a nonequilibrium phase of matter, the minimally disordered crystal phase, which we find exists between the maximally amorphous glasses and the ideal crystal. Even though these near crystals appear highly ordered, they display glassy and jamming features akin to those observed in amorphous solids. Structurally, they exhibit a power-law scaling in their probability distribution of weak forces and small interparticle gaps as well as a flat density of vibrational states. Dynamically, they display anomalous aging above a characteristic pressure. Quantitatively, this disordered crystal phase has much in common with the Gardner-like phase seen in maximally disordered solids. Near crystals should be amenable to experimental realizations in commercially available particulate systems and are to be indispensable in verifying the theory of amorphous materials.

A broader view on jamming: from spring networks to circle packings.

Jamming occurs when objects like grains are packed tightly together (e.g. grain silos). It is highly cooperative and can lead to phenomena like earthquakes, traffic jams, etc. In this paper we point out the paramount importance of the underlying contact network for jammed systems; the network must have one contact in excess of isostaticity and a finite bulk modulus. Isostatic means that the number of degrees of freedom is exactly balanced by the number of constraints. This defines a large class of networks that can be constructed without the necessity of packing particles together compressively (either in the lab or computationally). One such construction, which we explore here, involves setting up the Delaunay triangulation of a Poisson disk sampling and then removing edges to maximize the bulk modulus until the isostatic plus one edge is reached. This construction works in any dimensions and here we give results in 2D where we also show how such networks can be transformed into disk packs.

Degree product rule tempers explosive percolation in the absence of global information.

We introduce a guided network growth model, which we call the degree product rule process, that uses solely local information when adding new edges. For small numbers of candidate edges our process gives rise to a second-order phase transition, but becomes first order in the limit of global choice. We provide the set of critical exponents required to characterize the nature of this percolation transition. Such a process permits interventions which can delay the onset of percolation while tempering the explosiveness caused by cluster product rule processes.