Generalized discretization of the Kardar-Parisi-Zhang equation

TL;DR

This paper presents a generalized spatial discretization method for the 1+1 dimensional KPZ equation, deriving the steady state distribution and linking discretization schemes to specific models like ballistic deposition.

Contribution

It introduces a unified discretization framework for the KPZ equation and connects it to physical models, providing exact steady state solutions.

Findings

Exact steady state probability density function derived for any discretization scheme.

Discretization prescriptions are shown to be model-dependent.

Connection established between ballistic deposition and KPZ discretization.

Abstract

We introduce the generalized spatial discretization of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions. We solve exactly the steady state probability density function for the discrete heights of the interface, for any discretization scheme. We show that the discretization prescription is a consequence of each particular model. From the ballistic deposition model we derive the discretization prescription of the corresponding KPZ equation.