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  with spatial coordinate  and time coordinate :

 

Here,  is white Gaussian noise with average 

 

and second moment

, , and  are parameters of the model, and  is the dimension. In one spatial dimension, the KPZ equation corresponds to a stochastic version of Burgers' equation with field  via the substitution . 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. KPZ universality class 
Many interacting particle systems, such as the totally asymmetric simple exclusion process, lie in the KPZ universality class. This class is characterized by the following critical exponents in one spatial dimension (1 + 1 dimension): the roughness exponent , growth exponent , and dynamic exponent . 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  and  is the size of the system. For models within the KPZ class, the main properties of the surface  can be characterized by the Family–Vicsek scaling relation of the roughness 

 

with a scaling function  satisfying

 

In 2014, Hairer and Quastel showed that more generally, the following KPZ-like equations lie within the KPZ universality class:

where  is any even-degree polynomial. 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. Solving the KPZ equation 
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. Thus, the nonlinear term  is ill-defined in a classical sense. 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. In 2014, he was awarded the Fields Medal for this work on the KPZ equation, along with rough paths theory and regularity structures. Physical derivation 

This derivation is from  and. Suppose we want to describe a surface growth by some partial differential equation. Let  represent the height of the surface at position  and time . Their values are continuous. We expect that there would be a sort of smoothening mechanism. Then the simplest equation for the surface growth may be taken to be the diffusion equation,  

But this is a deterministic equation, implying the surface has no random fluctuations. The simplest way to include fluctuations is to add a noise term. Then we may employ the equation

with  taken to be the Gaussian white noise with mean zero and covariance . This is known as the Edwards–Wilkinson (EW) equation or stochastic heat equation with additive noise (SHE). Since this is a linear equation, it can be solved exactly by using Fourier analysis. But since the noise is Gaussian and the equation is linear, the fluctuations seen for this equation are still Gaussian. 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 , and the third is the white noise forcing . Therefore,

The key term , the deterministic part of the growth, is assumed to be a function only of the slope, and to be a symmetric function. A great observation of Kardar, Parisi, and Zhang (KPZ) 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. When the surface slope  is small, the effect takes the form , but this leads to a seemingly intractable equation. To circumvent this difficulty, one can take a general  and expand it as a Taylor series,

The first term can be removed from the equation by a time shift, since if  solves the KPZ equation, then  solves 

The second should vanish because of the symmetry of , but could anyway have been removed from the equation by a constant velocity shift of coordinates, since if  solves the KPZ equation, then  solves 

Thus the quadratic term is the first nontrivial contribution, and it is the only one kept. We arrive at the KPZ equation

See also 

 Fokker–Planck equation
 Fractal
 Quantum field theory
 Renormalization group
 Rough path
 Stochastic partial differential equation
 Surface growth
 Tracy–Widom distribution
 Universality (dynamical systems)

Sources

Notes 

 
 
 
 

Statistical mechanics
Stochastic differential equations
Partial differential equations