Ordering kinetics and steady state of Malthusian flock

Abstract

We study a collection of polar self-propelled particles or polar flock on a two dimensional substrate involving birth and death. Most of the previous studies of polar flock with birth and death have focused on the steady state characteristics of Malthusian flock. We emphasize on the significance of rate of birth and death on the kinetics as well as steady state of the system. Our system is modeled using coarse-grained hydrodynamic equations of motion for local density and velocity of the flock. Results are obtained for different birth and death rates by solving the hydrodynamic equations using numerical integration and linearized calculation about the broken symmetry state. The presence of finite birth and death rate affects the density field significantly, whereas the effect on velocity field is moderate. The early time growth of velocity field slows down in the presence of finite birth and death rate, whereas at late times it approaches the value of non-conserved growth kinetics for all birth and death rates. The density field shows the strong time dependent growth kinetics. The asymptotic growth law for density depends on the birth and death rates and shows a crossover from 5/6 for the immortal flock to 1/4 for large birth and death rates. In the steady state, the presence of birth and death rate leads to the suppression of speed of sound wave, velocity, and density fluctuations in the system.