Paradoxical diffusion: Discriminating between normal and anomalous random walks

TL;DR

This paper reveals that some anomalous diffusion processes can appear normal in their mean squared displacement but are actually non-Markovian and non-Gaussian, challenging traditional classification methods.

Contribution

It demonstrates that normal-looking diffusion can be non-Markovian and non-Gaussian, using a new Monte Carlo simulation framework to identify paradoxical diffusion behaviors.

Findings

Normal diffusive behavior can be non-Markovian and non-Gaussian.

Standard second moment analysis may be misleading for classifying diffusion.

Markovianity criteria can detect hidden memory effects.

Abstract

Commonly, normal diffusive behavior is characterized by a linear dependence of the second central moment on time, $< x^2(t) >\propto t$, while anomalous behavior is expected to show a different time dependence, $ < x^2(t) > \propto t^{\delta}$ with $\delta <1$ for subdiffusive and $\delta >1$ for superdiffusive motions. Here we demonstrate that this kind of qualification, if applied straightforwardly, may be misleading: There are anomalous transport motions revealing perfectly "normal" diffusive character ($< x^2(t) >\propto t$), yet being non-Markov and non-Gaussian in nature. We use recently developed framework \cite[Phys. Rev. E \textbf{75}, 056702 (2007)]{magdziarz2007b} 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 paradoxical 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).