Kardar–Parisi–Zhang equation

The Kardar–Parisi–Zhang (KPZ) equation^[1] (named after its creators Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang) is a non-linear stochastic partial differential equation. It describes the temporal change of the height $h(\vec x,t)$ at place $\vec x$ and time $t$. It is formally

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

where $\eta(\vec x,t)$ 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 well known Burgers' equation, in a field $u(x,t)$ say, via the substitution $u=-\lambda\, \partial h/\partial x$.

By use of renormalization group techniques it has been conjectured that the KPZ equation is 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^[2] in the case of the SOS model.

Many models in the field of interacting particle systems, such as the totally asymmetric simple exclusion process, also lie in the KPZ universality class. This class is characterised by models which, in one spatial dimension (1 + 1 dimension) have a 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, $W(L,t)$, defined as

$W(L,t)=\left\langle\frac1L\int_0^L \big( h(x,t)-\bar{h}(t)\big)^2 \, dx\right\rangle^{1/2},$

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 $h(x,t)$ can be characterized by the Family–Vicsek scaling relation^[3] of the roughness, where

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

with a scaling function satisfying

$f(u) \propto \begin{cases} u^{\beta} & \ u\ll 1 \\ 1 & \ u\gg1\end{cases}$

Due to the nonlinearity in the equation and the presence of space-time white-noise, the mathematical study of the KPZ equation has proven to be quite challenging: 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. A breakthrough in the mathematical study of the KPZ equation was achieved by Martin Hairer, whose work on the KPZ equation ^[4] earned him the 2014 Fields Medal. Hairer and Quastel^[5] have recently shown that equations of the type

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

where $P$ is any even polynomial, lie in the KPZ universality class.

Sources

1. Kardar, Mehran; Parisi, Giorgio; Zhang, Yi-Cheng (3 March 1986). "Dynamic Scaling of Growing Interfaces". Physical Review Letters 56 (9): 889–892. doi:10.1103/PhysRevLett.56.889. PMID 10033312.
2. Bertini, Lorenzo; Giacomin, Giambattista (1997). "Stochastic Burgers and KPZ equations from particle systems". Communications in Mathematical Physics 183 (3): 571–607. doi:10.1007/s002200050044.
3. 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. doi:10.1088/0305-4470/18/2/005.
4. Hairer, Martin (2013). "Solving the KPZ equation". Annals of Mathematics 178 (2): 559–664. doi:10.4007/annals.2013.178.2.4.
5. Hairer, Martin; Quastel, J (2014), Weak universality of the KPZ equation (PDF)

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 http://math.arizona.edu/~mathphys/school_2012/IntroKPZ-Arizona.pdf