{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T03:33:59Z","timestamp":1784259239961,"version":"3.55.0"},"reference-count":55,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,1]],"date-time":"2022-11-01T00:00:00Z","timestamp":1667260800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"NSF","doi-asserted-by":"publisher","award":["2103989"],"award-info":[{"award-number":["2103989"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"name":"University of Virginia","award":["2103989"],"award-info":[{"award-number":["2103989"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this review, motivated by the recent interest in high-temperature materials, we review our recent progress in theories of lattice dynamics in and out of equilibrium. To investigate thermodynamic properties of anharmonic crystals, the self-consistent phonon theory was developed, mainly in the 1960s, for rare gas atoms and quantum crystals. We have extended this theory to investigate the properties of the equilibrium state of a crystal, including its unit cell shape and size, atomic positions and lattice dynamical properties. Using the equation-of-motion method combined with the fluctuation\u2013dissipation theorem and the Donsker\u2013Furutsu\u2013Novikov (DFN) theorem, this approach was also extended to investigate the non-equilibrium case where there is heat flow across a junction or an interface. The formalism is a classical one and therefore valid at high temperatures.<\/jats:p>","DOI":"10.3390\/e24111585","type":"journal-article","created":{"date-parts":[[2022,11,2]],"date-time":"2022-11-02T06:49:02Z","timestamp":1667371742000},"page":"1585","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Equilibrium and Non-Equilibrium Lattice Dynamics of Anharmonic Systems"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1969-0956","authenticated-orcid":false,"given":"Keivan","family":"Esfarjani","sequence":"first","affiliation":[{"name":"Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904, USA"},{"name":"Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22904, USA"},{"name":"Department of Physics, University of Virginia, Charlottesville, VA 22904, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuan","family":"Liang","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Virginia, Charlottesville, VA 22904, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/j.actamat.2018.12.054","article-title":"Phase stability and mechanical properties of novel high entropy transition metal carbides","volume":"166","author":"Harrington","year":"2019","journal-title":"Acta Mater."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"448","DOI":"10.1016\/j.actamat.2016.08.081","article-title":"A critical review of high entropy alloys and related concepts","volume":"122","author":"Miracle","year":"2017","journal-title":"Acta Mater."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1080\/14786440408520575","article-title":"A new treatment of anharmonicity in lattice thermodynamics: I","volume":"46","author":"Hooton","year":"1955","journal-title":"Lond. 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