{"schema":"pubphys.bundle/1","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"},"content_withheld":"false","files":[],"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"7adce4a72faead8d2a4599283f2a55de6f84493e633deb21c370b2cf9765167a","schema":"pubphys.attested/1"},"id_token":null,"id_token_withheld":"false","envelope":{"attested_hash":"a3a507115e0253e57b872fdf11df3a95dffbb1ccd97798ac2e3b7ff251816582","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"9tZIBPTFJLh2v-SeUvq7iOZQZ-D4VjmFf_14Gev8i0vHmpZggIHizOnd7fGCJynUv9a2HslCuraERqk4qKkRAA"},"schema":"pubphys.envelope/1"},"ots":{"attested":["AE9wZW5UaW1lc3RhbXBzAABQcm9vZgC_ieLohOiSlAEIo6UHEV4CU-V7hy_fEd86ld_7sczZd5isLjt_8lGBZYLwIKPUW2iypSSF6ETBzDvITzHrOqcShaGCNJ8ObWp1cyb2CPAg2k40zQorngcxw7falofGzuXSXwejS11cnwiwseK-TGII8SAOch7NLFvgAn202rWccoCazHcmuammjxJTUnlBH0yavAjwIOEFDC1yNjjuEFqAyLYk2D7OUMQa4WUsioicijYWKvHUCPAgZEqoKv2BKUtW-iM3K1riO5k6kOR-ZYV-zeFIeWKcNRAI8CBhsTFtY7weC6tUnQlgn6pSJx3T28EYY7HR9yW1AJj_nwjwIJPsrh2yAs-Q-tN7WiRC5DpdLPxhgPCZUvw9VyDb326QCPEgrA74zq9OS_EO_vifvlqDPPUyYaAxJJae8fQw9_OOt1II8CB0Pk63gwyd_u5K2K1drJW3UxU3mXTBkDKQdutj6_nnEgjxIFf8EhaVa-ubjpYx3wEAcKGxkzfUQq6B-8jimpRd8IqnCPAgy6DmLQtIuWm-zrDRmaSy95vcFWsjvWyppQO7wIWrfC4I8SAQuuTPkAGyvS_wKMy9BtKvnki2puESdY2vXem8hT1uIQjwIBGYo5Vxy-d7U5zKv5vp3Ii7hJPH_BDLCzA13c47p9q7CP_wCHbghVgOFMuNCPAQPZK2lHbTMqcCYUotzOQcmQjwIJ7ruRIs6HnufKxCpmLUgeUl0aUu0ov4_3PqcppS-bddCPEgI530vHzdwz_3o9jYQ3WablSaV64Wfn7Pmby5WLyI-A0I8QRqwKw78AhITF7Tr3a1xP8Ag9_jDS75DI4sK2h0dHBzOi8vYm9iLmJ0Yy5jYWxlbmRhci5vcGVudGltZXN0YW1wcy5vcmcI8CBXWeNiWFWltfHE0nkDXsVKIykUUwuObhsnN-2fvBt98gjwIM5Vf7j68sOFgVysYMc5S09Hho-fqxYW3oQ4oKUX7JpPCPAgaKHHQB3sdd9w1hu4NMh7vmb-p0X2r4uyVKKCj1G7S28I8SAW2Mg6Bzi9ygjrrjIYB7w90fwMcxH24SsqLmzPyAMGfwjwIMQ863zlveYIpNKWVrZaDOa4vowwoOg2CcTW8gXtxK3dCPEgrZxGGAf-t8jhz_eLMUyBN1IhNrdYDRcNHWvQcwNG9woI8CCVOUHG1YXudbaFIgUYvrCIevRWvb2ubhQAo-z0duD0rQjxIDBeKerVGkO3J2eXDwdraSioXbIWsdIPEN1s8pIwJIS2CPAgV2kB3n3JZ5HwBQjpRL1CWKnqvdCjZk3RcALk6N2wHigI8CB-wbljNlfqElD25uZkNg6taS81YDRU2vfMqGVQ420O9AjxWQEAAAABJM43UU6MLMRrPlCUqossvu1q_knons5yylyWTgxZzWUAAAAAAP7___8CLgABAAAAAAAWABQrYt3E7OllKRCEmL3FL589l78_5wAAAAAAAAAAImog8ATiyw4ACAjxILmbcIO7ixv3pCtwUMZ1VQGKus54F9lZ8-w5ECD5ZuiICAjwIG6ynCyFTLPE7Rgk-0ZGCtBDyqNQx8YvaSZNHy8D8p8jCAjwIMX8zoD2Be1YWvcx_bexVPRmBX0hK-do8b8t9SarimQVCAjxIG6XtpqIEclLlhzcK1LVjqoPGrjjmLnyno8tcETMzECECAjwIF6mlfXOFY51JhrK4eze8zQlKV75OYZYqbtyi8BWxmAsCAjwIKKUcJmVoz-u-geiRV_-ZHdF0VEPXaEPFWf_JX_0e7G2CAjxIGMeD0Z5QWNKBcAEpHht007FFIILW6aSJ8f3CMkKNsneCAjwIBFr6Je8rkB7kwhPGbq0yRgFwZsvW9JhY5sXDRtrq0QwCAjwIAHAwXaus-yUgKICJB5mVvx8VZuJgjwnpb0qWPg1pi6DCAjxIAJhb8cefkVHTAMymdsxa-_i-tUua62LcpALQnPG7R9_CAjwIDvsFqBhPfAL6TCtXZd7gBWmMpx6iiZ-Ug0Jf_LlcstgCAjwIDJRBX6Q0Vge-ra-k1_g76yJWTEp6WWF13I3JpTnsZPJCAjwILykZCbkC9VQgl5gytPKKir6T9nTvKR6y05iWSHBTQgzCAgABYiWDXPXGQED45c78Ah-sGbmQAwfZwjwEAYqxtHnZiPZnJEGPx_JvKsI8SDXkqDg17YsZIrEz5acRx7mBXCKW5mKNgRYnw_-U2cpDgjxBGrArDrwCEfK6Mono428_wCD3-MNLvkMji4taHR0cHM6Ly9hbGljZS5idGMuY2FsZW5kYXIub3BlbnRpbWVzdGFtcHMub3JnCPAg10nyJ6Jn4kshQQQCKE0Lvj3hBC6nLRBVIzaPV4eh8VwI8CCQaAqWlxEbac_XOq1MOlf9iOKHEsxXspzbjK0vLpy8BwjwII3iRbRXUcDxrhAdkZmFlI15nySCSEZ7g922tF1Xln5DCPEgYYvZ1L4v4rPu_UqLfcHPtwSb2OH2GhPgVcVhhH6DpkQI8CDZCIoR20sC-uchP40oAmcLm9EJFV7xzwplj54AeVadXQjxILvLBjPFpCoz29m9Yf2qRnV7PI9GqxE8ozVQfk9QwCqrCPEgZXcMudvEoB_MJ8tqOW2PLTKAIeQBnE-1bfzptRc2gR8I8VkBAAAAAfXq-bmxWqOz7B5wKkwjzHGv2R4s7Y0Z76LqXWYNhmz0AAAAAAD-____Av0cAAAAAAAAFgAUl-skmdZDtn7Hnub6mF4HP2uVn7oAAAAAAAAAACJqIPAE4MsOAAgI8CBDmQsbEXc46Hu8DivhCBM1LGExa4TTM-wo7HpoCVjBTQgI8CCty5vkQtPHzzs8xqArOYhVSjgaDoG1ou3cojv5_P7wQQgI8SDQWB0kdXCTwbk-TqTt0QfRMyzDNLOtU1XXFgvGSRlW8wgI8CC2PKa_QCIBZtrXodq1GdEzppISMsMRDOnzr2gFjLKOAAgI8CCwQ_QF67bcFk_uJ2s0VCZmf5PGIjTmTCpt_qnVY5p_8AgI8SAbzyx7z-oxPMju6PvtQhzrJIQyH2ZAc9oyLuYemzbmZQgI8CC5bh7J74uUk91r8LwybHQFCNBC3PWeuwtMCCNFKWj19AgI8SAf4R2TUNPjaX_PWDUImpoITK0JoW89zaMKnD1wlEEi8AgI8SC2vNmdm7eyP5Ytl1yTzT0njliMdALl1ViKG-s3bG2zswgI8CCpbIVNz1GFVXdJk2zd82hBUoNFFemUPBt7IJdWEOciPwgI8CBtYdRQcrtX1FtCm1SxF7sx-5cO04U8KzbJh8fqBkyReAgI8CBHs2LGmX6gI-lV7yp1RrV7_tGfdiBAwnEFF6B6yXHzAQgI8CATotVmbXBMAixhZ-QKy5Q4lR0FusdA_wcIC1TbB7GmxAgIAAWIlg1z1xkBA-KXOw"],"envelope":["AE9wZW5UaW1lc3RhbXBzAABQcm9vZgC_ieLohOiSlAEIyLZ609hRhLgHgtzi5F2V9T4JYPub1-uR7QxwWhAq9H7xIMisp68K8MbIkZpCgRW-gSP2KxSO_oZkRppVvlyD9TriCPAg0xYLNGRyDKppAWrSGBdZBeSSkx5a8sxI-egwguO9ooQI8CDqcCszsHPSO66mjXzYfiWJawqW8AbEs6tyJ_5fGV2bdAjwIG5x7pisVUaAUf0Ow-pGHXyhEPd1eD16DNkF2EumvOEZCPEgIXwVXO_Fn6qMOuzDC4W_Jxpj58MVhPnhdKsgLIjlmRMI8SDYuizHGxSV-nucpFucDeOzYX71GG1k5L1jcKR3wUKzggjxIG3oNV_bFsbdmegJ-J7zCzLMZfopQQx82k36iQozIJYSCPEgD_N0RSyj69vGSrKrlwSNMl9O9vSUHCEATLrRs-xjrdgI8CCnyvApWpXSVFEeuVGtAaM1DHngynmyhNGrDXAgENc__QjwIG_qk_muz-jg8UolhZs34aKKg3vvCTOQfEn-hyT2gbIDCPEgUZCxxJx-baWw4JFRfzCCtSjPn4ATBx_3FQi1O2Yo4-8I8SAQuuTPkAGyvS_wKMy9BtKvnki2puESdY2vXem8hT1uIQjwIBGYo5Vxy-d7U5zKv5vp3Ii7hJPH_BDLCzA13c47p9q7CP_wCHbghVgOFMuNCPAQPZK2lHbTMqcCYUotzOQcmQjwIJ7ruRIs6HnufKxCpmLUgeUl0aUu0ov4_3PqcppS-bddCPEgI530vHzdwz_3o9jYQ3WablSaV64Wf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