arXiv subjects

Salvatore Torquato

Publications and source records attributed to Salvatore Torquato.

At least 145 records · Page 8Linked to original sources

arXiv2012

Microstructural Degeneracy associated with a Two-Point Correlation Function and its Information Content

Two-point correlation functions provide crucial yet incomplete characterization of microstructures because different microstructures may have the same correlation function. In an earlier Letter [Phys. Rev. Lett. 108, 080601 (2012)], we addressed the degeneracy question: What is the number of microstructures compatible with a specified correlation function? We computed this degeneracy, i.e., configurational entropy, in the framework of reconstruction methods, which enabled us to map the problem to the determination of ground-state degeneracies. Here, we provide a more comprehensive presentation and additional results. Since the configuration space of a reconstruction problem is a hypercube on which a Hamming distance is defined, we can calculate analytically an energy profile corresponding to the average energy of all microstructures at a given Hamming distance from a ground state. The steepness of this profile is a measure of the roughness of the energy landscape, which can be used as a proxy for ground-state degeneracy. The relationship between roughness metric and ground-state degeneracy is calibrated using a Monte Carlo algorithm for determining the degeneracy of a variety of microstructures, including hard disks and Poisson point processes as well as those with known degeneracies (single disks of various sizes and a particular crystalline microstructure). We show that our results can be expressed in terms of the information content of the two-point correlation functions. From this perspective, the a priori condition for a reconstruction to be accurate is that the information content, expressed in bits, should be comparable to the number of pixels in the unknown microstructure. We provide a formula to calculate the information content of any two-point correlation function, which makes our results broadly applicable to any field in which correlation functions are employed.

cond-mat.stat-mech↗

arXiv2012

Rigidity of spherical codes

A packing of spherical caps on the surface of a sphere (that is, a spherical code) is called rigid or jammed if it is isolated within the space of packings. In other words, aside from applying a global isometry, the packing cannot be deformed. In this paper, we systematically study the rigidity of spherical codes, particularly kissing configurations. One surprise is that the kissing configuration of the Coxeter-Todd lattice is not jammed, despite being locally jammed (each individual cap is held in place if its neighbors are fixed); in this respect, the Coxeter-Todd lattice is analogous to the face-centered cubic lattice in three dimensions. By contrast, we find that many other packings have jammed kissing configurations, including the Barnes-Wall lattice and all of the best kissing configurations known in four through twelve dimensions. Jamming seems to become much less common for large kissing configurations in higher dimensions, and in particular it fails for the best kissing configurations known in 25 through 31 dimensions. Motivated by this phenomenon, we find new kissing configurations in these dimensions, which improve on the records set in 1982 by the laminated lattices.

math.MG↗

arXiv2012

Nonequilibrium static diverging length scales on approaching a prototypical model glassy state

Maximally random jammed states of hard spheres are prototypical glasses. We study the small wavenumber $k$ behavior of the structure factor $S(k)$ of overcompressed million-sphere packings as a function of density up to the jammed state. We find both a precursor to the glassy jammed state evident long before the jamming density is reached and an associated growing length scale, extracted from the volume integral of the direct correlation function $c(r)$, which diverges at the "critical" jammed state. We also define a nonequilibrium index $X$ and use it to demonstrate that the packings studied are intrinsically nonequilibrium in nature well before the critical point is reached.

cond-mat.stat-mech↗

arXiv2012

Quantitative Characterization of the Microstructure and Transport Properties of Biopolymer Networks

Biopolymer networks are of fundamental importance to many biological processes in normal and tumorous tissues. In this paper, we employ the panoply of theoretical and simulation techniques developed for characterizing heterogeneous materials to quantify the microstructure and effective diffusive transport properties (diffusion coefficient $D_e$ and mean survival time $τ$) of collagen type I networks at various collagen concentrations. In particular, we compute the pore-size probability density function $P(δ)$ for the networks and present a variety of analytical estimates of the effective diffusion coefficient $D_e$ for finite-sized diffusing particles. The Hashin-Strikman upper bound on the effective diffusion coefficient $D_e$ and the pore-size lower bound on the mean survival time $τ$ are used as benchmarks to test our analytical approximations and numerical results. Moreover, we generalize the efficient first-passage-time techniques for Brownian-motion simulations in suspensions of spheres to the case of fiber networks and compute the associated effective diffusion coefficient $D_e$ as well as the mean survival time $τ$, which is related to nuclear magnetic resonance (NMR) relaxation times. Specifically, the Torquato approximation provides the most accurate estimates of $D_e$ for all collagen concentrations among all of the analytical approximations we consider. We formulate a universal curve for $τ$ for the networks at different collagen concentrations. We apply rigorous cross-property relations to estimate the effective bulk modulus of collagen networks from a knowledge of the effective diffusion coefficient computed here.

physics.bio-ph↗

arXiv2012

Densest binary sphere packings

The densest binary sphere packings in the alpha-x plane of small to large sphere radius ratio alpha and small sphere relative concentration x have historically been very difficult to determine. Previous research had led to the prediction that these packings were composed of a few known "alloy" phases including, for example, the AlB2 (hexagonal omega), HgBr2, and AuTe2 structures, and to XYn structures composed of close-packed large spheres with small spheres (in a number ratio of n to 1) in the interstices, e.g., the NaCl packing for n = 1. However, utilizing an implementation of the Torquato-Jiao sphere-packing algorithm [S. Torquato and Y. Jiao, Phys. Rev. E 82, 061302 (2010)], we have discovered that many more structures appear in the densest packings. For example, while all previously known densest structures were composed of spheres in small to large number ratios of one to one, two to one, and very recently three to one, we have identified densest structures with number ratios of seven to three and five to two. In a recent work [A. B. Hopkins, Y. Jiao, F. H. Stillinger, and S. Torquato, Phys. Rev. Lett. 107, 125501 (2011)], we summarized these findings. In this work, we present the structures of the densest-known packings and provide details about their characteristics. Our findings demonstrate that a broad array of different densest mechanically stable structures consisting of only two types of components can form without any consideration of attractive or anisotropic interactions. In addition, the novel structures that we have identified may correspond to currently unidentified stable phases of certain binary atomic and molecular systems, particularly at high temperatures and pressures.

cond-mat.stat-mech↗

arXiv2012

Density of States for a Specified Correlation Function and the Energy Landscape

The degeneracy of two-phase disordered microstructures consistent with a specified correlation function is analyzed by mapping it to a ground-state degeneracy. We determine for the first time the associated density of states via a Monte Carlo algorithm. Our results are described in terms of the roughness of the energy landscape, defined on a hypercubic configuration space. The use of a Hamming distance in this space enables us to define a roughness metric, which is calculated from the correlation function alone and related quantitatively to the structural degeneracy. This relation is validated for a wide variety of disordered systems.

cond-mat.stat-mech↗

arXiv2012

Emergent Behaviors from A Cellular Automaton Model for Invasive Tumor Growth in Heterogeneous Microenvironments

Understanding tumor invasion and metastasis is of crucial importance for both fundamental cancer research and clinical practice. In vitro experiments have established that the invasive growth of malignant tumors is characterized by the dendritic invasive branches composed of chains of tumor cells emanating from the primary tumor mass. The preponderance of previous tumor simulations focused on non-invasive (or proliferative) growth. The formation of the invasive cell chains and their interactions with the primary tumor mass and host microenvironment are not well understood. Here, we present a novel cellular automaton (CA) model that enables one to efficiently simulate invasive tumor growth in a heterogeneous host microenvironment. By taking into account a variety of microscopic-scale tumor-host interactions, including the short-range mechanical interactions between tumor cells and tumor stroma, degradation of extracellular matrix by the invasive cells and oxygen/nutrient gradient driven cell motions, our CA model predicts a rich spectrum of growth dynamics and emergent behaviors of invasive tumors. Besides robustly reproducing the salient features of dendritic invasive growth, such as least resistance and intrabranch homotype attraction, we also predict nontrivial coupling of the growth dynamics of the primary tumor mass and the invasive cells. In addition, we show that the properties of the host microenvironment can significantly affect tumor morphology and growth dynamics, emphasizing the importance of understanding the tumor-host interaction. The capability of our CA model suggests that well-developed in silico tools could eventually be utilized in clinical situations to predict neoplastic progression and propose individualized optimal treatment strategies.

physics.bio-ph↗

arXiv2012

Improved reconstructions of random media using dilation and erosion processes

By using the most sensitive two-point correlation functions introduced to date, we reconstruct the microstructures of two-phase random media with heretofore unattained accuracy. Such media arise in a host of contexts, including porous and composite media, ecological structures, biological media, and astrophysical structures. The aforementioned correlation functions are special cases of the so-called {\it canonical} $n$-point correlation function $H_n$ and generalize the ones that have been recently employed to advance our ability to reconstruct complex microstructures [Y. Jiao, F. H. Stillinger, and S. Torquato, Proc. Nat. Acad. Sci. {\bf 106}, 17634 (2009)]. The use of these generalized correlation functions is tantamount to dilating or eroding a reference phase of the target medium and incorporating the additional topological information of the modified media, thereby providing more accurate reconstructions of percolating, filamentary, and other topologically complex microstructures. We apply our methods to a multiply-connected "donut" medium and a dilute distribution of "cracks" (a set of essentially zero measure) demonstrating improved accuracy in both cases with implications for higher-dimensional and biconnected two-phase systems. The high information content of the generalized two-point correlation functions suggests that it would be profitable to explore their use to characterize the structural and physical properties of not only random media, but also molecular systems, including structural glasses.

cond-mat.mtrl-sci↗

arXiv2011

High-dimensional generalizations of the kagome and diamond crystals and the decorrelation principle for periodic sphere packings

In this paper, we introduce constructions of the high-dimensional generalizations of the kagome and diamond crystals. The two-dimensional kagome crystal and its three-dimensional counterpart, the pyrochlore crystal, have been extensively studied in the context of geometric frustration in antiferromagnetic materials. Similarly, the polymorphs of elemental carbon include the diamond crystal and the corresponding two-dimensional honeycomb structure, adopted by graphene. The kagome crystal in d Euclidean dimensions consists of vertex-sharing d-dimensional simplices in which all of the points are topologically equivalent. The d-dimensional generalization of the diamond crystal can then be obtained from the centroids of each of the simplices, and we show that this natural construction of the diamond crystal is distinct from the D_d^+ family of crystals for all dimensions d\neq 3. We analyze the structural properties of these high-dimensional crystals, including the packing densities, coordination numbers, void exclusion probability functions, covering radii, and quantizer errors. Our results demonstrate that the so-called decorrelation principle, which formally states that unconstrained correlations vanish in asymptotically high dimensions, remarkably applies to the case of periodic point patterns with inherent long-range order. We show that the decorrelation principle is already exhibited in periodic crystals in low dimensions via a "smoothed" pair correlation function obtained by convolution with a Gaussian kernel. These observations support the universality of the decorrelation principle for any point pattern in high dimensions, whether disordered or not. This universal property in turn suggests that the best conjectural lower bound on the maximal sphere-packing density in high Euclidean dimensions derived by Torquato and Stillinger [Expt. Math. 15, 307 (2006)] is in fact optimal.

cond-mat.stat-mech↗

arXiv2011

Phase diagram and structural diversity of the densest binary sphere packings

The densest binary sphere packings have historically been very difficult to determine. The only rigorously known packings in the alpha-x plane of sphere radius ratio alpha and relative concentration x are at the Kepler limit alpha = 1, where packings are monodisperse. Utilizing an implementation of the Torquato-Jiao sphere-packing algorithm [S. Torquato and Y. Jiao, Phys. Rev. E 82, 061302 (2010)], we present the most comprehensive determination to date of the phase diagram in (alpha,x) for the densest binary sphere packings. Unexpectedly, we find many distinct new densest packings.

cond-mat.stat-mech↗

arXiv2011

Inherent Structures for Soft Long-Range Interactions in Two-Dimensional Many-Particle Systems

We generate inherent structures, local potential-energy minima, of the "$k$-space overlap potential" in two-dimensional many-particle systems using a cooling and quenching simulation technique. The ground states associated with the $k$-space overlap potential are stealthy ({\it i.e.,} completely suppress single scattering of radiation for a range of wavelengths) and hyperuniform ({\it i.e.,} infinite wavelength density fluctuations vanish). However, we show via quantitative metrics that the inherent structures exhibit a range of stealthiness and hyperuniformity depending on the fraction of degrees of freedom that are constrained. Inherent structures in two dimensions typically contain five-particle rings, wavy grain boundaries, and vacancy-interstitial defects. The structural and thermodynamic properties of inherent structures are relatively insensitive to the temperature from which they are sampled, signifying that the energy landscape is relatively flat and devoid of deep wells. Using the nudged-elastic-band algorithm, we construct paths from ground-state configurations to inherent structures and identify the transition points between them. In addition, we use point patterns generated from a random sequential addition (RSA) of hard disks, which are nearly stealthy, and examine the particle rearrangements necessary to make the configurations absolutely stealthy. We introduce a configurational proximity metric to show that only small local, but collective, particle rearrangements are needed to drive initial RSA configurations to stealthy disordered ground states. These results lead to a more complete understanding of the unusual behaviors exhibited by the family of "collective-coordinate" potentials to which the $k$-space overlap potential belongs.

cond-mat.stat-mech↗

arXiv2011

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, 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-wavenumber nonanalytic behavior in the spectral density, resulting in quasi-long-range spatial correlations. A numerical and analytical analysis of the pore-size distribution for a binary MRJ 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.

cond-mat.stat-mech↗

arXiv2011

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 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. 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 for the first time 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.

cond-mat.stat-mech↗

arXiv2011

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 $\mathbb{R}^d$, 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.

cond-mat.stat-mech↗

arXiv2010

Anomalous local coordination, density fluctuations, and void statistics in disordered hyperuniform many-particle ground states

We provide numerical constructions of one-dimensional hyperuniform many-particle distributions that exhibit unusual clustering and asymptotic local number density fluctuations growing more slowly than the volume of an observation window but faster than the surface area. By targeting a specified form of the structure factor at small wavenumbers using collective density variables, we are able to tailor the form of asymptotic local density fluctuations while simultaneously measuring the effect of imposing weak and strong constraints on the available degrees of freedom within the system. Even in one dimension, the long-range effective interactions induce clustering and nontrivial phase transitions in the resulting ground-state configurations. We provide an analytical connection between the fraction of contrained degrees of freedom within the system and the disorder-order phase transition for a class of target structure factors by examining the realizability of the constrained contribution to the pair correlation function. Our results explicitly demonstrate that disordered hyperuniform many-particle ground states, and therefore also point distributions, with substantial clustering can be constructed. We directly relate the local coordination structure of our point patterns to the distribution of the void space external to the particles, and we provide a scaling argument for the configurational entropy of the systems when only a small fraction of the degrees of freedom are constrained. By emphasizing the intimate connection between geometrical constraints on the particle distribution and structural regularity, our work has direct implications for higher-dimensional systems, including an understanding of the appearance of hyperuniformity and quasi-long-range pair correlations in maximally random strictly jammed packings of hard spheres.

cond-mat.stat-mech↗

arXiv2010

Effects of Random Link Removal on the Photonic Band Gaps of Honeycomb Networks

We explore the effects of random link removal on the photonic band gaps of honeycomb networks. Missing or incomplete links are expected to be common in practical realizations of this class of connected network structures due to unavoidable flaws in the fabrication process. We focus on the collapse of the photonic band gap due to the defects induced by the link removal. We show that the photonic band gap is quite robust against this type of random decimation and survives even when almost 58% of the network links are removed.

physics.optics↗

arXiv2010

Toward an Ising Model of Cancer and Beyond

Theoretical and computational tools that can be used in the clinic to predict neoplastic progression and propose individualized optimal treatment strategies to control cancer growth is desired. To develop such a predictive model, one must account for the complex mechanisms involved in tumor growth. Here we review resarch work that we have done toward the development of an "Ising model" of cancer. The review begins with a description of a minimalist four-dimensional (three in space and one in time) cellular automaton (CA) model of cancer in which healthy cells transition between states (proliferative, hypoxic, and necrotic) according to simple local rules and their present states, which can viewed as a stripped-down Ising model of cancer. This model is applied to model the growth of glioblastoma multiforme, the most malignant of brain cancers. This is followed by a discussion of the extension of the model to study the effect on the tumor dynamics and geometry of a mutated subpopulation. A discussion of how tumor growth is affected by chemotherapeutic treatment is then described. How angiogenesis as well as the heterogeneous and confined environment in which a tumor grows is incorporated in the CA model is discussed. The characterization of the level of organization of the invasive network around a solid tumor using spanning trees is subsequently described. Then, we describe open problems and future promising avenues for future research, including the need to develop better molecular-based models that incorporate the true heterogeneous environment over wide range of length and time scales (via imaging data), cell motility, oncogenes, tumor suppressor genes and cell-cell communication. The need to bring to bear the powerful machinery of the theory of heterogeneous media to better understand the behavior of cancer in its microenvironment is presented.

q-bio.CB↗

arXiv2010

Optimized Monotonic Convex Pair Potentials Stabilize Low-Coordinated Crystals

We have previously used inverse statistical-mechanical methods to optimize isotropic pair interactions with multiple extrema to yield low-coordinated crystal classical ground states (e.g., honeycomb and diamond structures) in d-dimensional Euclidean space R^d. Here we demonstrate the counterintuitive result that no extrema are required to produce such low-coordinated classical ground states. Specifically, we show that monotonic convex pair potentials can be optimized to yield classical ground states that are the square and honeycomb crystals in R^2 over a non-zero number density range. Such interactions may be feasible to achieve experimentally using colloids and polymers.

cond-mat.stat-mech↗