Effective drift and diffusivity in non-Gaussian random gradient flows

TL;DR

This paper investigates the effective drift and diffusivity of particles in a non-Gaussian random medium, confirming the Einstein relation through perturbation theory and simulations, extending its validity beyond Gaussian assumptions.

Contribution

It provides a four-loop perturbation theory calculation for effective diffusivity in non-Gaussian fields and confirms the Einstein relation through simulations.

Findings

Perturbation theory agrees with simulations for diffusivity.

Effective drift and diffusivity satisfy the Einstein relation.

The Einstein relation holds for non-Gaussian drift fields.

Abstract

We study the long-range effective drift and diffusivity of a particle in a random medium moving subject to a given molecular diffusivity and a local drift. The local drift models the effect of a random electrostatic field on a neutral but polarizable molecule. Although the electrostatic field is assumed to obey Gaussian statistics the induced statistics of the drift velocity field are non-Gaussian. We show that a four-loop perturbation theory calculation of the effective diffusivity is in rather good agreement with the outcome of a numerical simulation for a reasonable range of the disorder parameter. We also measure the effective drift in our simulation and confirm the validity of the ``Einstein relation'' that expresses the equality of the renormalization factors, induced by the random medium, for the effective drift and effective diffusivity, relative to their molecular values. The Einstein relation has previously only been confirmed for Gaussian random drift fields. The simulation result, for our non-Gaussian drift model, is consistent with a previous theoretical analysis showing the Einstein relation should remain true, independently of the precise character of the statistics of the drift velocity field.