Kardar–Parisi–Zhang equation

Summary

The Kardar–Parisi–Zhang (KPZ) equation is the stochastic partial differential equation $h_t = \nu h_{xx} + \frac{\lambda}{2} (h_x)^2 + \xi$, where $\xi$ is space-time white noise. It was proposed by Kardar, Parisi and Zhang in 1986 as a model for a randomly growing interface. Its solutions are so rough that the square $(h_x)^2$ has no classical meaning, and giving the equation a meaning is a hard problem. It is the centre of the KPZ universality class, which contains many growth and particle models. The large-time fluctuations in this class are described by distributions from random matrix theory, such as the Tracy–Widom distribution.