{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"743bd82514f7452edecb40b369ca8297407fde06ee58a6b022c40685946cb830","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"49bbfcf747faecf3c02496edee322bdff699af3712a512c3f2b2990aaa71f472","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.kam-stability.diffusion-time","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"Nekhoroshev proved that drift in nearly regular systems takes at least an exponentially long time. Whether typical systems actually drift that fast, and with which exponent, is open.","posed_since":"","precise":"For analytic $H = h(I) + \\epsilon f$ with $h$ convex in $n$ degrees of freedom, Nekhoroshev gives $\\left|I(t) - I(0)\\right| \\le C \\epsilon^b$ for $\\left|t\\right| \\le \\exp(c \\epsilon^{-a})$ with optimal $a = 1/(2n)$. Prove that for generic $f$ there are orbits drifting by order one in time $\\exp(C \\epsilon^{-1/(2n)})$, matching the upper bound. Answer: a proof of the matching lower bound or a counter-argument.","problem_ref":null,"references":"","settled_by":"A construction of diffusing orbits with time $\\operatorname{exp}(C \\epsilon^{-1/(2n)})$ for an open set of perturbations.","status_note":"Specific analytic perturbations with diffusion time exponentially long in a power of 1/eps near the Nekhoroshev exponent were built by Ke Zhang (Inventiones Mathematicae, 2011); matching bounds for generic f are open.","title":"Optimal diffusion time compared with the Nekhoroshev stability 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