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When the H-theorem is satisfied, we demonstrate that different entropic forms may emerge due to the equation\u2019s nonlinearity. We obtain the entropy production related to these entropies and show that its form remains invariant. Furthermore, we investigate some behaviors for these equations from both numerical and analytical perspectives, showing a large class of behaviors connected with anomalous diffusion and their effects on entropy.<\/jats:p>","DOI":"10.3390\/e26080673","type":"journal-article","created":{"date-parts":[[2024,8,8]],"date-time":"2024-08-08T12:13:14Z","timestamp":1723119194000},"page":"673","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Generalized Kinetic Equations with Fractional Time-Derivative and Nonlinear Diffusion: H-Theorem and Entropy"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3853-1790","authenticated-orcid":false,"given":"Ervin K.","family":"Lenzi","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual de Ponta Grossa, Ponta Grossa 84030-900, PR, Brazil"},{"name":"National Institute of Science and Technology for Complex Systems, Centro Brasileiro de Pesquisas F\u00edsicas, Rio de Janeiro 22290-180, RJ, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6158-3068","authenticated-orcid":false,"given":"Michely P.","family":"Rosseto","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual de Ponta Grossa, Ponta Grossa 84030-900, PR, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5883-6404","authenticated-orcid":false,"given":"Derik W.","family":"Gryczak","sequence":"additional","affiliation":[{"name":"Independent Researcher, Irati 84507-012, PR, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7202-1386","authenticated-orcid":false,"given":"Luiz R.","family":"Evangelista","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual de Maring\u00e1, Maring\u00e1 87020-900, PR, Brazil"},{"name":"Istituto dei Sistemi Complessi (ISC\u2013CNR), Via dei Taurini, 19, 00185 Rome, Italy"},{"name":"Department of Molecular Science and Nanosystems, Ca\u2019Foscari University of Venice, Via Torino, 155, 30172 Mestre, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6875-1965","authenticated-orcid":false,"given":"Luciano R.","family":"da Silva","sequence":"additional","affiliation":[{"name":"National Institute of Science and Technology for Complex Systems, Centro Brasileiro de Pesquisas F\u00edsicas, Rio de Janeiro 22290-180, RJ, Brazil"},{"name":"Departamento de F\u00edsica, Universidade Federal do Rio Grande do Norte, Natal 59078-900, RN, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0103-9017","authenticated-orcid":false,"given":"Marcelo K.","family":"Lenzi","sequence":"additional","affiliation":[{"name":"Departamento de Engenharia Qu\u00edmica, Universidade Federal do Paran\u00e1, Curitiba 80060-000, PR, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4930-5262","authenticated-orcid":false,"given":"Rafael S.","family":"Zola","sequence":"additional","affiliation":[{"name":"Department of Physics, Universidade Tecnol\u00f3gica Federal do Paran\u00e1, Apucarana 86812-460, PR, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Boltzmann, L. 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