# A Deterministic Model for One-Dimensional Excluded Flow with Local Interactions

## TL;DR

This paper introduces a deterministic model for one-dimensional particle flow with local interactions, revealing how attractive and repulsive forces influence steady-state flow and traffic congestion.

## Contribution

The paper develops a novel deterministic compartmental model incorporating local interactions and proves convergence to a unique steady-state using contraction theory.

## Key findings

- Attracting forces increase steady-state flow.
- Repelling forces help alleviate traffic jams.
- Model predictions align with simulation results.

## Abstract

Natural phenomena frequently involve a very large number of interacting molecules moving in confined regions of space. Cellular transport by motor proteins is an example of such collective behavior. We derive a deterministic compartmental model for the unidirectional flow of particles along a one-dimensional lattice of sites with nearest-neighbor interactions between the particles. The flow between consecutive sites is governed by a soft simple exclusion principle and by attracting or repelling forces between neighboring particles. Using tools from contraction theory, we prove that the model admits a unique steady-state and that every trajectory converges to this steady-state. Analysis and simulations of the effect of the attracting and repelling forces on this steady-state highlight the crucial role that these forces may play in increasing the steady-state flow, and reveal that this increase stems from the alleviation of traffic jams along the lattice. Our theoretical analysis clarifies microscopic aspects of complex multi-particle dynamic processes.