{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"399592dd17beba452f2dcf26fcf96f1dc003bf035c249f97f6ab8972bca4b8b8","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"01d4bfd7a8b0e497656c11ddf7f86983eaa70a85216d853e8ad0862a29d52dd4","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.kinetic-limits.fourier-law","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"Heat flow through a solid is proportional to the temperature gradient, but nobody has proven this from the reversible equations of motion of the atoms. In one dimension, numerics show the law fails.","posed_since":"","precise":"Lattice of anharmonic oscillators in $d = 3$ (e.g. FPUT-beta) with Hamiltonian bulk dynamics and boundary heat baths at $T_{1}$ and $T_{2}$. Prove that the stationary heat current satisfies $J = \\kappa (T_{1} - T_{2})/L + o(1/L)$ with finite $\\kappa > 0$ as $L \\to \\infty$. Answer: a proof, or a proof of anomalous scaling.","problem_ref":null,"references":"","settled_by":"A proof of finite thermal conductivity for one deterministic 3D lattice model.","status_note":"Posed as a challenge by Bonetto, Lebowitz and Rey-Bellet (2000); results exist mainly for models with stochastic noise added to the dynamics.","title":"Fourier's law of heat conduction for a deterministic anharmonic 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