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

TL;DR

This paper models chemotaxis in porous media using a biased random walk inspired by Pac-Man, revealing that the movement exhibits anomalous diffusion with a continuously varying dynamical exponent depending on food attraction bias.

Contribution

It introduces a novel biased random walk model for chemotaxis on percolating clusters and characterizes its anomalous diffusion behavior.

Findings

Mean-squared displacement follows a power law with a non-standard dynamical exponent.

The dynamical exponent varies continuously with the bias towards food.

Movement slows down as bias towards food increases.

Abstract

We study chemotaxis in a porous medium using as a model a biased ("hungry") random walk on a percolating cluster. In close resemblance to the 1980s arcade game Pac-Man, the hungry random walker consumes food, which is initially distributed in the maze, and biases its movement towards food-filled sites. We observe that, on the percolating cluster, the mean-squared displacement of the pacman process shows anomalous dynamics, which 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, and results in slower differential growth when compared to the unbiased random walk.