{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"16ffd80165003eeb850e15dc6bc2feb992d785f572d0d6fcf1e23136c012eefe","created":"2026-10-03T07:17:56Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"98ec1426e7de851fba1543e789bf65ccfe1594244bc31fa5d4f749c22456f7e0","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"bio.active-matter.toner-tu-exponents","kind":"well-posed","literature_status":"contested","n":"1","parents":[],"plain":"Flocks of self-propelled particles that align with neighbours can move together in two dimensions, which equilibrium physics forbids for comparable systems. The exact laws governing the fluctuations of such a flock are disputed.","posed_since":"1995","precise":"In the ordered phase of compressible dry polar active matter (Vicsek model, Toner-Tu hydrodynamics) in $d = 2$, determine the roughness exponent $\\chi$, anisotropy exponent $\\zeta$ and dynamic exponent $z$ of velocity and density fluctuations. The incompressible 2D case is solved exactly by mapping to the 1D KPZ equation (Kardar-Parisi-Zhang, the standard equation for a randomly growing interface; Chen, Lee, Toner 2016). The 1995 conjecture $\\chi = -1/5, \\zeta = 3/5, z = 6/5$ for the compressible case is inconsistent with large-scale simulations published in 2019. An answer gives exact or controlled values confirmed by simulations that reach the asymptotic regime.","problem_ref":null,"references":"","settled_by":"An analytic determination of $\\chi$, $\\zeta$ and $z$ for $2\\mathrm{D}$ compressible flocks, matched by Vicsek-model simulations large enough to rule out crossovers.","status_note":"In 2024 Jentsch and Lee (PRL 133, 128301), Ikeda (PRL 133, 258301) and Amoretti et al. (PRE 110, 054108) each proposed exact exponents or a new universality class for the Vicsek ordered phase; these proposals are mutually inconsistent and none is established numerically (2026).","title":"Exact scaling exponents of two-dimensional compressible polar flocks","topic_ref":"6a008dc19304a289e116a6d6e6371cc6de0ef3d1a904e572d5454f039b2ac217"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"7adce4a72faead8d2a4599283f2a55de6f84493e633deb21c370b2cf9765167a","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"a3a507115e0253e57b872fdf11df3a95dffbb1ccd97798ac2e3b7ff251816582","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"9tZIBPTFJLh2v-SeUvq7iOZQZ-D4VjmFf_14Gev8i0vHmpZggIHizOnd7fGCJynUv9a2HslCuraERqk4qKkRAA"},"schema":"pubphys.envelope/1"},"record_hash":"7adce4a72faead8d2a4599283f2a55de6f84493e633deb21c370b2cf9765167a","leaf_index":724}