Chase E. Zachary

Statistical properties of determinantal point processes in high-dimensional Euclidean spaces

The goal of this paper is to quantitatively describe some statistical properties of higher-dimensional determinantal point processes with a primary focus on the nearest-neighbor distribution functions. Toward this end, we express these functions as determinants of NxN matrices and then extrapolate to N-->infinity . This formulation allows for a quick and accurate numerical evaluation of these quantities for point processes in Euclidean spaces of dimension d . We also implement an algorithm due to Hough for generating configurations of determinantal point processes in arbitrary Euclidean spaces, and we utilize this algorithm in conjunction with the aforementioned numerical results to characterize the statistical properties of what we call the Fermi-sphere point process for d=1-4 . This homogeneous, isotropic determinantal point process, discussed also in a companion paper [S. Torquato, A. Scardicchio, and C. E. Zachary, J. Stat. Mech.: Theory Exp. (2008) P11019.], is the high-dimensional generalization of the distribution of eigenvalues on the unit circle of a random matrix from the circular unitary ensemble. In addition to the nearest-neighbor probability distribution, we are able to calculate Voronoi cells and nearest-neighbor extrema statistics for the Fermi-sphere point process, and we discuss these properties as the dimension d is varied. The results in this paper accompany and complement analytical properties of higher-dimensional determinantal point processes developed in a prior paper.

Hyperuniform long-range correlations are a signature of disordered jammed hard-particle packings.

We show that quasi-long-range (QLR) pair correlations that decay asymptotically with scaling r(-(d+1)) in d-dimensional Euclidean space R(d), trademarks of certain quantum systems and cosmological structures, are a universal signature of maximally random jammed (MRJ) hard-particle packings. We introduce a novel hyperuniformity descriptor in MRJ packings by studying local-volume-fraction fluctuations and show that infinite-wavelength fluctuations vanish even for packings with size and shape distributions. Special void statistics induce hyperuniformity and QLR pair correlations.

Gaussian core model phase diagram and pair correlations in high Euclidean dimensions

The physical properties of a classical many-particle system with interactions given by a repulsive Gaussian pair potential are extended to arbitrarily high Euclidean dimensions. The goals of this paper are to characterize the behavior of the pair correlation function g(2) in various density regimes and to understand the phase properties of the Gaussian core model (GCM) as parametrized by dimension d. To this end, we explore the fluid (dilute and dense) and crystalline solid phases. For the dilute regime of the fluid phase, a cluster expansion of g(2) in reciprocal temperature beta is presented, the coefficients of which may be evaluated analytically due to the nature of the Gaussian potential. We present preliminary results concerning the convergence properties of this expansion. The analytical cluster expansion is related to numerical approximations for g(2) in the dense fluid regime by utilizing hypernetted chain, Percus-Yevick, and mean-field closures to the Ornstein-Zernike equation. Based on the results of these comparisons, we provide evidence in support of a decorrelation principle for the GCM in high Euclidean dimensions. In the solid phase, we consider the behavior of the freezing temperature T(f)(rho) in the limit rho-->+infinity and show T(f)(rho)-->0 in this limit for any d via a collective coordinate argument. Duality relations with respect to the energies of a lattice and its dual are then discussed, and these relations aid in the Maxwell double-tangent construction of phase coexistence regions between dual lattices based on lattice summation energies. The results from this analysis are used to draw conclusions about the ground-state structures of the GCM for a given dimension.

Hyperuniformity, quasi-long-range correlations, and void-space constraints in maximally random jammed particle packings. I. Polydisperse spheres.

Hyperuniform many-particle distributions possess a local number variance that grows more slowly than the volume of an observation window, implying that the local density is effectively homogeneous beyond a few characteristic length scales. Previous work on maximally random strictly jammed sphere packings in three dimensions has shown that these systems are hyperuniform and possess unusual quasi-long-range pair correlations decaying as r(-4), resulting in anomalous logarithmic growth in the number variance. However, recent work on maximally random jammed sphere packings with a size distribution has suggested that such quasi-long-range correlations and hyperuniformity are not universal among jammed hard-particle systems. In this paper, we show that such systems are indeed hyperuniform with signature quasi-long-range correlations by characterizing the more general local-volume-fraction fluctuations. We argue that the regularity of the void space induced by the constraints of saturation and strict jamming overcomes the local inhomogeneity of the disk centers to induce hyperuniformity in the medium with a linear small-wave-number nonanalytic behavior in the spectral density, resulting in quasi-long-range spatial correlations scaling with r(-(d+1)) in d Euclidean space dimensions. A numerical and analytical analysis of the pore-size distribution for a binary maximally random jammed system in addition to a local characterization of the n-particle loops governing the void space surrounding the inclusions is presented in support of our argument. This paper is the first part of a series of two papers considering the relationships among hyperuniformity, jamming, and regularity of the void space in hard-particle packings.

Hyperuniformity, quasi-long-range correlations, and void-space constraints in maximally random jammed particle packings. II. Anisotropy in particle shape.

We extend the results from the first part of this series of two papers by examining hyperuniformity in heterogeneous media composed of impenetrable anisotropic inclusions. Specifically, we consider maximally random jammed (MRJ) packings of hard ellipses and superdisks and show that these systems both possess vanishing infinite-wavelength local-volume-fraction fluctuations and quasi-long-range pair correlations scaling as r(-(d+1)) in d Euclidean dimensions. Our results suggest a strong generalization of a conjecture by Torquato and Stillinger [Phys. Rev. E 68, 041113 (2003)], namely, that all strictly jammed saturated packings of hard particles, including those with size and shape distributions, are hyperuniform with signature quasi-long-range correlations. We show that our arguments concerning the constrained distribution of the void space in MRJ packings directly extend to hard-ellipse and superdisk packings, thereby providing a direct structural explanation for the appearance of hyperuniformity and quasi-long-range correlations in these systems. Additionally, we examine general heterogeneous media with anisotropic inclusions and show unexpectedly that one can decorate a periodic point pattern to obtain a hard-particle system that is not hyperuniform with respect to local-volume-fraction fluctuations. This apparent discrepancy can also be rationalized by appealing to the irregular distribution of the void space arising from the anisotropic shapes of the particles. Our work suggests the intriguing possibility that the MRJ states of hard particles share certain universal features independent of the local properties of the packings, including the packing fraction and average contact number per particle.

Duality relations for the classical ground states of soft-matter systems

Bounded interactions are particularly important in soft-matter systems, such as colloids, microemulsions, and polymers. In this paper, we extend the results of a recent letter [S. Torquato and F. H. Stillinger, Phys. Rev. Lett., 2008, 100, 020602] on duality relations for ground states of pair interactions to include three-body and higher-order functions. Our novel and general relations link the energy of configurations associated with a real-space potential to the corresponding energy of the dual (Fourier-transformed) potential and can be applied to ordered and disordered classical ground states. We use the duality relations to demonstrate how information about the classical ground states of short-ranged potentials can be used to draw new conclusions about the ground states of long-ranged potentials and vice versa. The duality relations also lead to bounds on the T = 0 system energies in density intervals of phase coexistence. Additionally, we identify classes of “self-similar“ potentials, for which one can rigorously relate low- and high-density ground-state energies. We analyze the ground state configurations and thermodynamic properties of a one-dimensional system expected to exhibit an infinite number of structural phase transitions and comment on the known ground states of purely repulsive monotonic potentials in the context of our duality relations.