Anomalous diffusion in fast cellular flows at intermediate time scales

TL;DR

This paper investigates the intermediate time scale behavior of tracer particles in cellular flows, demonstrating that their variance grows like the square root of time, indicating anomalous diffusion before normal diffusion occurs.

Contribution

The paper provides a rigorous proof that variance grows as O(√t) on intermediate time scales in cellular flows, revealing anomalous diffusion prior to the onset of classical diffusion.

Findings

Variance grows like O(√t) on intermediate time scales

Anomalous diffusion behavior is observed before normal diffusion

Effective behavior deviates from classical Brownian motion at intermediate times

Abstract

It is well known that on long time scales the behaviour of tracer particles diffusing in a cellular flow is effectively that of a Brownian motion. This paper studies the behaviour on "intermediate" time scales before diffusion sets in. Various heuristics suggest that an anomalous diffusive behaviour should be observed. We prove that the variance on intermediate time scales grows like $O(\sqrt{t})$. Hence, on these time scales the effective behaviour can not be purely diffusive, and is consistent with an anomalous diffusive behaviour.