Spatial self-organization driven by temporal noise

TL;DR

Temporal noise with short memory can drive spatial self-organization and hyperuniformity in interacting particle systems, with anti-correlated noise ($c<0$) suppressing long-wavelength density fluctuations below a crossover length that diverges as $c \to -\tfrac{1}{2}$. A fluctuating-hydrodynamics framework derived from Dean's equation quantitatively predicts the structure factor $\tilde{S}(\tilde{k}) = (1+2c) + [B(1+2c) - 2c] \tilde{k}^2$, linking temporal correlations to hyperuniformity via a tunable $\tilde{l}_c = \sqrt{ B - \tfrac{2c}{1+2c} }$. Recasting the dynamics as a stochastic optimization problem shows temporal noise biases the system toward deeper and flatter minima, and combining temporal noise with SGD-like selection noise (SPGD) yields superior minima, mirroring perturbed gradient descent behavior in neural networks. The work establishes temporal correlations as a general mechanism for noise-driven self-organization with broad implications for materials design and privacy-preserving learning, and reveals deep connections between non-equilibrium statistical physics and optimization landscapes.

Abstract

The counterintuitive emergence of order from noise is a central phenomenon in science, ranging from pattern formation and synchronization to order-by-disorder in frustrated systems. While large-scale spatial self-organization induced by local spatial noise is well studied, whether temporal noise can also drive such organization remains an open question. Here, by studying interacting particle systems, we show that temporally correlated noise can lead to a self-organized state with suppressed long-range density fluctuations, or hyperuniformity. Further, we develop a fluctuating hydrodynamic theory that quantitatively explains the origin of this phenomenon. Finally, by casting the dynamics as a stochastic optimization problem, we show that temporal correlations lead to better solutions, akin to perturbed gradient descent in neural networks -- where noise is injected during training to escape poor minima. This reveals a striking correspondence between perturbed gradient descent dynamics on the energy landscapes of particle systems and the loss landscapes of neural networks. Our study establishes temporal correlations as a novel mechanism for noise-driven self-organization, with broad implications for self-assembling materials, biological systems, and optimization algorithms that leverage temporal noise for applications like differentially private learning.

Figures (3)

• Figure 1: Interacting particle systems driven by temporally correlated noise. (a) Schematic of the system showing passive particles immersed in a non-equilibrium bath (light blue). Black arrows denote deterministic pairwise interactions, while blue arrows represent noise from the bath. (b) Schematic of the correlation structure of the moving-average $q$ (MA($q$)) noise, where $q$ denotes the number of time steps with non-zero correlations. For $q=0$, the noise is white, reducing the system to an equilibrium interacting Brownian particle system. For $q>0$, the noise is temporally correlated, driving the system out of equilibrium. (c) Dynamics of a single particle driven by MA($1$) noise. Noise correlation structures for strongly anti-correlated ($c=-1/2$), uncorrelated ($c=0$), and strongly correlated ($c=1/2$) cases (Top). Exemplar particle trajectories from simulations for each case (Bottom). Here, $c$ is the Pearson correlation coefficient between noise at times $t$ and $t+1$ (Eq. \ref{['eq:pcc_maq']}). Zoomed in sections of the particle trajectories highlight that strongly correlated noise tends to produce persistent motion in one direction, whereas strongly anti-correlated noise causes frequent reversals.
• Figure 2: Emergent long-range structure. (a) Coarse-grained density fluctuations $\delta \rho(c)/|\delta \rho_{\text{avg}}(c=0)|$ from particle simulations, where $c$ is the noise correlation coefficient and $\delta \rho_{\text{avg}}$ is the average density fluctuation of the whole system. Panels show $c=-0.5$, $-0.4$, and $0$ (left to right). The system was coarse-grained with a Gaussian kernel of width $80R$, where $R$ is the particle radius. As $c$ decreases from $0$ (uncorrelated) to $-0.5$ (strongly anti-correlated), density fluctuations are progressively suppressed. (b) Normalized radially averaged structure factor $\Tilde{S}(\Tilde{k})$ versus normalized wave number $\Tilde{k}$. Here, $\tilde{S} = S(k)/S_{0}(2\pi/L)$, where $S_{0}(2\pi/L)$ is the structure factor for $c=0$ at $k = 2\pi/L$, and $L$ is the simulation box side length. $\tilde{k} = k/k_0$, where $k_0$ is the value at which $\tilde{S}(k_0) = 1$ for $c=-0.5$. Solid black lines show predictions of Eq. \ref{['eq:struc_fact']} for different $c$. Inset: Normalized crossover length scale ($l_c/l_0 = \tilde{l}_c = 1/\tilde{k}_c$) versus $c$. The normalized crossover wavenumber ($\tilde{k}_c$) is obtained in simulations from the intersection, on a log-log plot, between a slope $0$ fit near $\tilde{k} \to 0$, and a slope $2$ fit near $\tilde{k} \approx 1$. Solid black line in the inset shows the prediction of Eq. \ref{['eq:l_c']}. Gray shaded regions indicate short-range behavior ($\tilde{k}>1$). (c) Normalized number density $\tilde{\delta \rho}^2 (\tilde{l})$ versus normalized hypersphere diameter ($\Tilde{l}$) used for measuring density fluctuations. $\tilde{\delta \rho}^2 (\tilde{l}) = \delta \rho^2 (l)/\delta \rho^2 (l_0)$ where $\delta \rho^2 (l_0)$ is the density variance for $c=0$ at $l=l_0$, and $\tilde{l}=l/l_0$, with $l_0 = 2\pi/k_0$. Top inset: Data collapse of density variances upon rescaling $\tilde{l}$ by $\tilde{l}_c$ and $\tilde{\delta \rho}^2 (\tilde{l})$ by $\tilde{\delta \rho}^2 (\tilde{l}_c)$. Bottom inset: Infinite wavelength density fluctuations $\hat{\delta \rho}^2 (\tilde{l} \to \infty)$ versus $c$, where $\hat{\delta \rho}^2 (c) = [\tilde{\delta \rho}^2 (c) - \tilde{\delta \rho}^2 (c=-0.5)]/\tilde{\delta \rho}^2 (c=0)$. Solid black line denotes the prediction $1+2c$ from Eq. \ref{['eq:struc_fact']} in the $\tilde{k} \to 0$ limit. Gray shaded regions indicate short-range behavior ($\tilde{l}<1$). Circles in (b) and (c) denote particle simulations.
• Figure 3: Noise-driven exploration of energy landscape by perturbed gradient descent (PGD). (a) Schematic of time evolution towards an energy minimum. Starting from a random initial condition, the system undergoes noisy dynamics according to Eq. \ref{['eq:discrete_pgd']}, until it reaches a steady-state. The system is then quenched via (noiseless) gradient descent (GD), until it reaches an energy minimum. The system has an energy $E$ at the minimum and an energy $E + \Delta E$ after applying a small perturbation to the system. (b) Schematic of a generic energy landscape with multiple minima and maxima. "Good" minima are deeper and flatter, corresponding to low $E$ and $\Delta E$. (c) Normalized $E$ and $\Delta E$ versus noise correlation $c$ in particle simulations. Both quantities are normalized by the corresponding values from (noiseless) GD. Inset: Schematic of a single-particle motion without interactions. From time-step $t$ (black arrow) to $t+1$ (red arrow), for noise correlation: (i) $c = -0.5$, a particle typically reverses direction, (ii) $c = 0$, the motion is random, and (iii) $c=0.5$, the motion typically persists in the same direction. (d) Normalized $E$ and $\Delta E$ versus noise magnitude $\sigma$ in particle simulations. Both quantities are normalized by the corresponding values from (noiseless) GD ($\sigma = 0$). Insets: Schematic of a single-particle motion without interactions. (e) Comparison of (noiseless) GD and three different noisy dynamics (Left). Black arrows indicate deterministic pairwise interactions coming from the interaction potential, while blue arrows denote noise coming from the bath. Blue crosses indicate selection noise of stochastic gradient descent (SGD) (Eq. \ref{['eq:discrete_sgd_SI']}, SI Sec. IV). Stochastic perturbed gradient descent (SPGD) combines temporal noise from PGD with selection (or spatial) noise from SGD (Eq. \ref{['eq:discrete_spgd_SI']}, SI Sec. V). Light blue background denotes systems with temporal noise, originating from a non-equilibrium bath. Normalized $E$ and $\Delta E$ for the different dynamics in particle simulations (Right). Both quantities are normalized by the corresponding values from (noiseless) GD.