{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:17:11Z","timestamp":1760217431471,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2015,4,13]],"date-time":"2015-04-13T00:00:00Z","timestamp":1428883200000},"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>Assuming that the motions of a complex system structural units take place on continuous, but non-differentiable curves of a space-time manifold, the scale relativity model with arbitrary constant fractal dimension (the hydrodynamic and wave function versions) is built. For non-differentiability through stochastic processes of the Markov type, the non-differentiable entropy concept on a space-time manifold in the hydrodynamic version and its correspondence with motion variables (energy, momentum, etc.) are established. Moreover, for the same non-differentiability type, through a scale resolution dependence of a fundamental length and wave function independence with respect to the proper time, a non-differentiable Klein\u2013Gordon-type equation in the wave function version is obtained. For a phase-amplitude functional dependence on the wave function, the non-differentiable spontaneous symmetry breaking mechanism implies pattern generation in the form of Cooper non-differentiable-type pairs, while its non-differentiable topology implies some fractal logic elements (fractal bit, fractal gates, etc.).<\/jats:p>","DOI":"10.3390\/e17042184","type":"journal-article","created":{"date-parts":[[2015,4,14]],"date-time":"2015-04-14T03:11:46Z","timestamp":1428981106000},"page":"2184-2197","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Implications of Non-Differentiable Entropy on a Space-Time Manifold"],"prefix":"10.3390","volume":"17","author":[{"given":"Maricel","family":"Agop","sequence":"first","affiliation":[{"name":"Department of Physics, Gheorghe Asachi Technical University of Ia\u015fi, Carol I Blv. 11, Ia\u015fi 700050,Rom\u00e2nia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alina","family":"Gavrilu\u0163","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics, \"Al. I. Cuza\" University, Carol I Bd. 11, Ia\u015fi 700506, Rom\u00e2nia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gavril","family":"\u015etefan","sequence":"additional","affiliation":[{"name":"Faculty of Agriculture, Agroeconomy Department, University of Agricultural Sciences and Veterinary Medicine Ia\u015fi, Mihail Sadoveanu Alley 3, Ia\u015fi 700490, Rom\u00e2nia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bogdan","family":"Doroftei","sequence":"additional","affiliation":[{"name":"Origyn Fertility Center, Clinical Hospital of Obstetrics and Gynaecology, Grigore T. Popa University of Medicine and Pharmacy, Ia\u015fi 700032, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,4,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Nottale, L. (1993). 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