Interacting, running and tumbling: The active Dyson Brownian motion

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 still retains a Wigner semi-circular shape.

Active particle in a harmonic trap driven by a resetting noise: an approach via Kesten variables

“…A possible starting point could be the two RTPs model studied in [89]. It would also be interesting to use this framework to study the recently introduced active Dyson Brownian motion [23]. Similarly, it is natural to ask whether this approach can be extended to study models of active particles in two and higher dimensions, for which there exists very few analytical results.…”

“…These systems are composed of self-driven particles that can convert energy into directed motion, leading to a wide range of phenomena, including clustering and jamming [14][15][16][17], motility induced phase separation [18][19][20][21], and absence of well defined pressure [22]. However, these many-body phenomena are still difficult to describe analytically starting from microscopic models, for which very few exact results have been obtained (see however [17,23]).…”

Non-equilibrium phase transitions in active rank diffusions

“…We first investigate this question numerically for finite N . Next, using the Dean-Kawasaki approach [37,38], as in [36] for the passive case, and in [24] for the active Dyson Brownian motion, we show that the density fields evolve according to two coupled stochastic non-linear equations. In the large-N limit these equations become deterministic and of the Burgers type.…”

“…Using the Dean-Kawasaki approach [37,38], one can establish starting from (1), as in [36] and [24], an exact stochastic evolution equation for the density fields, which takes the following form:…”

“…where the O( 1 √ N ) term represents the passive and active noises, see the Supplementary Material Supplementarymaterial.pdf (SM) or [41], and sect. III A of [24]. It is convenient to define, as in [36], the rank fields…”