Probing rough surfaces: Markovian versus non-Markovian processes

We demonstrate that Markov processes play a fundamental role in probing rough surfaces and characterizing their topography. The surface topography obtained by a probe, as in the atomic force microscopy (AFM) technique, is based on the probe-surface interactions. When the size of the probe tip is comparable with the height of the surface fluctuation, the surface image can be aberrated from its origin. Due to the tip effect, there is a crossover in the structure function of the surface. For scales smaller than the Markov length the minimum length scale over which a process is Markovian-the stochastic function that describes a rough surface is non-Markovian, whereas for length scales larger than the Markov length, the surface may be described by a Markov process. Synthetic rough surfaces generated by fractional Guassian noise (FGN), as well as the rough surface of V2O5, obtained by AFM, confirm this.