Interacting, running and tumbling: The active Dyson Brownian motion

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 & PLUSMN; with velocity & PLUSMN;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 N & RARR; +& INFIN;. 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. Copyright & COPY; 2023 EPLA