@article{14710, abstract = {The self-assembly of complex structures from a set of non-identical building blocks is a hallmark of soft matter and biological systems, including protein complexes, colloidal clusters, and DNA-based assemblies. Predicting the dependence of the equilibrium assembly yield on the concentrations and interaction energies of building blocks is highly challenging, owing to the difficulty of computing the entropic contributions to the free energy of the many structures that compete with the ground state configuration. While these calculations yield well known results for spherically symmetric building blocks, they do not hold when the building blocks have internal rotational degrees of freedom. Here we present an approach for solving this problem that works with arbitrary building blocks, including proteins with known structure and complex colloidal building blocks. Our algorithm combines classical statistical mechanics with recently developed computational tools for automatic differentiation. Automatic differentiation allows efficient evaluation of equilibrium averages over configurations that would otherwise be intractable. We demonstrate the validity of our framework by comparison to molecular dynamics simulations of simple examples, and apply it to calculate the yield curves for known protein complexes and for the assembly of colloidal shells.}, author = {Curatolo, Agnese I. and Kimchi, Ofer and Goodrich, Carl Peter and Krueger, Ryan K. and Brenner, Michael P.}, issn = {20411723}, journal = {Nature Communications}, publisher = {Springer Nature}, title = {{A computational toolbox for the assembly yield of complex and heterogeneous structures}}, doi = {10.1038/s41467-023-43168-4}, volume = {14}, year = {2023}, } @article{9257, abstract = {The inverse problem of designing component interactions to target emergent structure is fundamental to numerous applications in biotechnology, materials science, and statistical physics. Equally important is the inverse problem of designing emergent kinetics, but this has received considerably less attention. Using recent advances in automatic differentiation, we show how kinetic pathways can be precisely designed by directly differentiating through statistical physics models, namely free energy calculations and molecular dynamics simulations. We consider two systems that are crucial to our understanding of structural self-assembly: bulk crystallization and small nanoclusters. In each case, we are able to assemble precise dynamical features. Using gradient information, we manipulate interactions among constituent particles to tune the rate at which these systems yield specific structures of interest. Moreover, we use this approach to learn nontrivial features about the high-dimensional design space, allowing us to accurately predict when multiple kinetic features can be simultaneously and independently controlled. These results provide a concrete and generalizable foundation for studying nonstructural self-assembly, including kinetic properties as well as other complex emergent properties, in a vast array of systems.}, author = {Goodrich, Carl Peter and King, Ella M. and Schoenholz, Samuel S. and Cubuk, Ekin D. and Brenner, Michael P.}, issn = {1091-6490}, journal = {Proceedings of the National Academy of Sciences}, number = {10}, publisher = {National Academy of Sciences}, title = {{Designing self-assembling kinetics with differentiable statistical physics models}}, doi = {10.1073/pnas.2024083118}, volume = {118}, year = {2021}, } @article{12667, abstract = {Unlike crystalline atomic and ionic solids, texture development due to crystallographically preferred growth in colloidal crystals is less studied. Here we investigate the underlying mechanisms of the texture evolution in an evaporation-induced colloidal assembly process through experiments, modeling, and theoretical analysis. In this widely used approach to obtain large-area colloidal crystals, the colloidal particles are driven to the meniscus via the evaporation of a solvent or matrix precursor solution where they close-pack to form a face-centered cubic colloidal assembly. Via two-dimensional large-area crystallographic mapping, we show that the initial crystal orientation is dominated by the interaction of particles with the meniscus, resulting in the expected coalignment of the close-packed direction with the local meniscus geometry. By combining with crystal structure analysis at a single-particle level, we further reveal that, at the later stage of self-assembly, however, the colloidal crystal undergoes a gradual rotation facilitated by geometrically necessary dislocations (GNDs) and achieves a large-area uniform crystallographic orientation with the close-packed direction perpendicular to the meniscus and parallel to the growth direction. Classical slip analysis, finite element-based mechanical simulation, computational colloidal assembly modeling, and continuum theory unequivocally show that these GNDs result from the tensile stress field along the meniscus direction due to the constrained shrinkage of the colloidal crystal during drying. The generation of GNDs with specific slip systems within individual grains leads to crystallographic rotation to accommodate the mechanical stress. The mechanistic understanding reported here can be utilized to control crystallographic features of colloidal assemblies, and may provide further insights into crystallographically preferred growth in synthetic, biological, and geological crystals.}, author = {Li, Ling and Goodrich, Carl Peter and Yang, Haizhao and Phillips, Katherine R. and Jia, Zian and Chen, Hongshun and Wang, Lifeng and Zhong, Jinjin and Liu, Anhua and Lu, Jianfeng and Shuai, Jianwei and Brenner, Michael P. and Spaepen, Frans and Aizenberg, Joanna}, issn = {1091-6490}, journal = {PNAS}, number = {32}, publisher = {Proceedings of the National Academy of Sciences}, title = {{Microscopic origins of the crystallographically preferred growth in evaporation-induced colloidal crystals}}, doi = {10.1073/pnas.2107588118}, volume = {118}, year = {2021}, } @article{7778, abstract = {Recent advances in synthetic posttranslational protein circuits are substantially impacting the landscape of cellular engineering and offer several advantages compared to traditional gene circuits. However, engineering dynamic phenomena such as oscillations in protein-level circuits remains an outstanding challenge. Few examples of biological posttranslational oscillators are known, necessitating theoretical progress to determine realizable oscillators. We construct mathematical models for two posttranslational oscillators, using few components that interact only through reversible binding and phosphorylation/dephosphorylation reactions. Our designed oscillators rely on the self-assembly of two protein species into multimeric functional enzymes that respectively inhibit and enhance this self-assembly. We limit our analysis to within experimental constraints, finding (i) significant portions of the restricted parameter space yielding oscillations and (ii) that oscillation periods can be tuned by several orders of magnitude using recent advances in computational protein design. Our work paves the way for the rational design and realization of protein-based dynamic systems.}, author = {Kimchi, Ofer and Goodrich, Carl Peter and Courbet, Alexis and Curatolo, Agnese I. and Woodall, Nicholas B. and Baker, David and Brenner, Michael P.}, journal = {Science Advances}, number = {51}, title = {{Self-assembly-based posttranslational protein oscillators}}, doi = {10.1126/sciadv.abc1939}, volume = {6}, year = {2020}, } @article{7754, abstract = {Creating a selective gel that filters particles based on their interactions is a major goal of nanotechnology, with far-reaching implications from drug delivery to controlling assembly pathways. However, this is particularly difficult when the particles are larger than the gel’s characteristic mesh size because such particles cannot passively pass through the gel. Thus, filtering requires the interacting particles to transiently reorganize the gel’s internal structure. While significant advances, e.g., in DNA engineering, have enabled the design of nano-materials with programmable interactions, it is not clear what physical principles such a designer gel could exploit to achieve selective permeability. We present an equilibrium mechanism where crosslink binding dynamics are affected by interacting particles such that particle diffusion is enhanced. In addition to revealing specific design rules for manufacturing selective gels, our results have the potential to explain the origin of selective permeability in certain biological materials, including the nuclear pore complex.}, author = {Goodrich, Carl Peter and Brenner, Michael P. and Ribbeck, Katharina}, issn = {2041-1723}, journal = {Nature Communications}, publisher = {Springer Nature}, title = {{Enhanced diffusion by binding to the crosslinks of a polymer gel}}, doi = {10.1038/s41467-018-06851-5}, volume = {9}, year = {2018}, } @article{7755, abstract = {We give a bird's-eye view of the plastic deformation of crystals aimed at the statistical physics community, as well as a broad introduction to the statistical theories of forced rigid systems aimed at the plasticity community. Memory effects in magnets, spin glasses, charge density waves, and dilute colloidal suspensions are discussed in relation to the onset of plastic yielding in crystals. Dislocation avalanches and complex dislocation tangles are discussed via a brief introduction to the renormalization group and scaling. Analogies to emergent scale invariance in fracture, jamming, coarsening, and a variety of depinning transitions are explored. Dislocation dynamics in crystals challenge nonequilibrium statistical physics. Statistical physics provides both cautionary tales of subtle memory effects in nonequilibrium systems and systematic tools designed to address complex scale-invariant behavior on multiple length scales and timescales.}, author = {Sethna, James P. and Bierbaum, Matthew K. and Dahmen, Karin A. and Goodrich, Carl Peter and Greer, Julia R. and Hayden, Lorien X. and Kent-Dobias, Jaron P. and Lee, Edward D. and Liarte, Danilo B. and Ni, Xiaoyue and Quinn, Katherine N. and Raju, Archishman and Rocklin, D. Zeb and Shekhawat, Ashivni and Zapperi, Stefano}, issn = {1531-7331}, journal = {Annual Review of Materials Research}, pages = {217--246}, publisher = {Annual Reviews}, title = {{Deformation of crystals: Connections with statistical physics}}, doi = {10.1146/annurev-matsci-070115-032036}, volume = {47}, year = {2017}, } @article{7756, abstract = {We study the shear jamming of athermal frictionless soft spheres, and find that in the thermodynamic limit, a shear-jammed state exists with different elastic properties from the isotropically-jammed state. For example, shear-jammed states can have a non-zero residual shear stress in the thermodynamic limit that arises from long-range stress-stress correlations. As a result, the ratio of the shear and bulk moduli, which in isotropically-jammed systems vanishes as the jamming transition is approached from above, instead approaches a constant. Despite these striking differences, we argue that in a deeper sense, the shear jamming and isotropic jamming transitions actually have the same symmetry, and that the differences can be fully understood by rotating the six-dimensional basis of the elastic modulus tensor.}, author = {Baity-Jesi, Marco and Goodrich, Carl Peter and Liu, Andrea J. and Nagel, Sidney R. and Sethna, James P.}, issn = {0022-4715}, journal = {Journal of Statistical Physics}, number = {3-4}, pages = {735--748}, publisher = {Springer Nature}, title = {{Emergent SO(3) symmetry of the frictionless shear jamming transition}}, doi = {10.1007/s10955-016-1703-9}, volume = {167}, year = {2017}, } @article{7757, abstract = {Recent advances in designing metamaterials have demonstrated that global mechanical properties of disordered spring networks can be tuned by selectively modifying only a small subset of bonds. Here, using a computationally efficient approach, we extend this idea to tune more general properties of networks. With nearly complete success, we are able to produce a strain between any two target nodes in a network in response to an applied source strain on any other pair of nodes by removing only ∼1% of the bonds. We are also able to control multiple pairs of target nodes, each with a different individual response, from a single source, and to tune multiple independent source/target responses simultaneously into a network. We have fabricated physical networks in macroscopic 2D and 3D systems that exhibit these responses. This work is inspired by the long-range coupled conformational changes that constitute allosteric function in proteins. The fact that allostery is a common means for regulation in biological molecules suggests that it is a relatively easy property to develop through evolution. In analogy, our results show that long-range coupled mechanical responses are similarly easy to achieve in disordered networks.}, author = {Rocks, Jason W. and Pashine, Nidhi and Bischofberger, Irmgard and Goodrich, Carl Peter and Liu, Andrea J. and Nagel, Sidney R.}, issn = {0027-8424}, journal = {Proceedings of the National Academy of Sciences}, number = {10}, pages = {2520--2525}, publisher = {Proceedings of the National Academy of Sciences}, title = {{Designing allostery-inspired response in mechanical networks}}, doi = {10.1073/pnas.1612139114}, volume = {114}, year = {2017}, } @article{7758, abstract = {Controlling motion at the microscopic scale is a fundamental goal in the development of biologically inspired systems. We show that the motion of active, self-propelled colloids can be sufficiently controlled for use as a tool to assemble complex structures such as braids and weaves out of microscopic filaments. Unlike typical self-assembly paradigms, these structures are held together by geometric constraints rather than adhesive bonds. The out-of-equilibrium assembly that we propose involves precisely controlling the 2D motion of active colloids so that their path has a nontrivial topology. We demonstrate with proof-of-principle Brownian dynamics simulations that, when the colloids are attached to long semiflexible filaments, this motion causes the filaments to braid. The ability of the active particles to provide sufficient force necessary to bend the filaments into a braid depends on a number of factors, including the self-propulsion mechanism, the properties of the filament, and the maximum curvature in the braid. Our work demonstrates that nonequilibrium assembly pathways can be designed using active particles.}, author = {Goodrich, Carl Peter and Brenner, Michael P.}, issn = {0027-8424}, journal = {Proceedings of the National Academy of Sciences}, number = {2}, pages = {257--262}, publisher = {Proceedings of the National Academy of Sciences}, title = {{Using active colloids as machines to weave and braid on the micrometer scale}}, doi = {10.1073/pnas.1608838114}, volume = {114}, year = {2017}, } @article{7763, abstract = {An orthogonal wavelet basis is characterized by its approximation order, which relates to the ability of the basis to represent general smooth functions on a given scale. It is known, though perhaps not widely known, that there are ways of exceeding the approximation order, i.e., achieving higher-order error in the discretized wavelet transform and its inverse. The focus here is on the development of a practical formulation to accomplish this first for 1D smooth functions, then for 1D functions with discontinuities and then for multidimensional (here 2D) functions with discontinuities. It is shown how to transcend both the wavelet approximation order and the 2D Gibbs phenomenon in representing electromagnetic fields at discontinuous dielectric interfaces that do not simply follow the wavelet-basis grid.}, author = {Lombardini, Richard and Acevedo, Ramiro and Kuczala, Alexander and Keys, Kerry P. and Goodrich, Carl Peter and Johnson, Bruce R.}, issn = {0021-9991}, journal = {Journal of Computational Physics}, pages = {244--262}, publisher = {Elsevier}, title = {{Higher-order wavelet reconstruction/differentiation filters and Gibbs phenomena}}, doi = {10.1016/j.jcp.2015.10.035}, volume = {305}, year = {2016}, } @article{7764, abstract = {States of self stress, organizations of internal forces in many-body systems that are in equilibrium with an absence of external forces, can be thought of as the constitutive building blocks of the elastic response of a material. In overconstrained disordered packings they have a natural mathematical correspondence with the zero-energy vibrational modes in underconstrained systems. While substantial attention in the literature has been paid to diverging length scales associated with zero- and finite-energy vibrational modes in jammed systems, less is known about the spatial structure of the states of self stress. In this work we define a natural way in which a unique state of self stress can be associated with each bond in a disordered spring network derived from a jammed packing, and then investigate the spatial structure of these bond-localized states of self stress. This allows for an understanding of how the elastic properties of a system would change upon changing the strength or even existence of any bond in the system.}, author = {Sussman, Daniel M. and Goodrich, Carl Peter and Liu, Andrea J.}, issn = {1744-683X}, journal = {Soft Matter}, number = {17}, pages = {3982--3990}, publisher = {Royal Society of Chemistry}, title = {{Spatial structure of states of self stress in jammed systems}}, doi = {10.1039/c6sm00094k}, volume = {12}, year = {2016}, } @article{7760, abstract = {We propose a Widom-like scaling ansatz for the critical jamming transition. Our ansatz for the elastic energy shows that the scaling of the energy, compressive strain, shear strain, system size, pressure, shear stress, bulk modulus, and shear modulus are all related to each other via scaling relations, with only three independent scaling exponents. We extract the values of these exponents from already known numerical or theoretical results, and we numerically verify the resulting predictions of the scaling theory for the energy and residual shear stress. We also derive a scaling relation between pressure and residual shear stress that yields insight into why the shear and bulk moduli scale differently. Our theory shows that the jamming transition exhibits an emergent scale invariance, setting the stage for the potential development of a renormalization group theory for jamming.}, author = {Goodrich, Carl Peter and Liu, Andrea J. and Sethna, James P.}, issn = {0027-8424}, journal = {Proceedings of the National Academy of Sciences}, number = {35}, pages = {9745--9750}, publisher = {Proceedings of the National Academy of Sciences}, title = {{Scaling ansatz for the jamming transition}}, doi = {10.1073/pnas.1601858113}, volume = {113}, year = {2016}, } @article{7761, abstract = {We study the effect of dilute pinning on the jamming transition. Pinning reduces the average contact number needed to jam unpinned particles and shifts the jamming threshold to lower densities, leading to a pinning susceptibility, χp. Our main results are that this susceptibility obeys scaling form and diverges in the thermodynamic limit as χp∝|ϕ−ϕ∞c|−γp where ϕ∞c is the jamming threshold in the absence of pins. Finite-size scaling arguments yield these values with associated statistical (systematic) errors γp=1.018±0.026(0.291) in d=2 and γp=1.534±0.120(0.822) in d=3. Logarithmic corrections raise the exponent in d=2 to close to the d=3 value, although the systematic errors are very large.}, author = {Graves, Amy L. and Nashed, Samer and Padgett, Elliot and Goodrich, Carl Peter and Liu, Andrea J. and Sethna, James P.}, issn = {0031-9007}, journal = {Physical Review Letters}, number = {23}, publisher = {American Physical Society}, title = {{Pinning susceptibility: The effect of dilute, quenched disorder on jamming}}, doi = {10.1103/physrevlett.116.235501}, volume = {116}, year = {2016}, } @article{7762, abstract = {Characterizing structural inhomogeneity is an essential step in understanding the mechanical response of amorphous materials. We introduce a threshold-free measure based on the field of vectors pointing from the center of each particle to the centroid of the Voronoi cell in which the particle resides. These vectors tend to point in toward regions of high free volume and away from regions of low free volume, reminiscent of sinks and sources in a vector field. We compute the local divergence of these vectors, where positive values correspond to overpacked regions and negative values identify underpacked regions within the material. Distributions of this divergence are nearly Gaussian with zero mean, allowing for structural characterization using only the moments of the distribution. We explore how the standard deviation and skewness vary with the packing fraction for simulations of bidisperse systems and find a kink in these moments that coincides with the jamming transition.}, author = {Rieser, Jennifer M. and Goodrich, Carl Peter and Liu, Andrea J. and Durian, Douglas J.}, issn = {0031-9007}, journal = {Physical Review Letters}, number = {8}, publisher = {American Physical Society}, title = {{Divergence of Voronoi cell anisotropy vector: A threshold-free characterization of local structure in amorphous materials}}, doi = {10.1103/physrevlett.116.088001}, volume = {116}, year = {2016}, } @article{7765, abstract = {We introduce a principle unique to disordered solids wherein the contribution of any bond to one global perturbation is uncorrelated with its contribution to another. Coupled with sufficient variability in the contributions of different bonds, this “independent bond-level response” paves the way for the design of real materials with unusual and exquisitely tuned properties. To illustrate this, we choose two global perturbations: compression and shear. By applying a bond removal procedure that is both simple and experimentally relevant to remove a very small fraction of bonds, we can drive disordered spring networks to both the incompressible and completely auxetic limits of mechanical behavior.}, author = {Goodrich, Carl Peter and Liu, Andrea J. and Nagel, Sidney R.}, issn = {0031-9007}, journal = {Physical Review Letters}, number = {22}, publisher = {American Physical Society}, title = {{The principle of independent bond-level response: Tuning by pruning to exploit disorder for global behavior}}, doi = {10.1103/physrevlett.114.225501}, volume = {114}, year = {2015}, } @article{7766, abstract = {We study the vibrational properties near a free surface of disordered spring networks derived from jammed sphere packings. In bulk systems, without surfaces, it is well understood that such systems have a plateau in the density of vibrational modes extending down to a frequency scale ω*. This frequency is controlled by ΔZ = 〈Z〉 − 2d, the difference between the average coordination of the spheres and twice the spatial dimension, d, of the system, which vanishes at the jamming transition. In the presence of a free surface we find that there is a density of disordered vibrational modes associated with the surface that extends far below ω*. The total number of these low-frequency surface modes is controlled by ΔZ, and the profile of their decay into the bulk has two characteristic length scales, which diverge as ΔZ−1/2 and ΔZ−1 as the jamming transition is approached.}, author = {Sussman, Daniel M. and Goodrich, Carl Peter and Liu, Andrea J. and Nagel, Sidney R.}, issn = {1744-683X}, journal = {Soft Matter}, number = {14}, pages = {2745--2751}, publisher = {Royal Society of Chemistry}, title = {{Disordered surface vibrations in jammed sphere packings}}, doi = {10.1039/c4sm02905d}, volume = {11}, year = {2015}, } @article{7767, abstract = {We present a model of soft active particles that leads to a rich array of collective behavior found also in dense biological swarms of bacteria and other unicellular organisms. Our model uses only local interactions, such as Vicsek-type nearest-neighbor alignment, short-range repulsion, and a local boundary term. Changing the relative strength of these interactions leads to migrating swarms, rotating swarms, and jammed swarms, as well as swarms that exhibit run-and-tumble motion, alternating between migration and either rotating or jammed states. Interestingly, although a migrating swarm moves slower than an individual particle, the diffusion constant can be up to three orders of magnitude larger, suggesting that collective motion can be highly advantageous, for example, when searching for food.}, author = {van Drongelen, Ruben and Pal, Anshuman and Goodrich, Carl Peter and Idema, Timon}, issn = {1539-3755}, journal = {Physical Review E}, number = {3}, publisher = {American Physical Society}, title = {{Collective dynamics of soft active particles}}, doi = {10.1103/physreve.91.032706}, volume = {91}, year = {2015}, } @unpublished{7779, abstract = {The fact that a disordered material is not constrained in its properties in the same way as a crystal presents significant and yet largely untapped potential for novel material design. However, unlike their crystalline counterparts, disordered solids are not well understood. One of the primary obstacles is the lack of a theoretical framework for thinking about disorder and its relation to mechanical properties. To this end, we study an idealized system of frictionless athermal soft spheres that, when compressed, undergoes a jamming phase transition with diverging length scales and clean power-law signatures. This critical point is the cornerstone of a much larger "jamming scenario" that has the potential to provide the essential theoretical foundation necessary for a unified understanding of the mechanics of disordered solids. We begin by showing that jammed sphere packings have a valid linear regime despite the presence of "contact nonlinearities." We then investigate the critical nature of the transition, focusing on diverging length scales and finite-size effects. Next, we argue that jamming plays the same role for disordered solids as the perfect crystal plays for crystalline solids. Not only can it be considered an idealized starting point for understanding disordered materials, but it can even influence systems that have a relatively high amount of crystalline order. The behavior of solids can thus be thought of as existing on a spectrum, with the perfect crystal and the jamming transition at opposing ends. Finally, we introduce a new principle wherein the contribution of an individual bond to one global property is independent of its contribution to another. This principle allows the different global responses of a disordered system to be manipulated independently and provides a great deal of flexibility in designing materials with unique, textured and tunable properties.}, author = {Goodrich, Carl Peter}, booktitle = {arXiv:1510.08820}, pages = {242}, title = {{Unearthing the anticrystal: Criticality in the linear response of disordered solids}}, year = {2015}, } @article{7768, abstract = {We investigate the vibrational modes of quasi-two-dimensional disordered colloidal packings of hard colloidal spheres with short-range attractions as a function of packing fraction. Certain properties of the vibrational density of states (vDOS) are shown to correlate with the density and structure of the samples (i.e., in sparsely versus densely packed samples). Specifically, a crossover from dense glassy to sparse gel-like states is suggested by an excess of phonon modes at low frequency and by a variation in the slope of the vDOS with frequency at low frequency. This change in phonon mode distribution is demonstrated to arise largely from localized vibrations that involve individual and/or small clusters of particles with few local bonds. Conventional order parameters and void statistics did not exhibit obvious gel-glass signatures as a function of volume fraction. These mode behaviors and accompanying structural insights offer a potentially new set of indicators for identification of glass-gel transitions and for assignment of gel-like versus glass-like character to a disordered solid material.}, author = {Lohr, Matthew A. and Still, Tim and Ganti, Raman and Gratale, Matthew D. and Davidson, Zoey S. and Aptowicz, Kevin B. and Goodrich, Carl Peter and Sussman, Daniel M. and Yodh, A. G.}, issn = {1539-3755}, journal = {Physical Review E}, number = {6}, publisher = {American Physical Society}, title = {{Vibrational and structural signatures of the crossover between dense glassy and sparse gel-like attractive colloidal packings}}, doi = {10.1103/physreve.90.062305}, volume = {90}, year = {2014}, } @article{7769, abstract = {Athermal packings of soft repulsive spheres exhibit a sharp jamming transition in the thermodynamic limit. Upon further compression, various structural and mechanical properties display clean power-law behavior over many decades in pressure. As with any phase transition, the rounding of such behavior in finite systems close to the transition plays an important role in understanding the nature of the transition itself. The situation for jamming is surprisingly rich: the assumption that jammed packings are isotropic is only strictly true in the large-size limit, and finite-size has a profound effect on the very meaning of jamming. Here, we provide a comprehensive numerical study of finite-size effects in sphere packings above the jamming transition, focusing on stability as well as the scaling of the contact number and the elastic response.}, author = {Goodrich, Carl Peter and Dagois-Bohy, Simon and Tighe, Brian P. and van Hecke, Martin and Liu, Andrea J. and Nagel, Sidney R.}, issn = {1539-3755}, journal = {Physical Review E}, number = {2}, publisher = {American Physical Society}, title = {{Jamming in finite systems: Stability, anisotropy, fluctuations, and scaling}}, doi = {10.1103/physreve.90.022138}, volume = {90}, year = {2014}, }