Avian photoreceptor patterns represent a disordered hyperuniform solution to a multiscale packing problem

Hyperuniformity and density fluctuations at a rigidity transition in a model of biological tissues

The suppression of density fluctuations at different length scales is the hallmark of hyperuniformity. Here, we explore the presence of this hidden order in a manybody interacting model of biological tissue, known to exhibit a transition, or sharp crossover, from a solid to a fluid like phase. We show that the density fluctuations in the rigid phase are only suppressed up to a finite lengthscale. This length scale monotonically increases and grows rapidly as we approach the fluid phase reminiscent to divergent behavior at a critical point, such that the system is effectively hyperuniform in the fluid phase. Furthermore, complementary behavior of the structure factor across the critical point also indicates that hyperuniformity found in the fluid phase is stealthy.

Disordered hyperuniformity in two-dimensional amorphous silica

Disordered hyperuniformity (DHU) is a recently proposed new state of matter, which has been observed in a variety of classical and quantum many-body systems. DHU systems are characterized by vanishing infinite-wavelength normalized density fluctuations and are endowed with unique novel physical properties. Here, we report the discovery of disordered hyperuniformity in atomic-scale two-dimensional materials, i.e., amorphous silica composed of a single layer of atoms, based on spectral-density analysis of high-resolution transmission electron microscopy images. Moreover, we show via large-scale density functional theory calculations that DHU leads to almost complete closure of the electronic bandgap compared to the crystalline counterpart, making the material effectively a metal. This is in contrast to the conventional wisdom that disorder generally diminishes electronic transport and is due to the unique electron wave localization induced by the topological defects in the DHU state.

Predicting maximally random jammed packing density of non-spherical hard particles via analytical continuation of fluid equation of state

We developed a formalism for accurately predicting the density of MRJ packing state of a wide spectrum of congruent non-spherical hard particles in 3D via analytical fluid EOS.

Hyperuniformity on spherical surfaces

We study and characterize local density fluctuations of ordered and disordered hyperuniform point distributions on spherical surfaces. In spite of the extensive literature on disordered hyperuniform systems in Euclidean geometries, to date few works have dealt with the problem of hyperuniformity in curved spaces. Indeed, some systems that display disordered hyperuniformity, like the spatial distribution of photoreceptors in avian retina, actually occur on curved surfaces. Here we will focus on the local particle number variance and its dependence on the size of the sampling window (which we take to be a spherical cap) for regular and uniform point distributions, as well as for equilibrium configurations of fluid particles interacting through Lennard-Jones, dipole-dipole, and charge-charge potentials. We show that the scaling of the local number variance as a function of the window size enables one to characterize hyperuniform and nonhyperuniform point patterns also on spherical surfaces.

Quantum jamming transition to a correlated electron glass in 1T-TaS2

Distinct many-body states may be created under non-equilibrium conditions through different ordering paths, even when their constituents are subjected to the same fundamental interactions. The phase-transition mechanism to such states remains poorly understood. Here, we show that controlled optical or electromagnetic perturbations can lead to an amorphous metastable state of strongly correlated electrons in a quasi-two-dimensional dichalcogenide. Scanning tunnelling microscopy reveals a hyperuniform pattern of localized charges, whereas multitip surface nanoscale conductivity measurements and tunnelling spectroscopy show an electronically gapless conducting state that is different from conventional Coulomb glasses and many-body localized systems. The state is stable up to room temperature and shows no signs of either local charge order or phase separation. The mechanism for its formation is attributed to a dynamical localization of electrons through mutual interactions. Theoretical calculations confirm the correlations between localized charges to be crucial for the state's unusual stability.

The Retinal Basis of Vertebrate Color Vision

The jawless fish that were ancestral to all living vertebrates had four spectral cone types that were probably served by chromatic-opponent retinal circuits. Subsequent evolution of photoreceptor spectral sensitivities is documented for many vertebrate lineages, giving insight into the ecological adaptation of color vision. Beyond the photoreceptors, retinal color processing is best understood in mammals, especially the blueON system, which opposes short- against long-wavelength receptor responses. For other vertebrates that often have three or four types of cone pigment, new findings from zebrafish are extending older work on teleost fish and reptiles to reveal rich color circuitry. Here, horizontal cells establish diverse and complex spectral responses even in photoreceptor outputs. Cone-selective connections to bipolar cells then set up color-opponent synaptic layers in the inner retina, which lead to a large variety of color-opponent channels for transmission to the brain via retinal ganglion cells.

Hyperuniform states generated by a critical friction field

Hyperuniform states are an efficient way to fill up space for disordered systems. In these states the particle distribution is disordered at the short scale but becomes increasingly uniform when looked at large scales. Hyperuniformity appears in several systems, in static or quasistatic regimes, as well as close to transitions to absorbing states. Here, we show that a vibrated granular layer, at the critical point of the liquid-to-solid transition, displays dynamic hyperuniformity. Prior to the transition, patches of the solid phase form, with length scales and mean lifetimes that diverge critically at the transition point. When reducing the wave number, density fluctuations encounter increasingly more patches that block their propagation, resulting in a static structure factor that tends to zero for small wave numbers at the critical point, which is a signature of hyperuniformity. A simple model demonstrates that this coupling of a density field to a highly fluctuating scalar friction field gives rise to dynamic hyperuniform states. Finally, we show that the structure factor detects better the emergence of hyperuniformity, compared to the particle number variance.

Disordered hyperuniform obstacles enhance sorting of dynamically chiral microswimmers

Disordered hyperuniformity, a brand new type of arrangement with novel physical properties, provides various practical applications in extensive fields. To highlight the great potential of applying disordered hyperuniformity to active systems, a practical example is reported here by an optimal sorting of dynamically chiral microswimmers in disordered hyperuniform obstacle environments in comparison with regular or disordered ones. This optimal chirality sorting stems from a competition between advantageous microswimmer-obstacle collisions and disadvantageous trapping of microswimmers by obstacles. Based on this mechanism, optimal chirality sorting is also realized by tuning other parameters including the number density of obstacles, the strength of driven force and the noise intensity. Our findings may open a new perspective on both theoretical and experimental investigations for further applications of disordered hyperuniformity in active systems.

Methodology to construct large realizations of perfectly hyperuniform disordered packings

Disordered hyperuniform packings (or dispersions) are unusual amorphous two-phase materials that are endowed with exotic physical properties. Such hyperuniform systems are characterized by an anomalous suppression of volume-fraction fluctuations at infinitely long-wavelengths, compared to ordinary disordered materials. While there has been growing interest in such singular states of amorphous matter, a major obstacle has been an inability to produce large samples that are perfectly hyperuniform due to practical limitations of conventional numerical and experimental methods. To overcome these limitations, we introduce a general theoretical methodology to construct perfectly hyperuniform packings in d-dimensional Euclidean space R^{d}. Specifically, beginning with an initial general tessellation of space by disjoint cells that meets a "bounded-cell" condition, hard particles of general shape are placed inside each cell such that the local-cell particle packing fractions are identical to the global packing fraction. We prove that the constructed packings with a polydispersity in size are perfectly hyperuniform in the infinite-sample-size limit, regardless of particle shapes, positions, and numbers per cell. We use this theoretical formulation to devise an efficient and tunable algorithm to generate extremely large realizations of such packings. We employ two distinct initial tessellations: Voronoi as well as sphere tessellations. Beginning with Voronoi tessellations, we show that our algorithm can remarkably convert extremely large nonhyperuniform packings into hyperuniform ones in R^{2} and R^{3}. Implementing our theoretical methodology on sphere tessellations, we establish the hyperuniformity of the classical Hashin-Shtrikman multiscale coated-spheres structures, which are known to be two-phase media microstructures that possess optimal effective transport and elastic properties. A consequence of our work is a rigorous demonstration that packings that have identical tessellations can either be nonhyperuniform or hyperuniform by simply tuning local characteristics. It is noteworthy that our computationally designed hyperuniform two-phase systems can easily be fabricated via state-of-the-art methods, such as 2D photolithographic and 3D printing technologies. In addition, the tunability of our methodology offers a route for the discovery of novel disordered hyperuniform two-phase materials.

Hyperuniformity of generalized random organization models

Studies of random organization models of monodisperse (i.e., identical) spherical particles have shown that a hyperuniform state is achievable when the system goes through an absorbing phase transition to a critical state. Here we investigate to what extent hyperuniformity is preserved when the model is generalized to particles with a size distribution and/or nonspherical shapes. We begin by examining binary disks in two dimensions and demonstrate that their critical states are hyperuniform as two-phase media, but not hyperuniform nor multihyperuniform as point patterns formed by the particle centroids. We further confirm the generality of our findings by studying particles with a continuous size distribution. Finally, to study the effect of rotational degrees of freedom, we extend our model to noncircular particles, namely, hard rectangles with various aspect ratios, including the hard-needle limit. Although these systems exhibit only short-range orientational order, hyperuniformity is still preserved. Our analysis reveals that the redistribution of the "mass" of the particles rather than the particle centroids is central to this dynamical process. The consideration of the "active volume fraction" of generalized random organization models may help to resolve which universality class they belong to and hence may lead to a deeper theoretical understanding of absorbing-state models. Our results suggest that general particle systems subject to random organization can be a robust way to fabricate a wide class of hyperuniform states of matter by tuning the structures via different particle-size and -shape distributions. This in turn potentially enables the creation of multifunctional hyperuniform materials with desirable optical, transport, and mechanical properties.

Universal hidden order in amorphous cellular geometries

Partitioning space into cells with certain extreme geometrical properties is a central problem in many fields of science and technology. Here we investigate the Quantizer problem, defined as the optimisation of the moment of inertia of Voronoi cells, i.e., similarly-sized ‘sphere-like’ polyhedra that tile space are preferred. We employ Lloyd’s centroidal Voronoi diagram algorithm to solve this problem and find that it converges to disordered states associated with deep local minima. These states are universal in the sense that their structure factors are characterised by a complete independence of a wide class of initial conditions they evolved from. They moreover exhibit an anomalous suppression of long-wavelength density fluctuations and quickly become effectively hyperuniform. Our findings warrant the search for novel amorphous hyperuniform phases and cellular materials with unique physical properties.

Disordered hyperuniformity implies a hidden order on length scales that can be found in various amorphous materials. Klatt et al. analyse the evolution of random point patterns using Llyod’s algorithm and show that they converge to an effectively hyperuniform state regardless of the initial conditions.

Hyperuniformity order metric of Barlow packings

The concept of hyperuniformity has been a useful tool in the study of density fluctuations at large length scales in systems ranging across the natural and mathematical sciences. One can rank a large class of hyperuniform systems by their ability to suppress long-range density fluctuations through the use of a hyperuniformity order metric Λ[over ¯]. We apply this order metric to the Barlow packings, which are the infinitely degenerate densest packings of identical rigid spheres that are distinguished by their stacking geometries and include the commonly known fcc lattice and hcp crystal. The "stealthy stacking" theorem implies that these packings are all stealthy hyperuniform, a strong type of hyperuniformity, which involves the suppression of scattering up to a wave vector K. We describe the geometry of three classes of Barlow packings, two disordered classes and small-period packings. In addition, we compute a lower bound on K for all Barlow packings. We compute Λ[over ¯] for the aforementioned three classes of Barlow packings and find that, to a very good approximation, it is linear in the fraction of fcc-like clusters, taking values between those of least-ordered hcp and most-ordered fcc. This implies that the value of Λ[over ¯] of all Barlow packings is primarily controlled by the local cluster geometry. These results highlight the special nature of anisotropic stacking disorder, which provides impetus for future research on the development of anisotropic order metrics and hyperuniformity properties.

Threshold response to stochasticity in morphogenesis

During development of biological organisms, multiple complex structures are formed. In many instances, these structures need to exhibit a high degree of order to be functional, although many of their constituents are intrinsically stochastic. Hence, it has been suggested that biological robustness ultimately must rely on complex gene regulatory networks and clean-up mechanisms. Here we explore developmental processes that have evolved inherent robustness against stochasticity. In the context of the Drosophila eye disc, multiple optical units, ommatidia, develop into crystal-like patterns. During the larva-to-pupa stage of metamorphosis, the centers of the ommatidia are specified initially through the diffusion of morphogens, followed by the specification of R8 cells. Establishing the R8 cell is crucial in setting up the geometric, and functional, relationships of cells within an ommatidium and among neighboring ommatidia. Here we study an PDE mathematical model of these spatio-temporal processes in the presence of parametric stochasticity, defining and applying measures that quantify order within the resulting spatial patterns. We observe a universal sigmoidal response to increasing transcriptional noise. Ordered patterns persist up to a threshold noise level in the model parameters. In accordance with prior qualitative observations, as the noise is further increased past a threshold point of no return, these ordered patterns rapidly become disordered. Such robustness in development allows for the accumulation of genetic variation without any observable changes in phenotype. We argue that the observed sigmoidal dependence introduces robustness allowing for sizable amounts of genetic variation and transcriptional noise to be tolerated in natural populations without resulting in phenotype variation.

Nonequilibrium strongly hyperuniform fluids of circle active particles with large local density fluctuations

Disordered hyperuniform structures are an exotic state of matter having vanishing long-wavelength density fluctuations similar to perfect crystals but without long-range order. Although its importance in materials science has been brought to the fore in past decades, the rational design of experimentally realizable disordered strongly hyperuniform microstructures remains challenging. Here we find a new type of nonequilibrium fluid with strong hyperuniformity in two-dimensional systems of chiral active particles, where particles perform independent circular motions of the radius R with the same handedness. This new hyperuniform fluid features a special length scale, i.e., the diameter of the circular trajectory of particles, below which large density fluctuations are observed. By developing a dynamic mean-field theory, we show that the large local density fluctuations can be explained as a motility-induced microphase separation, while the Fickian diffusion at large length scales and local center-of-mass-conserved noises are responsible for the global hyperuniformity.

Hidden Hyperuniformity in Soft Polymeric Materials

We investigate the nature of long-range density fluctuations in melts of model "soft" polymers, specifically stars and bottlebrushes, over a wide temperature range by molecular dynamics simulation. The cores of the stars and the backbones of bottlebrush polymers are found to have a hyperuniform distribution; i.e., they exhibit anomalously small density fluctuations over a wide temperature range above the glass transition temperature. The hyperuniformity of these substituent polymer subregions is hidden since the fluid as a whole does not exhibit this property. These findings offer a strategy for the practical design of hyperuniform polymeric materials.

Imaging using hyperuniform sampling with a single-pixel camera

Digital cameras use detector arrays with regular geometry for optical sampling. Though regular arrangement was demonstrated to be optimal for two-dimensional sampling, it causes aliasing at high frequencies exceeding its Nyquist limit. Here, we proposed a randomization procedure to generate 2D hyperuniform patterns that can be used to suppress aliasing in image retrieval. Experiments are performed using a single-pixel camera, where the sampling patterns do not necessarily follow a fixed Cartesian geometry. Results demonstrate that the images reconstructed by hyperuniform patterns have a lower root mean squared error and exhibit less moiré fringes at high frequencies than the images reconstructed by regular square patterns do. Furthermore, the same conclusion can be applied to the production of conventional detector arrays, where manufacturing imperfection could be utilized to suppress frequency aliasing in image retrieval.

A spatial frequency spectral peakedness model predicts discrimination performance of regularity in dot patterns

Subjective assessments of spatial regularity are common in everyday life and also in science, for example in developmental biology. It has recently been shown that regularity is an adaptable visual dimension. It was proposed that regularity is coded via the peakedness of the distribution of neural responses across receptive field size. Here, we test this proposal for jittered square lattices of dots. We examine whether discriminability correlates with a simple peakedness measure across different presentation conditions (dot number, size, and average spacing). Using a filter-rectify-filter model, we determined responses across scale. Consistently, two peaks are present: a lower frequency peak corresponding to the dot spacing of the regular pattern and a higher frequency peak corresponding to the pattern element (dot). We define the "peakedness" of a particular presentation condition as the relative heights of these two peaks for a perfectly regular pattern constructed using the corresponding dot size, number and spacing. We conducted two psychophysical experiments in which observers judged relative regularity in a 2-alternative forced-choice task. In the first experiment we used a single reference pattern of intermediate regularity and, in the second, Thurstonian scaling of patterns covering the entire range of regularity. In both experiments discriminability was highly correlated with peakedness for a wide range of presentation conditions. This supports the hypothesis that regularity is coded via peakedness of the distribution of responses across scale.

Evolutionary-Optimized Photonic Network Structure in White Beetle Wing Scales

Most studies of structural color in nature concern periodic arrays, which through the interference of light create color. The "color" white however relies on the multiple scattering of light within a randomly structured medium, which randomizes the direction and phase of incident light. Opaque white materials therefore must be much thicker than periodic structures. It is known that flying insects create "white" in extremely thin layers. This raises the question, whether evolution has optimized the wing scale morphology for white reflection at a minimum material use. This hypothesis is difficult to prove, since this requires the detailed knowledge of the scattering morphology combined with a suitable theoretical model. Here, a cryoptychographic X-ray tomography method is employed to obtain a full 3D structural dataset of the network morphology within a white beetle wing scale. By digitally manipulating this 3D representation, this study demonstrates that this morphology indeed provides the highest white retroreflection at the minimum use of material, and hence weight for the organism. Changing any of the network parameters (within the parameter space accessible by biological materials) either increases the weight, increases the thickness, or reduces reflectivity, providing clear evidence for the evolutionary optimization of this morphology.

Rational design of stealthy hyperuniform two-phase media with tunable order

Disordered stealthy hyperuniform materials are exotic amorphous states of matter that have attracted recent attention because of their novel structural characteristics (hidden order at large length scales) and physical properties, including desirable photonic and transport properties. It is therefore useful to devise algorithms that enable one to design a wide class of such amorphous configurations at will. In this paper, we present several algorithms enabling the systematic identification and generation of discrete (digitized) stealthy hyperuniform patterns with a tunable degree of order, paving the way towards the rational design of disordered materials endowed with novel thermodynamic and physical properties. To quantify the degree of order or disorder of the stealthy systems, we utilize the discrete version of the τ order metric, which accounts for the underlying spatial correlations that exist across all relevant length scales in a given digitized two-phase (or, equivalently, a two-spin state) system of interest. Our results impinge on a myriad of fields, ranging from physics, materials science and engineering, visual perception, and information theory to modern data science.

Characterization of maximally random jammed sphere packings. III. Transport and electromagnetic properties via correlation functions

In the first two papers of this series, we characterized the structure of maximally random jammed (MRJ) sphere packings across length scales by computing a variety of different correlation functions, spectral functions, hole probabilities, and local density fluctuations. From the remarkable structural features of the MRJ packings, especially its disordered hyperuniformity, exceptional physical properties can be expected. Here we employ these structural descriptors to estimate effective transport and electromagnetic properties via rigorous bounds, exact expansions, and accurate analytical approximation formulas. These property formulas include interfacial bounds as well as universal scaling laws for the mean survival time and the fluid permeability. We also estimate the principal relaxation time associated with Brownian motion among perfectly absorbing traps. For the propagation of electromagnetic waves in the long-wavelength limit, we show that a dispersion of dielectric MRJ spheres within a matrix of another dielectric material forms, to a very good approximation, a dissipationless disordered and isotropic two-phase medium for any phase dielectric contrast ratio. We compare the effective properties of the MRJ sphere packings to those of overlapping spheres, equilibrium hard-sphere packings, and lattices of hard spheres. Moreover, we generalize results to micro- and macroscopically anisotropic packings of spheroids with tensorial effective properties. The analytic bounds predict the qualitative trend in the physical properties associated with these structures, which provides guidance to more time-consuming simulations and experiments. They especially provide impetus for experiments to design materials with unique bulk properties resulting from hyperuniformity, including structural-color and color-sensing applications.

Binary mixtures of charged colloids: a potential route to synthesize disordered hyperuniform materials

A new route to fabricate large samples of 2D disordered hyperuniform materials via self-assembly of mixtures of charged colloids.

Disordered multihyperuniformity derived from binary plasmas

Disordered multihyperuniform many-particle systems are exotic amorphous states that allow exquisite color sensing capabilities due to their anomalous suppression of density fluctuations for distinct subsets of particles, as recently evidenced in photoreceptor mosaics in avian retina. Motivated by this biological finding, we present a statistical-mechanical model that rigorously achieves disordered multihyperuniform many-body systems by tuning interactions in binary mixtures of nonadditive hard-disk plasmas. We demonstrate that multihyperuniformity competes with phase separation and stabilizes a clustered phase. Our work provides a systematic means to generate disordered multihyperuniform solids, and hence lays the groundwork to explore their potentially unique photonic, phononic, electronic, and transport properties.

Equation of state of polydisperse hard-disk mixtures in the high-density regime

A proposal to link the equation of state of a monocomponent hard-disk fluid to the equation of state of a polydisperse hard-disk mixture is presented. Event-driven molecular dynamics simulations are performed to obtain data for the compressibility factor of the monocomponent fluid and of 26 polydisperse mixtures with different size distributions. Those data are used to assess the proposal and to infer the values of the compressibility factor of the monocomponent hard-disk fluid in the metastable region from those of mixtures in the high-density region. The collapse of the curves for the different mixtures is excellent in the stable region. In the metastable regime, except for two mixtures in which crystallization is present, the outcome of the approach exhibits a rather good performance. The simulation results indicate that a (reduced) variance of the size distribution larger than about 0.01 is sufficient to avoid crystallization and explore the metastable fluid branch.

Disordered hyperuniformity in two-component nonadditive hard-disk plasmas

We study the behavior of a classical two-component ionic plasma made up of nonadditive hard disks with additional logarithmic Coulomb interactions between them. Due to the Coulomb repulsion, long-wavelength total density fluctuations are suppressed and the system is globally hyperuniform. Short-range volume effects lead to phase separation or to heterocoordination for positive or negative nonadditivities, respectively. These effects compete with the hidden long-range order imposed by hyperuniformity. As a result, the critical behavior of the mixture is modified, with long-wavelength concentration fluctuations partially damped when the system is charged. It is also shown that the decrease of configurational entropy due to hyperuniformity originates from contributions beyond the two-particle level. Finally, despite global hyperuniformity, we show that in our system the spatial configuration associated with each component separately is not hyperuniform, i.e., the system is not "multihyperuniform."

Microstructure and mechanical properties of hyperuniform heterogeneous materials

A hyperuniform random heterogeneous material is one in which the local volume fraction fluctuations in an observation window decay faster than the reciprocal window volume as the window size increases. Recent studies show that this class of materials are endowed with superior physical properties such as large isotropic photonic band gaps and optimal transport properties. Here we employ a stochastic optimization procedure to systematically generate realizations of hyperuniform heterogeneous materials with controllable short-range order, which is partially quantified using the two-point correlation function S_{2}(r) associated with the phase of interest. Specifically, our procedure generalizes the widely used Yeong-Torquato reconstruction procedure by including an additional constraint for hyperuniformity, i.e., the volume integral of the autocovariance function χ(r)=S_{2}(r)-ϕ^{2} over the whole space is zero. In addition, we only require the reconstructed S_{2} to match the target function up to a certain cutoff distance γ, in order to give the system sufficient degrees of freedom to satisfy the hyperuniform condition. By systematically increasing the γ value for a given S_{2}, one can produce a spectrum of hyperuniform heterogeneous materials with varying degrees of partial short-range order compatible with the specified S_{2}. The mechanical performance including both elastic and brittle fracture behaviors of the generated hyperuniform materials is analyzed using a volume-compensated lattice-particle method. For the purpose of comparison, the corresponding nonhyperuniform materials with the same short-range order (i.e., with S_{2} constrained up to the same γ value) are also constructed and their mechanical performance is analyzed. Here we consider two specific S_{2} including the positive exponential decay function and the correlation function associated with an equilibrium hard-sphere system. For the constructed systems associated with these two specific functions, we find that although the hyperuniform materials are softer than their nonhyperuniform counterparts, the former generally possess a significantly higher brittle fracture strength than the latter. This superior mechanical behavior is attributed to the lower degree of stress concentration in the material resulting from the hyperuniform microstructure, which is crucial to crack initiation and propagation.

Large-Scale Structure and Hyperuniformity of Amorphous Ices

We investigate the large-scale structure of amorphous ices and transitions between their different forms by quantifying their large-scale density fluctuations. Specifically, we simulate the isothermal compression of low-density amorphous ice (LDA) and hexagonal ice to produce high-density amorphous ice (HDA). Both HDA and LDA are nearly hyperuniform; i.e., they are characterized by an anomalous suppression of large-scale density fluctuations. By contrast, in correspondence with the nonequilibrium phase transitions to HDA, the presence of structural heterogeneities strongly suppresses the hyperuniformity and the system becomes hyposurficial (devoid of "surface-area fluctuations"). Our investigation challenges the largely accepted "frozen-liquid" picture, which views glasses as structurally arrested liquids. Beyond implications for water, our findings enrich our understanding of pressure-induced structural transformations in glasses.

Characterizing pixel and point patterns with a hyperuniformity disorder length

We introduce the concept of a "hyperuniformity disorder length" h that controls the variance of volume fraction fluctuations for randomly placed windows of fixed size. In particular, fluctuations are determined by the average number of particles within a distance h from the boundary of the window. We first compute special expectations and bounds in d dimensions, and then illustrate the range of behavior of h versus window size L by analyzing several different types of simulated two-dimensional pixel patterns-where particle positions are stored as a binary digital image in which pixels have value zero if empty and one if they contain a particle. The first are random binomial patterns, where pixels are randomly flipped from zero to one with probability equal to area fraction. These have long-ranged density fluctuations, and simulations confirm the exact result h=L/2. Next we consider vacancy patterns, where a fraction f of particles on a lattice are randomly removed. These also display long-range density fluctuations, but with h=(L/2)(f/d) for small f, and h=L/2 for f→1. And finally, for a hyperuniform system with no long-range density fluctuations, we consider "Einstein patterns," where each particle is independently displaced from a lattice site by a Gaussian-distributed amount. For these, at large L,h approaches a constant equal to about half the root-mean-square displacement in each dimension. Then we turn to gray-scale pixel patterns that represent simulated arrangements of polydisperse particles, where the volume of a particle is encoded in the value of its central pixel. And we discuss the continuum limit of point patterns, where pixel size vanishes. In general, we thus propose to quantify particle configurations not just by the scaling of the density fluctuation spectrum but rather by the real-space spectrum of h(L) versus L. We call this approach "hyperuniformity disorder length spectroscopy".

Hyperuniformity disorder length spectroscopy for extended particles

The concept of a hyperuniformity disorder length h was recently introduced for analyzing volume fraction fluctuations for a set of measuring windows [Chieco et al., Phys. Rev. E 96, 032909 (2017).PLEEE81539-375510.1103/PhysRevE.96.032909]. This length permits a direct connection to the nature of disorder in the spatial configuration of the particles and provides a way to diagnose the degree of hyperuniformity in terms of the scaling of h and its value in comparison with established bounds. Here, this approach is generalized for extended particles, which are larger than the image resolution and can lie partially inside and partially outside the measuring windows. The starting point is an expression for the relative volume fraction variance in terms of four distinct volumes: that of the particle, the measuring window, the mean-squared overlap between particle and region, and the region over which particles have nonzero overlap with the measuring window. After establishing limiting behaviors for the relative variance, computational methods are developed for both continuum and pixelated particles. Exact results are presented for particles of special shape and for measuring windows of special shape, for which the equations are tractable. Comparison is made for other particle shapes, using simulated Poisson patterns. And the effects of polydispersity and image errors are discussed. For small measuring windows, both particle shape and spatial arrangement affect the form of the variance. For large regions, the variance scaling depends only on arrangement but particle shape sets the numerical proportionality. The combined understanding permit the measured variance to be translated to the spectrum of hyperuniformity lengths versus region size, as the quantifier of spatial arrangement. This program is demonstrated for a system of nonoverlapping particles at a series of increasing packing fractions as well as for an Einstein pattern of particles with several different extended shapes.

Structural Characterization and Statistical-Mechanical Model of Epidermal Patterns

In proliferating epithelia of mammalian skin, cells of irregular polygon-like shapes pack into complex, nearly flat two-dimensional structures that are pliable to deformations. In this work, we employ various sensitive correlation functions to quantitatively characterize structural features of evolving packings of epithelial cells across length scales in mouse skin. We find that the pair statistics in direct space (correlation function) and Fourier space (structure factor) of the cell centroids in the early stages of embryonic development show structural directional dependence (statistical anisotropy), which is a reflection of the fact that cells are stretched, which promotes uniaxial growth along the epithelial plane. In the late stages, the patterns tend toward statistically isotropic states, as cells attain global polarization and epidermal growth shifts to produce the skin's outer stratified layers. We construct a minimalist four-component statistical-mechanical model involving effective isotropic pair interactions consisting of hard-core repulsion and extra short-range soft-core repulsion beyond the hard core, whose length scale is roughly the same as the hard core. The model parameters are optimized to match the sample pair statistics in both direct and Fourier spaces. By doing this, the parameters are biologically constrained. In contrast with many vertex-based models, our statistical-mechanical model does not explicitly incorporate information about the cell shapes and interfacial energy between cells; nonetheless, our model predicts essentially the same polygonal shape distribution and size disparity of cells found in experiments, as measured by Voronoi statistics. Moreover, our simulated equilibrium liquid-like configurations are able to match other nontrivial unconstrained statistics, which is a testament to the power and novelty of the model. The array of structural descriptors that we deploy enable us to distinguish between normal, mechanically deformed, and pathological skin tissues. Our statistical-mechanical model enables one to generate tissue microstructure at will for further analysis. We also discuss ways in which our model might be extended to better understand morphogenesis (in particular the emergence of planar cell polarity), wound healing, and disease-progression processes in skin, and how it could be applied to the design of synthetic tissues.

Hexagonal Patterning of the Insect Compound Eye: Facet Area Variation, Defects, and Disorder

The regular hexagonal array morphology of facets (ommatidia) in the Drosophila compound eye is accomplished by regulation of cell differentiation and planar cell polarity during development. Mutations in certain genes disrupt regulation, causing a breakdown of this perfect symmetry, so that the ommatidial pattern shows onset of disorder in the form of packing defects. We analyze a variety of such mutants and compare them to normal (wild-type), finding that mutants show increased local variation in ommatidial area, which is sufficient to induce a significant number of defects. A model formalism based on Voronoi construction is developed to predict the observed correlation between ommatidium size variation and the number of defects, and to study the onset of disorder in this system with statistical tools. The model uncovers a previously unknown large-scale systematic size variation of the ommatidia across the eye of both wild-type and mutant animals. Such systematic variation of area, as well as its statistical fluctuations, are found to have distinct effects on eye disorder that can both be quantitatively modeled. Furthermore, the topological order is also influenced by the internal structure of the ommatidia, with cells of greater relative mechanical stiffness providing constraints to ommatidial deformation and thus to defect generation. Without free parameters, the simulation predicts the size-topology correlation for both wild-type and mutant eyes. This work develops formalisms of size-topology correlation that are very general and can be potentially applied to other cellular structures near the onset of disorder.

Particle localization and hyperuniformity of polymer-grafted nanoparticle materials

The properties of materials largely reflect the degree and character of the localization of the molecules comprising them so that the study and characterization of particle localization has central significance in both fundamental science and material design. Soft materials are often comprised of deformable molecules and many of their unique properties derive from the distinct nature of particle localization. We study localization in a model material composed of soft particles, hard nanoparticles with grafted layers of polymers, where the molecular characteristics of the grafted layers allow us to "tune" the softness of their interactions. Soft particles are particular interesting because spatial localization can occur such that density fluctuations on large length scales are suppressed, while the material is disordered at intermediate length scales; such materials are called "disordered hyperuniform". We use molecular dynamics simulation to study a liquid composed of polymer-grafted nanoparticles (GNP), which exhibit a reversible self-assembly into dynamic polymeric GNP structures below a temperature threshold, suggesting a liquid-gel transition. We calculate a number of spatial and temporal correlations and we find a significant suppression of density fluctuations upon cooling at large length scales, making these materials promising for the practical fabrication of "hyperuniform" materials.

Large-scale structure of randomly jammed spheres

We numerically analyze the density field of three-dimensional randomly jammed packings of monodisperse soft frictionless spherical particles, paying special attention to fluctuations occurring at large length scales. We study in detail the two-point static structure factor at low wave vectors in Fourier space. We also analyze the nature of the density field in real space by studying the large-distance behavior of the two-point pair correlation function, of density fluctuations in subsystems of increasing sizes, and of the direct correlation function. We show that such real space analysis can be greatly improved by introducing a coarse-grained density field to disentangle genuine large-scale correlations from purely local effects. Our results confirm that both Fourier and real space signatures of vanishing density fluctuations at large scale are absent, indicating that randomly jammed packings are not hyperuniform. In addition, we establish that the pair correlation function displays a surprisingly complex structure at large distances, which is however not compatible with the long-range negative correlation of hyperuniform systems but fully compatible with an analytic form for the structure factor. This implies that the direct correlation function is short ranged, as we also demonstrate directly. Our results reveal that density fluctuations in jammed packings do not follow the behavior expected for random hyperuniform materials, but display instead a more complex behavior.

Harmonic distortions enhance circular dichroism of dielectric single gyroids

The departure from strict periodic order in two-phase dielectric materials can offer properties that are otherwise inaccessible to perfectly ordered photonic crystals. Herewith, we investigate the circular dichroism of the single gyroid photonic crystal in the presence of spatial distortions. FDTD simulations and microwave transmission measurements on 3D-printed replicas show that certain harmonic long-wavelength spatial distortions ("sinusoidal chirp") nearly doubles the imbalance of the circular polarisation reflectances, as well as significantly strengthens polarisation-incoherent reflectance. The observed changes are partially rationalised by comparison with simpler distortion models (linear chirp and tetragonal deformation) of the Gyroid.

Disordered hyperuniform heterogeneous materials

Disordered hyperuniform many-body systems are distinguishable states of matter that lie between a crystal and liquid: they are like perfect crystals in the way they suppress large-scale density fluctuations and yet are like liquids or glasses in that they are statistically isotropic with no Bragg peaks. These systems play a vital role in a number of fundamental and applied problems: glass formation, jamming, rigidity, photonic and electronic band structure, localization of waves and excitations, self-organization, fluid dynamics, quantum systems, and pure mathematics. Much of what we know theoretically about disordered hyperuniform states of matter involves many-particle systems. In this paper, we derive new rigorous criteria that disordered hyperuniform two-phase heterogeneous materials must obey and explore their consequences. Two-phase heterogeneous media are ubiquitous; examples include composites and porous media, biological media, foams, polymer blends, granular media, cellular solids, and colloids. We begin by obtaining some results that apply to hyperuniform two-phase media in which one phase is a sphere packing in d-dimensional Euclidean space [Formula: see text]. Among other results, we rigorously establish the requirements for packings of spheres of different sizes to be 'multihyperuniform'. We then consider hyperuniformity for general two-phase media in [Formula: see text]. Here we apply realizability conditions for an autocovariance function and its associated spectral density of a two-phase medium, and then incorporate hyperuniformity as a constraint in order to derive new conditions. We show that some functional forms can immediately be eliminated from consideration and identify other forms that are allowable. Specific examples and counterexamples are described. Contact is made with well-known microstructural models (e.g. overlapping spheres and checkerboards) as well as irregular phase-separation and Turing-type patterns. We also ascertain a family of autocovariance functions (or spectral densities) that are realizable by disordered hyperuniform two-phase media in any space dimension, and present select explicit constructions of realizations. These studies provide insight into the nature of disordered hyperuniformity in the context of heterogeneous materials and have implications for the design of such novel amorphous materials.

Hyperuniformity and its generalizations

Disordered many-particle hyperuniform systems are exotic amorphous states of matter that lie between crystal and liquid: They are like perfect crystals in the way they suppress large-scale density fluctuations and yet are like liquids or glasses in that they are statistically isotropic with no Bragg peaks. These exotic states of matter play a vital role in a number of problems across the physical, mathematical as well as biological sciences and, because they are endowed with novel physical properties, have technological importance. Given the fundamental as well as practical importance of disordered hyperuniform systems elucidated thus far, it is natural to explore the generalizations of the hyperuniformity notion and its consequences. In this paper, we substantially broaden the hyperuniformity concept along four different directions. This includes generalizations to treat fluctuations in the interfacial area (one of the Minkowski functionals) in heterogeneous media and surface-area driven evolving microstructures, random scalar fields, divergence-free random vector fields, and statistically anisotropic many-particle systems and two-phase media. In all cases, the relevant mathematical underpinnings are formulated and illustrative calculations are provided. Interfacial-area fluctuations play a major role in characterizing the microstructure of two-phase systems (e.g., fluid-saturated porous media), physical properties that intimately depend on the geometry of the interface, and evolving two-phase microstructures that depend on interfacial energies (e.g., spinodal decomposition). In the instances of random vector fields and statistically anisotropic structures, we show that the standard definition of hyperuniformity must be generalized such that it accounts for the dependence of the relevant spectral functions on the direction in which the origin in Fourier space is approached (nonanalyticities at the origin). Using this analysis, we place some well-known energy spectra from the theory of isotropic turbulence in the context of this generalization of hyperuniformity. Among other results, we show that there exist many-particle ground-state configurations in which directional hyperuniformity imparts exotic anisotropic physical properties (e.g., elastic, optical, and acoustic characteristics) to these states of matter. Such tunability could have technological relevance for manipulating light and sound waves in ways heretofore not thought possible. We show that disordered many-particle systems that respond to external fields (e.g., magnetic and electric fields) are a natural class of materials to look for directional hyperuniformity. The generalizations of hyperuniformity introduced here provide theoreticians and experimentalists new avenues to understand a very broad range of phenomena across a variety of fields through the hyperuniformity "lens."

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.

Mathematical and computational models of the retina in health, development and disease

The retina confers upon us the gift of vision, enabling us to perceive the world in a manner unparalleled by any other tissue. Experimental and clinical studies have provided great insight into the physiology and biochemistry of the retina; however, there are questions which cannot be answered using these methods alone. Mathematical and computational techniques can provide complementary insight into this inherently complex and nonlinear system. They allow us to characterise and predict the behaviour of the retina, as well as to test hypotheses which are experimentally intractable. In this review, we survey some of the key theoretical models of the retina in the healthy, developmental and diseased states. The main insights derived from each of these modelling studies are highlighted, as are model predictions which have yet to be tested, and data which need to be gathered to inform future modelling work. Possible directions for future research are also discussed. Whilst the present modelling studies have achieved great success in unravelling the workings of the retina, they have yet to achieve their full potential. For this to happen, greater involvement with the modelling community is required, and stronger collaborations forged between experimentalists, clinicians and theoreticians. It is hoped that, in addition to bringing the fruits of current modelling studies to the attention of the ophthalmological community, this review will encourage many such future collaborations.

Confined disordered strictly jammed binary sphere packings

Disordered jammed packings under confinement have received considerably less attention than their bulk counterparts and yet arise in a variety of practical situations. In this work, we study binary sphere packings that are confined between two parallel hard planes and generalize the Torquato-Jiao (TJ) sequential linear programming algorithm [Phys. Rev. E 82, 061302 (2010)] to obtain putative maximally random jammed (MRJ) packings that are exactly isostatic with high fidelity over a large range of plane separation distances H, small to large sphere radius ratio α, and small sphere relative concentration x. We find that packing characteristics can be substantially different from their bulk analogs, which is due to what we term "confinement frustration." Rattlers in confined packings are generally more prevalent than those in their bulk counterparts. We observe that packing fraction, rattler fraction, and degree of disorder of MRJ packings generally increase with H, though exceptions exist. Discontinuities in the packing characteristics as H varies in the vicinity of certain values of H are due to associated discontinuous transitions between different jammed states. When the plane separation distance is on the order of two large-sphere diameters or less, the packings exhibit salient two-dimensional features; when the plane separation distance exceeds about 30 large-sphere diameters, the packings approach three-dimensional bulk packings. As the size contrast increases (as α decreases), the rattler fraction dramatically increases due to what we call "size-disparity" frustration. We find that at intermediate α and when x is about 0.5 (50-50 mixture), the disorder of packings is maximized, as measured by an order metric ψ that is based on the number density fluctuations in the direction perpendicular to the hard walls. We also apply the local volume-fraction variance σ(τ)(2)(R) to characterize confined packings and find that these packings possess essentially the same level of hyperuniformity as their bulk counterparts. Our findings are generally relevant to confined packings that arise in biology (e.g., structural color in birds and insects) and may have implications for the creation of high-density powders and improved battery designs.

A Geometric-Structure Theory for Maximally Random Jammed Packings

Maximally random jammed (MRJ) particle packings can be viewed as prototypical glasses in that they are maximally disordered while simultaneously being mechanically rigid. The prediction of the MRJ packing density ϕMRJ, among other packing properties of frictionless particles, still poses many theoretical challenges, even for congruent spheres or disks. Using the geometric-structure approach, we derive for the first time a highly accurate formula for MRJ densities for a very wide class of two-dimensional frictionless packings, namely, binary convex superdisks, with shapes that continuously interpolate between circles and squares. By incorporating specific attributes of MRJ states and a novel organizing principle, our formula yields predictions of ϕMRJ that are in excellent agreement with corresponding computer-simulation estimates in almost the entire α-x plane with semi-axis ratio α and small-particle relative number concentration x. Importantly, in the monodisperse circle limit, the predicted ϕMRJ = 0.834 agrees very well with the very recently numerically discovered MRJ density of 0.827, which distinguishes it from high-density "random-close packing" polycrystalline states and hence provides a stringent test on the theory. Similarly, for non-circular monodisperse superdisks, we predict MRJ states with densities that are appreciably smaller than is conventionally thought to be achievable by standard packing protocols.

Dynamical phase transitions in one-dimensional hard-particle systems

We analyze a one-dimensional model of hard particles, within ensembles of trajectories that are conditioned (or biased) to atypical values of the time-averaged dynamical activity. We analyze two phenomena that are associated with these large deviations of the activity: phase separation (at low activity) and the formation of hyperuniform states (at high activity). We consider a version of the model which operates at constant volume, and a version at constant pressure. In these nonequilibrium systems, differences arise between the two ensembles, because of the extra freedom available to the constant-pressure system, which can change its total density. We discuss the relationships between different ensembles, mechanical equilibrium, and the probability cost of rare density fluctuations.

Nanoscale engineering of two-dimensional disordered hyperuniform block-copolymer assemblies

Disordered hyperuniform (DH) media have been recognized as a new state of disordered matter that broadens our vision of material engineering. Here, long-range correlated disordered two-dimensional patterns are fabricated by self-assembling of spherical diblock-copolymer (BCP) micelles. Control of the self-assembling parameters leads to the formation of DH patterns of micelles that can host nanoscale material inclusions, therefore providing an effective strategy for fabricating multimaterial DH structures at molecular scale. Centroidal patterns are accurately determined by virtue of BCP micelles loaded with metal nanoparticles. Our analysis reveals the signature of nearly ideal DH BCP assemblies in the local density fluctuation and a dominant linear scaling in the local number fluctuation.

Ground states of stealthy hyperuniform potentials: I. Entropically favored configurations

Systems of particles interacting with "stealthy" pair potentials have been shown to possess infinitely degenerate disordered hyperuniform classical ground states with novel physical properties. Previous attempts to sample the infinitely degenerate ground states used energy minimization techniques, introducing algorithmic dependence that is artificial in nature. Recently, an ensemble theory of stealthy hyperuniform ground states was formulated to predict the structure and thermodynamics that was shown to be in excellent agreement with corresponding computer simulation results in the canonical ensemble (in the zero-temperature limit). In this paper, we provide details and justifications of the simulation procedure, which involves performing molecular dynamics simulations at sufficiently low temperatures and minimizing the energy of the snapshots for both the high-density disordered regime, where the theory applies, as well as lower densities. We also use numerical simulations to extend our study to the lower-density regime. We report results for the pair correlation functions, structure factors, and Voronoi cell statistics. In the high-density regime, we verify the theoretical ansatz that stealthy disordered ground states behave like "pseudo" disordered equilibrium hard-sphere systems in Fourier space. The pair statistics obey certain exact integral conditions with very high accuracy. These results show that as the density decreases from the high-density limit, the disordered ground states in the canonical ensemble are characterized by an increasing degree of short-range order and eventually the system undergoes a phase transition to crystalline ground states. In the crystalline regime (low densities), there exist aperiodic structures that are part of the ground-state manifold but yet are not entropically favored. We also provide numerical evidence suggesting that different forms of stealthy pair potentials produce the same ground-state ensemble in the zero-temperature limit. Our techniques may be applied to sample the zero-temperature limit of the canonical ensemble of other potentials with highly degenerate ground states.

Ground states of stealthy hyperuniform potentials. II. Stacked-slider phases

Stealthy potentials, a family of long-range isotropic pair potentials, produce infinitely degenerate disordered ground states at high densities and crystalline ground states at low densities in d-dimensional Euclidean space R^{d}. In the previous paper in this series, we numerically studied the entropically favored ground states in the canonical ensemble in the zero-temperature limit across the first three Euclidean space dimensions. In this paper, we investigate using both numerical and theoretical techniques metastable stacked-slider phases, which are part of the ground-state manifold of stealthy potentials at densities in which crystal ground states are favored entropically. Our numerical results enable us to devise analytical models of this phase in two, three, and higher dimensions. Utilizing this model, we estimated the size of the feasible region in configuration space of the stacked-slider phase, finding it to be smaller than that of crystal structures in the infinite-system-size limit, which is consistent with our recent previous work. In two dimensions, we also determine exact expressions for the pair correlation function and structure factor of the analytical model of stacked-slider phases and analyze the connectedness of the ground-state manifold of stealthy potentials in this density regime. We demonstrate that stacked-slider phases are distinguishable states of matter; they are nonperiodic, statistically anisotropic structures that possess long-range orientational order but have zero shear modulus. We outline some possible future avenues of research to elucidate our understanding of this unusual phase of matter.

Surface-enhanced Raman imaging of cell membrane by a highly homogeneous and isotropic silver nanostructure

Label-free chemical imaging of live cell membranes can shed light on the molecular basis of cell membrane functionalities and their alterations under membrane-related diseases. In principle, this can be done by surface-enhanced Raman scattering (SERS) in confocal microscopy, but requires engineering plasmonic architectures with a spatially invariant SERS enhancement factor G(x, y) = G. To this end, we exploit a self-assembled isotropic nanostructure with characteristics of homogeneity typical of the so-called near-hyperuniform disorder. The resulting highly dense, homogeneous and isotropic random pattern consists of clusters of silver nanoparticles with limited size dispersion. This nanostructure brings together several advantages: very large hot spot density (∼10(4) μm(-2)), superior spatial reproducibility (SD < 1% over 2500 μm(2)) and single-molecule sensitivity (Gav ∼ 10(9)), all on a centimeter scale transparent active area. We are able to reconstruct the label-free SERS-based chemical map of live cell membranes with confocal resolution. In particular, SERS imaging is here demonstrated on red blood cells in vitro in order to use the Raman-resonant heme of the cell as a contrast medium to prove spectroscopic detection of membrane molecules. Numerical simulations also clarify the SERS characteristics of the substrate in terms of electromagnetic enhancement and distance sensitivity range consistently with the experiments. The large SERS-active area is intended for multi-cellular imaging on the same substrate, which is important for spectroscopic comparative analysis of complex organisms like cells. This opens new routes for in situ quantitative surface analysis and dynamic probing of living cells exposed to membrane-targeting drugs.

Hyperuniform density fluctuations and diverging dynamic correlations in periodically driven colloidal suspensions

The emergence of particle irreversibility in periodically driven colloidal suspensions has been interpreted as resulting either from a nonequilibrium phase transition to an absorbing state or from the chaotic nature of particle trajectories. Using a simple model of a driven suspension, we show that a nonequilibrium phase transition is accompanied by hyperuniform static density fluctuations in the vicinity of the transition, where we also observe strong dynamic heterogeneities reminiscent of dynamics in glassy materials. We find that single particle dynamics becomes intermittent and strongly non-Fickian, and that collective dynamics becomes spatially correlated over diverging length scales. Our results suggest that the two theoretical scenarii can be experimentally discriminated using particle-resolved measurements of standard static and dynamic observables.

Hyperuniformity and phase separation in biased ensembles of trajectories for diffusive systems

We analyze biased ensembles of trajectories for diffusive systems. In trajectories biased either by the total activity or the total current, we use fluctuating hydrodynamics to show that these systems exhibit phase transitions into "hyperuniform" states, where large-wavelength density fluctuations are strongly suppressed. We illustrate this behavior numerically for a system of hard particles in one dimension and we discuss how it appears in simple exclusion processes. We argue that these diffusive systems generically respond very strongly to any nonzero bias, so that homogeneous states with "normal" fluctuations (finite compressibility) exist only when the bias is very weak.

Transcriptome profiling of developing photoreceptor subtypes reveals candidate genes involved in avian photoreceptor diversification

Avian photoreceptors are a diverse class of neurons, comprised of four single cones, the two members of the double cone, and rods. The signaling events and transcriptional regulators driving the differentiation of these diverse photoreceptors are largely unknown. In addition, many distinctive features of photoreceptor subtypes, including spectral tuning, oil droplet size and pigmentation, synaptic targets, and spatial patterning, have been well characterized, but the molecular mechanisms underlying these attributes have not been explored. To identify genes specifically expressed in distinct chicken (Gallus gallus) photoreceptor subtypes, we developed fluorescent reporters that label photoreceptor subpopulations, isolated these subpopulations by using fluorescence-activated cell sorting, and subjected them to next-generation sequencing. By comparing the expression profiles of photoreceptors labeled with rhodopsin, red opsin, green opsin, and violet opsin reporters, we have identified hundreds of differentially expressed genes that may underlie the distinctive features of these photoreceptor subtypes. These genes are involved in a variety of processes, including phototransduction, transcriptional regulation, cell adhesion, maintenance of intra- and extracellular structure, and metabolism. Of particular note are a variety of differentially expressed transcription factors, which may drive and maintain photoreceptor diversity, and cell adhesion molecules, which may mediate spatial patterning of photoreceptors and act to establish retinal circuitry. These analyses provide a framework for future studies that will dissect the role of these various factors in the differentiation of avian photoreceptor subtypes.