Spatial self-organization driven by temporal noise

The counterintuitive emergence of order from noise is a central phenomenon in science and mathematics, with examples ranging from pattern formation and synchronization to order-by-disorder in frustrated systems. While self-organization driven by spatial noise is well-studied, the role of temporal noise remains poorly understood. Can short-memory temporal noise---alone---drive long-range spatial self-organization? We study interacting particle systems driven by temporal noise and discover that tuning temporal correlations leads 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.