{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T08:25:18Z","timestamp":1790324718759,"version":"4.1.0"},"reference-count":94,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2015,5,13]],"date-time":"2015-05-13T00:00:00Z","timestamp":1431475200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We consider a system of Brownian particles with long-range interactions. We go beyond the mean field approximation and take fluctuations into account. We introduce a new class of stochastic Fokker-Planck equations associated with a generalized thermodynamical formalism. Generalized thermodynamics arises in the case of complex systems experiencing small-scale constraints. In the limit of short-range interactions, we obtain a generalized class of stochastic Cahn-Hilliard equations. Our formalism has application for several systems of physical interest including self-gravitating Brownian particles, colloid particles at a fluid interface, superconductors of type II, nucleation, the chemotaxis of bacterial populations, and two-dimensional turbulence. We also introduce a new type of generalized entropy taking into account anomalous diffusion and exclusion or inclusion constraints.<\/jats:p>","DOI":"10.3390\/e17053205","type":"journal-article","created":{"date-parts":[[2015,5,13]],"date-time":"2015-05-13T12:42:07Z","timestamp":1431520927000},"page":"3205-3252","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":33,"title":["Generalized Stochastic Fokker-Planck Equations"],"prefix":"10.3390","volume":"17","author":[{"given":"Pierre-Henri","family":"Chavanis","sequence":"first","affiliation":[{"name":"Laboratoire de Physique Th\u00e9orique, Universit\u00e9 Paul Sabatier, 118 route de Narbonne 31062 Toulouse, France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2015,5,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1002\/andp.19053220806","article-title":"\u00dcber die von der molekularkinetischen Theorie der W\u00e4rme geforderte Bewegung von in ruhenden Fl\u00fcssigkeiten suspendierten Teilchen","volume":"17","author":"Einstein","year":"1905","journal-title":"Ann. 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