Order Out of Noise and Disorder: Fate of the Frustrated Manifold

TL;DR

This paper investigates how random interactions and thermal noise cause particles on curved surfaces to spontaneously form quasi-one-dimensional structures, revealing disorder-induced symmetry breaking and connections to various physical theories.

Contribution

It introduces a model of frustrated Brownian particles on curved surfaces showing disorder-driven dimension reduction and symmetry breaking, with novel dynamical properties and broad theoretical implications.

Findings

Particles form bands, rings, and clusters depending on surface topology.

Spontaneous symmetry breaking involves slow evolution of the order parameter.

The system exhibits a diffusive precession of the structure normal, akin to Nambu-Goldstone modes.

Abstract

We study Langevin dynamics of $N$ Brownian particles on two-dimensional Riemannian manifolds, interacting through pairwise potentials linear in geodesic distance with quenched random couplings. These \emph{frustrated Brownian particles} experience competing demands of random attractive and repulsive interactions while confined to curved surfaces. We consider three geometries: the sphere $S^2$, torus $T^2$, and bounded cylinder. Our central finding is disorder-induced dimension reduction with spontaneous rotational symmetry breaking: order emerges from two sources of randomness (thermal noise and quenched disorder), with manifold topology determining the character of emerging structures. Glassy relaxation drives particles from 2D distributions to quasi-1D structures: bands on $S^2$, rings on $T^2$, and localized clusters on the cylinder. Unlike conventional symmetry breaking, the symmetry-breaking direction is not frozen but evolves slowly via thermal noise. On the sphere, the structure normal precesses diffusively on the Goldstone manifold with correlation time $\tau_c \approx 18$, a classical realization of type-A dissipative Nambu-Goldstone dynamics. The model requires no thermodynamic gradients, no fine-tuning, and no slow external input. We discuss connections to spin glass theory, quantum field theory, astrophysical structure formation, and self-organizing systems. The model admits a large-$N$ limit yielding statistical field theory on Riemannian surfaces, while remaining experimentally realizable in colloidal and soft matter systems.