Strong coupling probe for the Kardar-Parisi-Zhang equation

TL;DR

This paper provides an exact solution to the deterministic KPZ equation with a local force, revealing a critical force threshold and scaling behaviors in higher dimensions, and identifying an upper critical dimension.

Contribution

It introduces an exact solution for the deterministic KPZ equation with local driving force, highlighting new critical behaviors and the existence of an upper critical dimension at 4.

Findings

For d ≤ 2, interface velocity is non-zero for any positive force.

For d > 2, a critical force f_c is needed to induce interface motion.

Scaling laws for velocity near f_c and surface distortion are derived.

Abstract

We present an exact solution of the {\it deterministic} Kardar-Parisi-Zhang (KPZ) equation under the influence of a local driving force $f$. For substrate dimension $d \le 2$ we recover the well-known result that for arbitrarily small $f>0$, the interface develops a non-zero velocity $v(f)$. Novel behaviour is found in the strong-coupling regime for $d > 2$, in which $f$ must exceed a critical force $f_c$ in order to drive the interface with constant velocity. We find $v(f) \sim (f-f_c)^{\alpha (d)}$ for $f \searrow f_{c}$. In particular, the exponent $\alpha (d) = 2/(d-2)$ for $2<d<4$, but saturates at $\alpha(d)=1$ for $d>4$, indicating that for this simple problem, there exists a finite upper critical dimension $d_u=4$. For $d>2$ the surface distortion caused by the applied force scales logarithmically with distance within a critical radius $R_{c} \sim (f-f_{c})^{-\nu(d)}$, where $\nu(d) = \alpha (d)/2$. Connections between these results, and the critical properties of the weak/strong-coupling transition in the noisy KPZ equation are pursued.