Interacting, running and tumbling: the active Dyson Brownian motion

TL;DR

This paper introduces a one-dimensional active Dyson Brownian motion model with run-and-tumble particles, analyzing its steady states and density profiles through analytical and numerical methods, revealing deviations from classical models.

Contribution

It presents a novel active version of Dyson Brownian motion with two interaction models and analyzes their steady-state densities using Dean-Kawasaki approach and numerical simulations.

Findings

Steady states have finite support with singularities at finite N.

Model I's density deviates from Wigner semi-circle, vanishing with exponent 3/2 at edges.

Model II's density likely retains Wigner semi-circular shape in the large N limit.

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 $\pm$ with velocity $\pm v_0$. 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 $N \to +\infty$. 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 retains a Wigner semi-circular shape.