arXiv subjects

F. H. Stillinger

Publications and source records attributed to F. H. Stillinger.

28 records · Page 2Linked to original sources

arXiv2006

Packing Hyperspheres in High-Dimensional Euclidean Spaces

We present the first study of disordered jammed hard-sphere packings in four-, five- and six-dimensional Euclidean spaces. Using a collision-driven packing generation algorithm, we obtain the first estimates for the packing fractions of the maximally random jammed (MRJ) states for space dimensions $d=4$, 5 and 6 to be $ϕ_{MRJ} \simeq 0.46$, 0.31 and 0.20, respectively. To a good approximation, the MRJ density obeys the scaling form $ϕ_{MRJ}= c_1/2^d+(c_2 d)/2^d$, where $c_1=-2.72$ and $c_2=2.56$, which appears to be consistent with high-dimensional asymptotic limit, albeit with different coefficients. Calculations of the pair correlation function $g_{2}(r)$ and structure factor $S(k)$ for these states show that short-range ordering appreciably decreases with increasing dimension, consistent with a recently proposed ``decorrelation principle,'' which, among othe things, states that unconstrained correlations diminish as the dimension increases and vanish entirely in the limit $d \to \infty$. As in three dimensions (where $ϕ_{MRJ} \simeq 0.64$), the packings show no signs of crystallization, are isostatic, and have a power-law divergence in $g_{2}(r)$ at contact with power-law exponent $\simeq 0.4$. Across dimensions, the cumulative number of neighbors equals the kissing number of the conjectured densest packing close to where $g_{2}(r)$ has its first minimum. We obtain estimates for the freezing and melting desnities for the equilibrium hard-sphere fluid-solid transition, $ϕ_F \simeq 0.32$ and $ϕ_M \simeq 0.39$, respectively, for $d=4$, and $ϕ_F \simeq 0.19$ and $ϕ_M \simeq 0.24$, respectively, for $d=5$.

cond-mat.stat-mech↗

arXiv2006

Exactly Solvable Disordered Sphere-Packing Model in Arbitrary-Dimension Euclidean Spaces

We introduce a generalization of the well-known random sequential addition (RSA) process for hard spheres in $d$-dimensional Euclidean space $\mathbb{R}^d$. We show that all of the $n$-particle correlation functions of this nonequilibrium model, in a certain limit called the "ghost" RSA packing, can be obtained analytically for all allowable densities and in any dimension. This represents the first exactly solvable disordered sphere-packing model in arbitrary dimension. The fact that the maximal density $ϕ(\infty)=1/2^d$ of the ghost RSA packing implies that there may be disordered sphere packings in sufficiently high $d$ whose density exceeds Minkowski's lower bound for Bravais lattices, the dominant asymptotic term of which is $1/2^d$. Indeed, we report on a conjectural lower bound on the density whose asymptotic behavior is controlled by $2^{-(0.77865...) d}$, thus providing the putative exponential improvement on Minkowski's 100-year-old bound. Our results suggest that the densest packings in sufficiently high dimensions may be disordered rather than periodic, implying the existence of disordered classical ground states for some continuous potentials.

cond-mat.soft↗

arXiv2006

Do Binary Hard Disks Exhibit an Ideal Glass Transition?

We demonstrate that there is no ideal glass transition in a binary hard-disk mixture by explicitly constructing an exponential number of jammed packings with densities spanning the spectrum from the accepted ``amorphous'' glassy state to the phase-separated crystal. Thus the configurational entropy cannot be zero for an ideal amorphous glass, presumed distinct from the crystal in numerous theoretical and numerical estimates in the literature. This objection parallels our previous critique of the idea that there is a most-dense random (close) packing for hard spheres [Torquato et al, Phys. Rev. Lett., 84, 2064 (2000)].

cond-mat.stat-mech↗

arXiv2006

New Conjectural Lower Bounds on the Optimal Density of Sphere Packings

Sphere packings in high dimensions interest mathematicians and physicists and have direct applications in communications theory. Remarkably, no one has been able to provide exponential improvement on a 100-year-old lower bound on the maximal packing density due to Minkowski in $d$-dimensional Euclidean space $\Re^d$. The asymptotic behavior of this bound is controlled by $2^{-d}$ in high dimensions. Using an optimization procedure that we introduced earlier \cite{To02c} and a conjecture concerning the existence of disordered sphere packings in $\Re^d$, we obtain a provisional lower bound on the density whose asymptotic behavior is controlled by $2^{-0.7786... d}$, thus providing the putative exponential improvement of Minkowski's bound. The conjecture states that a hard-core nonnegative tempered distribution is a pair correlation function of a translationally invariant disordered sphere packing in $\Re^d$ for asymptotically large d if and only if the Fourier transform of the autocovariance function is nonnegative. The conjecture is supported by two explicit analytically characterized disordered packings, numerical simulations in low dimensions, known necessary conditions that only have relevance in very low dimensions, and the fact that we can recover the forms of known rigorous lower bounds. A byproduct of our approach is an asymptotic lower bound on the average kissing number whose behavior is controlled by $2^{0.2213... d}$, which is to be compared to the best known asymptotic lower bound on the individual kissing number of $2^{0.2075... d}$. Interestingly, our optimization procedure is precisely the dual of a primal linear program devised by Cohn and Elkies \cite{Co03} to obtain upper bounds on the density, and hence has implications for linear programming bounds.

math.MG↗

arXiv2005

Tetratic Order in the Phase Behavior of a Hard-Rectangle System

Previous Monte Carlo investigations by Wojciechowski \emph{et al.} have found two unusual phases in two-dimensional systems of anisotropic hard particles: a tetratic phase of four-fold symmetry for hard squares [Comp. Methods in Science and Tech., 10: 235-255, 2004], and a nonperiodic degenerate solid phase for hard-disk dimers [Phys. Rev. Lett., 66: 3168-3171, 1991]. In this work, we study a system of hard rectangles of aspect ratio two, i.e., hard-square dimers (or dominos), and demonstrate that it exhibits a solid phase with both of these unusual properties. The solid shows tetratic, but not nematic, order, and it is nonperiodic having the structure of a random tiling of the square lattice with dominos. We obtain similar results with both a classical Monte Carlo method using true rectangles and a novel molecular dynamics algorithm employing rectangles with rounded corners. It is remarkable that such simple convex two-dimensional shapes can produce such rich phase behavior. Although we have not performed exact free-energy calculations, we expect that the random domino tiling is thermodynamically stabilized by its degeneracy entropy, well-known to be $1.79k_{B}$ per particle from previous studies of the dimer problem on the square lattice. Our observations are consistent with a KTHNY two-stage phase transition scenario with two continuous phase transitions, the first from isotropic to tetratic liquid, and the second from tetratic liquid to solid.

cond-mat.stat-mech↗

arXiv2005

Comment on "Jamming at zero temperature and zero applied stress: The epitome of disorder"

O'Hern, Silbert, Liu and Nagel [Phys. Rev. E. 68, 011306 (2003)] (OSLN) claim that a special point $J$ of a "jamming phase diagram" (in density, temperature, stress space) is related to random close packing of hard spheres, and that it represents, for their suggested definitions of jammed and random, the recently introduced maximally random jammed state. We point out several difficulties with their definitions and question some of their claims. Furthermore, we discuss the connections between their algorithm and other hard-sphere packing algorithms in the literature.

cond-mat.mtrl-sci↗

arXiv2005

Unexpected Density Fluctuations in Jammed Disordered Sphere Packings

We computationally study jammed disordered hard-sphere packings as large as a million particles. We show that the packings are saturated and hyperuniform, i.e., that local density fluctuations grow only as a logarithmically-augmented surface area rather than the volume of the window. The structure factor shows an unusual non-analytic linear dependence near the origin, $S(k)\sim|k|$. In addition to exponentially damped oscillations seen in liquids, this implies a weak power-law tail in the total correlation function, $h(r)\sim-r^{-4}$, and a long-ranged direct correlation function.

cond-mat.stat-mech↗

arXiv2004

Pair Correlation Function Characteristics of Nearly Jammed Disordered and Ordered Hard-Sphere Packings

We study the approach to jamming in hard-sphere packings, and, in particular, the pair correlation function $g_{2}(r)$ around contact, both theoretically and computationally. Our computational data unambiguously separates the narrowing delta-function contribution to $g_{2}$ due to emerging interparticle contacts from the background contribution due to near contacts. The data also shows with unprecedented accuracy that disordered hard-sphere packings are strictly isostatic, i.e., the number of exact contacts in the jamming limit is exactly equal to the number of degrees of freedom, once rattlers are removed. For such isostatic packings, we derive a theoretical connection between the probability distribution of interparticle forces $P_{f}(f)$, which we measure computationally, and the contact contribution to $g_{2}$. We verify this relation for computationally-generated isostatic packings that are representative of the maximally jammed random state. We clearly observe a maximum in $P_{f}$ and a nonzero probability of zero force. We computationally observe an unusual power-law divergence in the near-contact contribution to $g_{2}$, persistent even in the jamming limit, with an exponent of -0.4. We also present the first computational data on the contact-contribution to $g_{2}$ for vacancy-diluted FCC crystal packings and also investigate partially crystallized packings along the transition from maximally disordered to fully ordered packings. Unlike previous studies, we find that ordering has a significant impact on the shape of $P_{f}$ for small forces.

cond-mat.stat-mech↗

arXiv2002

Controlling the Short-Range Order and Packing Densities of Many-Particle Systems

Questions surrounding the spatial disposition of particles in various condensed-matter systems continue to pose many theoretical challenges. This paper explores the geometric availability of amorphous many-particle configurations that conform to a given pair correlation function g(r). Such a study is required to observe the basic constraints of non-negativity for g(r) as well as for its structure factor S(k). The hard sphere case receives special attention, to help identify what qualitative features play significant roles in determining upper limits to maximum amorphous packing densities. For that purpose, a five-parameter test family of g's has been considered, which incorporates the known features of core exclusion, contact pairs, and damped oscillatory short-range order beyond contact. Numerical optimization over this five-parameter set produces a maximum-packing value for the fraction of covered volume, and about 5.8 for the mean contact number, both of which are within the range of previous experimental and simulational packing results. However, the corresponding maximum-density g(r) and S(k) display some unexpected characteristics. A byproduct of our investigation is a lower bound on the maximum density for random sphere packings in $d$ dimensions, which is sharper than a well-known lower bound for regular lattice packings for d >= 3.

cond-mat.stat-mech↗

arXiv2001

Multiplicity of Generation, Selection, and Classification Procedures for Jammed Hard-Particle Packings

Hard-particle packings have served as useful starting points to study the structure of diverse systems such as liquids, living cells, granular media, glasses, and amorphous solids. Howard Reiss has played a major role in helping to illuminate our understanding of hard-particle systems, which still offer scientists many interesting conundrums. Jammed configurations of hard particles are of great fundamental and practical interest. What one precisely means by a "jammed" configuration is quite subtle and considerable ambiguity remains in the literature on this question. We will show that there is a multiplicity of generation, selection, and classification procedures for jammed configurations of identical d-dimensional spheres. We categorize common ordered lattices according to our definitions and discuss implications for random disk and sphere packings. We also show how the concept of rigidity percolation (which has been used to understand the mechanical properties of network glasses) can be generalized to further characterize hard-sphere packings.

cond-mat.stat-mech↗