{"paper":{"title":"The Brownian motion as the limit of a deterministic system of hard-spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.PR"],"primary_cat":"math.AP","authors_text":"Isabelle Gallagher, Laure Saint-Raymond, Thierry Bodineau","submitted_at":"2013-05-15T09:12:54Z","abstract_excerpt":"We provide a rigorous derivation of the brownian motion as the limit of a deterministic system of hard-spheres as the number of particles $N$ goes to infinity and their diameter $\\varepsilon$ simultaneously goes to $0$, in the fast relaxation limit $\\alpha = N\\varepsilon^{d-1}\\to \\infty $ (with a suitable diffusive scaling of the observation time). As suggested by Hilbert in his sixth problem, we rely on a kinetic formulation as an intermediate level of description between the microscopic and the fluid descriptions: we use indeed the linear Boltzmann equation to describe one tagged particle in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.3397","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}