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 :
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.
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 α = 1/2, growth exponent β = 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 Family–Vicsek scaling relation of the roughness
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:
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 Hölder 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. In 2014 he was awarded the Fields Medal for this work, along with rough paths theory and regularity structures.
- Fokker–Planck equation
- Stochastic partial differential equation
- Universality (dynamical systems)
- rough path
- Renormalization group
- surface growth
- quantum field theory
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- Hairer, Martin; Quastel, J (2014), Weak universality of the KPZ equation (PDF)
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- "Solving the KPZ equation | Annals of Mathematics". Retrieved 2019-05-06.
- Hairer, Martin (2013). "Solving the KPZ equation". Annals of Mathematics. 178 (2): 559–664. arXiv:1109.6811. doi:10.4007/annals.2013.178.2.4.
- 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 http://math.arizona.edu/~mathphys/school_2012/IntroKPZ-Arizona.pdf