Strong Coupling Probe for the Kardar-Parisi-Zhang Equation

T.J. Newman; Harald Kallabis

doi:doi:10.1051/jp1:1996162

Issue		J. Phys. I France Volume 6, Number 3, March 1996


Page(s)		373 - 383
DOI		https://doi.org/10.1051/jp1:1996162

DOI: 10.1051/jp1:1996162
J. Phys. I France 6 (1996) 373-383

Strong Coupling Probe for the Kardar-Parisi-Zhang Equation

T.J. Newman¹ and Harald Kallabis²

¹ Institut für Theoretische Physik, Universität zu Köln, D-50937 Köln, Germany
² Höchstleistungsrechenzentrum, Forschungszentrum Jülich, 52425 Jülich, Germany

(Received 13 November 1995, received in final form 20 November 1995, accepted 27 November 1995)

Abstract
We present an exact solution of the deterministic Kardar-Parisi-Zhang (KPZ) equation under the influence of a local driving force f. For substrate dimension $d \leq 2$ 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 $f_{\rm c}$ in order to drive the interface with constant velocity. We find $v(f)\sim (f-f_{\rm c})^{\alpha (d)}$ for $f\searrow f_{\rm c}$ . 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 $d_{\rm u}=4$ . For d>2 the surface distortion caused by the applied force scales logarithmically with distance within a critical radius $R_{\rm c}\sim (f-f_{\rm c})^{-\nu (d)}$ , 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.