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 two-dimensional 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 $\varphi$, until complete order $\varphi=1.0$ is reached (polar consensus). We find that $\varphi$ increases as $t^{1/2}$ for short times and approaches exponentially fast to $1.0$ for long times. Also, the mean time to consensus $\tau$ varies non-monotonically with the density of particles $\rho$, reaching a minimum at some intermediate density $\rho_{\tiny \mbox{min}}$. At $\rho_{\tiny \mbox{min}}$, the mean consensus time scales with the system size $N$ as $\tau_{\tiny \mbox{min}} \sim N^{0.765}$, and thus the consensus is faster than in the case of all-to-all interactions (large $\rho$) where $\tau=2N$. 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.