Discriminating between normal and anomalous random walks

Abstract

Commonly, normal diffusive behavior is characterized by a linear dependence of the second central moment on time, {x2(t) proportional t, while anomalous behavior is expected to show a different time dependence, x2(t) proportional t{delta} with delta<1 for subdiffusive and delta>1 for superdiffusive motions. Here we explore in details the fact that this kind of qualification, if applied straightforwardly, may be misleading: there are anomalous transport motions revealing perfectly "normal" diffusive character (x2(t) proportional t) yet being non-Markov and non-Gaussian in nature. We use recently developed framework of Monte Carlo simulations which incorporates anomalous diffusion statistics in time and space and creates trajectories of such an extended random walk. For special choice of stability indices describing statistics of waiting times and jump lengths, the ensemble analysis of anomalous diffusion is shown to hide temporal memory effects which can be properly detected only by examination of formal criteria of Markovianity (fulfillment of the Chapman-Kolmogorov equation).