Mode coupling and renormalization group results for the noisy Burgers equation

TL;DR

This paper uses renormalization group and mode-coupling techniques to analyze the noisy Burgers equation, confirming the validity of mode-coupling approximations and providing accurate scaling functions and finite-size effects.

Contribution

It demonstrates that mode-coupling theory is a reliable self-consistent approximation for the noisy Burgers equation, supported by two-loop renormalization group analysis and vertex correction evaluations.

Findings

Mode-coupling approach yields accurate scaling functions.

Vertex corrections are of order one but have small impact on correlations.

Finite-size effects can be effectively studied.

Abstract

We investigate the noisy Burgers equation (Kardar--Parisi--Zhang equation in 1+1 dimensions) using the dynamical renormalization group (to two--loop order) and mode--coupling techniques. The roughness and dynamic exponent are fixed by Galilean invariance and a fluctuation--dissipation theorem. The fact that there are no singular two--loop contributions to the two--point vertex functions supports the mode--coupling approach, which can be understood as a self--consistent one--loop theory where vertex corrections are neglected. Therefore, the numerical solution of the mode coupling equations yields very accurate results for the scaling functions. In addition, finite--size effects can be studied. Furthermore, the results from exact Ward identities, as well as from second--order perturbation theory permit the quantitative evaluation of the vertex corrections, and thus provide a quantitative test for the mode--coupling approach. It is found that the vertex corrections themselves are of the order one. Surprisingly, however, their effect on the correlation function is substantially smaller.