Anomalous Diffusion in Aperiodic Environments

TL;DR

This paper investigates how a particle diffuses in aperiodic environments, revealing that fluctuation properties of the medium drastically influence diffusion behavior, from normal to ultra-slow diffusion, with analytical insights derived from a correspondence with the Ising model.

Contribution

It provides new analytical results linking the fluctuation properties of aperiodic environments to diffusion dynamics, extending understanding beyond random media.

Findings

Normal diffusion in environments with bounded fluctuations

Ultra-slow logarithmic diffusion in unbounded fluctuation environments

Continuous variation of diffusion and persistence exponents at marginal fluctuations

Abstract

We study the Brownian motion of a classical particle in one-dimensional inhomogeneous environments where the transition probabilities follow quasiperiodic or aperiodic distributions. Exploiting an exact correspondence with the transverse-field Ising model with inhomogeneous couplings we obtain many new analytical results for the random walk problem. In the absence of global bias the qualitative behavior of the diffusive motion of the particle and the corresponding persistence probability strongly depend on the fluctuation properties of the environment. In environments with bounded fluctuations the particle shows normal diffusive motion and the diffusion constant is simply related to the persistence probability. On the other hand in a medium with unbounded fluctuations the diffusion is ultra-slow, the displacement of the particle grows on logarithmic time scales. For the borderline situation with marginal fluctuations both the diffusion exponent and the persistence exponent are continuously varying functions of the aperiodicity. Extensions of the results to disordered media and to higher dimensions are also discussed.