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

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.