Enhanced hyperuniformity from random reorganization

Significance

Particles in sheared suspensions rearrange due to collisions, until they find positions where they no longer collide. At this point, the system stops evolving. Above a critical density/strain, they can no longer find such an absorbing state, and activity never ceases. At this critical point, the systems are hyperuniform, with number fluctuations in a volume of size ${\mathrm{\ell }}^d$ scaling as $<\delta N^2>\sim {\mathrm{\ell }}^{\lambda },\lambda <d$, slower than random (diffusive), where $<\delta N^2>\sim {\mathrm{\ell }}^d$. Does hyperuniformity survive noise and thermal diffusion? We find that noise enhances the hyperuniformity toward the most uniform, $<\delta N^2>\sim {\mathrm{\ell }}^{d-1}$ near that of crystals. Practically, this provides a simple means of fabricating noncrystalline materials of arbitrary density and high homogeneity.

Abstract

Diffusion relaxes density fluctuations toward a uniform random state whose variance in regions of volume $v={\mathrm{\ell }}^d$ scales as ${\sigma }_{\rho }^2\equiv ⟨{\rho }^2(\mathrm{\ell })⟩-{⟨\rho ⟩}^2\sim {\mathrm{\ell }}^{-d}$. Systems whose fluctuations decay faster, ${\sigma }_{\rho }^2\sim {\mathrm{\ell }}^{-\lambda }$ with $d<\lambda \le d+1$, are called hyperuniform. The larger $\lambda$, the more uniform, with systems like crystals achieving the maximum value: $\lambda =d+1$. Although finite temperature equilibrium dynamics will not yield hyperuniform states, driven, nonequilibrium dynamics may. Such is the case, for example, in a simple model where overlapping particles are each given a small random displacement. Above a critical particle density ${\rho }_c$, the system evolves forever, never finding a configuration where no particles overlap. Below ${\rho }_c$, however, it eventually finds such a state, and stops evolving. This “absorbing state” is hyperuniform up to a length scale $\xi$, which diverges at ${\rho }_c$. An important question is whether hyperuniformity survives noise and thermal fluctuations. We find that hyperuniformity of the absorbing state is not only robust against noise, diffusion, or activity, but that such perturbations reduce fluctuations toward their limiting behavior, $\lambda \to d+1$, a uniformity similar to random close packing and early universe fluctuations, but with arbitrary controllable density.

One example of fascinating behavior exhibited by periodically driven systems is a phase transition between an active phase, where dynamics persist forever, and an absorbing phase, in which the system ceases to evolve. Such behavior has been studied extensively in a variety of theoretical models (1, 2), as well as in an experiment, in a sheared suspension of non-Brownian particles (3). For a given particle density in the experiment, if the shear amplitude is small enough, particles self-organize such that their trajectories during a cycle of the driving are reversible; when the system is strobed, the particles appear static. For large strain amplitude, the particles remain “active” and undergo diffusive-like random motion forever. This behavior is captured well by a model (4) in which active particles—those that would collide during a strain cycle—are displaced by a random amount at each time step.

The phase transition observed in these systems has the characteristics of a continuous phase transition with diverging length and time scales at the critical point (4). However, in contrast to the usual large fluctuations observed at equilibrium critical points, in this case, the density fluctuations at criticality are greatly diminished (5), with the system becoming hyperuniform (6, 7). This is revealed in the small $k$ behavior of the structure factor $S(\overrightarrow{k})=\frac{1}{N}{|{\sum }_{i}exp(-i\overrightarrow{k}\cdot \overrightarrow{r_i})|}^2$, since $S(k\to \mathrm{\hspace{0.17em}0})=\frac{⟨N^2⟩-{⟨N⟩}^2}{⟨N⟩}\to \mathrm{\hspace{0.17em}0}$ for hyperuniform systems. It follows (8) that if $S(k\to 0)\sim k^{\lambda -d}$, then ${\sigma }_{\rho }^2\sim {\mathrm{\ell }}^{-\lambda }$. In the case where $\lambda =d+1$, so that $S(k)\sim k$, the density fluctuations develop a logarithmic correction (8), with ${\sigma }_{\rho }^2\sim {\mathrm{\ell }}^{-(d+1)}\log \mathrm{\ell }$.

Two models with absorbing-state transitions are shown schematically in Fig. 1. In the Manna model (Fig. 1A), we begin from some distribution of particles on the sites of a lattice (in arbitrary dimension). A site is considered active if it has more than a defined number of particles (e.g., two), and at each time step the dynamics moves all of the particles in active sites randomly to neighboring sites. The random organization (RandOrg) model, shown in Fig. 1B, is a continuous model where, at each time step, particles that overlap are given a small random displacement $|\delta r|<\mathrm{\hspace{0.17em}2}a$, where $a$ is a particle diameter. In both models, there is a critical density, ${\rho }_c$ (or strain, ${\gamma }_c$) which separates absorbing states from active ones. In these models, there is also a density, ${\rho }_{geom}$, above which no absorbing states exist. Whereas ${\rho }_{geom}$ is only determined by geometry, ${\rho }_c$ is determined both by geometry and model dynamics, and, importantly, ${\rho }_c<{\rho }_{geom}$. Even in the active phase, for ${\rho }_c<\rho <{\rho }_{geom}$, there is an extensive number of absorbing states that are, however inaccessible to the system; in this sense, the system is not ergodic (9).

Fig. 1.

Two absorbing-state models in the same universality class. (A) Manna model in one dimension. A site is considered active if it has $z>\mathrm{\hspace{0.17em}2}$ particles. All particles in an active site (shown in green) are moved randomly to the right or left, resulting in a new configuration. The dynamics are then repeated. For this model, ${\rho }_{geom}=\mathrm{\hspace{0.17em}2}$, and ${\rho }_c\approx \mathrm{\hspace{0.17em}1.6718}$. (B) One-dimensional RandOrg model with no strain. Each overlapping particle is given a small random displacement, resulting in a new configuration, and the dynamics are repeated. Here, ${\rho }_{geom}=\mathrm{\hspace{0.17em}1}$ and ${\rho }_c=\mathrm{\hspace{0.17em}.91}$. Both models can be immediately extended to any dimension, and the RandOrg model may be generalized to activity when overlaps occur on the application of periodic strain.

The goal of this work is to study the effects of weak noise on these models. By weak, we mean that when the system is active, particle motion is dominated by the activity rather than the noise. With noise, the strict notion of absorbing states no longer exists. We will return to the question of when noise is “weak” toward the end of the paper. Our central question is: How does noise affect the density fluctuations, and, in particular, is the hyperuniformity destroyed? Surprisingly, we find that not only does weak noise not destroy hyperuniformity, it enhances it in the absorbing phase. In fact, the disordered configurations obtained tend toward the limiting value $\lambda \to d+1$, with fluctuations only logarithmically different from those of a slightly randomized periodic lattice.

The analysis we present here is based on numerical study of the Manna model (1, 2), with local activity if a site is occupied by more than two particles, and the isotropic zero strain (that is, overlapping disks are active) version of the RandOrg model (4), both in two dimensions. For the former, ${\rho }_{geom}=\mathrm{\hspace{0.17em}2},{\rho }_c\simeq \mathrm{\hspace{0.17em}1.306}$, whereas for the latter, ${\rho }_{geom}=\pi /\sqrt{12},{\rho }_c\simeq \mathrm{\hspace{0.17em}0.3262}$, in units where the disk diameter is unity. In the absence of noise, as $\rho \to {\rho }_c$, both systems become hyperuniform (5), with ${\sigma }_{\rho }^2\sim {\mathrm{\ell }}^{-2.45}$ and $S(k\to \mathrm{\hspace{0.17em}0})\sim k^{0.45}$.

The main result of this work is that, rather than increasing the fluctuations in the system, noise makes the absorbing phase more uniform. To simulate the effect of weak noise in the Manna model, we allow evolution to an absorbing state, and then, at a fraction $f$ of randomly chosen occupied sites, a particle is moved to a random neighboring site. This typically reactivates the system, which we then let evolve under its dynamics to a new absorbing state. We repeat this reactivation again and again, and study the order of the system as a function of the number of reactivations. Changing the value of $f$ from 0.01 to 1 has little effect on the statistics of the particle distribution, so here we use $f=\mathrm{\hspace{0.17em}1}$. For the RandOrg model (4), we simulated noise in two ways: (i) by allowing particles to diffuse with a random step $\delta$ after each update cycle, where $\delta \ll \delta r$; and (ii) by reactivating after an absorbing state is reached. Both methods lead to the same results, but because the latter converges much faster, we present data from this procedure. Note that the reactivation process itself occurs on small length scales and does not modify the fluctuations on large scales.

In Fig. 2, the structure factor $S(k)$ and the variance ${\sigma }_{\rho }^2$ are shown for different numbers of reactivations. As previously found (5), for $\rho <{\rho }_c$, the system is hyperuniform on length scales up to a $\rho$-dependent correlation length ${\xi }_1$ and random at larger scales. As expected, applying a small number of reactivations affects only short length scales, with fluctuations on larger length scales being reduced as more iterations are applied. Reactivating the system not only leads to hyperuniformity on longer scales, but reduces the fluctuations: The system becomes more hyperuniform, with $\lambda -d$ approaching its maximum value, 1. Fig. 2 indicates an increase in uniformity as the number of reactivations increases, finally arriving at an asymptotic form where additional iterations have no effect. In particular, after many reorganization cycles, the density fluctuations decay, as ${\sigma }_{\rho }^2\sim \log (\mathrm{\ell }){\mathrm{\ell }}^{-d-1}\sim \log (\mathrm{\ell }){\mathrm{\ell }}^{-3.0}$. We term this iterated reactivation process Random Reorganization, or ReOrg. Although the Manna model is considerably faster for computation, the RandOrg model well describes a physical process that can be used to create isotropic metamaterials.

Fig. 2.

Structure factor $S(k)$ (A) and density fluctuations ${\sigma }_{\rho }^2$ (B). $S(k)$ and ${\sigma }_{\rho }^2$ as a function of length scale $\mathrm{\ell }$ for a 2D Manna model with activity for $>$ two particles per site for $\rho =\mathrm{\hspace{0.17em}1.28}<{\rho }_c\approx \mathrm{\hspace{0.17em}1.306}$. The system is reactivated by random particle displacement after an absorbing configuration is reached. Data are shown for 1, 10, 100, and 1,000 reactivation cycles, averaged over many windows in each of 50 samples. The system becomes more uniform the more reactivation cycles are performed.

Using the reactivation protocol, we can now create highly uniform isotropic particle packings over a range of densities. It is therefore worthwhile comparing the different sample preparations suggested by the present work. Fig. 3 shows a comparison of 2D particle patterns from a random Poisson process (allowing overlaps), isotropic RandOrg (zero strain), isotropic ReOrg (zero strain), and a square crystal. It is very difficult to discern the difference between RandOrg and ReOrg from their point patterns (Fig. 3, row 1) or their pair distribution functions $g(r)$ (Fig. 3, row 5). The nature of the fluctuations becomes more apparent when the density is averaged over a scale comparable to the correlation length,* ${\xi }_1$, as seen in Fig. 3 (row 2). The effects of reactivation are most clearly seen in the large-scale behavior of ${\sigma }_{\rho }^2$ (Fig. 3, row 4) and the small $k$ behavior of $S(k)$ (Fig. 3, row 3). At small $\mathrm{\ell }$, RandOrg (for $\rho <{\rho }_c$) is hyperuniform, with $\lambda =2.45$, crossing over at $\mathrm{\ell }>{\xi }_1$ into larger, Poissonian fluctuations characterized by $\lambda =2$. At the same overall density $\rho$, the reactivated system is hyperuniform, with ${\sigma }_{\rho }^2\sim {\mathrm{\ell }}^{-2.45}$ to larger $\mathrm{\ell }$, and then crosses over to ${\sigma }_{\rho }^2\sim {\mathrm{\ell }}^{-3}\log (\mathrm{\ell })$.

Fig. 3.

Comparison of random and three hyperuniform particle patterns. Rows from top to bottom are as follows: point pattern, density fluctuations on a scale of 100 particle spacings, structure factor, S(q), density fluctuations as a function of the size of the sampling circle, pair distribution function, g(r). Columns from left to right are as follows: random point pattern allowing overlaps, an absorbing-state configuration for the RandOrg model with no strain, a configuration of ReOrg model with no strain after 100 reactivation kicks and a crystal system. Note the comparison in row 4 between ${\sigma }_{\rho }^2$ column 2, and column 3. For the absorbing state just below threshold, the hyperuniformity seen at short length scales gives way to Poissonian fluctuations at large scales. For the reactivated system, the fluctuations at long length scales are further suppressed toward their minimum behavior.

The small $k$ behavior of $S(k)$ for the reactivated systems is particularly revealing, ultimately approaching a form with three regions, delineated by two values of the wavevector, $k_1$ and $k_2$, as seen in Figs. 4 and 5. (A) For $k>k_1$, $S(k)\sim k^{0.45}$, as occurs at criticality. (B) For $k_2<k<k_1$ $S(k)\sim k$. (C) For $k<k_2$, $S\left(k\right)$ reaches a plateau indicating that “$\sqrt{N}$” fluctuations occur at very large length scales. The length scale corresponding to $k_1^{-1}$ is the previously identified (5) correlation length, $k_1^{-1}={\xi }_1\sim {|{\rho }_c-\rho |}^{-0.8}$ familiar from the noiseless systems. The new length scale $k_2^{-1}={\xi }_2\sim {|{\rho }_c-\rho |}^{-1.1\pm 0.1}$ marks the maximal length scale for which the reactivation dynamics have an effect, and Fig. 5 indicates that ${\xi }_2$ diverges with a larger exponent than ${\xi }_1$ as the critical point is approached. Similar behavior is shown in Fig. 4 for both the discrete Manna model “kicked” and the continuum ReOrg models. In Fig. 6, we compare reactivation with weak diffusion for a 2D RandOrg system, and we see that the behavior of $S(k)$ is similar, with a similar enhancement of the hyperuniformity.

Fig. 4.

Data collapse of $S(k)$. Wavevector $k$ is scaled with ${\xi }_1\propto {\left|\mathrm{\Delta }\rho \right|}^{-{\nu }_{⟂}}$, with ${\nu }_{⟂}\approx \mathrm{\hspace{0.17em}0.8}$. (Left) For the discrete Manna model at different densities below ${\rho }_c\approx \mathrm{\hspace{0.17em}1.306}$ with activity for more than two particles per site after reactivations with 100 kicks. (Center) For the discrete Manna model in the slightly active region just above ${\rho }_c$. (Right) For the continuum RandOrg for densities slightly above ${\rho }_c$, where activity persists with no reactivation. The similarity of the three images indicates that, independent of model near ${\rho }_c$, the presence of active particles has a similar effect, whether from a small amount of activity, noise, or diffusion.

Fig. 5.

Scaling of the ${\xi }_1$ and ${\xi }_2$. (Upper) At high $k$ above and below (with kicks) the critical density, simulations of $S(k)$ collapse on scaling with $k\delta {\rho }^{-0.8}$, $\delta \rho =|\rho -{\rho }_c|$, indicating that ${\xi }_1\sim \delta {\rho }^{-0.8}$ as in Fig. 4. (Lower) A similar collapse occurs for small $k$, but with ${\xi }_2\sim \delta {\rho }^{-1.1}$. Note that the range of q between ${\xi }_1$ and ${\xi }_2$ with the higher value of $\lambda$ increases as $\rho \to {\rho }_c$. The results are for the Manna model in 2D.

Fig. 6.

Comparison of diffusion and discrete reactivation. $S(k)$ for RandOrg systems under discrete reactivation (blue) and diffusion (red). The diffusive system was 1,600 $\times$ 1,600, and the reactivated system was $800\times \mathrm{\hspace{0.17em}800}$, both at relative density $\delta \rho =-0.062$. The reactivated system was reactivated 300 times, whereas for the diffusive system, at each step, every particle was given a random displacement $\eta ={\mathrm{\hspace{0.17em}10}}^{-3}$ in units, where the particle diameter is unity. The system was given $8\times {\mathrm{\hspace{0.17em}10}}^5$ such steps.

Interestingly, as pointed out in refs. 10 and 11, above the critical density, where a small amount of activity persists, fluctuations are suppressed, and a linear scaling regime has been observed, $S(k)\sim k^{1.0}$. (As discussed above, we find $S(k)\sim k^{1.0}$ in the range $k_2<k<k_1$.) The similarities between the three panels of Fig. 4 suggest that, near the critical point, the small fraction of active particles act as an effective noise. Indeed, this highlights the different ways the system approaches the critical point from above and below the critical density, as demonstrated by looking at $S(k\to \mathrm{\hspace{0.17em}0})$. For $\rho <{\rho }_c$, $S(k\to \mathrm{\hspace{0.17em}0})$ depends on the initial configuration. If the initial configuration is random, then for length scales greater than ${\xi }_1,k<k_1$, $S(k\to \mathrm{\hspace{0.17em}0})\sim k^0\sim$ constant. If the initial configuration is hyperuniform, the $S(k\to \mathrm{\hspace{0.17em}0})$ behavior reflects the initial state, e.g., for a slightly disordered lattice, $S(k)\sim k^2$ for $k<k_1$ (8). For $\rho >{\rho }_c$, where the system is always active, there is no memory of the initial configuration. As in Figs. 4 and 5, above the largest length scale, ${\xi }_2,k<k_2,S(k\to \mathrm{\hspace{0.17em}0})\sim k^0$. The presence of noise below the transition, $\rho <{\rho }_c$, causes the density fluctuations to mimic the behavior of the activity above the transition and erase the memory of the initial configuration. At criticality, both length scales, ${\xi }_1$ and ${\xi }_2$, diverge, and the systems are hyperuniform, with $\lambda =\mathrm{\hspace{0.17em}2.45}$, $S(k\to \mathrm{\hspace{0.17em}0})\sim k^{0.45}$. Consider the hysteresis that results from these memory effects in the absence of noise. Starting from a random initial configuration, we take two paths to a final state, with $\rho \lesssim {\rho }_c$. Approaching this state from below ${\rho }_c$, we find $S(k)\sim k^{0.45},k_1<k,S(k)\sim k^0,k<k_1$. Approaching this state from above ${\rho }_c$, we find $S(k\to \mathrm{\hspace{0.17em}0})\sim k^{0.45}$ for both $k>k_1$ and for $k<k_1$.

In Fig. 6, we show $S(k)$ for 2D RandOrg systems with reactivation and with weak diffusion, with both showing similar enhancement of hyperuniformity. This leads to the question of how large diffusion can be before it has the opposite effect. As a rule of thumb, for the diffusion to enhance the hyperuniformity, there should be no more than one pair of particles excited by the noise in a correlation volume ${\xi }_1^d$ in the time $\tau$ that it takes for such noise-induced activity to die out. (This may be taken to be the sense in which diffusion is weak.) Otherwise, strong diffusion will lead to a steady-state fraction of active particles similar to what is seen deep in the active phase, with hyperuniformity being similarly washed out. We note that the above does not apply to noise, which does not continuously reactivate the system, such as if we were to move some fraction of the particles each time a new absorbing state is reached. This sort of discrete reactivation will lead to enhanced hyperuniformity, as discussed above.

A general feature of systems with absorbing-state dynamics is that they are hyperuniform at criticality. Hyperuniformity has been demonstrated to influence other physical properties, such as the existence of band gaps in excitation spectra (12, 13). There are now several examples of physical systems where periodic strain or flow have been shown to organize the systems into absorbing states (14, 15). The present study indicates that intrinsic or imposed noise reduces, rather than enhances, the density fluctuations, and can produce particle configurations with near-maximal large-scale isotropic uniformity comparable to that observed in random close packing (16) and early universe fluctuations (7). In periodically sheared suspensions, the control parameter (at given density) is the strain amplitude that allows one to create such highly uniform structure at arbitrary density. Although such systems are intrinsically diffusive, they produce configurations and dynamics (such as nonergodicity) that differ greatly from thermal diffusion. We expect that further investigations will show similar striking differences in the power spectra, temporal correlations, and other nonequilibrium statistics.