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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.18653 |
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| _version_ | 1866915758523547648 |
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| author | Halperin, Igor |
| author_facet | Halperin, Igor |
| contents | 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 $τ_c \approx 18$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_18653 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Order Out of Noise and Disorder: Fate of the Frustrated Manifold Halperin, Igor Adaptation and Self-Organizing Systems Disordered Systems and Neural Networks Soft Condensed Matter Statistical Mechanics High Energy Physics - Theory 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 $τ_c \approx 18$, 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. |
| title | Order Out of Noise and Disorder: Fate of the Frustrated Manifold |
| topic | Adaptation and Self-Organizing Systems Disordered Systems and Neural Networks Soft Condensed Matter Statistical Mechanics High Energy Physics - Theory |
| url | https://arxiv.org/abs/2601.18653 |