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


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

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

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

and second moment

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

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

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

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.

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 $$\alpha=\tfrac{1}{2}$$, growth exponent $$\beta=\tfrac{1}{3}$$, and dynamic exponent $$z=\tfrac{3}{2}$$. In order to check if a growth model is within the KPZ class, one can calculate the width of the surface:


 * $$W(L,t)=\left\langle\frac1L\int_0^L \big( h(x,t)-\bar{h}(t)\big)^2 \, dx\right\rangle^{1/2},$$

where $$ \bar{h}(t) $$ is the mean surface height at time $$ t $$ and $$ L $$ is the size of the system. For models within the KPZ class, the main properties of the surface $$ h(x,t) $$ can be characterized by the Family–Vicsek scaling relation of the roughness



W(L,t) \approx L^{\alpha} f(t/L^z), $$

with a scaling function f(u) satisfying



f(u) \propto \begin{cases} u^{\beta} & \ u\ll 1 \\ 1 & \ u\gg1\end{cases} $$

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


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

where $$P$$ 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 and the KPZ fixed point.

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 $$ \left(\nabla h\right)^2 $$ 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. There were 6 different analytic self-similar solutions found for the (1+1) KPZ equation with different analytic noise terms.

Physical derivation
This derivation is from and. Suppose we want to describe a surface growth by some partial differential equation. Let $$ h(x, t)$$ represent the height of the surface at position $$ x $$ and time $$ t $$. 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,


 * $$\frac{\partial h(x,t)}{\partial t }=\frac{1}{2}\frac{\partial^{2} h(x,t)}{\partial x^2 } $$

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


 * $$\frac{\partial h(x,t)}{\partial t }=\frac{1}{2}\frac{\partial^{2} h(x,t)}{\partial^{2} x }+\eta(x,t), $$

with $$\eta $$ taken to be the Gaussian white noise with mean zero and covariance $$E[\eta(x,t)\eta(x',t')]=\delta(x-x')\delta(t-t') $$. 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 $$F\left(\frac{\partial h(x,t)}{\partial x }\right)$$. The second is a relaxation, or regularization, through the diffusion term $$ \frac{\partial^{2} h(x,t)}{\partial^{2} x }$$, and the third is the white noise forcing $$ \eta(x,t) $$. Therefore,


 * $$\frac{\partial h(x,t)}{\partial t }=-\lambda F\left(\frac{\partial h(x,t)}{\partial x }\right)+ \frac{1}{2}\frac{\partial^{2} h(x,t)}{\partial^{2} x }+\eta(x,t) $$

The key term $$ F\left(\frac{\partial h(x,t)}{\partial x }\right)$$, 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 $$\partial_x h = \tfrac{\partial h}{\partial x} $$ is small, the effect takes the form $$F(\partial_{x}h)=(1+|\partial_{x}h|^{2})^{-\frac{1}{2}} $$, but this leads to a seemingly intractable equation. To circumvent this difficulty, one can take a general $$ F $$ and expand it as a Taylor series,


 * $$ F(s)=F(0)+F'(0)s+\frac{1}{2}F''(0)s^{2}+... $$

The first term can be removed from the equation by a time shift, since if $$ h(x,t)$$ solves the KPZ equation, then $$ \tilde{h}(x,t):=h(x,t)-\lambda F(0)t$$ solves


 * $$\frac{\partial h(x,t)}{\partial t }=-\lambda F(0)+ \frac{1}{2}\frac{\partial^{2} h(x,t)}{\partial^{2} x }+\eta(x,t). $$

The second should vanish because of the symmetry of $$ F $$, but could anyway have been removed from the equation by a constant velocity shift of coordinates, since if $$ h(x,t)$$ solves the KPZ equation, then $$ \tilde{h}(x,t):=h(x- \lambda F'(0)t,t-\lambda F'(0)x)$$ solves


 * $$\frac{\partial \tilde{h}(x,t)}{\partial t }=-\lambda F'(0)\frac{\partial \tilde{h}(x,t)}{\partial x }+ \frac{1}{2}\frac{\partial^{2} \tilde{h}(x,t)}{\partial^{2} x }+\eta(x,t). $$

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


 * $$\frac{\partial h(x,t)}{\partial t }=-\lambda\left(\frac{\partial h(x,t)}{\partial x }\right)^{2} + \frac{1}{2}\frac{\partial^{2} h(x,t)}{\partial^{2} x }+\eta(x,t). $$