Nonuniqueness of the steady state for run-and-tumble particles with a double-well interaction potential

We study N run-and-tumble particles (RTPs) in one dimension interacting via a double-well pairwise potential W(r)=-k_{0}r^{2}/2+gr^{4}/4, which is repulsive at short interparticle distance r and attractive at large distance. At large time, the system forms a bound state where the density of particles has a finite support. We focus on the determination of the total density of particles in the stationary state ρ_{s}(x), in the limit N→+∞. We obtain an explicit expression for ρ_{s}(x) as a function of the "renormalized" interaction parameter k=k_{0}-3m_{2} where m_{2} is the second moment of ρ_{s}(x). Interestingly, this stationary solution exhibits a transition between a connected and a disconnected support for a certain value of k, which has no equivalent in the case of Brownian particles. Analyzing in detail the expression of the stationary density in the two cases, we find a variety of regimes characterized by different behaviors near the edges of the support and around x=0. Furthermore, by studying the relation between k and k_{0}, we find that the mapping k_{0}→k becomes multivalued below a certain value of the tumbling rate γ of the RTPs for some range of values of k_{0} near the transition, implying the existence of two stable solutions. Finally, we show that in the case of a disconnected support, it is possible to observe steady states where the density ρ_{s}(x) is not symmetric, characterized by a third moment m_{3} which can take a continuous range of values. All our analytical predictions are in good agreement with numerical simulations already for systems of N=100 particles. The nonuniqueness of the stationary state is a particular feature of this model in the presence of active (RTP) noise, which contrasts with the uniqueness of the Gibbs equilibrium for Brownian particles. We argue that these results are also relevant for a class of more realistic interactions with both an attractive and a repulsive part but which decay at infinity.