Random Perturbations of 2-dimensional Hamiltonian Flows

TL;DR

This paper investigates how small molecular diffusion influences the long-term behavior of particles in periodic 2D Hamiltonian flows, deriving asymptotic formulas for effective diffusivity as diffusion vanishes.

Contribution

It provides the first detailed asymptotic analysis of effective diffusivity in 2D Hamiltonian flows with small molecular diffusion.

Findings

Asymptotic formulas for effective diffusivity as molecular diffusion approaches zero

Characterization of long-term particle dispersion in perturbed Hamiltonian flows

Insights into homogenization limits for periodic divergence-free flows

Abstract

We consider the motion of a particle in a periodic two dimensional flow perturbed by small (molecular) diffusion. The flow is generated by a divergence free zero mean vector field. The long time behavior corresponds to the behavior of the homogenized process - that is diffusion process with the constant diffusion matrix (effective diffusivity). We obtain the asymptotics of the effective diffusivity when the molecular diffusion tends to zero.