Following Your Nose: Autochemotaxis and Other Mechanisms for Spinodal Decomposition in Flocks

We develop the hydrodynamic theory of dry, polar ordered, active matter (“flocking”) with autochemotaxis; i.e., self-propelled entities moving in the same direction, each emitting a substance which attracts the others (e.g., ants). We find that sufficiently strong autochemotaxis leads to an instability to phase separation into one high and one low density band. This is very analogous to both equilibrium phase separation, and “motility induced phase separation” and can occur in flocks due to any microscopic mechanism (e.g., sufficiently strong attractive interactions) that makes the entities cohere.

Figure 1

(a) The region of instability in $\mathbf{q}$ space, for autochemotaxic systems very close to the instability threshold on the unstable side. The instability is toward forming “bands” in real space running parallel to the direction of mean flock motion, as illustrated in (b) and Fig. 2. (b) The band structure of the instability at intermediate times. The density is only modulated along one of the directions (which we call $y$, and which is indicated in the figure) perpendicular to the direction $\widehat{x}$ of mean flock motion. The mean speeds within each band also differ.

Figure 2

A plot of the initial (dashed blue curve) and final (solid orange curve) density for a typical numerical solution of the hydrodynamic equations for 1D configurations, as described in the text, at intermediate times. Here, $\ell$ is the linear spatial extent of our system in the $y$ direction (which is periodic).

Figure 3

Phase diagram of flock phase-separation. Solid blue and dashed orange lines are the binodal and spinodal lines, respectively. The orange filled region, labeled “unstable,” corresponds to the coexistence of the “liquid” (high density) and “gas” (low density) phases.

Figure 4

Heuristic illustration of why the bands always form perpendicular to the direction of mean flock motion. See the text after Eq. (4) for explanation.

Figure 5

(a) Illustration of the common tangent construction. (b) Illustration of the uncommon tangent construction.