Survey of roughness by stochastic oscillations

TL;DR

This paper explores the relationship between surface roughness and stochastic oscillations using models like random walks and Ornstein-Uhlenbeck processes, providing analytical solutions and insights into surface profiles.

Contribution

It introduces a novel approach linking surface roughness to stochastic oscillations through analytical models and the KPZ equation, enhancing understanding of rough surface dynamics.

Findings

Continuous random walk leads to a KPZ equation solution.

Ornstein-Uhlenbeck process provides surface profile information.

Analytical solutions connect stochastic models to surface roughness.

Abstract

In this paper, connections between surface roughness and directed polymers in random medium are studied, when the surface is considered as a directed line undergoing stochastic oscillations. This is performed by studying the influence of a stochastic elastic forcing term $-\kappa y+\eta(s)$, on a particle moving along a rough surface. Two models are proposed and analysed in this way: the random-walk process (RW) in its discrete and continuous form, and a Markovian process via the Ornstein-Uhlenbeck (O-U) process. It is shown that the continuous RW leads to an oscillator equation, via an effective action obeying a KPZ equation which is solved analytically. The O-U process allows to obtain information on the profile of surface for a long size substrate. The analogy with the roughness is achieved by introducing a quantity suited to directed line formalism: the height velocity variation $\partial h/\partial s$.