{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"fe59f1859f0ebbc9a53fa98ecb7cf4da1117979cc8f0f962021e66b6cc14fc16","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["2c9684024cce20a0c9050aaf7f0a9e029a707487b6951500e5a44cd0504281b4","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"6fbe03f303789d63c5641766543282c3548ad52b1d25e2399aa6e96c5c6e5fe5","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":"2","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":"2c9684024cce20a0c9050aaf7f0a9e029a707487b6951500e5a44cd0504281b4","references":"","settled_by":"A construction of diffusing orbits with time $\\exp(C \\epsilon^{-1/(2n)})$ 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 bound","topic_ref":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"a9f1dff11f55f9a56aadd6948cb0c3ffa90045f0b6ffae0a84a8fd05222fc599","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"ccef23ef263980fa92a9f783c96d443169228096b1b128f029ef05b2cf32db00","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"12KZe2YXNDbq_sLs0qDds60H9_Y-UYKW0YGanfNjLhu7OMTKoYLwyjsbMVt0xCD1JCc9t4Np3VYswiSOKwjuCw"},"schema":"pubphys.envelope/1"},"record_hash":"a9f1dff11f55f9a56aadd6948cb0c3ffa90045f0b6ffae0a84a8fd05222fc599","leaf_index":3430}