Interacting, running and tumbling: The active Dyson Brownian motion

¹ Laboratoire de Physique de l'École Normale Supérieure, CNRS, ENS & PSL University, Sorbonne Université, Université de Paris - 75005 Paris, France
² Sorbonne Université, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589 4 Place Jussieu, 75252 Paris Cedex 05, France

^(a) E-mail: schehr@lpthe.jussieu.fr (corresponding author)

Received: 22 March 2023
Accepted: 1 June 2023

Abstract

We introduce and study a model in one dimension of N run-and-tumble particles (RTP) which repel each other logarithmically in the presence of an external quadratic potential. This is an “active” version of the well-known Dyson Brownian motion (DBM) where the particles are subjected to a telegraphic noise, with two possible states ± with velocity ±v₀. We study analytically and numerically two different versions of this model. In model I a particle only interacts with particles in the same state, while in model II all the particles interact with each other. In the large time limit, both models converge to a steady state where the stationary density has a finite support. For finite N, the stationary density exhibits singularities, which disappear when . In that limit, for model I, using a Dean-Kawasaki approach, we show that the stationary density of + (respectively −) particles deviates from the DBM Wigner semi-circular shape, and vanishes with an exponent 3/2 at one of the edges. In model II, the Dean-Kawasaki approach fails but we obtain strong evidence that the density in the large N limit (still) retains a Wigner semi-circular shape.