# Kardarâ€“Parisiâ€“Zhang equation Information

*https://en.wikipedia.org/wiki/Kardarâ€“Parisiâ€“Zhang_equation*

This article needs attention from an expert in Mathematics. The specific problem is: Disputed section. (February 2021) |

In
mathematics, the **Kardarâ€“Parisiâ€“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 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 the 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
SOS model. A rigorous proof has been given by Bertini and Giacomin in the case of the SOS model.^{
[3]}

## 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 *Î±* = 1/2, growth exponent *Î²* = 1/3, and dynamic exponent *z* = 3/2. 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 t and L is the size of the system. For models within the KPZ class, the main properties of the surface can be characterized by the
Familyâ€“
Vicsek
scaling relation of the
roughness^{
[4]}

with a scaling function satisfying

In 2014, Hairer and Quastel have shown that more generally the following KPZ-like equations lie within the KPZ universality class:^{
[2]}

Here is any even-degree polynomial.

## Solving the KPZ equation

Due to the nonlinearity in the equation and the presence of space-time white-noise, the solutions to the KPZ equation are known not to be smooth or regular but rather 'fractal' or 'rough.' Indeed, even without the nonlinear term, the equation reduces to the stochastic heat equation, whose solution is not differentiable in the space variable but verifies a HÃ¶lder condition with exponent < 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 constructing approximations using
Feynman diagrams.^{
[5]} In 2014 he was awarded the
Fields Medal for this work, along with
rough paths theory and
regularity structures.

## Physical derivation

This section's factual accuracy is
disputed. (February 2021) |

This derivation is from ^{
[6]} and.^{
[7]} Suppose we want to describe a
surface growth by some
partial differential equation. Let represent the height of the surface at position x and at 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

But this is a deterministic equation ( heat equation) and the surface has no 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 Edwardsâ€“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. 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, Zhang (KPZ) ^{
[1]} was that, while a surface grows in a normal (to the surface) direction, we are measuring the height on the height axis, which is perpendicular to the space x 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 however this leads to a seemingly intractable equation. In fact what is done is to take a general F and expand it

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

The second should vanish because of the symmetry, but could anyway have been removed from the equation by a constant velocity shift of coordinates: 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

- Fokkerâ€“Planck equation
- Stochastic partial differential equation
- Universality (dynamical systems)
- rough path
- fractal
- Renormalization group
- surface growth
- quantum field theory

## Sources

- ^
^{a}^{b}Kardar, Mehran; Parisi, Giorgio; Zhang, Yi-Cheng (3 March 1986). "Dynamic Scaling of Growing Interfaces".*Physical Review Letters*.**56**(9): 889â€“892. Bibcode: 1986PhRvL..56..889K. doi: 10.1103/PhysRevLett.56.889. PMID 10033312. - ^
^{a}^{b}Hairer, Martin; Quastel, J (2014),*Weak universality of the KPZ equation*(PDF) **^**Bertini, Lorenzo; Giacomin, Giambattista (1997). "Stochastic Burgers and KPZ equations from particle systems".*Communications in Mathematical Physics*.**183**(3): 571â€“607. Bibcode: 1997CMaPh.183..571B. CiteSeerX 10.1.1.49.4105. doi: 10.1007/s002200050044. S2CID 122139894.**^**Family, F.; Vicsek, T. (1985). "Scaling of the active zone in the Eden process on percolation networks and the ballistic deposition model".*Journal of Physics A: Mathematical and General*.**18**(2): L75â€“L81. Bibcode: 1985JPhA...18L..75F. doi: 10.1088/0305-4470/18/2/005.**^**Hairer, Martin (2013). "Solving the KPZ equation".*Annals of Mathematics*.**178**(2): 559â€“664. arXiv: 1109.6811. doi: 10.4007/annals.2013.178.2.4. S2CID 119247908.**^**"Lecture Notes by Jeremy Quastel" (PDF).**^**Tomohiro, Sasamoto (2016). "The 1D Kardarâ€“Parisiâ€“Zhang equation: height distribution and universality".*Progress of Theoretical and Experimental Physics*.**2016**(2). doi: 10.1093/ptep/ptw002.

## Notes

- Barabasi, Albert-Laszlo; Stanley, Harry Eugene (1995).
*Fractal concepts in surface growth*. Cambridge University Press. ISBN 978-0-521-48318-6. - Corwin, Ivan (2011). "The Kardar-Parisi-Zhang equation and universality class". arXiv: 1106.1596 [ math.PR].
- "Lecture Notes by Jeremy Quastel" (PDF).
- Tomohiro, Sasamoto (2016).
"The 1D Kardarâ€“Parisiâ€“Zhang equation: height distribution and universality".
*Progress of Theoretical and Experimental Physics*.**2016**(2). doi: 10.1093/ptep/ptw002.