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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.[1][2][3] 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 the 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 SOS model. A rigorous proof has been given by Bertini and Giacomin in the case of the SOS model.[4]

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:

$$W(L,t)=\left\langle {\frac 1L}\int _{0}^{L}{\big (}h(x,t)-{\bar {h}}(t){\big )}^{2}\,dx\right\rangle ^{{1/2}},$$

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 ) {\displaystyle h(x,t)} h(x,t) can be characterized by the Family–Vicsek scaling relation of the roughness[5]

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

with a scaling function f(u) satisfying

$${\begin{cases}u^{{\beta }}&\ u\ll 1\\1&\ u\gg 1\end{cases}}$$

In 2014, Hairer and Quastel have shown that more generally the following KPZ-like equations lie within the KPZ universality class:[3]

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

Here P 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 $$\left(\nabla h\right)^2$$ is ill-defined in a classical sense.

In 2013, Martin Hairer made a breakthrough in solving the KPZ equation by constructing approximations using Feynman diagrams.[6] In 2014 he was awarded the Fields Medal for this work, along with rough paths theory and regularity structures.[7]

Fokker–Planck equation
Stochastic partial differential equation
Universality (dynamical systems)
rough path
fractal
Renormalization group
surface growth
quantum field theory

Sources

Kardar, Mehran; Parisi, Giorgio; Zhang, Yi-Cheng (3 March 1986). "Dynamic Scaling of Growing Interfaces". Physical Review Letters. 56 (9): 889–892. Bibcode:1986PhRvL..56..889K. doi:10.1103/PhysRevLett.56.889. PMID 10033312.
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): 571–607. 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): L75–L81. Bibcode:1985JPhA...18L..75F. doi:10.1088/0305-4470/18/2/005.
"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. S2CID 119247908.

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).

Mathematics Encyclopedia