Rectification and Non-Gaussian Diffusion in Heterogeneous Media

Paolo Malgaretti,J. Rubi,Ignacio Pagonabarraga

doi:10.3390/e18110394

Abstract

We show that when Brownian motion takes place in a heterogeneous medium, the presence of local forces and transport coefficients leads to deviations from a Gaussian probability distribution that make that the ratio between forward and backward probabilities depends on the nature of the host medium, on local forces and also on time. We have applied our results to two situations: diffusion in a disordered medium and diffusion in a confined system. For such scenarios we have shown that our theoretical predictions are in very good agreement with numerical results. Moreover we have shown that the deviations from the Gaussian solution lead to the onset of rectification. Our predictions could be used to detect the presence of local forces and to characterize the intrinsic short-scale properties of the host medium, a problem of current interest in the study of micro and nano-systems.

Highlights

The symmetry of the probability distribution of a system in equilibrium, expressed through the detailed balance condition, breaks down when a driving force is applied [1]
In order to study the accuracy of the perturbative expression Equation (13), we will consider two scenarios where different physical mechanisms lead to a local force and diffusion coefficient
We have shown that the diffusion of particles is strongly affected by heterogeneities resulting from irregularities of the boundaries or from the intrinsic nature of the host medium

Summary

Introduction

The symmetry of the probability distribution of a system in equilibrium, expressed through the detailed balance condition, breaks down when a driving force is applied [1]. In a variety of situations, such as for particles diffusing in porous media or displacing through ion channels or membrane pores, the assumption of a constant force and/or transport coefficients is not justified For these local transport scenarios, Equation (1) cannot be applied. The local nature of these quantities leads to a non-Gaussian pdf for particle displacement, due to a coupling between particle advection and diffusion This behavior has not been reported in previously studied systems, where particles diffuse in a homogeneous medium and are subjected to uniform forces [2,3,4] (a case of nonuniform forces where an analytical solution of the associated Smoluchowski equation exists is the Ornstein–Uhlenbeck process; in this case, the probability distribution converges rapidly to a Gaussian due to the confining nature of the potential [9]). In the last section, we present our main conclusions

Diffusion in Heterogeneous Systems

Results

Diffusion in an Inhomogeneous Unbounded Medium

Diffusion in a Periodic Channel

Discussion