Clearing out a maze: The hungry random walker and its anomalous diffusion

Tanja Schilling, Thomas Voigtmann

Published 2016-07-05Version 1

We study chemotaxis in a porous medium using as a model a biased ("hungry") random walk on a percolating cluster. The model closely resembles the 1980s arcade game Pac-Man\textsuperscript{\textregistered}, in which the player moves a particle through a maze which is filled with food. We observe that, on the percolating cluster, the hungry random walker's mean-squared displacement shows anomalous dynamics that follow a power law with a dynamical exponent different from both that of a self avoiding random walk as well as that of an unbiased random walk. The change in dynamics with the propensity to move towards food is well described by a dynamical exponent that depends continuously on this propensity. It crosses over from the exponent given by the walk dimension of the unbiased random walk on the percolating cluster, to the exponent of a random walk on all clusters in the system.