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

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.

27 pages, 4 figures, Mathematica notebook as supplementary material

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• Tuning attraction and repulsion between active particles through persistence
• From a microscopic solution to a continuum description of active particles with a recoil interaction in one dimension
• Dynamical crossovers and correlations in a harmonic chain of active particles
• Time-dependent properties of run-and-tumble particles: Density relaxation
• Geometric and Nonequilibrium Criticality in Run-and-Tumble Particles with Competing Motility and Attraction
• Emergent short-range repulsion for attractively coupled active particles
• Non-equilibrium phase transitions in active rank diffusions
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• Active stop and go motion: a strategy to improve spatial exploration and survival
• From Attraction to Repulsion: Emergent Interactions in Harmonically Coupled Active Binary System