Yoav Kallus

Free energy of singular sticky-sphere clusters

Networks of particles connected by springs model many condensed-matter systems, from colloids interacting with a short-range potential and complex fluids near jamming, to self-assembled lattices and various metamaterials. Under small thermal fluctuations the vibrational entropy of a ground state is given by the harmonic approximation if it has no zero-frequency vibrational modes, yet such singular modes are at the epicenter of many interesting behaviors in the systems above. We consider a system of N spherical particles, and directly account for the singularities that arise in the sticky limit where the pairwise interaction is strong and short ranged. Although the contribution to the partition function from singular clusters diverges in the limit, its asymptotic value can be calculated and depends on only two parameters, characterizing the depth and range of the potential. The result holds for systems that are second-order rigid, a geometric characterization that describes all known ground-state (rigid) sticky clusters. To illustrate the applications of our theory we address the question of emergence: how does crystalline order arise in large systems when it is strongly disfavored in small ones? We calculate the partition functions of all known rigid clusters up to N≤21 and show the cluster landscape is dominated by hyperstatic clusters (those with more than 3N-6 contacts); singular and isostatic clusters are far less frequent, despite their extra vibrational and configurational entropies. Since the most hyperstatic clusters are close to fragments of a close-packed lattice, this underlies the emergence of order in sticky-sphere systems, even those as small as N=10.

The random packing density of nearly spherical particles

Obtaining general relations between macroscopic properties of random assemblies, such as density, and the microscopic properties of their constituent particles, such as shape, is a foundational challenge in the study of amorphous materials. By leveraging existing understanding of the random packing of spherical particles, we estimate the random packing density for all sufficiently spherical shapes. Our method uses the ensemble of random packing configurations of spheres as a reference point for a perturbative calculation, which we carry to linear order in the deformation. A fully analytic calculation shows that all sufficiently spherical shapes pack more densely than spheres. Additionally, we use simulation data for spheres to calculate numerical estimates for nonspherical particles and compare these estimates to simulations.

Paradoxes in leaky microbial trade

Microbes produce metabolic resources that are important for cell growth yet leak into the environment. Other microbes can use these resources, adjust their own metabolic production accordingly, and alter the resources available for others. We analyze a model in which metabolite concentrations, production regulation, and population frequencies coevolve in the simple case of two cell types producing two metabolites. We identify three paradoxes where changes that should intuitively benefit a cell type actually harm it. For example, a cell type can become more efficient at producing a metabolite and its relative frequency can decrease—or alternatively the total population growth rate can decrease. Another paradox occurs when a cell type manipulates its counterpart’s production so as to maximize its own instantaneous growth rate, only to achieve a lower final growth rate than had it not manipulated. These paradoxes highlight the complex and counterintuitive dynamics that emerge in simple microbial economies.Microbes live in communities and exchange metabolites, but the resulting dynamics are poorly understood. Here, the authors study the interplay between metabolite production strategies and population dynamics, and find that complex and unexpected dynamics emerge even in simple microbial economies.

Pessimal packing shapes

We address the question of which convex shapes, when packed as densely as possible under certain restrictions, fill the least space and leave the most empty space. In each different dimension and under each different set of restrictions, this question is expected to have a different answer or perhaps no answer at all. As the problem of identifying global minima in most cases appears to be beyond current reach, in this paper we focus on local minima. We review some known results and prove these new results: in two dimensions, the regular heptagon is a local minimum of the double-lattice packing density, and in three dimensions, the directional derivative (in the sense of Minkowski addition) of the double-lattice packing density at the point in the space of shapes corresponding to the ball is in every direction positive.

Marginal stability in jammed packings: quasicontacts and weak contacts.

Maximally random jammed (MRJ) sphere packing is a prototypical example of a system naturally poised at the margin between underconstraint and overconstraint. This marginal stability has traditionally been understood in terms of isostaticity, the equality of the number of mechanical contacts and the number of degrees of freedom. Quasicontacts, pairs of spheres on the verge of coming in contact, are irrelevant for static stability, but they come into play when considering dynamic stability, as does the distribution of contact forces. We show that the effects of marginal dynamic stability, as manifested in the distributions of quasicontacts and weak contacts, are consequential and nontrivial. We study these ideas first in the context of MRJ packing of d-dimensional spheres, where we show that the abundance of quasicontacts grows at a faster rate than that of contacts. We reexamine a calculation of Jin et al. [Phys. Rev. E 82, 051126 (2010)], where quasicontacts were originally neglected, and we explore the effect of their inclusion in the calculation. This analysis yields an estimate of the asymptotic behavior of the packing density in high dimensions. We argue that this estimate should be reinterpreted as a lower bound. The latter part of the paper is devoted to Bravais lattice packings that possess the minimum number of contacts to maintain mechanical stability. We show that quasicontacts play an even more important role in these packings. We also show that jammed lattices are a useful setting for studying the Edwards ensemble, which weights each mechanically stable configuration equally and does not account for dynamics. This ansatz fails to predict the power-law distribution of near-zero contact forces, P(f)∼f(θ).

Statistical mechanics of the lattice sphere packing problem.

We present an efficient Monte Carlo method for the lattice sphere packing problem in d dimensions. We use this method to numerically discover de novo the densest lattice sphere packing in dimensions 9 through 20. Our method goes beyond previous methods, not only in exploring higher dimensions but also in shedding light on the statistical mechanics underlying the problem in question. We observe evidence of a phase transition in the thermodynamic limit d→∞. In the dimensions explored in the present work, the results are consistent with a first-order crystallization transition but leave open the possibility that a glass transition is manifested in higher dimensions.

Jammed lattice sphere packings

We generate and study an ensemble of isostatic jammed hard-sphere lattices. These lattices are obtained by compression of a periodic system with an adaptive unit cell containing a single sphere until the point of mechanical stability. We present detailed numerical data about the densities, pair correlations, force distributions, and structure factors of such lattices. We show that this model retains many of the crucial structural features of the classical hard-sphere model and propose it as a model for the jamming and glass transitions that enables exploration of much higher dimensions than are usually accessible.

Scaling collapse at the jamming transition.

The jamming transition of particles with finite-range interactions is characterized by a variety of critical phenomena, including power-law distributions of marginal contacts. We numerically study a recently proposed simple model of jamming, which is conjectured to lie in the same universality class as the jamming of spheres in all dimensions. We extract numerical estimates of the critical exponents, θ=0.451±0.006 and γ=0.404±0.004, that match the exponents observed in sphere packing systems. We analyze finite-size scaling effects that manifest in a subcritical cutoff regime and size-independent but protocol-dependent scaling curves. Our results support the conjectured link with sphere jamming, provide more precise measurements of the critical exponents than previously reported, and shed light on the finite-size scaling behavior of continuous constraint satisfiability transitions.

The Local Optimality of the Double Lattice Packing

This paper introduces a new technique for proving the local optimality of packing configurations of Euclidean space. Applying this technique to a general convex polygon, we prove that under mild assumptions satisfied generically, the construction of the optimal double lattice packing by Kuperberg and Kuperberg is also locally optimal in the full space of packings.

When is the Ball a Local Pessimum for Covering

We consider the problem of identifying the worst point-symmetric shape for covering