Abstract
Since the pioneering work by Vicsek and his collaborators on the motion of self-propelled particles, most of the subsequent studies have focused on the onset of ordered states through a phase transition driven by particle density and noise. Usually, the particles in these systems are placed within periodic boundary conditions and interact via short-range velocity alignment forces. However, when the periodic boundaries are eliminated, letting the particles move in open space, the system is not able to organize into a coherently moving group since even small amounts of noise cause the flock to break apart. While the phase transition has been thoroughly studied, the conditions to keep the flock cohesive in open space are still poorly understood. Here we extend the Vicsek model of collective motion by introducing long-range alignment interactions between the particles. We show that just a small number of these interactions is enough for the system to build up long lasting ordered states of collective motion in open space and in the presence of noise. This finding was verified for other models in addition to the Vicsek one, suggesting its generality and revealing the importance that long-range interactions can have for the cohesion of the flock.
Highlights
Collective motion is one of the most spectacular displays of coordinated behavior in nature, exhibited by systems of very different kinds, ranging from cell populations to various species of insects and vertebrates, such as flocks of starlings, sheep herds, fish shoals and human crowds[1,2]
The main problem has been to determine the nature of the interactions between particles and the mechanisms needed for the system to build up states of collective motion
L0 = 3 N /ρ0, being ρ0 the initial density. This density was chosen in such a way that the system would have reached a high degree of order in the standard Vicsek model with periodic boundary conditions (see Fig. 3(a))
Summary
Collective motion is one of the most spectacular displays of coordinated behavior in nature, exhibited by systems of very different kinds, ranging from cell populations to various species of insects and vertebrates, such as flocks of starlings, sheep herds, fish shoals and human crowds[1,2]. The main problem has been to determine the nature of the interactions between particles and the mechanisms needed for the system to build up states of collective motion Most of these models are based on local (or short-range) alignment interactions where individuals modify their direction of motion to match the average direction of their immediate neighbors, plus some noise[11,22,23]. These short-range interactions can be defined either metrically, where each particle interacts with all others within some distance r0 (like in the standard Vicsek model, see Fig. 1(a)), or topologically where particles interact with a fixed number αl of their first neighbors regardless of their relative distance (like in Ballerini et al, see Fig. 1(b)). Other models introduce repulsion and attraction between particles in order to avoid collisions and prevent the system from breaking apart[16,27,28,29,30]
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