The Fast Wandering of Slow Birds

TL;DR

This paper analyzes the transverse wandering behavior of a slow bird within a flock, revealing characteristic speeds and anomalous diffusion scaling laws depending on the bird's speed relative to a specific characteristic speed.

Contribution

It introduces the concept of a characteristic speed for the slow bird and derives novel scaling laws for its transverse displacement in different dimensions.

Findings

At characteristic speed, the slow bird exhibits superdiffusive transverse wandering with specific power-law exponents.

The transverse mean-squared displacement scales differently depending on whether the bird's speed equals or differs from the characteristic speed.

Crossover times between different scaling regimes depend on the difference between the bird's speed and the characteristic speed.

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 $\gamma\ne v_0$, where $v_0$ is the mean speed of the flock, such that, if the speed $v_s$ of the "slow" bird equals $\gamma$, 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 $v_s\neq \gamma$, 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 $t_c \propto|v_s-\gamma|^{-2}$.