{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,6]],"date-time":"2026-01-06T08:52:12Z","timestamp":1767689532196,"version":"3.48.0"},"reference-count":24,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2026,1,6]],"date-time":"2026-01-06T00:00:00Z","timestamp":1767657600000},"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>A description of the Kibble\u2013Zurek mechanism with linear response theory has been done previously, but ad hoc hypotheses were used, such as the rate-dependent impulse window via the Zurek equation in the context of no driving in the relaxation time. In this work, I present a new framework where such hypotheses are unnecessary while preserving all the characteristics of the phenomenon. The Kibble-Zurek scaling obtained for the excess work is close to 2\/5, a result that holds for open and thermally isolated systems with relaxation time that diverges at the critical point and the first zero of the relaxation function is finite. I exemplify the results using four different but significant types of scaling functions.<\/jats:p>","DOI":"10.3390\/e28010066","type":"journal-article","created":{"date-parts":[[2026,1,6]],"date-time":"2026-01-06T08:42:58Z","timestamp":1767688978000},"page":"66","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Unifying Kibble\u2013Zurek Mechanism in Weakly Driven Processes"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2684-9147","authenticated-orcid":false,"given":"Pierre","family":"Naz\u00e9","sequence":"first","affiliation":[{"name":"Instituto de Ci\u00eancias Exatas e Naturais, Faculdade de F\u00edsica, Universidade Federal do Par\u00e1, Av. Augusto Correa, 1, Guam\u00e1, Bel\u00e9m 66075-110, PA, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Deffner, S., and Campbell, S. (2019). Quantum Thermodynamics: An Introduction to the Thermodynamics of Quantum Information, Morgan & Claypool Publishers.","DOI":"10.1088\/2053-2571\/ab21c6"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1387","DOI":"10.1088\/0305-4470\/9\/8\/029","article-title":"Topology of cosmic domains and strings","volume":"9","author":"Kibble","year":"1976","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/S0370-1573(96)00009-9","article-title":"Cosmological experiments in condensed matter systems","volume":"276","author":"Zurek","year":"1996","journal-title":"Phys. Rep."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1038\/s41586-019-1070-1","article-title":"Quantum Kibble\u2013Zurek mechanism and critical dynamics on a programmable Rydberg simulator","volume":"568","author":"Keesling","year":"2019","journal-title":"Nature"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"022002","DOI":"10.1088\/1361-6455\/50\/2\/022002","article-title":"Exploring the Kibble\u2013Zurek mechanism with homogeneous Bose gases","volume":"50","author":"Beugnon","year":"2017","journal-title":"J. Phys. B At. Mol. Opt. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"6925","DOI":"10.1073\/pnas.1500763112","article-title":"Kibble\u2013Zurek mechanism in colloidal monolayers","volume":"112","author":"Dillmann","year":"2015","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_7","first-page":"021015","article-title":"Defect formation beyond Kibble-Zurek mechanism and holography","volume":"5","author":"Chesler","year":"2015","journal-title":"Phys. Rev. X"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2254","DOI":"10.1038\/s41467-019-10048-9","article-title":"The Kibble-Zurek mechanism at exceptional points","volume":"10","author":"Heyl","year":"2019","journal-title":"Nat. Commun."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"155301","DOI":"10.1103\/PhysRevLett.116.155301","article-title":"Quantum Kibble-Zurek mechanism in a spin-1 Bose-Einstein condensate","volume":"116","author":"Anquez","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"227001","DOI":"10.1103\/PhysRevLett.129.227001","article-title":"Kibble-Zurek mechanism for dynamical ordering in a driven vortex system","volume":"129","author":"Maegochi","year":"2022","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"134108","DOI":"10.1103\/PhysRevB.90.134108","article-title":"Kibble-Zurek mechanism and finite-time scaling","volume":"90","author":"Huang","year":"2014","journal-title":"Phys. Rev. B"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"055","DOI":"10.21468\/SciPostPhys.9.4.055","article-title":"Kibble\u2013Zurek mechanism in the Ising field theory","volume":"9","author":"Kormos","year":"2020","journal-title":"SciPost Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1038\/s42005-022-00952-w","article-title":"Kibble-Zurek mechanism for nonequilibrium phase transitions in driven systems with quenched disorder","volume":"5","author":"Reichhardt","year":"2022","journal-title":"Commun. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"075701","DOI":"10.1103\/PhysRevLett.105.075701","article-title":"Structural Defects in Ion Chains by Quenching the External Potential: The Inhomogeneous Kibble-Zurek Mechanism","volume":"105","author":"Morigi","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"095301","DOI":"10.1103\/PhysRevLett.125.095301","article-title":"Kibble-Zurek mechanism in driven dissipative systems crossing a nonequilibrium phase transition","volume":"125","author":"Zamora","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"032020","DOI":"10.1103\/PhysRevResearch.2.032020","article-title":"Kibble-Zurek scaling in quantum speed limits for shortcuts to adiabaticity","volume":"2","author":"Puebla","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"ref_17","unstructured":"Kubo, R., Toda, M., and Hashitsume, N. (2012). Statistical Physics II: Nonequilibrium Statistical Mechanics, Springer Science & Business Media."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Naz\u00e9, P., Bonan\u00e7a, M.V., and Deffner, S. (2022). Kibble\u2013Zurek Scaling from Linear Response Theory. Entropy, 24.","DOI":"10.3390\/e24050666"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"113037","DOI":"10.1088\/1367-2630\/aca177","article-title":"Failure of the geometric approach prediction of excess work scaling for open and isolated quantum systems","volume":"24","author":"Soriani","year":"2022","journal-title":"New J. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"064114","DOI":"10.1103\/PhysRevE.107.064114","article-title":"Adiabatic processes like isothermal processes","volume":"107","year":"2023","journal-title":"Phys. Rev. E"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"013206","DOI":"10.1088\/1742-5468\/ab54ba","article-title":"Compatibility of linear-response theory with the second law of thermodynamics and the emergence of negative entropy production rates","volume":"2020","year":"2020","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_22","first-page":"123204","article-title":"Optimal driving of isothermal processes close to equilibrium","volume":"140","author":"Deffner","year":"2014","journal-title":"J. Chem. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"024114","DOI":"10.1063\/1.2948948","article-title":"Optimal protocols for minimal work processes in underdamped stochastic thermodynamics","volume":"129","author":"Schmiedl","year":"2008","journal-title":"J. Chem. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"294","DOI":"10.1063\/1.1703954","article-title":"Time-dependent statistics of the Ising model","volume":"4","author":"Glauber","year":"1963","journal-title":"J. Math. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/1\/66\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,6]],"date-time":"2026-01-06T08:47:53Z","timestamp":1767689273000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/1\/66"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,6]]},"references-count":24,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["e28010066"],"URL":"https:\/\/doi.org\/10.3390\/e28010066","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,6]]}}}