Consider a diffusive passive scalar advected by a two dimensional incompressible flow. If the flow is cellular (i.e. has a periodic Hamiltonian with no unbounded trajectories), then classical homogenization results show that the long time behaviour is an effective Brownian motion. We show that on intermediate time scales, the effective behaviour is instead a fractional kinetic process. At the PDE level this means that while the long time scaling limit is the heat equation, the intermediate time scaling limit is a time fractional heat equation. We will also describe the expected intermediate behaviour in the presence of open channels.

In the last part of the talk we will describe a few other trap models that arise in PDE homogenization limits that exhibit a similar behaviour on intermediate time scales.