Flocking transition within the framework of Kuramoto paradigm for synchronization: Clustering and the role of the range of interaction

Abstract

A Kuramoto-type approach to address flocking phenomena is presented. First, we analyze a simple generalization of the Kuramoto model for interacting active particles, which is able to show the flocking transition (the emergence of coordinated movements in a group of interacting self-propelled agents). In the case of all-to-all interaction, the proposed model reduces to the Kuramoto model for phase synchronization of identical motionless noisy oscillators. In general, the nature of this non-equilibrium phase transition depends on the range of interaction between the particles. Namely, for a small range of interaction, the transition is first order, while for a larger range of interaction, it is a second order transition. Moreover, for larger interaction ranges, the system exhibits the same features as in the case of all-to-all interaction, showing a spatially homogeneous flux when flocking phenomenon has emerged, while for lower interaction ranges, the flocking transition is characterized by cluster formation. We compute the phase diagram of the model, where we distinguish three phases as a function of the range of interaction and the effective coupling strength: a disordered phase, a spatially homogeneous flocking phase, and a cluster-flocking phase. Then, we present a general discussion about the applicability of this way of modeling to more realistic and general situations, ending with a brief presentation of a second example (a second model with a conservative interaction) where the flocking transition may be studied within the framework that we are proposing.

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