{"ID":2895903,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.07343","arxiv_id":"2507.07343","title":"Way More Than the Sum of Their Parts: From Statistical to Structural Mixtures","abstract":"We show that mixtures comprised of multicomponent systems typically are much more structurally complex than the sum of their parts; sometimes, infinitely more complex. We contrast this with the more familiar notion of statistical mixtures, demonstrating how statistical mixtures miss key aspects of emergent hierarchical organization. This leads us to identify a new kind of structural complexity inherent in multicomponent systems and to draw out broad consequences for system ergodicity.","short_abstract":"We show that mixtures comprised of multicomponent systems typically are much more structurally complex than the sum of their parts; sometimes, infinitely more complex. We contrast this with the more familiar notion of statistical mixtures, demonstrating how statistical mixtures miss key aspects of emergent hierarchical...","url_abs":"https://arxiv.org/abs/2507.07343","url_pdf":"https://arxiv.org/pdf/2507.07343v1","authors":"[\"James P. Crutchfield\"]","published":"2025-07-10T00:01:53Z","proceeding":"cond-mat.stat-mech","tasks":"[\"cond-mat.stat-mech\",\"cs.LG\",\"math.DS\",\"math.ST\",\"nlin.CD\"]","methods":"[\"Generative Adversarial Network\"]","has_code":false}