{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T15:23:00Z","timestamp":1774452180210,"version":"3.50.1"},"reference-count":32,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2018,11,16]],"date-time":"2018-11-16T00:00:00Z","timestamp":1542326400000},"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>The entropy production rate is a well established measure for the extent of irreversibility in a process. For irreversible processes, one thus usually expects that the entropy production rate approaches zero in the reversible limit. Fractional diffusion equations provide a fascinating testbed for that intuition in that they build a bridge connecting the fully irreversible diffusion equation with the fully reversible wave equation by a one-parameter family of processes. The entropy production paradox describes the very non-intuitive increase of the entropy production rate as that bridge is passed from irreversible diffusion to reversible waves. This paradox has been established for time- and space-fractional diffusion equations on one-dimensional continuous space and for the Shannon, Tsallis and Renyi entropies. After a brief review of the known results, we generalize it to time-fractional diffusion on a finite chain of points described by a fractional master equation.<\/jats:p>","DOI":"10.3390\/e20110881","type":"journal-article","created":{"date-parts":[[2018,11,16]],"date-time":"2018-11-16T11:48:31Z","timestamp":1542368911000},"page":"881","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Between Waves and Diffusion: Paradoxical Entropy Production in an Exceptional Regime"],"prefix":"10.3390","volume":"20","author":[{"given":"Karl Heinz","family":"Hoffmann","sequence":"first","affiliation":[{"name":"Institute of Physics, Chemnitz University of Technology, D-09107 Chemnitz, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kathrin","family":"Kulmus","sequence":"additional","affiliation":[{"name":"Institute of Physics, Chemnitz University of Technology, D-09107 Chemnitz, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christopher","family":"Essex","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, University of Western Ontario, Middlesex College, London, ON N6A 5B7, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8463-4611","authenticated-orcid":false,"given":"Janett","family":"Prehl","sequence":"additional","affiliation":[{"name":"Institute of Physics, Chemnitz University of Technology, D-09107 Chemnitz, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,11,16]]},"reference":[{"key":"ref_1","first-page":"108","article-title":"Fractional Differential Equations and Initial Value Problems","volume":"23","author":"Davison","year":"1998","journal-title":"Math. 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