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Hyperuniformity is a Signature of Maximally Random Jammed Packings
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Growing Length Scales in Supercooled Liquids and Glasses
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Disordered Multihyperuniformity in Avian Photoreceptor Cell Patterns
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Diagnosing Hyperuniformity in Two-Dimensional, Disordered, Jammed Packings of Soft Spheres
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Characterizing the local density fluctuations in a many-body system represents a fundamental problem in the physical and biological sciences. Examples include the large-scale structure of the Universe, condensed phases of matter, the structure and collective motion of grains in vibrated granular media, energy levels in integrable quantum systems, and the structure of living cells. In each of these cases, one is interested in characterizing the variance in the local number of points of a general point pattern (henceforth known as the number variance), and this problem extends naturally to higher dimensions with applications to number theory.

Of particular importance in this regard is the notion of **hyperuniformity** in a point pattern or many-particle configuration.
A hyperuniform configuration is one in which the number variance \(\sigma^2\) associated with the number of points (particles) in some local observation window grows more slowly than the window volume as the size of the window increases.
In the case of a spherical window of radius \(R\) in \(d\)-dimensional Euclidean space \(\mathcal R^d\) (see figure), hyperuniformity means that the number variance \(\sigma^2(R)\) grows more slowly than \(R^d\).
This implies that hyperuniform point patterns are characterized by vanishing infinite-wavelength density fluctuations (when appropriately scaled) and encompass all crystals, quasicrystals, and special disordered many-particle systems [1].
The degree to which large-scale density fluctuations are suppressed enables one to rank order crystals, quasicrystals and the aforementioned special disordered systems [1,2].
Hyperuniform disordered structures can be regarded as a **new state of disordered matter** in that they behave more like crystals or quasicrystals in the manner in which they suppress density fluctuations on large length scales, and yet are also like liquids and glasses in that they are statistically isotropic structures with no Bragg peaks.
Thus, hyperuniform disordered materials can be regarded to possess a "**hidden order**" that is not apparent on short length scales.

We have also demonstrated that hyperuniformity also signals the onset of an "inverted" critical point in which the direct correlation function (rather than the standard pair correlation function) becomes long-ranged [1]. We extended the notion of hyperuniformity to include also two-phase random heterogeneous media [2]. Hyperuniform random media do not possess infinite-wavelength volume fraction fluctuations, implying that the variance in the local volume fraction in an observation window decays faster than the reciprocal of the window volume \(r^{-d}\) as \(R\) increases.

Figure 1 shows typical periodic and nonperiodic point patterns and two-phase media along with the corresponding observation windows.

In an initial study we showed that so-called maximally random jammed (MRJ)
packings of identical three-dimensional spheres, which can be viewed
as a prototypical glass [6], are hyperuniform such that
pair correlations decay asymptotically with scaling \(r^{-4}\), which
we call *quasi-long-range* correlations [7]. Such correlations
are to be contrasted with typical disordered systems in which
pair correlaitons decay exponentially fast.
More recently, we have shown that quasi-long-range pair correlations that decay
asymptotically with scaling \(r^{-(d+1)}\) in \(d\)-dimensional Euclidean space \(\mathcal R^d\),
trademarks of certain quantum systems and cosmological
structures, are a universal signature of a wide
class of maximally random jammed (MRJ) hard-particle packings, including
nonspherical particles with a polydispersity in size [8-10].

We have recently demonstrated that quasi-long-range pair correlations
are present well before overcompressed hard-sphere systems [11]
or supercooled liquids reach their glass transition [12].
These quasi-long-range correlations translate into a long-ranged
direct correaltion function, which enables us extract a length
scale that is *nonequilibrium* in nature. We show that this nonequilibrium
static length scale grows on approach to the glassy state.
This provides an alternative view of the nature of the
glass transition.

The evolution of animal eyes has been an intense subject of research since Darwin. The purpose of a visual system is to sample light in such a way as to provide an animal with actionable knowledge of its surroundings that will permit it to survive and reproduce. Cone photoreceptor cells in the retina are responsible for detecting colors and they are often spatially arranged in a regular array (e.g., insects, some fish and reptiles), which is often a superior arrangement to sample light. In the absence of any other constraints, classical sampling theory tells us that the triangular lattice (i.e., a hexagonal array) is the best arrangement.

Diurnal birds have one of the most sophisticated cone visual systems of any vertebrate, consisting of four types of single cone (violet, blue, green and red) which mediate color vision and double cones involved in luminance detection; see Fig. 2a. Given the utility of the perfect triangular-lattice arrangement of photoreceptors for vision, the presence of disorder in the spatial arrangement of avian cone patterns was puzzling.

Our recent investigation in collaboration with Joseph Corbo at Washington University presents a stunning example of how fundamental physical principles can constrain and limit optimization in a biological system [13]. By analyzing the chicken cone photoreceptor system consisting of five different cell types using a variety of sensitive microstructural descriptors, we found that the disordered photoreceptor patterns are ``hyperuniform'' (as defined above), a property that had heretofore been identified in a unique subset of physical systems, but had never been observed in any living organism.

Disordered hyperuniform structures can be regarded as a new exotic state of disordered matter in that they behave more like crystals or quasicrystals in the manner in which they suppress density fluctuations on large length scales, and yet are also like liquids and glasses in that they are statistically isotropic structures with no Bragg peaks. Thus, hyperuniform disordered materials can be regarded to possess a "hidden order" that is not apparent on short length scales.

Remarkably, the patterns of both the total population and the individual cell types are simultaneously hyperuniform, which has never been observed in any system before, physical or not. We term such patterns ``multi-hyperuniform'' because multiple distinct subsets of the overall point pattern are themselves hyperuniform. We devised a unique multiscale cell packing model in two dimensions that suggests that photoreceptor types interact with both short- and long-ranged repulsive forces and that the resultant competition between the types gives rise to the aforementioned singular spatial features characterizing the system, including multi-hyperuniformity; see Fig. 2b. These findings suggest that a disordered hyperuniform pattern may represent the most uniform sampling arrangement attainable in the avian system, given intrinsic packing constraints within the photoreceptor epithelium. In addition, they show how fundamental physical constraints can change the course of a biological optimization process. Our results suggest that multi-hyperuniform disordered structures have implications for the design of materials with novel physical properties and therefore may represent a fruitful area for future research.

The task of determining whether or not an image of an experimental system is hyperuniform is experimentally challenging due to finite-resolution, noise, and sample-size effects that influence characterization measurements. We have explored these issues, employing video optical microscopy to study hyperuniformity phenomena in disordered two-dimensional jammed packings of soft spheres [14]. Using a combination of experiment and simulation we have characterized the possible adverse effects of particle polydispersity, image noise, and finite-size effects on the assignment of hyperuniformity, and we have developed a methodology that permits improved diagnosis of hyperuniformity from real-space measurements. The key to this improvement is a simple packing reconstruction algorithm that incorporates particle polydispersity to minimize the free volume. In addition, simulations show that hyperuniformity in finite-sized samples can be ascertained more accurately in direct space than in reciprocal space. Finally, our experimental colloidal packings of soft polymeric spheres were shown to be effectively hyperuniform.

We have shown that so-called *stealthy* hyperuniform states can be created
as disordered ground states [3,4] and that they possess novel scattering properties [3]. This has
led to the discovery of the existence disordered two-phase dielectric
materials with complete photonic band gaps that are comparable in
size to those in photonic crystals [5]. Thus, the significance of hyperuniform materials for photonics
applications has enabled us, for the first time, to broaden the class of 2D dielectric materials
possessing complete and large photonic band gaps to include not only crystal and quasicrystal structures but
certain hyperuniform disordered ones. The interested reader can refer to the following article that describes
potential technological applications of this capability: *Advancing
photonic functionalities*.

Information on additional studies into stealthy hyperuniform systems is available here.

Invited presentation on stealthy disordered ground states given at the APS March Meeting 2016 on March 18, 2016 (pdf)

References:

- S. Torquato and F. H. Stillinger,
*Local Density Fluctuations, Hyperuniform Systems, and Order Metrics*,**Physical Review E**,**68**, 041113 (2003). - C. E. Zachary and S. Torquato,
*Hyperuniformity in Point Patterns and Two-Phase Random Heterogeneous Media*,**Journal of Statistical Mechanics: Theory and Experiment**, P12015 (2009). - R. D. Batten, F. H. Stillinger and S. Torquato,
*Classical Disordered Ground States: Super-Ideal Gases, and Stealth and Equi-Luminous Materials*,**Journal of Applied Physics**,**104**, 033504 (2008). - C. E. Zachary and S. Torquato,
*Anomalous Local Coordination, Density Fluctuations, and Void Statistics in Disordered Hyperuniform Many-Particle Ground States*,**Physical Review E**,**83**, 051133 (2011). - M. Florescu, S. Torquato and P. J. Steinhardt,
*Designer Disordered Materials with Large, Complete Photonic Band Gaps*,**Proceedings of the National Academy of Sciences**,**106**, 20658 (2009). - S. Torquato and F. H. Stillinger,
*Jammed Hard-Particle Packings: From Kepler to Bernal and Beyond*,**Reviews of Modern Physics**,**82**, 2633 (2010). - A. Donev , F. H. Stillinger, and S. Torquato,
*Unexpected Density Fluctuations in Disordered Jammed Hard-Sphere Packings*,**Physical Review Letters**,**95**, 090604 (2005). - C. E. Zachary, Y. Jiao, and S. Torquato,
*Hyperuniform Long-Range Correlations are a Signature of Disordered Jammed Hard-Particle Packings*,**Physical Review Letters**,**106**, 178001 (2011). - C. E. Zachary, Y. Jiao, and S. Torquato,
*Hyperuniformity, Quasi-Long-Range Correlations, and Void Space Constraints in Maximally Random Jammed Particle Packings. I. Polydisperse Spheres*,**Physical Review E**,**83**, 051308 (2011). - C. E. Zachary, Y. Jiao, and S. Torquato,
*Hyperuniformity, Quasi-Long-Range Correlations, and Void Space Constraints in Maximally Random Jammed Particle Packings. II. Anisotropy in Particle Shape*,**Physical Review E**,**83**, 051309 (2011). - A. B. Hopkins, F. H. Stillinger, and S. Torquato,
*Nonequilibrium Static Diverging Length Scales on Approaching a Prototypical Model Glassy State*,**Physical Review E**,**86**, 021505 (2012). - É. Marcotte, F. H. Stillinger, and S. Torquato,
*Nonequilibrium Static Growing Length Scales in Supercooled Liquids on Approaching the Glass Transition*,**Journal of Chemical Physics**,**138**, 12A508 (2013). - Y. Jiao, T. Lau, H. Hatzikirou, M. Meyer-Hermann, J. C. Corbo, and S. Torquato,
*Avian Photoreceptor Patterns Represent a Disordered Hyperuniform Solution to a Multiscale Packing Problem*,**Physical Review E**,**89**, 022721 (2014). - R. Dreyfus, Y. Xu, T. Still, L. A. Hough, A. G. Yodh, and S. Torquato
*Diagnosing Hyperuniformity in Two-Dimensional, Disordered, Jammed Packings of Soft Spheres*,**Physical Review E**,**91**, 012302 (2015).

Questions concerning this work should be directed to Professor Torquato.

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