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.
[1][2] It describes the temporal change of a height field
is white Gaussian noise with average
In one spatial dimension, the KPZ equation corresponds to a stochastic version of Burgers' equation with field
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.
[3] Many interacting particle systems, such as the totally asymmetric simple exclusion process, lie in the KPZ universality class.
In order to check if a growth model is within the KPZ class, one can calculate the width of the surface: where
is the mean surface height at time
For models within the KPZ class, the main properties of the surface
satisfying In 2014, Hairer and Quastel showed that more generally, the following KPZ-like equations lie within the KPZ universality class:[2] where
A family of processes that are conjectured to be universal limits in the (1+1) KPZ universality class and govern the long time fluctuations are the Airy processes and the KPZ fixed point.
Due to the nonlinearity in the equation and the presence of space-time white noise, solutions to the KPZ equation are known to not be smooth or regular, but rather 'fractal' or 'rough.'
Even without the nonlinear term, the equation reduces to the stochastic heat equation, whose solution is not differentiable in the space variable but satisfies a Hölder condition with exponent less than 1/2.
In 2013, Martin Hairer made a breakthrough in solving the KPZ equation by an extension of the Cole–Hopf transformation and constructing approximations using Feynman diagrams.
[5] In 2014, he was awarded the Fields Medal for this work on the KPZ equation, along with rough paths theory and regularity structures.
[8] Suppose we want to describe a surface growth by some partial differential equation.
represent the height of the surface at position
The simplest way to include fluctuations is to add a noise term.
taken to be the Gaussian white noise with mean zero and covariance
Since this is a linear equation, it can be solved exactly by using Fourier analysis.
This means the EW equation is not enough to describe the surface growth of interest, so we need to add a nonlinear function for the growth.
Therefore, surface growth change in time has three contributions.
The first models lateral growth as a nonlinear function of the form
The second is a relaxation, or regularization, through the diffusion term
A great observation of Kardar, Parisi, and Zhang (KPZ)[1] was that while a surface grows in a normal direction (to the surface), we are measuring the height on the height axis, which is perpendicular to the space axis, and hence there should appear a nonlinearity coming from this simple geometric effect.
is small, the effect takes the form
and expand it as a Taylor series, The first term can be removed from the equation by a time shift, since if
, but could anyway have been removed from the equation by a constant velocity shift of coordinates, since if
solves Thus the quadratic term is the first nontrivial contribution, and it is the only one kept.