Transport and fluctuation-dissipation relations in asymptotic and pre-asymptotic diffusion across channels with variable section

TL;DR

This paper investigates the diffusive behavior of Brownian particles in periodically varying channels, deriving effective diffusion coefficients and analyzing the validity of fluctuation-dissipation relations in both asymptotic and pre-asymptotic regimes.

Contribution

It introduces a new approximation for the effective diffusion coefficient that surpasses perturbation methods and compares it with numerical simulations, including pre-asymptotic effects.

Findings

Effective diffusion coefficient matches numerical results.

Pre-asymptotic diffusion persists up to large crossover times.

Fluctuation-dissipation relation holds in both regimes.

Abstract

We study the asymptotic and pre-asymptotic diffusive properties of Brownian particles in channels whose section varies periodically in space. The effective diffusion coefficient $D_{\mathrm{eff}}$ is numerically determined by the asymptotic behavior of the root mean square displacement in different geometries, considering even cases of steep variations of the channel boundaries. Moreover, we compared the numerical results to the predictions from the various corrections proposed in the literature to the well known Fick-Jacobs approximation. Building an effective one dimensional equation for the longitudinal diffusion, we obtain an approximation for the effective diffusion coefficient. Such a result goes beyond a perturbation approach, and it is in good agreement with the actual values obtained by the numerical simulations. We discuss also the pre-asymptotic diffusion which is observed up to a crossover time whose value, in the presence of strong spatial variation of the channel cross section, can be very large. In addition, we show how the Einstein's relation between the mean drift induced by a small external field and the mean square displacement of the unperturbed system is valid in both asymptotic and pre-asymptotic regimes.