Direct Numerical Observation of Anomalous Diffusion in Random Media

TL;DR

This paper reports computer simulations of anomalous diffusion in two-dimensional random media, emphasizing the importance of high-quality pseudo-random number generators to accurately observe the phenomenon.

Contribution

It demonstrates the necessity of using high-quality pseudo-random number generators in simulations of anomalous diffusion, aligning Monte Carlo results with theoretical predictions.

Findings

Monte Carlo simulations agree with renormalization group calculations

High-quality pseudo-random number generators are essential for accurate results

Low-quality generators can produce incorrect ballistic motion

Abstract

We present computer simulations of anomalous diffusion, $< r^2(t) > \sim a t^{1-\delta}$, in two dimensions. The Monte Carlo calculations are in excellent agreement with previous renormalization group calculations. Interestingly, use of a high quality pseudo-random number generator is necessary to observe the anomalous diffusion. A linear-feedback, shift-register method leads to incorrect, ballistic motion of the random walkers.