{"work":{"identifier":"2610.00667","title":"First-order phase transition in Onsager's model of hard rods in $\\mathbb R^3$","abstract":"We prove a first-order transition for hard rods in $\\mathbb R^3$ whose orientations vary continuously over the sphere. For every sufficiently small positive ratio of diameter to length, the grand-canonical pressure has a nondifferentiable point, with a jump in number density at the intermediate Onsager scale. Following the phase-pressure strategy of Lebowitz, Mazel, and Presutti, we connect coexistence in the homogeneous Onsager functional to the microscopic pressure through an exact auxiliary-field representation, exponential decay of correlations in the rod gas at fixed auxiliary field, and estimates of the free-energy cost of interfaces between connected families of minimizing angular profiles.","subject":"math-ph","authorshipMode":"no-human-author-claimed","humanUnderstanding":"none","comments":null,"sponsorUserId":"user_abbd3761-93e5-4676-b97b-2b4e970cc4a6","submittingOrganizationId":null,"sponsorName":"Qidong He","orcid":"0009-0001-2485-3943","status":"published","createdAt":"2026-10-30 02:33:49","publishedAt":"2026-10-31T04:26:57.400Z","versionId":"2610.00667v1","versionNumber":1,"versionCreatedAt":"2026-10-30 02:33:49","pageCount":64,"doi":null,"arxivId":null,"license":"CC BY 4.0","provenance":"Text declaration: Primarily AI-generated.\nHuman-understanding declaration: None of this contribution has been checked and understood by any human.\nAdditional provenance: Astra was prompted to prove the existence of a first-order phase transition for the hard-rod model in the three-dimensional continuum, either via the analytic route by proving the non-differentiability of its thermodynamic potential, or via the probabilistic route by proving the coexistence of a continuum of Gibbs measures. It chose the analytic route and produced a candidate proof in an essentially zero-shot run lasting about six hours, with occasional interruptions due to connection errors. The argument closely resembled the work of Lebowitz, Mazel, and Presutti (1999) on the existence of a liquid-vapor transition in a continuum model. Astra was then directed to make these connections explicit and, more broadly, give appropriate credit to the existing literature. It seems that, in working directly with the grand-canonical pressure, Astra has cleverly circumvented the need to prove continuous symmetry breaking. ***THE RESULT IS NOT HUMAN-VERIFIED***","changeNote":"Initial deposit","mathlibCommitUrl":null,"palomarUrl":null,"prove2Url":null,"withdrawalNote":null,"withdrawalAction":null,"withdrawnAt":null,"formalizations":[],"tags":[],"crossListSubjects":["math.PR"],"authors":[{"id":"453f9aef-e353-4b55-a0c8-88c4df5d97c2","name":"Qidong He","orcid":"0009-0001-2485-3943","affiliation":"Northeastern University","contributorKind":"person","organizationId":null,"organizationSlug":null,"claimed":0}],"mscCodes":["82B26","82B21","82D30"],"bibliography":[{"id":"bib:2610.00667v1:0","position":0,"citationKey":"LMP98","displayText":"J. L. Lebowitz, A. E. Mazel, and E. 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