Selected topics in transport phenomena: deterministic and probabilistic aspects
Anomalous diffusion in passive scalar transport
Carnegie Mellon University
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.