The Kardar-Parisi-Zhang Equation with Temporally Correlated Noise - A Self Consistent Approach

TL;DR

This paper introduces a self-consistent approach to analyze the KPZ equation with temporally correlated noise, deriving scaling exponents and behavior of the dynamic structure factor, extending previous methods and highlighting the need for further numerical studies.

Contribution

It develops a generalized self-consistent method based on the Langevin form for the KPZ equation with correlated noise, surpassing traditional Fokker-Planck based approaches.

Findings

Derived scaling exponents for KPZ with correlated noise

Analyzed the dynamic structure factor $\,\Phi_q(t)$ behavior

Compared with other analytical and numerical methods

Abstract

In this paper we discuss the well known Kardar Parisi Zhang (KPZ) equation driven by temporally correlated noise. We use a self consistent approach to derive the scaling exponents of this system. We also draw general conclusions about the behavior of the dynamic structure factor $\Phi_q(t)$ as a function of time. The approach we use here generalizes the well known self consistent expansion (SCE) that was used successfully in the case of the KPZ equation driven by white noise, but unlike SCE, it is not based on a Fokker-Planck form of the KPZ equation, but rather on its Langevin form. A comparison to two other analytical methods, as well as to the only numerical study of this problem is made, and a need for an updated extensive numerical study is identified. We also show that a generalization of this method to any spatio-temporal correlations in the noise is possible, and two examples of this kind are considered.