Survey of roughness by stochastic oscillations

In this paper, connection 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 −κy+η(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 ∂h/∂s. PACS numbers: 05.40.-a, 02.50.Ey, 68.35.Ja