Stirred Kardar-Parisi-Zhang equation with quenched random noise: Emergence of induced nonlinearity

TL;DR

This paper investigates how turbulent environmental effects induce new nonlinearities in the stochastic Kardar-Parisi-Zhang equation with quenched noise, significantly impacting its scaling behavior and universality classes.

Contribution

It reveals the emergence of an induced nonlinearity due to turbulent advection, altering the critical behavior of the KPZ equation with quenched noise, and provides first-order critical exponent calculations.

Findings

Induced nonlinearity affects universality classes.

Turbulent advection influences scaling even when irrelevant.

Critical exponents calculated to first order.

Abstract

We study the stochastic Kardar-Parisi-Zhang equation for kinetic roughening where the time-independent (columnar or spatially quenched) Gaussian random noise $f(t,{\bf x})$ is specified by the pair correlation function $\langle f(t,{\bf x})f(t',{\bf x'}) \rangle \propto \delta^{(d)} ({\bf x-x'})$, $d$ being the dimension of space. The field-theoretic renormalization group analysis shows that the effect of turbulent motion of the environment (modelled by the coupling with the velocity field described by the Kazantsev-Kraichnan statistical ensemble for an incompressible fluid) gives rise to a new nonlinear term, quadratic in the velocity field. It turns out that this "induced" nonlinearity strongly affects the scaling behaviour in several universality classes (types of long-time, large-scale asymptotic regimes) even when the turbulent advection appears irrelevant in itself. Practical calculation of the critical exponents (that determine the universality classes) is performed to the first order of the double expansion in $\varepsilon=4-d$ and the velocity exponent $\xi$ (one-loop approximation). As is the case with most "descendants" of the Kardar-Parisi-Zhang model, some relevant fixed points of the renormalization group equations lie in "forbidden zones," i.e. in those corresponding to negative kinetic coefficients or complex couplings. This persistent phenomenon in stochastic non-equilibrium models requires careful and inventive physical interpretation.