Transport and nonequilibrium phase transitions in polygonal urn models

We study the deterministic dynamics of N point particles moving at a constant speed in a 2D table made of two polygonal urns connected by an active rectangular channel, which applies a feedback control on the particles, inverting the horizontal component of their velocities when their number in the channel exceeds a fixed threshold. Such a bounce-back mechanism is non-dissipative: it preserves volumes in phase space. An additional passive channel closes the billiard table forming a circuit in which a stationary current may flow. Under specific constraints on the geometry and on the initial conditions, the large N N limit allows nonequilibrium phase transitions between homogeneous and inhomogeneous phases. The role of ergodicity in making a probabilistic theory applicable is discussed for both rational and irrational urns. The theoretical predictions are compared with the numerical simulation results. Connections with the dynamics of feedback-controlled biological systems are highlighted. Nonequilibrium phase transitions take place in an N N -particles 2D non-dissipative billiard, constituted by two polygonal urns connected by two rectangular channels. One of the channels inverts the velocities of particles when their number is too high, making possible a stationary inhomogeneous density distribution. A probabilistic approach is developed to predict the realization of such phenomena. Numerical simulations for irrational urns and narrow channels accurately reproduce the theoretical predictions. In that case, in fact, the billiard dynamics is expected to be ergodic, justifying the probabilistic approach. That is not the case of rational urns.