Simulating diffusion and disorder-induced localization in random walks and transmission lines

TL;DR

This paper introduces two simulations demonstrating Anderson localization, one using classical random walks and the other modeling electromagnetic wave propagation in disordered transmission lines, illustrating how disorder inhibits diffusion.

Contribution

It provides accessible simulation tools for exploring Anderson localization through classical and electromagnetic models, enhancing understanding of wave interference effects in disordered media.

Findings

Transition from normal diffusion to localization observed

Exponential confinement of wave energy demonstrated

Quantitative measures confirm localization phenomena

Abstract

We present two complementary simulations that lead to an exploration of Anderson localization, a phenomenon in which wave diffusion is suppressed in disordered media by interference from multiple scattering. To build intuition, the first models the random walk of classical, non-interacting point-like particles, providing a clear analogy to the way disorder can limit transport. The second examines the propagation of an electromagnetic pulse through a one-dimensional, lossless transmission line with randomly varying propagation constant and characteristic impedance along its length, a system that captures the interference effects essential for true Anderson localization. We evaluate quantitative measures that reveal the transition from normal diffusion to localization of particles in one case, and the exponential confinement of wave energy in the other. Together, these simulations offer a pair of accessible tools for investigating localization phenomena in an instructional setting.