{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,3]],"date-time":"2026-02-03T18:36:25Z","timestamp":1770143785994,"version":"3.49.0"},"reference-count":47,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T00:00:00Z","timestamp":1767139200000},"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>Interfaces of rather different natures\u2014as, e.g., bacterial colony or forest fire boundaries, or semiconductor layers grown by different methods (MBE, sputtering, etc.)\u2014are self-affine fractals, and feature scaling with universal exponents (depending on the substrate\u2019s dimensionality d and global topology, as well as on the driving randomness\u2019 spatial and temporal correlations but not on the underlying mechanisms). Adding lateral growth as an essential (non-equilibrium) ingredient to the known equilibrium ones (randomness and interface relaxation), the Kardar\u2013Parisi\u2013Zhang (KPZ) equation succeeded in finding (via the dynamic renormalization group) the correct exponents for flat d=1 substrates and (spatially and temporally) uncorrelated randomness. It is this interplay which gives rise to the unique, non-Gaussian scaling properties characteristic of the specific, universal type of non-equilibrium roughening. Later on, the asymptotic statistics of process h(x) fluctuations in the scaling regime was also analytically found for d=1 substrates. For d&gt;1 substrates, however, one has to rely on numerical simulations. Here we review a variational approach that allows for analytical progress regardless of substrate dimensionality. After reviewing our previous numerical results in d=1, 2, and 3 on the time evolution of one of the functionals\u2014which we call the non-equilibrium potential (NEP)\u2014as well as its scaling behavior with the nonlinearity parameter \u03bb, we discuss the stochastic thermodynamics of the roughening process and the memory of process h(x) in KPZ and in the related Golubovi\u0107\u2013Bruinsma (GB) model, providing numerical evidence for the significant dependence on initial conditions of the NEP\u2019s asymptotic behavior in both models. Finally, we highlight some open questions.<\/jats:p>","DOI":"10.3390\/e28010055","type":"journal-article","created":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T16:34:23Z","timestamp":1767198863000},"page":"55","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The KPZ Equation of Kinetic Interface Roughening: A Variational Perspective"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6183-9617","authenticated-orcid":false,"given":"Horacio S.","family":"Wio","sequence":"first","affiliation":[{"name":"Institute for Cross-Disciplinary Physics and Complex Systems (IFISC), UIB-CSIC, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2469-3302","authenticated-orcid":false,"given":"Roberto R.","family":"Deza","sequence":"additional","affiliation":[{"name":"IFIMAR, FCEyN-UNMdP and CONICET, Mar del Plata B7602AYL, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2135-3510","authenticated-orcid":false,"given":"Jorge A.","family":"Revelli","sequence":"additional","affiliation":[{"name":"IFEG, FaMAF-UNC and CONICET, Cordoba X5000HUA, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8277-6026","authenticated-orcid":false,"given":"Rafael","family":"Gallego","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Gijon Campus, University of Oviedo, E-33203 Gijon, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5491-6036","authenticated-orcid":false,"given":"Reinaldo","family":"Garc\u00eda-Garc\u00eda","sequence":"additional","affiliation":[{"name":"Department of Physics and Applied Mathematics, School of Sciences, University of Navarra, E-31008 Pamplona, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4184-0463","authenticated-orcid":false,"given":"Miguel A.","family":"Rodr\u00edguez","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica de Cantabria (IFCA), CSIC, University of Cantabria, E-39005 Santander, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"889","DOI":"10.1103\/PhysRevLett.56.889","article-title":"Dynamic scaling of growing interfaces","volume":"56","author":"Kardar","year":"1986","journal-title":"Phys. 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