{"paper":{"title":"On the asymptotic behavior at the kinetic time of a weakly interacting Fermi gas","license":"http://creativecommons.org/licenses/by/4.0/","headline":"For a weakly interacting Fermi gas starting near equilibrium, the leading decay of two-point time correlations at kinetic times is fixed exactly by the collision frequency of the Boltzmann-Nordheim operator.","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Herbert Spohn, Minh-Binh Tran, Peter S. Madsen, Phan Th\\`anh Nam","submitted_at":"2026-05-13T13:21:43Z","abstract_excerpt":"This paper is devoted to the dynamics of a weakly interacting Fermi gas at the kinetic time regime $t\\sim \\lambda^{-2}$ where $\\lambda \\ll 1$ is the strength of the interaction potential. We prove that if the initial state is close to equilibrium, then the two-point time correlation function of the many-body quantum dynamics can be computed effectively. In fact, we show that its leading order behavior is determined completely by the collisional frequency of the Boltzmann-Nordheim collision operator at equilibrium. This settles a prediction by Lukkarinen-Spohn, and thus gives a justification of"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we show that its leading order behavior is determined completely by the collisional frequency of the Boltzmann-Nordheim collision operator at equilibrium. This settles a prediction by Lukkarinen-Spohn, and thus gives a justification of the quantum Boltzmann equation from many-body quantum mechanics.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The initial state is close to equilibrium; the interaction strength λ is small, and the analysis is restricted to the kinetic time regime t ∼ λ^{-2}.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The leading-order asymptotic behavior of the two-point correlation function for a weakly interacting Fermi gas at kinetic times is exactly the collisional frequency of the equilibrium Boltzmann-Nordheim collision operator.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For a weakly interacting Fermi gas starting near equilibrium, the leading decay of two-point time correlations at kinetic times is fixed exactly by the collision frequency of the Boltzmann-Nordheim operator.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"797c8b2592fed9cb782be81b7cc42c2c34146586b30c638ac00c462ce5c59eeb"},"source":{"id":"2605.13499","kind":"arxiv","version":1},"verdict":{"id":"ab77bea6-f944-42db-a7a7-d47bf1941ec0","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:03:30.901948Z","strongest_claim":"we show that its leading order behavior is determined completely by the collisional frequency of the Boltzmann-Nordheim collision operator at equilibrium. This settles a prediction by Lukkarinen-Spohn, and thus gives a justification of the quantum Boltzmann equation from many-body quantum mechanics.","one_line_summary":"The leading-order asymptotic behavior of the two-point correlation function for a weakly interacting Fermi gas at kinetic times is exactly the collisional frequency of the equilibrium Boltzmann-Nordheim collision operator.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The initial state is close to equilibrium; the interaction strength λ is small, and the analysis is restricted to the kinetic time regime t ∼ λ^{-2}.","pith_extraction_headline":"For a weakly interacting Fermi gas starting near equilibrium, the leading decay of two-point time correlations at kinetic times is fixed exactly by the collision frequency of the Boltzmann-Nordheim operator."},"references":{"count":21,"sample":[{"doi":"","year":1975,"title":"R. Balescu. Equilibrium and nonequilibrium statistical mechanics.NASA STI Recon Technical Report A, 76, 1975","work_id":"d0710276-b2de-4507-b4ff-cfe60eb0f0fd","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"Y. Deng and Z. Hani. Full derivation of the wave kinetic equation.Inventiones Mathematicae, 233(2):543–724, 2023","work_id":"5bba6081-39c6-464a-bf1e-594bc80d3b19","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"Y. Deng and Z. Hani. Long time justification of wave turbulence theory. arXiv e-print, arXiv:2311.10082, 2023","work_id":"34492de4-92c1-42a8-ab55-4035d73130f0","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Long time derivation of the boltzmann equation from hard sphere dynamics","work_id":"a505e4c3-d5ab-4437-bff0-13e1bd253fd3","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2004,"title":"L. Erd¨ os, M. Salmhofer, and H.-T. Yau. On the quantum boltzmann equation.Journal of Statistical Physics, 116:367–380, 08 2004","work_id":"23e4e7c6-68ef-44b1-9d64-e4d1e18acc2f","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":21,"snapshot_sha256":"7c3afb20a18ca28cdcb36418929a29686da8020775340638f5dad93e5674b3b0","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"}