Scaling behavior of randomly alternating surface growth processes

TL;DR

This paper analyzes how the roughness of surfaces grown by two processes that randomly alternate in time scales, revealing how the growth exponent depends on the distribution of application durations and extending results to both linear and non-linear processes.

Contribution

It provides an analytical framework for understanding the scaling behavior of surface roughness under random alternating processes, including effects of distribution tail properties.

Findings

Growth exponent depends on the tail of the duration distribution.

For finite mean durations, behavior matches cyclical process with mean application times.

Power-law tail distributions lead to continuously varying growth exponents.

Abstract

The scaling properties of the roughness of surfaces grown by two different processes randomly alternating in time, are addressed. The duration of each application of the two primary processes is assumed to be independently drawn from given distribution functions. We analytically address processes in which the two primary processes are linear and extend the conclusions to non-linear processes as well. The growth scaling exponent of the average roughness with the number of applications is found to be determined by the long time tail of the distribution functions. For processes in which both mean application times are finite, the scaling behavior follows that of the corresponding cyclical process in which the uniform application time of each primary process is given by its mean. If the distribution functions decay with a small enough power law for the mean application times to diverge, the growth exponent is found to depend continuously on this power law exponent. In contrast, the roughness exponent does not depend on the timing of the applications. The analytical results are supported by numerical simulations of various pairs of primary processes and with different distribution functions. Self-affine surfaces grown by two randomly alternating processes are common in nature (e.g., due to randomly changing weather conditions) and in man-made devices such as rechargeable batteries.