Transient phases in the Vicsek model of flocking

Abstract

The Vicsek model of flocking is studied by computer simulation. We confined our studies here to the morphologies and the lifetimes of transient phases. In our simulation, we have identified three distinct transient phases, namely, vortex phase, colliding phase and multi-domain phase. The mapping of Vicsek model to XY model in the $v \to 0$ limit prompted us to explore the possibility of finding any vortex kind of phases in the Vicsek model. We have obtained rotating vortex phase and measured the lifetime of this vortex phase. We have also measured the lifetimes of other two transient phases, i.e., colliding phase and multi-domain phase. We have measured the integrated lifetime ($\tau_t$) of all these transient phases and studied this as function of density ($\rho$) and noise ($\eta$). In the low noise regime, we proposed here a scaling law $\tau_t N^{-a} = F(\rho N^{-b})$ where $F(x)$ is a scaling function like $F(x) \sim x^{-s}$. By the method of data collapse, we have estimated the exponents as $a=-0.110\pm0.010$, $b=0.950\pm0.010$ and $s=1.027\pm0.008$. The integrated lifetime $\tau$ (defined in the text differently) was observed to decrease as the noise approaches the critical noise from below. This behaviour is quite unusual and contrary to the critical slowing down observed in the case of equilibrium phase transitions. We have provided a possible explanation from the time evolution of the distribution of the directions of velocities.