Kardar–Parisi–Zhang equation

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 ${\displaystyle h({\vec {x}},t)}$ with spatial coordinate ${\displaystyle {\vec {x}}}$ and time coordinate ${\displaystyle t}$:

${\displaystyle {\frac {\partial h({\vec {x}},t)}{\partial t}}=\nu \nabla ^{2}h+{\frac {\lambda }{2}}\left(\nabla h\right)^{2}+\eta ({\vec {x}},t)\;.}$

Here, ${\displaystyle \eta ({\vec {x}},t)}$ is white Gaussian noise with average

${\displaystyle \langle \eta ({\vec {x}},t)\rangle =0}$

and second moment

${\displaystyle \langle \eta ({\vec {x}},t)\eta ({\vec {x}}',t')\rangle =2D\delta ^{d}({\vec {x}}-{\vec {x}}')\delta (t-t'),}$

${\displaystyle \nu }$, ${\displaystyle \lambda }$, and ${\displaystyle D}$ are parameters of the model, and ${\displaystyle d}$ is the dimension.

In one spatial dimension, the KPZ equation corresponds to a stochastic version of Burgers' equation with field ${\displaystyle u(x,t)}$ via the substitution ${\displaystyle u=-\lambda \,\partial h/\partial x}$.

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.