{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T09:09:01Z","timestamp":1762074541558,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,4]],"date-time":"2022-11-04T00:00:00Z","timestamp":1667520000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["21534002","21973018"],"award-info":[{"award-number":["21534002","21973018"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This work is devoted to deriving the entropy of a single photon in a beam of light from first principles. Based on the quantum processes of light\u2013matter interaction, we find that, if the light is not in equilibrium, there are two different ways, depending on whether the photon is being added or being removed from the light, of defining the single-photon entropy of this light. However, when the light is in equilibrium at temperature T, the two definitions are equivalent and the photon entropy of this light is h\u03bd\/T. From first principles, we also re-derive the J\u00fcttner velocity distribution showing that, even without interatomic collisions, two-level atoms will relax to the state satisfying the Maxwell\u2013J\u00fcttner velocity distribution when they are moving in blackbody radiation fields.<\/jats:p>","DOI":"10.3390\/e24111609","type":"journal-article","created":{"date-parts":[[2022,11,8]],"date-time":"2022-11-08T11:46:43Z","timestamp":1667908003000},"page":"1609","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["First-Principle Derivation of Single-Photon Entropy and Maxwell\u2013J\u00fcttner Velocity Distribution"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2869-3991","authenticated-orcid":false,"given":"Changhao","family":"Li","sequence":"first","affiliation":[{"name":"The State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8811-0030","authenticated-orcid":false,"given":"Jianfeng","family":"Li","sequence":"additional","affiliation":[{"name":"The State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China"}]},{"given":"Yuliang","family":"Yang","sequence":"additional","affiliation":[{"name":"The State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1103\/PhysRev.98.887","article-title":"Entropy of radiation","volume":"98","author":"Ore","year":"1955","journal-title":"Phys. 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