{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"4f82fd695ba308bdeecd2371e1cadb428435f7dbc5e073236713c0905c4c3049","created":"2026-10-03T07:18:11Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"b0d30f47149c6b4626b2484fdfba1dadce0ea4d77b73558de4ed42ef3899a5ab","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"classification","assisted_by":[],"external_id":"stat.long-range-interactions.dynamical-long-range-threshold","kind":"well-posed","literature_status":"partially-resolved","n":"1","parents":[],"plain":"Interactions that fall off with distance as $1/r^\\alpha$ behave as long-range in thermodynamics when $\\alpha$ is below the space dimension, but the threshold for long-lived non-equilibrium states may differ. The exact boundary is not established.","posed_since":"2010","precise":"For particles in $d$ dimensions with pair potential $V(r) \\sim 1/r^{\\alpha}$, determine the range of $\\alpha$ for which the $N \\to \\infty$ dynamics follows a Vlasov equation with quasi-stationary states whose lifetime diverges with $N$, as opposed to collisional relaxation on $N$-independent times; the proposed criterion is $\\alpha < d - 1$ (force integrable at large distance). An answer is a classification with proof or with simulations controlling $N \\to \\infty$.","problem_ref":null,"references":"","settled_by":"A kinetic-theory derivation of the $N$ dependence of relaxation times for general $\\alpha$, confirmed by simulations in $d = 1, 2, 3$.","status_note":"The criterion was proposed by Gabrielli, Joyce and Marcos (Phys. Rev. Lett. 105, 210602, 2010); a derivation for general $\\alpha$ and $d$ is incomplete.","title":"Which power-law decays make interactions dynamically 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