Velocity jump process with volume exclusions in a narrow channel

TL;DR

This study models how volume exclusions and narrow channel constraints affect particle movement and interactions, deriving a nonlinear transport equation from particle dynamics and validating it through numerical comparisons.

Contribution

It introduces a systematic derivation of a nonlinear transport equation for particles with volume exclusions in a narrow channel from a velocity jump process.

Findings

Good agreement between kinetic model and particle simulations at low occupancy.

Volume exclusions influence particle speeds and passing probabilities.

Channel width and particle size affect the passing behavior and overall dynamics.

Abstract

This paper analyses the impact of collisions in a system of $N$ identical hard-core particles driven according to a velocity jump process. The physical space is essentially a channel in $\mathbb{R}$ with a probability of occupants being able to pass each other. The system mimics what nature does, where individuals pass one another in a narrow channel while making incidental contact with those moving in the opposite direction. The passing probability may depend on the particles' size and the channel's width. Starting from the particle level model, we systematically derive a nonlinear transport equation based on an asymptotic expansion. Under low-occupied fractions, numerical solutions of both the kinetic model and the stochastic particle system are compared well during biased and unbiased random velocity changes. Analysis of the subpopulation motility within a large population exhibits the consequences of volume exclusions and channel confinements on the travelling speeds.