Disordered Hyperuniformity

Hyperuniform materials are mixed-component many-particle systems with unusually low fluctuations in component density at large scales, when compared to the distribution of constituents in common disordered systems, like a mixed ideal gas (air) or typical liquids or amorphous solids: A disordered ''hyperuniform'' system is statistically isotropic, like a liquid, but exhibits reduced long-wavelength density fluctuations, similar to crystals.

All perfect crystals, perfect quasicrystals and special disordered systems are ''hyperuniform''.

Quantitatively, a many-particle system is ''hyperuniform'' if the variance of the number of points within a spherical observation window grows more slowly than the volume of the observation window. This definition is equivalent to a vanishing of the structure factor in the long-wavelength limit.
Disordered hyperuniform systems, were shown to be poised at an "inverted" critical point. They can be obtained via equilibrium or nonequilibrium routes, and are found in both classical physical and quantum-mechanical systems.

''Disordered hyperuniform'' systems are exotic ideal states of matter that lie between a crystal and liquid: They are like perfect crystals, in that their large-scale density fluctuations are unusually low, and yet are like liquids or glasses in that they are statistically isotropic, with no Bragg peaks, and hence lack any conventional long-range order. These peculiar organizational characteristics are now known to endow hyperuniform materials with novel physical properties itations

Originally coined to describe materials, the concept has been abstracted to collections of mathematical objects like the prime numbers, so that now the concept of ''hyperuniformity'' connects a broad range of topics in physics, mathematics, biology, and materials science.

History

The term ''hyperuniformity'' (also independently called super-homogeneity'' in the context of cosmology ) was coined and studied by Salvatore Torquato and Frank Stillinger in a 2003 paper in which, they showed, that among other things, hyperuniformity provides a unified framework to classify and structurally characterize crystals, quasicrystals, and the exotic disordered varieties. Thus, hyperuniformity is a long-range property that can be viewed as generalizing the traditional notion of long-range order (e.g., translational / orientational order of crystals or orientational order of quasicrystals) to also encompass exotic disordered systems.

Hyperuniformity was first introduced for point processes and later generalized to two-phase materials (or porous media) and random scalar or vectors fields.

Definition

A stationary (or homogeneous) point process $\mathcal$ of $\mathbb^d$ of intensity $\rho>0$ is often said to be hyperuniform or super-homogeneous if the variance of the number of points of $\mathcal$ that fall in a Euclidean ball scales slower than the volume of that ball, i.e.,
:

$\lim_ \frac = 0,$

where $\mathcal( B(\mathbf, R))$ stands for the cardinality of $\mathcal \cap B(\mathbf, R)$, $\mathrm$ the variance, and $, B(\mathbf, R),$ the volume of the Euclidean ball centered at the origin and of radius $R$.
Although hyperuniformity ''a priori'' depends on the shape of the window, mild technical assumptions allow considering any growing window.
An equivalent definition of hyperuniformity goes through the structure factor $S$ of the point process defined for any $\mathbf \in \mathbb^d$ by
:

$S(\mathbf) = 1+ \rho \mathcal(g-1)(\mathbf),$

where, $g$ is the pair correlation function of $\mathcal$ and $\mathcal$ stands for the Fourier transform.
Under mild assumptions, $\mathcal$ is hyperuniform iff $S(\mathbf) = 0$.
Moreover, if the structure factor undergoes a power decay $\vert S(\mathbf)\vert\sim c \Vert \mathbf \Vert_2^\alpha$ in the neighborhood of zero, the process can be classified into three categories depending on how $\alpha$ compares to 1.

Hyperuniformity diagnostics

Given a realization of a stationary (translation-invariant) point process $\mathcal$, the standard empirical diagnostic of hyperuniformity involves estimating the structure factor of the underlying point process.
Various estimators of the structure factor $S$ exist and the statistical diagnostics of hyperuniformity depend often on each field folklore.

Structure factor estimators

The ''scattering intensity'' $\widehat_$ is an estimator of the structure factor widely used in the physics literature and is defined as follows,
:

$\widehat_(\mathbf)= \frac \left , \sum_ e^ \^2$

where, $\rho$ is the intensity of $\mathcal$, W=\prod_^d \limits L_j/2,L_j/2/math> is a centered rectangular window, and $\mathbf \in \mathbb_$ defined by
:

$\mathbb_= \.$

Note that it's common to use a self-normalized version of $\widehat_$, where $\frac$ is replaced by $1/\mathcal(W)$.
The restriction to $\mathbb_$ is used to remove an important part of the bias of the estimator $\widehat_$ highly pronounced near the origin.
Although $\widehat_$ is the most known estimator, it's just a particular case of the family of tapered estimators $\widehat_$ defined by
:

$\widehat_(t, \mathbf) = \frac \left , \sum_ t(\mathbf, W) e^ \right , ^2,$

where $t$ is a uniformly (in $W$) square-integrable function supported on the observation window $W$, called a taper.
Variants of the estimator $\widehat_$ exist and can be used to reduce the bias. One can also use a family of orthogonal tapers (multitaper) instead of a single taper.
If the point process is also isotropic (rotation-invariant) radial estimators can be used.

Examples

Examples of disordered hyperuniform systems in physics are disordered ground states, jammed disordered sphere packings, amorphous speckle patterns, certain fermionic systems, random self-organization, perturbed lattices, and avian photoreceptor cells.

In mathematics, disordered hyperuniformity has been studied in the context of probability theory, geometry, and number theory, where the prime numbers have been found to be effectively limit periodic and hyperuniform in a certain scaling limit. Further examples include certain random walks and stable matchings of point processes. amorphous speckle patterns, certain fermionic systems, random self-organization, perturbed lattices, and avian photoreceptor cells.

Ordered hyperuniformity

While weakly correlated noise typically preserves hyperuniformity, correlated excitations at finite temperature tend to destroy hyperuniformity.

Examples of ordered, hyperuniform systems include all crystals, all quasicrystals, and limit-periodic sets.

Hyperuniformity was also reported for fermionic quantum matter in correlated electron systems as a result of cramming.

Disordered hyperuniformity

Torquato (2014) gives an illustrative example of the hidden order found in a "shaken box of marbles", which fall into an arrangement, called ''maximally random jammed packing''. Such hidden order may eventually be used for self-organizing colloids or optics with the ability to transmit light with an efficiency like a crystal but with a highly flexible design.

It has been found that disordered hyperuniform systems possess unique optical properties. For example, disordered hyperuniform photonic networks have been found to exhibit complete photonic band gaps that are comparable in size to those of photonic crystals but with the added advantage of isotropy, which enables free-form waveguides not possible with crystal structures. Moreover, in stealthy hyperuniform systems, light of any wavelength longer than a value specific to the material is able to propagate forward without loss (due to the correlated disorder) even for high particle density.

By contrast, in conditions where light is propagated through an uncorrelated, disordered material of the same density, the material would appear opaque due to multiple scattering. “Stealthy” hyperuniform materials can be theoretically designed for light of any wavelength, and the applications of the concept cover a wide variety of fields of wave physics and materials engineering.

Disordered hyperuniformity was found in the photoreceptor cell patterns in the eyes of chickens. This is thought to be the case because the light-sensitive cells in chicken or other bird eyes cannot easily attain an optimal crystalline arrangement but instead form a disordered configuration that is as uniform as possible. Indeed, it is the remarkable property of "mulithyperuniformity" of the avian cone patterns, that enables birds to achieve acute color sensing.

Disordered hyperuniformity was recently discovered in amorphous 2‑D materials, which was shown to enhance electronic transport in the material.
It may also emerge in the mysterious biological patterns known as fairy circles - circle and patterns of circles that emerge in arid places.

Making disordered, but highly uniform, materials

The challenge of creating disordered hyperuniform materials is partly attributed to the inevitable presence of imperfections, such as defects and thermal fluctuations. For example, the fluctuation-compressibility relation dictates that any compressible one-component fluid in thermal equilibrium cannot be strictly hyperuniform at finite temperature.

Recently Chremos & Douglas (2018) proposed a design rule for the practical creation of hyperuniform materials at the molecular level.
Specifically, effective hyperuniformity as measured by the hyperuniformity index is achieved by specific parts of the molecules (e.g., the core of the star polymers or the backbone chains in the case of bottlebrush polymers).

The combination of these features leads to molecular packings that are highly uniform at both small and large length scales.

Non-equilibrium hyperuniform fluids and length scales

Disordered hyperuniformity implies a long-ranged direct correlation function (the Ornstein–Zernike equation). In an equilibrium many-particle system, this requires delicately designed effectively long-ranged interactions, which are not necessary for the dynamic self-assembly of non-equilibrium hyperuniform states. In 2019, Ni and co-workers theoretically predicted a non-equilibrium strongly hyperuniform fluid phase that exists in systems of circularly swimming active hard spheres, which was confirmed experimentally in 2022.

This new hyperuniform fluid features a special length scale, i.e., the diameter of the circular trajectory of active particles, below which large density fluctuations are observed. Moreover, based on a generalized random organising model, Lei and Ni (2019) formulated a hydrodynamic theory for non-equilibrium hyperuniform fluids, and the length scale above which the system is hyperuniform is controlled by the inertia of the particles. The theory generalizes the mechanism of fluidic hyperuniformity as the damping of the stochastic harmonic oscillator, which indicates that the suppressed long-wavelength density fluctuation can exhibit as either acoustic (resonance) mode or diffusive (overdamped) mode.

See also

* Crystal
* Quasicrystal
* Amorphous solid
* State of matter

References

External links

*
* {{cite magazine , url=https://www.quantamagazine.org/a-chemist-shines-light-on-a-surprising-prime-number-pattern-20180514/ , title=A chemist shines light on a surprising prime number pattern , first=Natalie , last=Wolchover , magazine=Quanta Magazine

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