Diffusion in a Prandtl-Tomlinson potential: Analytical and Monte Carlo results

The Prandtl-Tomlinson model is one of the most simple and efficient approaches used to describe the nanoscale frictional behavior of an atomic force microscope (AFM) tip moving (“diffusing”) across a flat crystal surface. In its classical version, this model incorporates a sinusoidal energy potential as a function of lateral distance, where the tip slides between minima in a finite system (particle in box like system) and the corresponding transition rates are controlled by thermally activated processes that depend on the tip position over the energy profile. This work analyzes the parameters that characterize these type of diffusing systems, such as mean square displacement and roughness, by means of analytical solutions and kinetic Monte Carlo simulations. An analytical expression that can be numerically solved is obtained for the occupation probabilities. This expression validates the Monte Carlo algorithm, which allows the study of larger systems and their temporal evolution. It is shown that the distribution of these probabilities follows a normal distribution determined by parameters that are intrinsic to the PT potential. Thus allowing an analytical expression for this normal probability distribution to be propose. The results of such analysis could be used to rationalize AFM results, in particular in the field of nanotribology, and could also be used to analyze other diffusion controlled processes.