Interacting, running and tumbling: The active Dyson Brownian motion
Résumé
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 ± 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 . 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.
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