Flocking dynamics with voter-like interactions

Abstract

We study the collective motion of a large set of self-propelled particles subject to voter-like interactions. Each particle moves on a 2D space at a constant speed in a direction that is randomly assigned initially. Then, at every step of the dynamics, each particle adopts the direction of motion of a randomly chosen neighboring particle. We investigate the time evolution of the global alignment of particles measured by the order parameter j, until complete order j=1.0 is reached (polar consensus). We find that j increases as t^ 1/2 for short times and approaches 1.0 exponentially fast for longer times. Also, the mean time to consensus τ varies non-monotonically with the density of particles ρ, reaching a minimum at some intermediate density ρmin. At ρmin , the mean consensus time scales with the system size N as τmin ≈ N^0.765 , and thus the consensus is faster than in the case of all-to-all interactions (large ρ) where τ=2N . We show that the fast consensus, also observed at intermediate and high densities, is a consequence of the segregation of the system into clusters of equally-oriented particles which breaks the balance of transitions between directional states in well mixed systems.