{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:19:42Z","timestamp":1760235582631,"version":"build-2065373602"},"reference-count":46,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2021,9,2]],"date-time":"2021-09-02T00:00:00Z","timestamp":1630540800000},"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>We study the statistical mechanics of binary systems under the gravitational interaction of the Modified Newtonian Dynamics (MOND) in three-dimensional space. Considering the binary systems in the microcanonical and canonical ensembles, we show that in the microcanonical systems, unlike the Newtonian gravity, there is a sharp phase transition, with a high-temperature homogeneous phase and a low-temperature clumped binary one. Defining an order parameter in the canonical systems, we find a smoother phase transition and identify the corresponding critical temperature in terms of the physical parameters of the binary system.<\/jats:p>","DOI":"10.3390\/e23091158","type":"journal-article","created":{"date-parts":[[2021,9,2]],"date-time":"2021-09-02T21:13:36Z","timestamp":1630617216000},"page":"1158","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Phase Transition in Modified Newtonian Dynamics (MONDian) Self-Gravitating Systems"],"prefix":"10.3390","volume":"23","author":[{"given":"Mohammad Hossein","family":"Zhoolideh Haghighi","sequence":"first","affiliation":[{"name":"Department of Physics, K.N. Toosi University of Technology, Tehran P.O. Box 15875-4416, Iran"},{"name":"School of Astronomy, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5746, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7084-5725","authenticated-orcid":false,"given":"Sohrab","family":"Rahvar","sequence":"additional","affiliation":[{"name":"Department of Physics, Sharif University of Technology, Tehran P.O. Box 11365-9161, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohammad Reza","family":"Rahimi Tabar","sequence":"additional","affiliation":[{"name":"Department of Physics, Sharif University of Technology, Tehran P.O. Box 11365-9161, Iran"},{"name":"Institute of Physics and ForWind, Carl von Ossietzky University of Oldenburg, Carl-von-Ossietzky-Stra\u00dfe 9\u201311, 26111 Oldenburg, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,2]]},"reference":[{"key":"ref_1","first-page":"517","article-title":"Formation of galaxies and large-scale structure with cold dark matter","volume":"311","author":"Blumenthal","year":"1984","journal-title":"Nat. Cell Biol."},{"key":"ref_2","unstructured":"Planck Collaboration, Ade, P., Aghanim, N., Alves, M.I.R., Armitage-Caplan, C., Arnaud, M., Ashdown, M., Atrio-Barandela, F., Aumont, J., and Aussel, H. (2014). Planck2013 results. I. Overview of products and scientific results. Astron. 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