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Concept
Kardar–Parisi–Zhang equation
Summary
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.
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.
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.
Official source
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