# Exact Solution of Two Interacting Run-and-Tumble Random Walkers with Finite Tumble Duration

## TL;DR

This paper provides an exact solution for a two-particle run-and-tumble model with finite tumble duration, revealing complex stationary states characterized by two distinct lengthscales.

## Contribution

It introduces a solvable model with finite tumble times and derives the exact nonequilibrium stationary state for two interacting particles.

## Key findings

- Stationary state characterized by two lengthscales.
- One lengthscale vanishes in the continuous limit.
- Stationary state exhibits jammed, attractive, and extended regions.

## Abstract

We study a model of interacting run-and-tumble random walkers operating under mutual hardcore exclusion on a one-dimensional lattice with periodic boundary conditions. We incorporate a finite, Poisson-distributed, tumble duration so that a particle remains stationary whilst tumbling, thus generalising the persistent random walker model. We present the exact solution for the nonequilibrium stationary state of this system in the case of two random walkers. We find this to be characterised by two lengthscales, one arising from the jamming of approaching particles, and the other from one particle moving when the other is tumbling. The first of these lengthscales vanishes in a scaling limit where the continuous-space dynamics is recovered whilst the second remains finite. Thus the nonequilibrium stationary state reveals a rich structure of attractive, jammed and extended pieces.