Growing Surfaces with Anomalous Diffusion - Results for the Fractal Kardar-Parisi-Zhang Equation

TL;DR

This paper investigates a fractional Laplacian-based model for surface growth with anomalous diffusion, deriving exact and predicted critical exponents across dimensions, and discusses the model's behavior and critical dimension.

Contribution

It provides an exact solution for the one-dimensional case and uses Self-Consistent Expansion to predict exponents in arbitrary dimensions for the Fractal KPZ equation.

Findings

Exact critical exponents in 1D for FKPZ

SCE predictions match exact results in 1D

Discussion of upper critical dimension

Abstract

In this paper I study a model for a growing surface in the presence of anomalous diffusion, also known as the Fractal Kardar-Parisi-Zhang equation (FKPZ). This equation includes a fractional Laplacian that accounts for the possibility that surface transport is caused by a hopping mechanism of a Levy flight. I show that for a specific choice of parameters of the FKPZ equation, the equation can be solved exactly in one dimension, so that all the critical exponents, which describe the surface that grows under FKPZ, can be derived for that case. Afterwards, I use a Self-Consistent Expansion (SCE) to predict the critical exponents for the FKPZ model for any choice of the parameters and any spatial dimension. It is then verified that the results obtained using SCE recover the exact result in one dimension. At the end a simple picture for the behavior of the Fractal KPZ equation is suggested, and the upper critical dimension of this model is discussed.