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Metallic glasses are frequently used as structural materials. Therefore, it is important to develop methods to predict their mechanical response as a function of the microstructure prior to loading. We develop a coarse-grained spring network model, which describes the mechanical response of metallic glasses using an equivalent series network of springs, which can break and re-form to mimic atomic rearrangements during deformation. To validate the model, we perform simulations of quasistatic, uniaxial tension of Lennard-Jones and embedded atom method (EAM) potentials for Cu$_{50}$Zr$_{50}$ metallic glasses. We consider samples prepared using a wide range of cooling rates and with different amounts of crystalline order. We show that both the Lennard-Jones and EAM models possess qualitatively similar stress $\sigma$ versus strain $\gamma$ curves. By specifying five parameters in the spring network model (ultimate strength, strain at ultimate strength, slopes of $\sigma(\gamma)$ at $\gamma=0$ and at large strain, and strain at fracture where $\sigma=0$), we can accurately describe the form of the stress-strain curves during uniaxial tension for the computational studies of Cu$_{50}$Zr$_{50}$, as well as recent experimental studies of several Zr-based metallic glasses. For the computational studies of Cu$_{50}$Zr$_{50}$, we find that the yield strain distribution is shifted to larger strains for slowly cooled glasses compared to rapidly cooled glasses. In addition, the average number of new springs and their rate of formation decreases with decreasing cooling rate. These effects offset each other at large strains, causing the stress-strain curve to become independent of the sample preparation protocol in this regime. In future studies, we will extract the parameters that define the spring network model directly from atomic rearrangements that occur during uniaxial deformation., Comment: 16 pages, 13 figures

Jammed packings of granular materials display complex mechanical response. For example, the ensemble-averaged shear modulus $\left\langle G \right\rangle$ increases as a power-law in pressure $p$ for static packings of spherical particles that can rearrange during compression. We seek to design granular materials with shear moduli that can either increase {\it or} decrease with pressure without particle rearrangements even in the large-system limit. To do this, we construct {\it tessellated} granular metamaterials by joining multiple particle-filled cells together. We focus on cells that contain a small number of bidisperse disks in two dimensions. We first study the mechanical properties of individual disk-filled cells with three types of boundaries: periodic boundary conditions, fixed-length walls, and flexible walls. Hypostatic jammed packings are found for disk-filled cells with flexible walls, but not in cells with periodic boundary conditions and fixed-length walls, and they are stabilized by quartic modes of the dynamical matrix. The shear modulus of a single cell depends linearly on $p$. We find that the slope of the shear modulus with pressure, $\lambda_c < 0$ for all packings in single cells with periodic boundary conditions where the number of particles per cell $N \ge 6$. In contrast, single cells with fixed-length and flexible walls can possess $\lambda_c > 0$, as well as $\lambda_c < 0$, for $N \le 16$. We show that we can force the mechanical properties of multi-cell granular metamaterials to possess those of single cells by constraining the endpoints of the outer walls and enforcing an affine shear response. These studies demonstrate that tessellated granular metamaterials provide a novel platform for the design of soft materials with novel mechanical properties.

Most granular packings possess shear moduli ($G$) that increase with the applied external pressure, and bulk moduli ($B$) that increase or remain constant with pressure. This paper presents "tessellated" granular metamaterials for which both $G$ and the ratio $G/B$ decrease with increasing pressure. The granular metamaterials are made from flexible tessellations forming a ring of closed cells, each containing a small number of solid particles. For under-constrained tessellations, the dominant contributions to $G$ and $B$ are the particle-particle and particle-cell interactions. With specific particle configurations in the cells, we limit the number of possible particle rearrangements to achieve decreasing $G$ as we increase the pressure difference between the inside and outside of the tessellation, leading to $G/B \ll 1$ at large pressures. We further study tessellated granular metamaterials with cells containing a single particle and many particles to determine the variables that control the mechanical response of particle-filled tessellations as a function of pressure., Comment: 16 pages, 6 figures

During epithelial wound healing, cell morphology near the healed wound and the healing rate vary strongly among different developmental stages even for a single species like \textit{Drosophila}. We develop deformable particle (DP) model simulations to understand how variations in cell mechanics give rise to distinct wound closure phenotypes in the \textit{Drosophila} embryonic ectoderm and larval wing disc epithelium. We find that plastic deformation of the cell membrane can generate large changes in cell shape consistent with wound closure in the embryonic ectoderm. Our results show that the embryonic ectoderm is best described by cell membranes with an elasto-plastic response, whereas the larval wing disc is best described by cell membranes with an exclusively elastic response. By varying the mechanical response of cell membranes in DP simulations, we recapitulate the wound closure behavior of both the embryonic ectoderm and the larval wing disc.

The spongy mesophyll is a complex, porous tissue found in plant leaves that enables carbon capture and provides mechanical stability. Unlike many other biological tissues, which remain confluent throughout development, the spongy mesophyll must develop from an initially confluent tissue into a tortuous network of cells with a large proportion of intercellular airspace. How the airspace in the spongy mesophyll develops while the cells remain mechanically stable remains unknown. Here, we used computer simulations of deformable particles to develop a purely mechanical model for the development of the spongy mesophyll tissue. By stipulating that (1) cell perimeter grows only near voids, (2) cells both form and break adhesive bonds, and (3) the tissue pressure remains constant, the computational model was able to recapitulate the developmental trajectory of the microstructure of the spongy mesophyll observed in Arabidopsis thaliana leaves. Robust generation of pore space in the spongy mesophyll requires a balance of cell growth, adhesion, stiffness and tissue pressure to ensure cell networks remain both porous yet mechanically robust. The success of this mechanical model of tissue growth and porosity evolution suggests that simple physical principles can coordinate and drive the development of complex plant tissues like the spongy mesophyll., 28 pages, 6 figures, 1 table, 9 pages of supplementary information

Numerous studies have investigated the differences and similarities between protein structures determined by solution NMR spectroscopy and those determined by x-ray crystallography. A fundamental question is whether any observed differences are due to differing methodologies, or to differences in the behavior of proteins in solution versus in the crystalline state. Here, we compare the properties of the hydrophobic cores of high-resolution protein crystal structures and those in NMR structures, determined using increasing numbers and types of restraints. Prior studies have reported that many NMR structures have denser cores compared to those of high-resolution x-ray crystal structures. Our current work investigates this result in more detail, and finds that these NMR structures tend to violate basic features of protein stereochemistry, such as small non-bonded atomic overlaps and few Ramachandran and side chain dihedral angle outliers. We find that NMR structures solved with more restraints, and which do not significantly violate stereochemistry, have hydrophobic cores that have a similar size and packing fraction as their counterparts determined by x-ray crystallography at high-resolution. These results lead us to conclude that, at least regarding the core packing properties, high-quality structures determined by NMR and x-ray crystallography are the same, and the differences reported earlier are most likely a consequence of methodology, rather than fundamental differences between the protein in the two different environments., 12 pages, 6 figures

Experimental data on the compressive strength $\sigma_{\rm max}$ versus strain rate ${\dot \varepsilon}_{\rm eng}$ for metallic glasses undergoing uniaxial compression shows significantly different behavior for different alloys. For some metallic glasses, $\sigma_{\rm max}$ decreases with increasing ${\dot \varepsilon}_{\rm eng}$, for others, $\sigma_{\rm max}$ increases with increasing ${\dot \varepsilon}_{\rm eng}$, and for others $\sigma_{\rm max}$ versus ${\dot \varepsilon}_{\rm eng}$ is nonmonotonic. Using numerical simulations of metallic glasses undergoing uniaxial compression at nonzero strain rate and temperature, we show that they obey a universal relation for the compressive strength versus temperature, which determines their mechanical response. At low ${\dot \varepsilon}_{\rm eng}$, increasing strain rate leads to increases in temperature and decreases in $\sigma^*_{\rm max}$, whereas at high ${\dot \varepsilon}_{\rm eng}$, increasing strain rate leads to decreases in temperature and increases in $\sigma^*_{\rm max}$. This non-monotonic behavior of $\sigma^*_{\rm max}$ versus temperature causes the nonmonotonic behavior of $\sigma^*_{\rm max}$ versus ${\dot \varepsilon}_{\rm eng}$. Variations in the internal dissipation change the characteristic strain rate at which the nonmonotonic behavior occurs. These results are general for a wide range of metallic glasses with different atomic interactions, damping coefficients, and chemical compositions., Comment: 5 pages, 3 figures

We review the physics of jamming from the theoretical, experimental and numerical perspectives. We summarize the mean-field theory of jamming and the marginally stable solid phase, with particular emphasis on the connection with the Replica Symmetry Breaking theory of glasses. We report validations of the mean-field theory of jamming from experimental and numerical studies of critical behaviors near the transition. In particular, we describe the physics of jamming of frictionless, spherical particles, as well as more recent work on jamming of frictionless, non-spherical particles and frictional, nearly spherical particles. We also present current efforts in expanding the mean-field theory to systems that more closely resemble externally driven granular media, cell aggregates, and active colloidal suspensions., Comment: To appear as a contribution to the edited volume "Spin Glass Theory & Far Beyond - Replica Symmetry Breaking after 40 Years", World Scientific

Globular proteins undergo thermal fluctuations in solution, while maintaining an overall well-defined folded structure. In particular, studies have shown that the core structure of globular proteins differs in small, but significant ways when they are solved by x-ray crystallography versus solution-based NMR spectroscopy. Given these discrepancies, it is unclear whether molecular dynamics (MD) simulations can accurately recapitulate protein conformations. We therefore perform extensive MD simulations across multiple force fields and sampling techniques to investigate the degree to which computer simulations can capture the ensemble of conformations observed in experiments. By analyzing fluctuations in the atomic coordinates and core packing, we show that conformations sampled in MD simulations both move away from and sample a larger conformational space than the ensemble of structures observed in NMR experiments. However, we find that adding inter-residue distance restraints that match those obtained via Nuclear Overhauser Effect measurements enables the MD simulations to sample more NMR-like conformations, though significant differences between the core packing features in restrained MD and the NMR ensemble remain. Given that the protein structures obtained from the MD simulations possess smaller and less dense protein cores compared to those solved by NMR, we suggest that future improvements to MD forcefields should aim to increase the packing of hydrophobic residues in protein cores., Comment: 17 pages, 9 figures

We carry out numerical studies of static packings of frictionless superellipsoidal particles in three spatial dimensions. We consider more than $200$ different particle shapes by varying the three shape parameters that define superellipsoids. We characterize the structural and mechanical properties of both disordered and ordered packings using two packing-generation protocols. We perform athermal quasi-static compression simulations starting from either random, dilute configurations (Protocol 1) or thermalized, dense configurations (protocol $2$), which allows us to tune the orientational order of the packings. In general, we find that the contact numbers at jamming onset for superellipsoid packings are hypostatic, with $z_J < z_{\rm iso}$, where $z_{\rm iso} = 2d_f$ and $d_f = 5$ or $6$ depending on whether the particles are axi-symmetric or not. Over the full range of orientational order, we find that the number of quartic modes of the dynamical matrix for the packings always matches the number of missing contacts relative to the isostatic value. This result suggests that there are no mechanically redundant contacts for ordered, yet hypostatic packings of superellipsoidal particles. Additionally, we find that the packing fraction at jamming onset for disordered packings of superellipsoidal particles can be collapsed using two particle shape parameters, e.g. the asphericity $\mathcal{A}$ and reduced aspect ratio $��$ of the particles., 10 pages, 14 figures

We employ numerical simulations to study active transistor-like switches made from two-dimensional (2D) granular crystals containing two types of grains with the same size, but different masses. We tune the mass contrast and arrangement of the grains to maximize the width of the frequency band gap in the device. The input signal is applied to a single grain on one side of the device, and the output signal is measured from another grain on the other side of the device. Changing the size of one or many grains tunes the pressure, which controls the vibrational response of the device. Switching between the on and off states is achieved using two mechanisms: 1) pressure-induced switching where the interparticle contact network is the same in the on and off states, and 2) switching through contact breaking. In general, the performance of the acoustic switch, as captured by the gain ratio and switching time between the on and off states, is better for pressure-induced switching. We show that in these acoustic switches the gain ratio between the on and off states can be larger than $10^4$ and the switching time (multiplied by the driving frequency) is comparable to that obtained recently for sonic crystals and less than that for photonic transistor-like switches. Since the self-assembly of grains with different masses into 2D granular crystals is challenging, we describe simulations of circular grains with small circular knobs placed symmetrically around the perimeter mixed with circular grains without knobs. Using umbrella sampling techniques, we show that devices with grains with $3$ knobs most efficiently form the hexagonal crystals that yield the largest band gap., Comment: 16 pages, 23 figures

We develop a theoretical description for mechanically stable frictional packings in terms of the difference between the total number of contacts required for isostatic packings of frictionless disks and the number of contacts in frictional packings, $m=N_c^0-N_c$. The saddle order $m$ represents the number of unconstrained degrees of freedom that a static packing would possess if friction were removed. Using a novel numerical method that allows us to enumerate disk packings for each $m$, we show that the probability to obtain a packing with saddle order $m$ at a given static friction coefficient $\mu$, $P_m(\mu)$, can be expressed as a power-series in $\mu$. Using this form for $P_m(\mu)$, we quantitatively describe the dependence of the average contact number on friction coefficient for static disk packings obtained from direct simulations of the Cundall-Strack model for all $\mu$ and $N$., Comment: 4 pages, 4 figures

We describe numerical simulations and analyses of a quasi-one-dimensional (Q1D) model of glassy dynamics. In this model, hard rods undergo Brownian dynamics through a series of narrow channels connected by $J$ intersections. We do not allow the rods to turn at the intersections, and thus there is a single, continuous route through the system. This Q1D model displays caging behavior, collective particle rearrangements, and rapid growth of the structural relaxation time, which are also found in supercooled liquids and glasses. The mean-square displacement $\Sigma(t)$ for this Q1D model displays several dynamical regimes: 1) short-time diffusion $\Sigma(t) \sim t$, 2) a plateau in the mean-square displacement caused by caging behavior, 3) single-file diffusion characterized by anomalous scaling $\Sigma(t) \sim t^{0.5}$ at intermediate times, and 4) a crossover to long-time diffusion $\Sigma(t) \sim t$ for times $t$ that grow with the complexity of the circuit. We develop a general procedure for determining the structural relaxation time $t_D$, beyond which the system undergoes long-time diffusion, as a function of the packing fraction $\phi$ and system topology. This procedure involves several steps: 1) define a set of distinct microstates in configuration space of the system, 2) construct a directed network of microstates and transitions between them, 3) identify minimal, closed loops in the network that give rise to structural relaxation, 4) determine the frequencies of `bottleneck' microstates that control the slow dynamics and time required to transition out of them, and 5) use the microstate frequencies and lifetimes to deduce $t_D(\phi)$. We find that $t_D$ obeys power-law scaling, $t_D\sim (\phi^* - \phi)^{-\alpha}$, where both $\phi^*$ (signaling complete kinetic arrest) and $\alpha>0$ depend on the system topology., Comment: 16 pages, 18 figures

We perform molecular dynamics simulations to compress binary hard spheres into jammed packings as a function of the compression rate $R$, size ratio $\alpha$, and number fraction $x_S$ of small particles to determine the connection between the glass-forming ability (GFA) and packing efficiency in bulk metallic glasses (BMGs). We define the GFA by measuring the critical compression rate $R_c$, below which jammed hard-sphere packings begin to form "random crystal" structures with defects. We find that for systems with $\alpha \gtrsim 0.8$ that do not de-mix, $R_c$ decreases strongly with $\Delta \phi_J$, as $R_c \sim \exp(-1/\Delta \phi_J^2)$, where $\Delta \phi_J$ is the difference between the average packing fraction of the amorphous packings and random crystal structures at $R_c$. Systems with $\alpha \lesssim 0.8$ partially de-mix, which promotes crystallization, but we still find a strong correlation between $R_c$ and $\Delta \phi_J$. We show that known metal-metal BMGs occur in the regions of the $\alpha$ and $x_S$ parameter space with the lowest values of $R_c$ for binary hard spheres. Our results emphasize that maximizing GFA in binary systems involves two competing effects: minimizing $\alpha$ to increase packing efficiency, while maximizing $\alpha$ to prevent de-mixing., Comment: 5 pages, 4 figures

Contact breaking and Hertzian interactions between grains can both give rise to nonlinear vibrational response of static granular packings. We perform molecular dynamics simulations at constant energy in 2D of frictionless bidisperse disks that interact via Hertzian spring potentials as a function of energy and measure directly the vibrational response from the Fourier transform of the velocity autocorrelation function. We compare the measured vibrational response of static packings near jamming onset to that obtained from the eigenvalues of the dynamical matrix to determine the temperature above which the linear response breaks down. We compare packings that interact via single-sided (purely repulsive) and double-sided Hertzian spring interactions to disentangle the effects of the shape of the potential from contact breaking. Our studies show that while Hertzian interactions lead to weak nonlinearities in the vibrational behavior (e.g. the generation of harmonics of the eigenfrequencies of the dynamical matrix), the vibrational response of static packings with Hertzian contact interactions is dominated by contact breaking as found for systems with repulsive linear spring interactions., 8 pages, 6 figures

We propose a `phase diagram' for particulate systems that interact via purely repulsive contact forces, such as granular media and colloidal suspensions. We identify and characterize two distinct classes of behavior as a function of the input kinetic energy per degree of freedom $T_0$ and packing fraction deviation above and below jamming onset $\Delta \phi=\phi - \phi_J$ using numerical simulations of purely repulsive frictionless disks. Iso-coordinated solids (ICS) only occur above jamming for $\Delta \phi > \Delta \phi_c(T_0)$; they possess average coordination number equal to the isostatic value ($< z> = z_{\rm iso}$) required for mechanically stable packings. ICS display harmonic vibrational response, where the density of vibrational modes from the Fourier transform of the velocity autocorrelation function is a set of sharp peaks at eigenfrequencies $\omega_k^d$ of the dynamical matrix evaluated at $T_0=0$. Hypo-coordinated solids (HCS) occur both above and below jamming onset within the region defined by $\Delta \phi > \Delta \phi^*_-(T_0)$, $\Delta \phi < \Delta \phi^*_+(T_0)$, and $\Delta \phi > \Delta \phi_{cb}(T_0)$. In this region, the network of interparticle contacts fluctuates with $< z> \approx z_{\rm iso}/2$, but cage-breaking particle rearrangements do not occur. The HCS vibrational response is nonharmonic, {\it i.e} the density of vibrational modes $D(\omega)$ is not a collection of sharp peaks at $\omega_k^d$, and its precise form depends on the measurement method. For $\Delta \phi > \Delta \phi_{cb}(T_0)$ and $\Delta \phi < \Delta \phi^*_{-}(T_0)$, the system behaves as a hard-particle liquid., Comment: 5 pages, 4 figures

This is a response to the comment on our manuscript "Repulsive contact interactions make jammed particulate systems inherently nonharmonic" (Physical Review Letters 107 (2011) 078301) by C. P. Goodrich, A. J. Liu, and S. R. Nagel., 6 pages, 1 figure

We perform extensive molecular dynamics simulations of dense liquids composed of bidisperse dimer- and ellipse-shaped particles in 2D that interact via repulsive contact forces. We measure the structural relaxation times obtained from the long-time decay of the self-part of the intermediate scattering function for the translational and rotational degrees of freedom (DOF) as a function of packing fraction \phi, temperature T, and aspect ratio \alpha. We are able to collapse the \phi and T-dependent structural relaxation times for disks, and dimers and ellipses over a wide range of \alpha, onto a universal scaling function {\cal F}_{\pm}(|\phi-\phi_0|,T,\alpha), which is similar to that employed in previous studies of dense liquids composed of purely repulsive spherical particles in 3D. {\cal F_{\pm}} for both the translational and rotational DOF are characterized by the \alpha-dependent scaling exponents \mu and \delta and packing fraction \phi_0(\alpha) that signals the crossover in the scaling form {\cal F}_{\pm} from hard-particle dynamics to super-Arrhenius behavior for each aspect ratio. We find that the fragility at \phi_0, m(\phi_0), decreases monotonically with increasing aspect ratio for both ellipses and dimers. Moreover, the results for the slow dynamics of dense liquids composed of dimer- and ellipse-shaped particles are qualitatively the same, despite the fact that zero-temperature static packings of dimers are isostatic, while static packings of ellipses are hypostatic., Comment: 10 pages, 17 figures, and 1 table

Typical quasistatic compression algorithms for generating jammed packings of athermal, purely repulsive particles begin with dilute configurations and then apply successive compressions with relaxation of the elastic energy allowed between each compression step. It is well-known that during isotropic compression athermal systems with purely repulsive interactions undergo a jamming transition at packing fraction $\phi_J$ from an unjammed state with zero pressure to a jammed, rigid state with nonzero pressure. Using extensive computer simulations, we show that a novel second-order-like transition, the contact percolation transition, which signals the formation of a system-spanning cluster of mutually contacting particles, occurs at $\phi_P < \phi_J$, preceding the jamming transition. By measuring the number of non-floppy modes of the dynamical matrix, and the displacement field and time-dependent pressure following compression, we find that the contact percolation transition also heralds the onset of complex spatiotemporal response to applied stress. Thus, highly heterogeneous, cooperative, and non-affine particle motion occurs in unjammed systems significantly below the jamming transition for $\phi_P < \phi < \phi_J$, not only for jammed systems with $\phi > \phi_J$., Comment: 4 pages, 4 figures

We present a systematic study of photonic band gaps (PBGs) in three-dimensional (3D) photonic amorphous structures (PAS) with short-range order. From calculations of the density of optical states (DOS) for PAS with different topologies, we find that tetrahedrally connected dielectric networks produce the largest isotropic PBGs. Local uniformity and tetrahedral order are essential to the formation of PBGs in PAS, in addition to short-range geometric order. This work demonstrates that it is possible to create broad, isotropic PBGs for vector light fields in 3D PAS without long-range order., 6 pages, 8 figures

Jammed particulate systems, such as granular media, colloids, and foams, interact via one-sided forces that are nonzero only when particles overlap. We find that systems with one-sided repulsive interactions possess no linear response regime in the large system limit ($N\rightarrow \infty$) for all pressures $p$ (or compressions $\Delta \phi$), and for all $N$ near jamming onset $p\rightarrow 0$. We perform simulations on 2D frictionless bidisperse mechanically stable disk packings over a range of packing fractions $\Delta \phi = \phi-\phi_J$ above jamming onset $\phi_J$. We apply perturbations with amplitude $\delta$ to the packings along each eigen-direction from the dynamical matrix and determine whether the response of the system evolving at constant energy remains in the original eigenmode of the perturbation. For $\delta > \delta_c$, which we calculate analytically, a single contact breaks and fluctuations abruptly spread to all harmonic modes. As $\delta$ increases further all discrete harmonic modes disappear into a continuous frequency band. We find that $\sim \Delta \phi/N^{\lambda}$, where $1 > \lambda > 0.5$, and thus jammed particulate systems are inherently nonharmonic with no linear vibrational response regime as $N\rightarrow \infty$ over the full range of $\Delta \phi$, and as $\Delta \phi \rightarrow 0$ at any $N$., Comment: 4 pages, 3 figures

We study the response of granular materials to external stress using experiment, simulation, and theory. We derive an entropic, Ginzburg-Landau functional that enforces mechanical stability and positivity of contact forces. In this framework, the elastic moduli depend only on the applied stress. A combination of this feature and the positivity constraint leads to stress correlations whose shape and magnitude are extremely sensitive to the applied stress. The predictions from the theory describe the stress correlations for both simulations and experiments semiquantitatively., Comment: 4 pages, 3 figures

Granular matter is comprised of a large number of particles whose collective behavior determines macroscopic properties such as flow and mechanical strength. A comprehensive theory of the properties of granular matter, therefore, requires a statistical framework. In molecular matter, equilibrium statistical mechanics, which is founded on the principle of conservation of energy, provides this framework. Grains, however, are small but macroscopic objects whose interactions are dissipative since energy can be lost through excitations of the internal degrees of freedom. In this work, we construct a statistical framework for static, mechanically stable packings of grains, which parallels that of equilibrium statistical mechanics but with conservation of energy replaced by the conservation of a function related to the mechanical stress tensor. Our analysis demonstrates the existence of a state function that has all the attributes of entropy. In particular, maximizing this state function leads to a well-defined granular temperature for these systems. Predictions of the ensemble are verified against simulated packings of frictionless, deformable disks. Our demonstration that a statistical ensemble can be constructed through the identification of conserved quantities other than energy is a new approach that is expected to open up avenues for statistical descriptions of other non-equilibrium systems., Comment: 5 pages, 4 figures

Relative frequencies of mechanically stable (MS) packings of frictionless bidisperse disks are studied numerically in small systems. The packings are created by successively compressing or decompressing a system of soft purely repulsive disks, followed by energy minimization, until only infinitesimal particle overlaps remain. For systems of up to 14 particles most of the MS packings were generated. We find that the packings are not equally probable as has been assumed in recent thermodynamic descriptions of granular systems. Instead, the frequency distribution, averaged over each packing-fraction interval $\Delta \phi$, grows exponentially with increasing $\phi$. Moreover, within each packing-fraction interval MS packings occur with frequencies $f_k$ that differ by many orders of magnitude. Also, key features of the frequency distribution do not change when we significantly alter the packing-generation algorithm--for example frequent packings remain frequent and rare ones remain rare. These results indicate that the frequency distribution of MS packings is strongly influenced by geometrical properties of the multidimensional configuration space. By adding thermal fluctuations to a set of the MS packings, we were able to examine a number of local features of configuration space near each packing including the time required for a given packing to break to a distinct one, which enabled us to estimate the energy barriers that separate one packing from another. We found a positive correlation between the packing frequencies and the heights of the lowest energy barriers $\epsilon_0$. We also examined displacement fluctuations away from the MS packings to correlate the size and shape of the local basins near each packing to the packing frequencies., Comment: 21 pages, 20 figures, 1 table

We create collectively jammed (CJ) packings of 50-50 bidisperse mixtures of smooth disks in 2d using an algorithm in which we successively compress or expand soft particles and minimize the total energy at each step until the particles are just at contact. We focus on small systems in 2d and thus are able to find nearly all of the collectively jammed states at each system size. We decompose the probability $P(\phi)$ for obtaining a collectively jammed state at a particular packing fraction $\phi$ into two composite functions: 1) the density of CJ packing fractions $\rho(\phi)$, which only depends on geometry and 2) the frequency distribution $\beta(\phi)$, which depends on the particular algorithm used to create them. We find that the function $\rho(\phi)$ is sharply peaked and that $\beta(\phi)$ depends exponentially on $\phi$. We predict that in the infinite system-size limit the behavior of $P(\phi)$ in these systems is controlled by the density of CJ packing fractions--not the frequency distribution. These results suggest that the location of the peak in $P(\phi)$ when $N \to \infty$ can be used as a protocol-independent definition of random close packing., Comment: 9 pages, 14 figures