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Transport and nonequilibrium phase transitions in polygonal urn models
Emilio N M Cirillo1, Matteo Colangeli2, Antonio Di Francesco2
1Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, via A. Scarpa 16, 00161 Roma, Italy.
Abstract:
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 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.
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