{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T15:22:59Z","timestamp":1774452179234,"version":"3.50.1"},"reference-count":43,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2016,7,23]],"date-time":"2016-07-23T00:00:00Z","timestamp":1469232000000},"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 discovery of the entropy production paradox (Hoffmann et al., 1998) raised basic questions about the nature of irreversibility in the regime between diffusion and waves. First studied in the form of spatial movements of moments of H functions, pseudo propagation is the pre-limit propagation-like movements of skewed probability density function (PDFs) in the domain between the wave and diffusion equations that goes over to classical partial differential equation propagation of characteristics in the wave limit. Many of the strange properties that occur in this extraordinary regime were thought to be connected in some manner to this form of proto-movement. This paper eliminates pseudo propagation by employing a similar evolution equation that imposes spatial unimodal symmetry on evolving PDFs. Contrary to initial expectations, familiar peculiarities emerge despite the imposed symmetry, but they have a distinct character.<\/jats:p>","DOI":"10.3390\/e18070275","type":"journal-article","created":{"date-parts":[[2016,7,25]],"date-time":"2016-07-25T10:04:26Z","timestamp":1469441066000},"page":"275","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Symmetric Fractional Diffusion and Entropy Production"],"prefix":"10.3390","volume":"18","author":[{"given":"Janett","family":"Prehl","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Physik, Technische Universit\u00e4t Chemnitz, Chemnitz D-09107, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Frank","family":"Boldt","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Physik, Technische Universit\u00e4t Chemnitz, Chemnitz D-09107, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Karl","family":"Hoffmann","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Physik, Technische Universit\u00e4t Chemnitz, Chemnitz D-09107, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christopher","family":"Essex","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, University of Western Ontario, London, ON N6A 5B7, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,7,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1080\/00018738700101072","article-title":"Diffusion in Disordered Media","volume":"36","author":"Havlin","year":"1987","journal-title":"Adv. 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