Dynamics of radial fractional growing surfaces

Abstract

Fractional derivatives in the form of a Laplacian operator −(−∇2)ζ/2 appear in the stochastic differential equations of many physical systems. For a stochastically growing interface, the evolution of the height of the interface is usually written as the Langevin form of a fractional diffusion equation. This fractional Langevin equation has been recently generalized to radial geometries, and this has raised questions regarding the validity of the application of the standard scaling analysis to the study of radially growing interfaces. Here, we study the fractional Langevin equation in polar coordinate and calculate the correlation functions. We show that, for ζ > 1, the correlation increases as t1−(1/ζ) for small angles and in the long-time limit. This means that the interface width grows as t1/4 for ζ = 2 (radial Edwards-Wilkinson equation) and as t3/8 for ζ = 4 (radial Mullins-Herring equation). These results are the same as those obtained in the corresponding planar geometry. Finally, we obtain a logarithmic behavior with time for ζ ⩽ 1. In addition, we numerically solve the fractional stochastic equation and obtain the interface width for different values of ζ, arriving at results which confirm our analytical calculations.