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

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, $μ$, 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 $μ$, 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.