# Diffusion in Quasi-One Dimensional Channels: A Small System $n,p,T,$ Transition State Theory for Hopping Times

## TL;DR

This paper develops a transition state theory to calculate hopping times for particles in quasi-one-dimensional channels, bridging single file and Fickian diffusion regimes, validated by simulations but limited by interaction neglect in wider channels.

## Contribution

It introduces a transition state theory approach using the small system isobaric-isothermal ensemble to accurately compute hopping times in confined particle systems.

## Key findings

- Theory accurately predicts hopping times for ideal gases and hard discs in narrow channels.
- Simulation scheme effectively calculates free energy barriers for particle hopping.
- Method underestimates barriers in wider channels due to interaction neglect.

## Abstract

Particles confined to a single file, in a narrow quasi-one dimensional channel, exhibit a dynamic crossover from single file diffusion to Fickian diffusion as the channel radius increases and the particles can begin to pass each other. The long time diffusion coefficient for a system in the crossover regime can be described in terms of a hopping time, which measures the time it takes for a particle to escape the cage formed by its neighbours. In this paper, we develop a transition state theory approach to the calculation of the hopping time, using the small system isobaric--isothermal ensemble to rigorously account for the volume fluctuations associated with the size of the cage. We also describe a Monte Carlo simulation scheme that can be used to calculate the free energy barrier for particle hopping. The theory and simulation method correctly predict the hopping times for a two dimensional confined ideal gas system and a system of confined hard discs over a range of channel radii, but the method breaks down for wide channels in the hard discs case, underestimating the height of the hopping barrier due to the neglect of interactions between the small system and its surroundings.