Taming L\'evy flights in confined crowded geometries

TL;DR

This study investigates how obstacles in confined geometries affect the superdiffusive Cauchy random walk of a tracer particle, revealing sublinear growth of mean squared displacement and complex anomalous diffusion properties.

Contribution

It demonstrates the impact of obstacles on Le9vy flights, showing a transition from superdiffusive to subdiffusive behavior and analyzing the memory properties of the process.

Findings

Obstacles cause the mean squared displacement to grow sublinearly.

The diffusion remains memoryless over long times despite inhomogeneity.

Anomalous diffusion properties depend on obstacle configuration.

Abstract

We study a two-dimensional diffusive motion of a tracer particle in restricted, crowded anisotropic geometries. The underlying medium is the same as in our previous work [J. Chem. Phys. 140, 044706 (2014)] in which standard, gaussian diffusion was studied. Here, a tracer is allowed to perform Cauchy random walk with uncorrelated steps. Our analysis shows that presence of obstacles significantly influences motion, which in an obstacle-free space would be of a superdiffusive type. At the same time, the selfdiffusive process reveals different anomalous properties, both at the level of a single trajectory realization and after the ensemble averaging. In particular, due to obstacles, the sample mean squared displacement asymptotically grows sublinearly in time, suggesting non-Markov character of motion. Closer inspection of survival probabilities indicates however that underlying diffusion is memoryless over long time scales despite strong inhomogeneity of motion induced by orientational ordering.