Self diffusion in a system of interacting Langevin particles

TL;DR

This paper develops a perturbative and renormalization group approach to analyze the self-diffusion constant of interacting Langevin particles, revealing conditions for diverging relaxation times and validating results with numerical simulations.

Contribution

It introduces a systematic double expansion method and a semi-phenomenological renormalization group technique for studying diffusion in interacting particle systems.

Findings

Exact summation of one-loop diagrams in certain limits

Prediction of diverging relaxation times under specific conditions

Quantitative agreement with two-dimensional numerical simulations

Abstract

The behavior of the self diffusion constant of Langevin particles interacting via a pairwise interaction is considered. The diffusion constant is calculated approximately within a perturbation theory in the potential strength about the bare diffusion constant. It is shown how this expansion leads to a systematic double expansion in the inverse temperature $\beta$ and the particle density $\rho$. The one-loop diagrams in this expansion can be summed exactly and we show that this result is exact in the limit of small $\beta$ and $\rho\beta$ constant. The one-loop result can also be re-summed using a semi-phenomenological renormalization group method which has proved useful in the study of diffusion in random media. In certain cases the renormalization group calculation predicts the existence of a diverging relaxation time signalled by the vanishing of the diffusion constant -- possible forms of divergence coming from this approximation are discussed. Finally, at a more quantitative level, the results are compared with numerical simulations, in two-dimensions, of particles interacting via a soft potential recently used to model the interaction between coiled polymers.