Perturbation theory for the effective diffusion constant in a medium of random scatterer

TL;DR

This paper develops perturbation theory and renormalization group methods to analyze the effective diffusion constant of a tracer particle in a medium with randomly distributed scatterers, comparing predictions with numerical simulations.

Contribution

It introduces a perturbative and renormalization approach to approximate the diffusion in random media with point scatterers, extending methods used for Gaussian potentials.

Findings

Perturbation theory identifies conditions for Gaussian approximation.

Renormalization group scheme agrees with known exact results.

Numerical simulations validate the theoretical predictions.

Abstract

We develop perturbation theory and physically motivated resummations of the perturbation theory for the problem of a tracer particle diffusing in a random media. The random media contains point scatterers of density $\rho$ uniformly distributed through out the material. The tracer is a Langevin particle subjected to the quenched random force generated by the scatterers. Via our perturbative analysis we determine when the random potential can be approximated by a Gaussian random potential. We also develop a self-similar renormalisation group approach based on thinning out the scatterers, this scheme is similar to that used with success for diffusion in Gaussian random potentials and agrees with known exact results. To assess the accuracy of this approximation scheme its predictions are confronted with results obtained by numerical simulation.