{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,12]],"date-time":"2026-03-12T07:17:17Z","timestamp":1773299837669,"version":"3.50.1"},"reference-count":67,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,13]],"date-time":"2023-08-13T00:00:00Z","timestamp":1691884800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["21-72-20029"],"award-info":[{"award-number":["21-72-20029"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["075-15-2022-311"],"award-info":[{"award-number":["075-15-2022-311"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Ministry of Science and Hgher Education of Russian Federation","award":["21-72-20029"],"award-info":[{"award-number":["21-72-20029"]}]},{"name":"Ministry of Science and Hgher Education of Russian Federation","award":["075-15-2022-311"],"award-info":[{"award-number":["075-15-2022-311"]}]},{"DOI":"10.13039\/501100012528","name":"Kazan Federal University","doi-asserted-by":"crossref","award":["21-72-20029"],"award-info":[{"award-number":["21-72-20029"]}],"id":[{"id":"10.13039\/501100012528","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100012528","name":"Kazan Federal University","doi-asserted-by":"crossref","award":["075-15-2022-311"],"award-info":[{"award-number":["075-15-2022-311"]}],"id":[{"id":"10.13039\/501100012528","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["www.mdpi.com"],"crossmark-restriction":true},"short-container-title":["Symmetry"],"abstract":"<jats:p>The out-of-thermal-equilibrium Casimir\u2013Polder force between nanoparticles and dielectric substrates coated with gapped graphene is considered in the framework of the Dirac model using the formalism of the polarization tensor. This is an example of physical phenomena violating the time-reversal symmetry. After presenting the main points of the used formalism, we calculate two contributions to the Casimir\u2013Polder force acting on a nanoparticle on the source side of a fused silica glass substrate coated with gapped graphene, which is either cooler or hotter than the environment. The total nonequilibrium force magnitudes are computed as a function of separation for different values of the energy gap and compared with those from an uncoated plate and with the equilibrium force in the presence of graphene coating. According to our results, the presence of a substrate increases the magnitude of the nonequlibrium force. The force magnitude becomes larger with higher and smaller with lower temperature of the graphene-coated substrate as compared to the equilibrium force at the environmental temperature. It is shown that, with increasing energy gap, the magnitude of the nonequilibrium force becomes smaller, and the graphene coating makes a lesser impact on the force acting on a nanoparticle from the uncoated substrate. Possible applications of the obtained results are discussed.<\/jats:p>","DOI":"10.3390\/sym15081580","type":"journal-article","created":{"date-parts":[[2023,8,14]],"date-time":"2023-08-14T10:20:14Z","timestamp":1692008414000},"page":"1580","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Nonequilibrium Casimir\u2013Polder Interaction between Nanoparticles and Substrates Coated with Gapped Graphene"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0495-5907","authenticated-orcid":false,"given":"Galina","family":"Klimchitskaya","sequence":"first","affiliation":[{"name":"Central Astronomical Observatory at Pulkovo of the Russian Academy of Sciences, 196140 Saint Petersburg, Russia"},{"name":"Peter the Great Saint Petersburg Polytechnic University, 195251 Saint Petersburg, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6227-5542","authenticated-orcid":false,"given":"Constantine","family":"Korikov","sequence":"additional","affiliation":[{"name":"Huawei Noah\u2019s Ark Laboratory, Krylatskaya str. 17, 121614 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8560-998X","authenticated-orcid":false,"given":"Vladimir","family":"Mostepanenko","sequence":"additional","affiliation":[{"name":"Central Astronomical Observatory at Pulkovo of the Russian Academy of Sciences, 196140 Saint Petersburg, Russia"},{"name":"Peter the Great Saint Petersburg Polytechnic University, 195251 Saint Petersburg, Russia"},{"name":"Kazan Federal University, 420008 Kazan, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8308-059X","authenticated-orcid":false,"given":"Oleg","family":"Tsybin","sequence":"additional","affiliation":[{"name":"Peter the Great Saint Petersburg Polytechnic University, 195251 Saint Petersburg, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"360","DOI":"10.1103\/PhysRev.73.360","article-title":"The influence of retardation on the London-van der Waals forces","volume":"73","author":"Casimir","year":"1948","journal-title":"Phys. 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