{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T10:55:53Z","timestamp":1760784953427,"version":"build-2065373602"},"reference-count":51,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,2,9]],"date-time":"2021-02-09T00:00:00Z","timestamp":1612828800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["W911NF1901"],"award-info":[{"award-number":["W911NF1901"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We study two forms of anomalous diffusion, one equivalent to replacing the ordinary time derivative of the standard diffusion equation with the Caputo fractional derivative, and the other equivalent to replacing the time independent diffusion coefficient of the standard diffusion equation with a monotonic time dependence. We discuss the joint use of these prescriptions, with a phenomenological method and a theoretical projection method, leading to two apparently different diffusion equations. We prove that the two diffusion equations are equivalent and design a time series that corresponds to the anomalous diffusion equation proposed. We discuss these results in the framework of the growing interest in fractional derivatives and the emergence of cognition in nature. We conclude that the Caputo fractional derivative is a signature of the connection between cognition and self-organization, a form of cognition emergence different from the other source of anomalous diffusion, which is closely related to quantum coherence. We propose a criterion to detect the action of self-organization even in the presence of significant quantum coherence. We argue that statistical analysis of data using diffusion entropy should help the analysis of physiological processes hosting both forms of deviation from ordinary scaling.<\/jats:p>","DOI":"10.3390\/e23020211","type":"journal-article","created":{"date-parts":[[2021,2,9]],"date-time":"2021-02-09T23:43:16Z","timestamp":1612914196000},"page":"211","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Caputo Fractional Derivative and Quantum-Like Coherence"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3096-4834","authenticated-orcid":false,"given":"Garland","family":"Culbreth","sequence":"first","affiliation":[{"name":"Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, TX 76201, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mauro","family":"Bologna","sequence":"additional","affiliation":[{"name":"Departamento de Ingenier\u00eda El\u00e9ctrica-Electr\u00f3nica, Universidad de Tarapac\u00e1, 1000000 Arica, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4961-1542","authenticated-orcid":false,"given":"Bruce J.","family":"West","sequence":"additional","affiliation":[{"name":"Army Research Office, DEVCOM US Army Research Laboratory, Research Triangle Park, NC 27708, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2762-9856","authenticated-orcid":false,"given":"Paolo","family":"Grigolini","sequence":"additional","affiliation":[{"name":"Center for Nonlinear Science, University of North Texas, P.O. 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