Spatial self-organization driven by temporal noise

TL;DR

This paper demonstrates that temporally correlated noise can induce spatial self-organization and hyperuniformity in particle systems, with implications for materials, biological systems, and optimization algorithms.

Contribution

It introduces the concept that temporal noise correlations can drive self-organization, supported by a hydrodynamic theory and analogy to neural network training.

Findings

Temporal noise leads to hyperuniformity in particle systems.

A fluctuating hydrodynamic theory explains the phenomenon.

Temporal correlations improve optimization solutions, similar to neural network training.

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.