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 $\phi_{MRJ} \simeq 0.46$, 0.31 and 0.20, respectively. To a good approximation, the MRJ density obeys the scaling form $\phi_{MRJ}= c_1/2^d+(c_2 d)/2^d$, 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 $\phi_{MRJ} \simeq 0.64$), 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, $\phi_F \simeq 0.32$ and $\phi_M \simeq 0.39$, respectively, for $d=4$, and $\phi_F \simeq 0.19$ and $\phi_M \simeq 0.24$, respectively, for $d=5$.Comment: 28 pages, 9 figures. To appear in Physical Review

Black Hole-Neutron Star Binaries in General Relativity: Quasiequilibrium Formulation

We present a new numerical method for the construction of quasiequilibrium models of black hole-neutron star binaries. We solve the constraint equations of general relativity, decomposed in the conformal thin-sandwich formalism, together with the Euler equation for the neutron star matter. We take the system to be stationary in a corotating frame and thereby assume the presence of a helical Killing vector. We solve these coupled equations in the background metric of a Kerr-Schild black hole, which accounts for the neutron star's black hole companion. In this paper we adopt a polytropic equation of state for the neutron star matter and assume large black hole--to--neutron star mass ratios. These simplifications allow us to focus on the construction of quasiequilibrium neutron star models in the presence of strong-field, black hole companions. We summarize the results of several code tests, compare with Newtonian models, and locate the onset of tidal disruption in a fully relativistic framework.Comment: 17 pages, 7 figures; added discussion, tables; PRD in pres

Gauge conditions for binary black hole puncture data based on an approximate helical Killing vector

We show that puncture data for quasicircular binary black hole orbits allow a special gauge choice that realizes some of the necessary conditions for the existence of an approximate helical Killing vector field. Introducing free parameters for the lapse at the punctures we can satisfy the condition that the Komar and ADM mass agree at spatial infinity. Several other conditions for an approximate Killing vector are then automatically satisfied, and the 3-metric evolves on a timescale smaller than the orbital timescale. The time derivative of the extrinsic curvature however remains significant. Nevertheless, quasicircular puncture data are not as far from possessing a helical Killing vector as one might have expected.Comment: 11 pages, 6 figures, 2 table

Conformal-thin-sandwich initial data for a single boosted or spinning black hole puncture

Sequences of initial-data sets representing binary black holes in quasi-circular orbits have been used to calculate what may be interpreted as the innermost stable circular orbit. These sequences have been computed with two approaches. One method is based on the traditional conformal-transverse-traceless decomposition and locates quasi-circular orbits from the turning points in an effective potential. The second method uses a conformal-thin-sandwich decomposition and determines quasi-circular orbits by requiring the existence of an approximate helical Killing vector. Although the parameters defining the innermost stable circular orbit obtained from these two methods differ significantly, both approaches yield approximately the same initial data, as the separation of the binary system increases. To help understanding this agreement between data sets, we consider the case of initial data representing a single boosted or spinning black hole puncture of the Bowen-York type and show that the conformal-transverse-traceless and conformal-thin-sandwich methods yield identical data, both satisfying the conditions for the existence of an approximate Killing vector.Comment: 13 pages, 2 figure

Can a combination of the conformal thin-sandwich and puncture methods yield binary black hole solutions in quasi-equilibrium?

We consider combining two important methods for constructing quasi-equilibrium initial data for binary black holes: the conformal thin-sandwich formalism and the puncture method. The former seeks to enforce stationarity in the conformal three-metric and the latter attempts to avoid internal boundaries, like minimal surfaces or apparent horizons. We show that these two methods make partially conflicting requirements on the boundary conditions that determine the time slices. In particular, it does not seem possible to construct slices that are quasi-stationary and avoid physical singularities and simultaneously are connected by an everywhere positive lapse function, a condition which must obtain if internal boundaries are to be avoided. Some relaxation of these conflicting requirements may yield a soluble system, but some of the advantages that were sought in combining these approaches will be lost.Comment: 8 pages, LaTeX2e, 2 postscript figure

Photon Management in Two-Dimensional Disordered Media

Elaborating reliable and versatile strategies for efficient light coupling between free space and thin films is of crucial importance for new technologies in energy efficiency. Nanostructured materials have opened unprecedented opportunities for light management, notably in thin-film solar cells. Efficient coherent light trapping has been accomplished through the careful design of plasmonic nanoparticles and gratings, resonant dielectric particles and photonic crystals. Alternative approaches have used randomly-textured surfaces as strong light diffusers to benefit from their broadband and wide-angle properties. Here, we propose a new strategy for photon management in thin films that combines both advantages of an efficient trapping due to coherent optical effects and broadband/wide-angle properties due to disorder. Our approach consists in the excitation of electromagnetic modes formed by multiple light scattering and wave interference in two-dimensional random media. We show, by numerical calculations, that the spectral and angular responses of thin films containing disordered photonic patterns are intimately related to the in-plane light transport process and can be tuned through structural correlations. Our findings, which are applicable to all waves, are particularly suited for improving the absorption efficiency of thin-film solar cells and can provide a novel approach for high-extraction efficiency light-emitting diodes

Chemotactic response and adaptation dynamics in Escherichia coli

Adaptation of the chemotaxis sensory pathway of the bacterium Escherichia coli is integral for detecting chemicals over a wide range of background concentrations, ultimately allowing cells to swim towards sources of attractant and away from repellents. Its biochemical mechanism based on methylation and demethylation of chemoreceptors has long been known. Despite the importance of adaptation for cell memory and behavior, the dynamics of adaptation are difficult to reconcile with current models of precise adaptation. Here, we follow time courses of signaling in response to concentration step changes of attractant using in vivo fluorescence resonance energy transfer measurements. Specifically, we use a condensed representation of adaptation time courses for efficient evaluation of different adaptation models. To quantitatively explain the data, we finally develop a dynamic model for signaling and adaptation based on the attractant flow in the experiment, signaling by cooperative receptor complexes, and multiple layers of feedback regulation for adaptation. We experimentally confirm the predicted effects of changing the enzyme-expression level and bypassing the negative feedback for demethylation. Our data analysis suggests significant imprecision in adaptation for large additions. Furthermore, our model predicts highly regulated, ultrafast adaptation in response to removal of attractant, which may be useful for fast reorientation of the cell and noise reduction in adaptation.Comment: accepted for publication in PLoS Computational Biology; manuscript (19 pages, 5 figures) and supplementary information; added additional clarification on alternative adaptation models in supplementary informatio

Overview of mathematical approaches used to model bacterial chemotaxis I: the single cell

Mathematical modeling of bacterial chemotaxis systems has been influential and insightful in helping to understand experimental observations. We provide here a comprehensive overview of the range of mathematical approaches used for modeling, within a single bacterium, chemotactic processes caused by changes to external gradients in its environment. Specific areas of the bacterial system which have been studied and modeled are discussed in detail, including the modeling of adaptation in response to attractant gradients, the intracellular phosphorylation cascade, membrane receptor clustering, and spatial modeling of intracellular protein signal transduction. The importance of producing robust models that address adaptation, gain, and sensitivity are also discussed. This review highlights that while mathematical modeling has aided in understanding bacterial chemotaxis on the individual cell scale and guiding experimental design, no single model succeeds in robustly describing all of the basic elements of the cell. We conclude by discussing the importance of this and the future of modeling in this area

Structural transitions in granular packs: statistical mechanics and statistical geometry investigations

We investigate equal spheres packings generated from several experiments and from a large number of different numerical simulations. The structural organization of these disordered packings is studied in terms of the network of common neighbours. This geometrical analysis reveals sharp changes in the network's clustering occurring at the packing fractions (fraction of volume occupied by the spheres respect to the total volume, $\rho$) corresponding to the so called Random Loose Packing limit (RLP, $\rho \sim 0.555$) and Random Close Packing limit (RCP, $\rho \sim 0.645$). At these packing fractions we also observe abrupt changes in the fluctuations of the portion of free volume around each sphere. We analyze such fluctuations by means of a statistical mechanics approach and we show that these anomalies are associated to sharp variations in a generalized thermodynamical variable which is the analogous for these a-thermal systems to the specific heat in thermal systems.Comment: 7 pages, 6 figure

Chemotaxis of Cell Populations through Confined Spaces at Single-Cell Resolution

Cell migration is crucial for both physiological and pathological processes. Current in vitro cell motility assays suffer from various drawbacks, including insufficient temporal and/or optical resolution, or the failure to include a controlled chemotactic stimulus. Here, we address these limitations with a migration chamber that utilizes a self-sustaining chemotactic gradient to induce locomotion through confined environments that emulate physiological settings. Dynamic real-time analysis of both population-scale and single-cell movement are achieved at high resolution. Interior surfaces can be functionalized through adsorption of extracellular matrix components, and pharmacological agents can be administered to cells directly, or indirectly through the chemotactic reservoir. Direct comparison of multiple cell types can be achieved in a single enclosed system to compare inherent migratory potentials. Our novel microfluidic design is therefore a powerful tool for the study of cellular chemotaxis, and is suitable for a wide range of biological and biomedical applications