A stochastic model for anomalous diffusion in confined nano-films near a strain-induced critical point

Abstract

A diffusive process is said to be anomalous if in any given direction the average square of the separation a particle experiences from its origin grows nonlinearly with time. Any diffusive process is anomalous if viewed on a short enough time scale, but interestingly, many diffusive processes remain anomalous over longer times. As a canonical example we study one such process here, diffusion in a laterally-confined nano-film as a function of the strain induced critical point. For this example we motivate and illustrate how a simple but novel process, Brownian motion run with a nonlinear clock (Bm-nlc), statistically mimics trajectories generated via Newton’s force law. The model is easily generalized to more complicated random processes and has application in many fields, including but not limited to, random conductivity field or terrain generation, Richardson turbulence in the atmosphere, and time dependent dispersion in hydrology.