The KPZ equation of kinetic interface roughening: A variational perspective

Interfaces of rather different natures-as, e.g., bacterial colony or forest fire boundaries, or semiconductor layers grown by different methods (MBE, sputtering, etc.)-are self-affine fractals, and feature scaling with universal exponents (depending on the substrate's dimensionality d and globa...

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Autores: Wio, Horacio S., Deza, Roberto R., Revelli, Jorge A., Gallego, Rafael, García-García, Reinaldo, Rodríguez, Miguel A.
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2026
País:España
Institución:Consejo Superior de Investigaciones Científicas (CSIC)
Repositorio:DIGITAL.CSIC. Repositorio Institucional del CSIC
OAI Identifier:oai:digital.csic.es:10261/424974
Acceso en línea:http://hdl.handle.net/10261/424974
https://api.elsevier.com/content/abstract/scopus_id/105028506175
Access Level:acceso abierto
Palabra clave:Variational approach
KPZ equation
Kinetic interface roughening
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spelling The KPZ equation of kinetic interface roughening: A variational perspectiveWio, Horacio S.Deza, Roberto R.Revelli, Jorge A.Gallego, RafaelGarcía-García, ReinaldoRodríguez, Miguel A.Variational approachKPZ equationKinetic interface rougheningInterfaces of rather different natures-as, e.g., bacterial colony or forest fire boundaries, or semiconductor layers grown by different methods (MBE, sputtering, etc.)-are self-affine fractals, and feature scaling with universal exponents (depending on the substrate's dimensionality d and global topology, as well as on the driving randomness' 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-Parisi-Zhang (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>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-which we call the non-equilibrium potential (NEP)-as well as its scaling behavior with the nonlinearity parameter λ, we discuss the stochastic thermodynamics of the roughening process and the memory of process h(x) in KPZ and in the related Golubović-Bruinsma (GB) model, providing numerical evidence for the significant dependence on initial conditions of the NEP's asymptotic behavior in both models. Finally, we highlight some open questions.Peer reviewedMultidisciplinary Digital Publishing InstituteWio, Horacio S. [0000-0001-6183-9617]Deza, Roberto R. [0000-0002-2469-3302]Revelli, Jorge A. [0000-0003-2135-3510]Gallego, Rafael [0000-0002-8277-6026]García-García, Reinaldo [0000-0002-5491-6036]Rodríguez, Miguel A. [0000-0003-4184-0463]Consejo Superior de Investigaciones Científicas [https://ror.org/02gfc7t72]202620262026info:eu-repo/semantics/articlehttp://purl.org/coar/resource_type/c_6501Publisher's versioninfo:eu-repo/semantics/publishedVersionapplication/pdfhttp://hdl.handle.net/10261/424974https://api.elsevier.com/content/abstract/scopus_id/105028506175reponame:DIGITAL.CSIC. Repositorio Institucional del CSICinstname:Consejo Superior de Investigaciones Científicas (CSIC)Ingléshttps://doi.org/10.3390/e28010055Síinfo:eu-repo/semantics/openAccessoai:digital.csic.es:10261/4249742026-05-22T06:33:51Z
dc.title.none.fl_str_mv The KPZ equation of kinetic interface roughening: A variational perspective
title The KPZ equation of kinetic interface roughening: A variational perspective
spellingShingle The KPZ equation of kinetic interface roughening: A variational perspective
Wio, Horacio S.
Variational approach
KPZ equation
Kinetic interface roughening
title_short The KPZ equation of kinetic interface roughening: A variational perspective
title_full The KPZ equation of kinetic interface roughening: A variational perspective
title_fullStr The KPZ equation of kinetic interface roughening: A variational perspective
title_full_unstemmed The KPZ equation of kinetic interface roughening: A variational perspective
title_sort The KPZ equation of kinetic interface roughening: A variational perspective
dc.creator.none.fl_str_mv Wio, Horacio S.
Deza, Roberto R.
Revelli, Jorge A.
Gallego, Rafael
García-García, Reinaldo
Rodríguez, Miguel A.
author Wio, Horacio S.
author_facet Wio, Horacio S.
Deza, Roberto R.
Revelli, Jorge A.
Gallego, Rafael
García-García, Reinaldo
Rodríguez, Miguel A.
author_role author
author2 Deza, Roberto R.
Revelli, Jorge A.
Gallego, Rafael
García-García, Reinaldo
Rodríguez, Miguel A.
author2_role author
author
author
author
author
dc.contributor.none.fl_str_mv Wio, Horacio S. [0000-0001-6183-9617]
Deza, Roberto R. [0000-0002-2469-3302]
Revelli, Jorge A. [0000-0003-2135-3510]
Gallego, Rafael [0000-0002-8277-6026]
García-García, Reinaldo [0000-0002-5491-6036]
Rodríguez, Miguel A. [0000-0003-4184-0463]
Consejo Superior de Investigaciones Científicas [https://ror.org/02gfc7t72]
dc.subject.none.fl_str_mv Variational approach
KPZ equation
Kinetic interface roughening
topic Variational approach
KPZ equation
Kinetic interface roughening
description Interfaces of rather different natures-as, e.g., bacterial colony or forest fire boundaries, or semiconductor layers grown by different methods (MBE, sputtering, etc.)-are self-affine fractals, and feature scaling with universal exponents (depending on the substrate's dimensionality d and global topology, as well as on the driving randomness' 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-Parisi-Zhang (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>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-which we call the non-equilibrium potential (NEP)-as well as its scaling behavior with the nonlinearity parameter λ, we discuss the stochastic thermodynamics of the roughening process and the memory of process h(x) in KPZ and in the related Golubović-Bruinsma (GB) model, providing numerical evidence for the significant dependence on initial conditions of the NEP's asymptotic behavior in both models. Finally, we highlight some open questions.
publishDate 2026
dc.date.none.fl_str_mv 2026
2026
2026
dc.type.none.fl_str_mv info:eu-repo/semantics/article
http://purl.org/coar/resource_type/c_6501
Publisher's version
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/10261/424974
https://api.elsevier.com/content/abstract/scopus_id/105028506175
url http://hdl.handle.net/10261/424974
https://api.elsevier.com/content/abstract/scopus_id/105028506175
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv https://doi.org/10.3390/e28010055
Sí
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Multidisciplinary Digital Publishing Institute
publisher.none.fl_str_mv Multidisciplinary Digital Publishing Institute
dc.source.none.fl_str_mv reponame:DIGITAL.CSIC. Repositorio Institucional del CSIC
instname:Consejo Superior de Investigaciones Científicas (CSIC)
instname_str Consejo Superior de Investigaciones Científicas (CSIC)
reponame_str DIGITAL.CSIC. Repositorio Institucional del CSIC
collection DIGITAL.CSIC. Repositorio Institucional del CSIC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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