Current of interacting particles inside a channel of exponential cavities: Application of a modified Fick--Jacobs equation

TL;DR

This paper demonstrates that a modified Fick--Jacobs equation accurately models the current, concentration, and mobility of interacting particles in a channel with exponential cavities, aligning well with Monte Carlo simulations.

Contribution

It provides an exact solution for a system with interacting particles using the modified Fick--Jacobs equation and compares its predictions with numerical simulations.

Findings

Modified Fick--Jacobs equation accurately predicts particle behavior with interactions.

Results agree with Monte Carlo simulations for both non-interacting and interacting particles.

The modified equation outperforms the exact non-interacting solution in certain force scenarios.

Abstract

The Fick--Jacobs equation has been widely studied, because of its applications in the diffusion and transport of non-interacting particles in narrow channels. It is also known that a modified version of this equation can be used to describe the same system with particles interacting through a hard-core potential. In this work we present a system that can be exactly solved using the Fick--Jacobs equation. The exact results of the particle concentration profile along the channel $n$, the current, $J$, and the mobility, $\mu$, of particles as a function of an external force are contrasted with Monte Carlo simulations results of non-interacting particles. For interacting particles the behavior of $n$, $J$ and $\mu$, obtained from the modified Fick--Jacobs equation are in agreement with numerical simulations, where the hard-core interaction is taken into account. Even more, for interacting particles the modified Fick--Jacobs equation gives comparatively more accurate results of the current difference (when a force is applied in opposite directions) than the exact result for the non-interacting ones.