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Kardar–Parisi–Zhang equation — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Kardar%E2%80%93Parisi%E2%80%93Zhang_equation
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-07T14:31:39.426Z
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Kardar–Parisi–Zhang equation — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Kardar%E2%80%93Parisi%E2%80%93Zhang_equation (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_44e93ff697ee (Metraipolis archive dc_44e93ff697ee, sha256 e416ad2768ad97b59011ac99cd599beb4242f609ab72867757a4fd61fd6c4cac)

BibTeX

@misc{dc44e93ff697ee,
  title = {Kardar–Parisi–Zhang equation — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Kardar%E2%80%93Parisi%E2%80%93Zhang_equation},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_44e93ff697ee; sha256 e416ad2768ad97b59011ac99cd599beb4242f609ab72867757a4fd61fd6c4cac},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Kardar%E2%80%93Parisi%E2%80%93Zhang_equation. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In mathematics, the Kardar–Parisi–Zhang (KPZ) equation is a non-linear stochastic partial differential equation, introduced by Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang in 1986. It describes the temporal change of a height field h ( x → , t ) {\displaystyle h({\vec {x}},t)} with spatial coordinate x → {\displaystyle {\vec {x}}} and time coordinate t {\displaystyle t} : ∂ h ( x → , t ) ∂ t = ν ∇ 2 h + λ 2 ( ∇ h ) 2 + η ( x → , t ) . {\displaystyle {\frac {\partial h({\vec {x}},t)}{\partial t}}=\nu \nabla ^{2}h+{\frac {\lambda }{2}}\left(\nabla h\right)^{2}+\eta ({\vec {x}},t)\;.} Here, η ( x → , t ) {\displaystyle \eta ({\vec {x}},t)} is white Gaussian noise with average ⟨ η ( x → , t ) ⟩ = 0 {\displaystyle \langle \eta ({\vec {x}},t)\rangle =0} and second moment ⟨ η ( x → , t ) η ( x → ′ , t ′ ) ⟩ = 2 D δ d ( x → − x → ′ ) δ ( t − t ′ ) , {\displaystyle \langle \eta ({\vec {x}},t)\eta ({\vec {x}}',t')\rangle =2D\delta ^{d}({\vec {x}}-{\vec {x}}')\delta (t-t'),} ν {\displaystyle \nu } , λ {\displaystyle \lambda } , and D {\displaystyle D} are parameters of the model, and d {\displaystyle d} is the dimension. In one spatial dimension, the KPZ equation corresponds to a stochastic version of Burgers' equation with field u ( x , t ) {\displaystyle u(x,t)} via the substitution u = − λ ∂ h / ∂ x {\displaystyle u=-\lambda \,\partial h/\partial x} . Via the renormalization group, the KPZ equation is conjectured to be the field theory of many surface growth models, such as the Eden model, ballistic deposition, and the weakly asymmetric single step solid on solid process (SOS) model. A rigorous proof has been given by Bertini and Giacomin in the case of the SOS model.

Tema degli esploratori: fisica e astronomia (deposito sistematico; categoria «Statistical mechanics»).
Fonte: Wikipedia in inglese, voce «Kardar–Parisi–Zhang equation» — https://en.wikipedia.org/wiki/Kardar%E2%80%93Parisi%E2%80%93Zhang_equation
Wikidata Q6369122 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q6369122
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Kardar%E2%80%93Parisi%E2%80%93Zhang_equation?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1367951586, consultata il 2026-10-07T14:08:51.833Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 69b074167bd36864dbc882644a7ef794c543c2f8e28ae10a97a9e1455fd56a9c