The Fast Wandering of Slow Birds

Abstract

I study a single "slow" bird moving with a flock of birds of a different, and faster (or slower) species. I find that every "species" of flocker has a characteristic speed $γ\ne v_0$, where $v_0$ is the mean speed of the flock, such that, if the speed $v_s$ of the "slow" bird equals $γ$, it will randomly wander transverse to the mean direction of flock motion far faster than the other birds will: its mean-squared transverse displacement will grow in $d=2$ with time $t$ like $t^{5/3}$, in contrast to $t^{4/3}$ for the other birds. In $d=3$, the slow bird's mean squared transverse displacement grows like $t^{5/4}$, in contrast to $t$ for the other birds. If $v_s\neq γ$, the mean-squared displacement of the "slow" bird crosses over from $t^{5/2}$ to $t^{4/3}$ scaling in $d=2$, and from $t^{5/4}$ to $t$ scaling in $d=3$, at a time $t_c$ that scales according to $t_c \propto|v_s-γ|^{-2}$.