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, , 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.[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 , 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–
Vicsekscaling relation of the
roughness[4]
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:[2]
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 is
ill-defined in a classical sense.
This section's factual accuracy is
disputed. The term is not small. In fact, it is huge. So one needs to subtract a huge term reflecting the small scale fluctuations. Relevant discussion may be found on the
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This derivation is from [7] and.[8] 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
Gaussianwhite 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)[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. 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