Variational Formulation for the KPZ and Related Kinetic Equations

TL;DR

This paper introduces a variational approach to the KPZ equation, providing a thermodynamic-like potential that reveals invariance properties and insights into stationary distributions, extending to related nonlinear kinetic equations.

Contribution

It develops a variational formulation for KPZ and related equations, uncovering invariance properties and stationary distribution forms in arbitrary dimensions.

Findings

Proved global shift invariance properties of KPZ using the potential.

Derived stationary probability distribution functions for various dimensions.

Extended the variational approach to other nonlinear kinetic equations.

Abstract

We present a variational formulation for the Kardar-Parisi-Zhang (KPZ) equation that leads to a thermodynamic-like potential for the KPZ as well as for other related kinetic equations. For the KPZ case, with the knowledge of such a potential we prove some global shift invariance properties previously conjectured by other authors. We also show a few results about the form of the stationary probability distribution function for arbitrary dimensions. The procedure used for KPZ was extended in order to derive more general forms of such a functional leading to other nonlinear kinetic equations, as well as cases with density dependent surface tension.